Key concepts CFA 2018 level 2 schweser note book

Ethics All trade allocations to client accounts shall be made on a pro rata basis prior to or immediately following part or all of a block trade Supervisors have a responsibility to ensure that compliance policies are clear and well developed Supervisors and managers must document the procedures and disseminate them to staff In addition to distributing the policy and procedures manual, they have a responsibility to ensure adequate training of each new employee concerning the key policies and procedures of the firm Periodic refresher training sessions for all staff are also recommended Generally, determining whether an individual has supervisory responsibilities depends on whether employees are subject to that individual’s control or influence In other words, does the individual have the authority, for example, to hire, fire, reward, and punish an employee If a firm claims compliance with the CFA Institute Code of Ethics and Standards of Professional Conduct, This claim has to be verified by CFA Institute A company should not discriminate among analysts in the provision of information or “blackball” particular analysts who have given negative reports on the company in the past Issue Press Releases Companies should consider issuing press releases prior to analyst meetings and conference calls and scripting those meetings and calls to decrease the chance that further information will be disclosed If material non-public information is disclosed for the first time in an analyst meeting or call, the company should promptly issue a press release or otherwise make the information publicly available Employers are not obliged to adhere to CFA code and standards - they should not develop conflicting practices Prevention of Personnel Overlap When an analyst assist the investment banking side with a project, he will be treated as an investment banker until the project is over and all non-public information is disclosed to the public - hence not allowed to use information gained for any research purposes Having analysts work with investment bankers is appropriate only when the conflicts are adequately and effectively managed and disclosed Compliance Procedure: Member / Candidate must understand what makes an adequate system - make reasonable effort to close any gaps between current to adequate If violations occur, during investigation place employee on restricted activities and monitor their actions Proprietary Trading Procedures The most prudent course for firms is to suspend arbitrage activity when a security is placed on the watch list Inadequate Procedure: if member / candidate cant discharge supervisory duties due to inadequate systems - must decline role Adequate compliance procedure includes procedures for reporting violations Record Retention: Inputs and outputs - meeting briefs - Retain for of years - Regulators may have different period Recommended Procedure: member/candidate must archive notes - maintaining the records is the firms responsibility GIPS: When a firm claims compliance with the GIPS standards, it must also comply with the GIPS Guidance Statement on Error Correction in relation to the error When claiming compliance with the GIPS standards, firms must meet all of the requirements, make mandatory disclosures, and meet any other requirements that apply to that firm’s specific situation Disclosure of conflict: Disclosure should be made to both employers and clients Duties to Employers: Disclosure of Additional Compensation Arrangements, CFA Institute members and candidates must not accept gifts, benefits, compensation, or consideration that competes with, or might reasonably be expected to create a conflict of interest with, their employer’s interest unless they obtain written consent from all parties involved Under Standard VI: Client interest > Employer Interest > Members or Candidates Employees must make full and fair disclosure of all matters that could reasonably be expected to impair their independence and objectivity or interfere with respective duties to their employer, clients, and prospective clients Members and candidates must ensure that such disclosures are prominent, are delivered in plain language, and communicate the relevant information effectively The relationship between the analyst and the company through a relative is so tangential that it does not create a conflict of interest necessitating disclosure Proxy Voting: A cost–benefit analysis may show that voting all proxies may not benefit the client, so voting proxies may not be necessary in all instances Members and candidates should disclose to clients their proxy voting policies Directed Brokerage: A client will direct a manager to use the client’s brokerage to purchase goods or services for the client, a practice that is commonly called “directed brokerage Because brokerage commission is an asset of the client and is used to benefit that client, not the manager, such a practice does not violate any duty of loyalty Proper Usage of the CFA Marks Must be used as a subjective not as a noun - not put periods between The CFA logo certification mark is a certification mark, it must be used only to directly refer to an individual charterholder or group of charter holders The only appropriate use of the CFA logo is on the business card or letterhead of each individual CFA charterholder Research Objectivity: Best practice is for independent analysts, prior to writing their reports, to negotiate only a flat fee for their work that is not linked to their conclusions or recommendations Actions undertaken through social media that knowingly misrepresent investment recommendations or professional activities are considered a violation of Standard I(C) Conflict of Interest: Best practices dictate updating disclosures when the nature of a conflict of interest changes materially Independence and Objectivity Travel Funding: To avoid the appearance of compromising their independence and objectivity, best practice dictates that members and candidates always use commercial transportation at their expense or at the expense of their firm Should commercial transportation be unavailable, modestly arranged travel to participate in appropriate information-gathering events, such as a property tour are ok Best practice dictates that members and candidates reject any offer of gift or entertainment that could be expected to threaten their independence and objectivity Solicitations not have to benefit members and candidates personally to conflict with Standard I(B) Requesting contributions to a favourite charity or political organization may also be perceived as an attempt to influence the decision-making process Buy-side members and candidates should disclose their procedures for reporting requirements for personal transactions Members or candidates should disclose special compensation arrangements with the employer that might conflict with client interests, such as bonuses based on short-term performance criteria, commissions, incentive fees, performance fees, and referral fees If the member’s or candidate’s firm does not permit such disclosure, the member or candidate should document the request and may consider dissociating from the activity Agent Options: Disclose amount and time until expiration CFA Institute recommends that firms work to achieve the following objectives when designing policies and procedures to implement the CFA Institute-ROS: 1.To prepare research reports, make investment recommendations, and take investment actions; and develop policies, procedures, and disclosures that always place the interests of investing clients before their employees’ or the firm’s interests To facilitate full, fair, meaningful, and specific disclosures of potential and actual conflicts of interest of the firm or its employees to its current and prospective clients To promote the creation and maintenance of effective policies and procedures that would minimize and manage conflicts of interest that may jeopardize the independence and objectivity of research To support self-regulation through voluntary industry development of, and adherence to, specific, measurable, and demonstrable standards that promote and reward independent and objective research To provide a work environment for all investment professionals that supports, encourages, and rewards ethical behavior and supports CFA Institute members, CFA charterholders, and CFA candidates in their adherence to the CFA Institute Code and Standards Covered employee: Firm employee who 1) conducts research, writes research reports, and/or makes investment recommendations; or assists in the research process; 2) takes investment action on behalf of clients or the firm, or who comes in contact with investment recommendations or decisions during the decision-making process; or 3) may benefit, personally or professionally, from influencing research reports or recommendations Immediate family: Individual(s) whose principal residence is the same as the principal residence of the subject person Quiet period: Period during which covered employees are prohibited from issuing research reports or recommendations on, and publicly speaking about, a specific subject company Restricted period: A period of time during which a firm prohibits its covered employees from trading specified securities Fixed income Term Structure and Interest Rate Dynamics: Forward Rates: T* = When loan is initiated T = Tenor of the loan Forward rate = (when loan is initiated, tenor of loan) = (time until initiation + tenor) / (tenor of loan) The forward rate (Breakeven Rate): Par: The rate to discount multiple cash flows to get the present value of the bond (market price) this is also the YTM for Coupon paying bonds Spot: The rate to discount individual cash flows at given maturities – it is the YTM for a zero coupon bond The key that links the spot curve to the par curve is that you have to get the same price whether you use the par curve or the spot curve (otherwise, there would be an arbitrage opportunity) The spot curve is derived from the par curve with this relationship in mind; the process to derive the spot curve is called bootstrapping The first observation is that the forward contract price remains unchanged as long as future spot rates evolve as predicted by today’s forward curve The level movement refers to an upward or downward shift in the yield curve The steepness movement refers to a non-parallel shift in the yield curve when either shortterm rates change more than long-term rates or long-term rates change more than short-term rates The curvature movement refers to movement in segments of the yield curve: the short-term and long-term segments rise while the middle-term segment falls or vice versa Principal Component Analysis: The method to determine the number of factors—and their economic interpretation—begins with a measurement of the change of key rates on the yield curve, The next step is to try to discover a number of independent factors (not to exceed the number of variables—in this case, selected points along the yield curve) that can explain the observed variance/covariance matrix PCA creates a number of synthetic factors defined as (and calculated to be) statistically independent of each other Zero Spread or Z – Spread: Constant basis point spread that would need to be added to the implied spot yield curve so that the discounted cash flows of a bond is equal to its current market price For credit / Liquidity risk The N square root = the tenor of the loan f(1,2) is this example is the Breakeven Rate = the rate at which an investor impartial to investing for years The Forward pricewould from abeforward rate: or for years starting in year It’s a forward rate which we derive from the sport rate curve Forward Price: Swap Spread = (Swap Rate – On the run Gov Bond Yield) It’s added on to the Government Bond Yield – covers liquidity and credit risk Ted Spread = (Libor – T-bill) this measures the overall credit risk of the economy (related to economic cycles) Libor–OIS spread is considered an indicator of the risk and liquidity of money market securities / Measure of counterparty risk and risk in banking system Equilibrium term structure models One-factor or multifactor models - Assume that a single observable factor (state variable) drives all yield curve movements Both the Vasicek and CIR models assume a single factor, the short-term interest rate Note that because both models model require the short-term rate to follow a certain process, the estimated yield curve may not match the observed yield curve But if the parameters of the models are believed to be correct, then investors can use these models to determine mispricing Modern Term Structure Models: (Explains how interest rates evolve) The Cox–Ingersoll–Ross Model: CIR = dr=a(b−r)dt+σ√rd b = mean reverting level T* = Period when loan is initiated = this is a discount factor now T = Tenor of the loan = (this is a discount factor now) = (1/1+r) Following the same principles above Bootstrapping: 100 = (coupon / 1+spot1) + (Par and coupon / 1+ spot2^2) Spot1 = Par1 hence It is given Spot2 = [Par + coupon / (1 – coupon/1+spot1)] - Then take nth square root which In this case Once identified repeat steps to find spot etc… The coupons will differ as the bonds on the par curve will differ Assumes Economy has a constant long-run interest rate that the short-term interest rate converges to over time  Interest Rates are non-negative  Volatility increases with level on interest rates  Explains interest rate movements in terms of an individual’s preferences for investment and consumption as well as the risks and returns of the productive processes of the economy  Assuming that an individual requires a term premium on the long-term rate, the model shows that the short-term rate can determine the entire term structure of interest rates and the valuation of interest rate–contingent claims  σ√rdz = stochastic or volatility term which follows the random normal distribution for which the mean is zero, the standard deviation is 1, and the standard deviation factor is σ√r Term Structure of the Interest Rate Volatility: Short End: More linked to uncertainty about monetary policy Long End: More linked with uncertainty about real economy and inflation CIR and Vsicek - both have the same drift tern but differ in terms of stochastic terms The Vasicek Model: dr = a(b – r)dt + σdz Also captures mean reversion  Interest rates are calculated assuming that volatility remains constant over the period of analysis Interest Rates can be negative  The stochastic or volatility term, σdz follows the random normal distribution for which the mean is zero & the standard deviation is σdz = volatility term Arbitrage-free models, the analysis begins with the observed market prices An assumed random process with a drift term and volatility factor is used for the generation of the yield curve This calibration is typically performed via a binomial lattice-based model in which at each node the yield curve can move up or down with equal probability This probability is called the “implied risk-neutral or “risk-neutral probability Similar to Black-Scholes Model Term Structure Model: Explain the shape of the curve at a point in time  Unbiased expectations theory - Forward rates are an unbiased predictor of future spot rates Also known as the pure expectations theory  Local expectations theory - Bond maturity does not influence returns for short term holding periods Returns for all Bonds are the same over short term  Liquidity preference theory - Investors demand a liquidity premium that is positively related to a bond's maturity Long-term rates will be higher than investors' expectations of future rates  Segmented markets theory - The shape of the yield curve is the result of the interactions of supply and demand for funds in different market (i.e., maturity) segments o Segmented markets theory contends that asset/liability management constraints force investors to buy securities whose maturities match the maturities of their liabilities  Preferred habitat theory - Similar to the segmented markets theory, but recognizes that market participants will deviate from their preferred maturity habitat if compensated  Arbitrage Free Valuation Framework The first type of arbitrage opportunity is often called value additivity or, put simply, the value of the whole equals the sum of the values of the parts Binomeial Tree: Not appropriate for MBS because it is not path dependent Dominance Arbitrage Opportunity: Value 1L = (½) [(V2U + C) / (1 + r1L)] + [(V2L + C) / (1 + r1L)] T0: T1: Bond A 100 105 Bond B 200 220 Bond B is dominant as I can sell (2*bond) to finance Bond B which will return 10% at maturity High Node = Middle Node x ( e N x volatility) Low Node = Middle Node / ( e N x volatility) N = Maturity of Bond Pathwise Valuation: Compare current market price vs No Arbitrage price No Arb price as PV, with coupon = PMT and F = zer0 and N = Maturity Pathwise valuation calculates the present value of a bond for each possible interest rate path and takes the average of these values across paths One of the benefits of a lognormal distribution is that if interest rates get too close to zero, the absolute change in interest rates becomes smaller and smaller Negative interest rates are not possible Number of paths = (number of cash flows – 1) Use Binomial if bonds have options Ratchet bonds are floating-rate bonds with both issuer and investor options Ratchet bonds, in a nutshell:  Coupon rate starts very high: much higher than the borrowing rate for the issuer  There’s a formula for resetting the coupon rate periodically  The rate can remain the same or decrease at a reset date; it cannot increase  Whenever the coupon is reset lower, the bondholder has an option to put the bond at par You can think of the reset as a sort of call option: the issuer calls the existing bond and replaces it with another callable bond that has a lower coupon rate, and the same maturity as the existing bond Steps: Specify a list of all potential paths through the tree Determine present value of a bond along each potential path Calculate the average PV Bond Price across all possible paths Monte Carlo Simulation: Used to simulate sufficinetly large number of intrest rates paths Use it when cash flows are path dependent e.g when the cash flow change depending on where the intres rates are In MBS as interest rate drops , prepayment goes up Parallel shift in the yield curve will impact the effective duration of a portfolio hence portfolios with the same effective duration should have the same change in price 1.Simulate numerous paths based on probability assumption 2.Generate sport rates from the simulation 3.Determine cf for all paths and compute present value Add a constant to all spot rates at each paths so value = market value When you have a long position in a forward or futures contract, you have, in essence, already purchased the underlying asset (you simply haven’t paid for it yet); therefore, you have positive duration If you have a short position in a forward or futures contract, you have, in essence, already sold the underlying asset (you simply haven’t been paid for it yet); therefore, you have negative duration Furthermore, the duration will be comparable to that of the underlying bonds (Slightly different because these contracts don’t transfer ownership of the interim coupon payments, but comparable.) The same holds true for options: if you’re long calls or short puts, you have positive duration; if you’re short calls or long puts, you have negative duration Here, however, the duration will be shorter – possibly much shorter – that that of the underlying bonds; it will depend on the option’s delta Valuation: Bonds with embedded options Convertibles Forced Conversion: Issuer will call the bond when the underlying share price increases above the conversion price in order to avoid paying further coupons Conversion Ratio: Par Value / Conversion Price Conversion value: Current Market Share Price × Conversion ratio Minimum Value of a Convertible Bond: Conversion value or the option-free straight bond value Market conversion price: Current Convertible Bond Market Price / Conversion ratio Market conversion premium per share: Market conversion price – Current Market Share Price Market conversion premium ratio: Market conversion price / Underlying share price Premium over straight Value: Current Convertible Bond Market Price / Straight Bond Price Underlying Share Price > Conversion Share: Bond exhibits mostly stock risk–return characteristics The call on the underlying is ITM and hence the price movements of the stock have significant impact on the bond The bond is more likely to be exercised by the bondholder Binominal Calibration: Iteration of discount rates until the value of the bond = market value of the bond Soft Put: The issuer may redeem the convertible bond for cash, common stock, subordinated notes, or a combination of the three What Is Volatility and How Is It Estimated Estimating historical interest rate volatility by using data from the recent past with the assumption that what has happened recently is indicative of the future Estimate interest rate volatility is based on observed market prices of interest rate derivatives (e.g., swaptions, caps, floors) This approach is called implied volatility Callable Convertibles: The issuer may the call option and redeem the bond early if interest rates are falling or if its credit rating is revised upward and issue new debt at a lower cost Busted Convertible = Call is OTM Constructing the Binomial Interest Rate Tree Describe the process of calibrating a binomial interest rate tree to match a specific term structure Assume that volatility Underlying Share Price < Conversion Share Price Bond exhibits mostly bond risk– return characteristics Call is OTM and hence share price movements have limited impact This is stronger when there is less time to maturity OAS: Added to the one-period forward rates on the tree to produce a value or price for a bond OAS = Average spreads over the Treasury spot rate curve Identify the impact the change in interest rate volatility will have on the option and Bond Price at the old and new volatility assumption Assuming r is constant (1+r+OAS), compare the OAS that needs to be added to get market price using (Bond Price / (1+ r) + OAS) Market Price will be lower because it is discount by (1+r+OAS) If two bonds have the same characteristics and credit quality, they should have the same OAS If not the bond with the largest OAS is likely to be underpriced relative to the bond with the smallest OAS Z-Spread: = OAS - Option Cost Option Free Bond = [Credit Spread +Liquidity Spread] Callable Bond = [Credit Spread + Liquidity Spread] - Option Cost Putable Bond = [Credit Spread+ Liquidity Spread] - Option Cost As interest rates decline, the value of the straight bond rises, but the rise is partially offset by the increase in the value of the call option Call option will be ITM because it is likely to be called by Issuer Rate of Price Rises when Interest Rates Decline: Straight Bond > Callable Bond > Putable Bond Rate of Price Rises when Interest Rates Rise: Straight Bond > Putable Bond > Callable Bond The decrease in the bond price is partially offset by the put, but net the price of the putable will decrease Shape of Yield Curve: Flattens/ Invents = call option is ITM and gains in value (Bond is less likely to be called) Upward sloping = put is ITM gains in value (Bond is more likely to be put) The Option cost for a put is negative and the option cost for a call is positive hence the OAS for all will be the same It is the change in Option cost that increases or decreases the Z –spread and hence the discount rate for the Bond Estate Puts (Death Puts): Putable by the heirs after the death of the bondholder Bond Value hence depends on life expectancy as well as interest rate movements A prime example is a sinking fund bond (sinker), which requires the issuer to set aside funds over time to retire the bond issue, thus reducing credit risk From the issuer’s perspective, the combination of the call option and the delivery option is effectively a long straddle One-sided durations—that is, the effective durations when interest rates go up or down—are better at capturing the interest rate sensitivity of a callable or putable bond than the (two-sided) effective duration, particularly when the embedded option is near the money Bond will be more sensitive to one direction of interest movement – which is the direction with the highest durations Key rate durations (partial durations), which reflect the sensitivity of the bond’s price to changes in specific maturities on the benchmark yield curve Thus, key rate durations help identify the “shaping risk” for bonds—that is, the bonds sensitivity to changes in the shape of the yield curve (e.g., steepening and flattening) Can be -ve Key rate duration for a Par Bond = rate matching the tenor of the Bond So a 10 year par bond would only be affected by 10 year key rate Callable Bond = Bond - Call Puttable Bond = Bond + Put Callable and Puttable = Bond – Call + Put Convertible = Bond + Call Value of floored floater = Value of straight bond + Value of embedded floor Bondholder option Protects against rate decreases Positive Convexity: When interest rates are high and the value of the call option is low, the callable and straight bond both has positive convexity Negative Convexity: Only call option near the money The reason is because when interest rates decline, the price of the callable bond is capped by the price of the call option if it is near the exercise date Putable bonds have more upside potential than otherwise identical callable bonds when interest rates decline In contrast, when interest rates rise, callable bonds have more upside potential than otherwise identical putable bonds Zero-volatility spread is a commonly used measure of relative value for MBS and ABS Value of capped floater = Value of straight bond - Value of embedded cap (Issuer option) Protects against rate increases because the coupon rate is capped at a specific rate As a consequence, a sinking fund bond benefits the issuer not only if interest rates decline but also if they rise Putable and extendible bonds are equivalent, except that their underlying optionfree bonds are different Credit Analysis Models The expected loss is equal to the probability of default multiplied by the loss given default The present value of the expected loss is the largest price to pay on a bond to a third party (e.g., an insurer) to entirely remove the credit risk of purchasing and holding the bond Uses Risk-neutral probability The present value of the expected loss is the preferred measure because it includes the: -Probability of default -Loss given default -Time value of money -Risk premium in its computation Importance PV of expected loss Expected loss Default probability Traditional Credit Models: Credit scoring – Ordinal: Ranks most risky to least risky Credit Rating- does not provide an estimate of the loan’s default probability Negative: not explicitly depend on the business cycle Asset Backed Securities: Do no default when payment is missed Unlike corporate debt, an ABS does not go into default when an interest payment is missed Structural or Reduced Form or Monte Carlo to Value The credit risk measures used for corporate or sovereign bonds can be applied: probability of loss, expected loss, and present value of the expected loss Probability of Default does not apply Monte Carlo: A constant is added to all interest rates on all paths such that the average present value for each benchmark bond equals its market value Structural Models: Balance sheet structure: debt (zero-coupon bond) and assets Equity owners: Equivalent to holding a long European call option on firm’s assets Debt owners: Equivalent to holding risk free bond selling European put on firm’s assets Reduced Form Models: Nothing is constant Reduced form models were originated to overcome a key weakness of the structural model the assumption that the company’s assets trade  The riskless rate of interest is stochastic  The state of the economy can be described stochastic variables that represent the macroeconomic factors influencing the economy  Default Probability is also stochastic The second assumption allows interest rates to be stochastic Allowing for this possibility is essential to capture the interest rate risk inherent in the pricing of fixed-income securities Only the term structure evolution must be arbitrage free For a given state of the economy, whether a company defaults depends only on company-specific considerations, because given the macroeconomic state variables, a company's default represents idiosyncratic risk A company-specific action could be that the company’s management made an error in their debt choice in years past, which results in their defaulting now Management error is idiosyncratic risk, not economy-wide or systematic risk The loss given default explicitly depends on the business cycle through the macroeconomic state variables This allows, for example, that in a recession the loss given default is larger than it is in a healthy economy This assumption is also very general and not restrictive (PV expected loss > Expected Loss) = Risk Premium is dominant In other words, in the absence of a risk premium; the present value of the expected loss will be less than the expected loss Can use Historical Estimation = Hazard rate estimation is a technique for estimating the probability of a binary event, like default/no default More flexible than implicit estimation Using the reduced form model, the value of debt is determined by calculating the expected discounted value of debt after adjusting for risk Value of the company’s debt = Expected discounted payoff of company debt, if there is no default +Expected discounted payoff of debt if default occurs Historical Estimation: one uses past time-series observations of the underlying asset’s price and standard statistical procedures to estimate the parameters Assumptions  the company’s assets trade in frictionless markets that are arbitrage free,  the riskless rate of interest, r, is constant over time ( means there is no interest rate risk)  the time T value of the company’s assets has a lognormal distribution Implicit Estimation: also called calibration, uses market prices of the options themselves to find the value of the parameter that equates the market price to the formula’s price Uses Black-Scholes model – hence no risk premium has to be estimated The problem with implicit estimation, of course, is that if one uses a misspecified model that is inconsistent with the market structure then the resulting estimates will be biased This problem can be avoided by historical estimation The probability of default depends explicitly on the company’s assumed liability structure This explicit dependency of the probability of default on the company’s liability structure is a limitation of the structural model The fact that one needs to estimate the markets equity risk premium in the computation of the company default probability is a weakness of the structural model Expected Loss: Notional * 1-e- default density * loss given default * N Probability of Default: – e- default density * N Negatives: - Can’t use historical estimation – because asset prices are not observable (they not trade) - Balance sheet will have a liability structure much more complex than zero-coupon bond - Interest rates are not constant over time - The asset’s return volatility is constant, independent of changing economic conditions/ business cycles -Credit risk measures are biased because implicit estimation procedures inherit errors in the model's formulation Credit Default Swaps Credit Default Swaps: The designated instrument is usually a senior unsecured obligation, which is often referred to as a senior CDS, but the reference obligation is not the only instrument covered by the CDS Any debt obligation issued by the borrower that is pari passu (ranked equivalently in priority of claims) or higher relative to the reference obligation is covered CDS Index: ↑credit correlation = ↑cost of insurance CDS Spread / Credit Spread: Periodic payments from buyer to seller ≈ Probability of default x loss given default bps Long / Short: Hence, the CDS industry views the credit protection seller as the long and the buyer as the short This point can lead to confusion because we effectively say the credit protection buyer is short and the credit protection seller is long Credit Events (decided by a committee): Filing of Bankruptcy Failure to Pay scheduled interest or principle Restructuring change in seniority or reduction in coupon / principle forced on the borrower (not a default event in the US) Succession Event: Event that changes the debt issuer or structure of the reference entity is and their obligations Payout ratio = (1 – Recovery rate (%) Payout amount = (pay-out ratio × Notional) (N / N of Index Entities)* Notional = Exposure to Specific Entity If a firm in the Index defaults, remove (Notional/ N of Index) * Notional from Notional Index CDS are typically more liquid than single-name CDS The hazard rate is the probability that an event will occur given that it has not already occurred Once the event occurs, there is no further likelihood of its occurrence Probability of survival: p (%) in year * p (%) in year n Upfront payment = (PV of protection leg – PV of premium leg) Upfront premium % ≈ (Credit spread – Fixed coupon) × Duration = (100 – Price of CDS in currency per 100 Par) Credit Spread = Upfront Payment / Duration + Fixed Coupon Price of CDS in currency per 100 par = (100 – Upfront premium %) Profit for the buyer of protection ≈ (Change in spread in bps * Duration * Notional) Alternatively % Change in CDS price = (Change in spread in bps × Duration) The CDS indices also permit some opportunities for a type of arbitrage trade If the cost of the index is not equivalent to the aggregate cost of the index components, the opportunity exists to go long the cheaper instrument and short the more expensive instrument Basis Trading: CDS is the base CDS spread > Bond Credit Spread Positive Basis = short bond (pay 1.5%) and sell CDS to profit (receive 3%) Profit = ∆ CDS spread < Bond Credit Spread Negative Basis = long bond (receive 3%) and buy CDS (pay 1.5%) Profit = ∆ Don’t Forget: (Bond Yield – Libor) = Credit Spread of the Bond CDS: Sell = long Buy = short Curve Trading: Buying a CDS of one maturity (deteriorating credit) and selling a CDS on the same reference entity with a different maturity (improving credit) Cheapest to deliver = Bond trading at lowest of notional meaning where the % par value is the lowest If bond trades 25% of par and there was another bond trading at 40% of par the 25% of par would be cheapest to deliver The maturity is not relevant in choosing the cheapest to deliver it’s the priority if claims that’s important index CDS are typically more liquid than single-name CDS INTERCORPORATE INVESTMENTS AFS HFT FVPL DFV HTM Balance Sheet Income Statement OCI Market Value Interest Payments Unrecognised P/L FV – Amortised Value (for debt) Market Value Unrecognised P/L and Interest Payments Amortised Cost Effective rate * cost Equity Method: ROA and ROE are Higher because Assets and Equity (Denominator is lower)  Liabilities and leverage = Lower  Net Profit Margin, ROE, and ROA = Higher  D/E is also ↓because Equity under Acquisition method includes the non-controlling stake so numerator is ↑  Current Ratio is ↑due to lower denominator US GAAP: IFRS Allows to value at FV under Equity Method Only Allows FV for VC’s, mutual funds etc The choice of equity method or proportionate consolidation does not affect reported shareholders’ equity Consolidation Process: Non-controlling investments: Reversals are prohibited under both IFRs and US GAAP Acquirer takes 100% of Targets:  Tangible and Intangible Assets at FV (includes measureable and probable contingent assets and liabilities)  Current Assets at FV  Liabilities at FV  Revenues  Expenses  Remove costs paid for target from cash on Acquirer’s IS Among the other factors that are considered in determining whether the Group has significant influence are representation on the board of directors (supervisory board in the case of German stock corporations) and material intercompany transactions The existence of these factors could require the application of the equity method of accounting for a particular investment even though the Groups investment is for less than 20% of the voting stock Under US GAAP, an acquirer is identified, but the business combinations are categorized as merger, acquisition, or consolidation based on the legal structure after the combination Minority interest = (the proportion not owned of Target) x (Equity of Target) If Acquirer does not owe 100%, remove proportion of income not owed from Income Statement Control is present when 1) The investor has the ability to exert influence on the financial and 2) Operating policy of the entity is exposed, or has rights, to variable returns from its involvement with the investee Assuming Consolidation: Acquirer Shareholder Equity Post Merger = (Capital Stock + Value of stock paid for target + Retained Earnings) Value of stock paid for target = Portion bought + non-controlling interest Contingent Assets and Liabilities: Both IFRS and US GAAP not re-measure equity classified contingent consideration; instead, settlement is accounted for within equity Acquisition Method Equity will be higher by the amount of the minority interest Assets and Liabilities are highest here Under IFRS, the cost of an acquisition is allocated to the fair value of assets, liabilities, and contingent liabilities Contingent liabilities are recorded separately as part of the cost allocation process, provided that their fair values can be measured reliably Subsequently, the contingent liability is measured at the higher of the amount initially recognized or the best estimate of the amount required to settle ROA and ROA will be different under Partial Goodwill compared to Full Goodwill This will change over time as the results of change in equity Differences between IFRS and U.S GAAP treatment of intercorporate investments include:  Unrealized foreign exchange gains and losses on AFS securities are recognized on the IS under IFRS and as OCI under U.S GAAP  IFRS permits either the partial goodwill or full goodwill method to value goodwill and non-controlling interest in business combinations U.S GAAP requires the full goodwill method Restructuring Cost IFRS and US GAAP not recognize restructuring costs that are associated with the business combination as part of the cost of the acquisition Instead, they are recognized as an expense in the periods the restructuring costs are incurred IFRS FVPL or FVOCI HTM Reclassification from HFT is severely restricted → Reclassification from HFT is severely restricted ← ← → Under the full method (IFRS and US GAAP), minority interest is (% x Fair Value of Target) Under the partial method (only IFRS), minority interest is (% x Fair Value of Net Assets) US GAAP Off Balance sheet financing: SPE / SPV have to be consolidated if a firm (sponsor) has significant beneficial interest in the SPE even if no voting rights Move gains & loss from OCI into IS on date of transfer ∆ in Value (FV – Amor Cost) reported in IS ← ` Contingent assets: Measured at the lower of acquisition date fair value or the best estimate of the future settlement amount AFS → ← → Reclassification from HFT is severely restricted Previous ∆ in Value from OCI are amortised using effective rate method in IS ∆ in Value (FV – Amor Cost) reported in OCI HFT Under US GAAP, contractual contingent assets and liabilities are recognized and recorded at their fair values at the time of acquisition Non-contractual contingent assets and liabilities must also be recognized and recorded only if it is more likely than not and they meet the definition of an asset or a liability at the acquisition date Subsequently, a contingent liability is measured at the higher of the amount initially recognized or the best estimate of the amount of the loss Cumulative P/L previously recognized in other OCI is recognized in IS Previous ∆ in Value from OCI are amortised using effective rate method in IS ∆ in Value (FV – Amor Cost) reported in OCI If an SPE is reversed, on the consolidated balance sheet, the accounts receivable balance will be the same since the sale to the SPE will be reversed upon consolidation on the balance sheet Hence consolidating an SPV reverses it US GAAP only: VIE When below points are present Has to be consolidated when total equity at risk is insufficient to finance equity without outside support Equity investors lack ability to make decisions and lack obligation to absorb losses Sponsor = party absorbing majority of losses / retains risk of default Under IFRS, SPEs cannot be classified as qualifying Acquisition Method Goodwill: Measurement Goodwill Impairments IFRS Partial Goodwill Fair value of consideration 80% of Fair value of identifiable Net assets Goodwill recognized 800,000 720,000 80 IFRS and US GAAP Full Goodwill Fair value of entity Fair value of identifiable Net Assets 1,000,000 900,000 Goodwill recognized 100,000 Partial Goodwill = lower Equity hence higher Debt / Full Goodwill = higher Equity hence lower Debt Bond Amortisation: Amortisation of coupon = (Interest Paid – Interest Income) Interest Paid = (Coupon x begin Par) Interest Income = (Effective market rate x Cost) what we actually get If Bond is purchased at a premium, deduct amortisation every year until it reaches par at maturity If Bond is purchased at a discount, add amortisation every year until it reaches par at maturity HTM Ending BS Value = Amortised Bond Value Unrealised P/L = (MV – Amortised Value) which goes into OCI Effective Interest Rate Method: Beginning Value of a Bond issued at Premium or Discount after N periods PMT => Coupon FV => Notional I/Y => Effective Rate N => (Years to maturity at issuance – years gone by since) PV = Bond Amortised Value at N period after issuance IFRS: Cash generating Units Recoverable Amount < Carrying Value = Impairment Record ∆ and deduct from Goodwill Amount allocated until its zero Then reduce non-cash assets US GAAP: Reporting Unit (Fair Value < Carrying Amount) = Impairment (Carrying Amount of Goodwill - Implied FV of Goodwill) = Cash Amount of Impairment (Fair Value – Net Identifiable Assets) = Implied Goodwill Fair value of reporting unit 1,300,000 Less: net assets 1,200,000 Implied goodwill 100,000 Current carrying value of goodwill 300,000 Less: implied goodwill 100,000 Impairment loss 200,000 After the goodwill of the reporting unit has been eliminated, no other adjustments are made automatically to the carrying values of any of the reporting unit’s other assets or liabilities Under both IFRS and US GAAP, the impairment loss is recorded as a separate line item in the consolidated income statement IFRS and US GAAP recognize in-process research and development acquired in a business combination as a separate intangible asset and measure it at fair value (if it can be measured reliably) Reversal of impairments Financial Assets: IFRS Available for sale = Equities No reversals = Debt can be reversed if increase in FV can be objectively related to an event occurring after the impairment loss HTM = Reversals allowed up to original value Reclassification of equity securities under the new standards is not permitted as the initial designation (FVPL or FVOCI) is irrevocable Equity Method Goodwill in Purchase Price when Investment in Excess of Book Value Purchase price 100,000.00 - Ownership % * book value of target equity 66,000.00 = Excess purchase price 34,000.00 Attributable to net assets (MV – BV) of Plant and equipment * % ownership 9,000.00 (MV – BV) of Land * % ownership 6,000.00 Goodwill (residual) 19,000.00 Excess purchase price 34,000.00 This method is used when (BV ≠ MV) and there is an excess over book value The additional step is to amortise the value of the excess purchase price over the relevant period and exclude it from the income that is going to the acquirer The excess will be attributable to PPE or Land most likely Value of Investment in Associate using Equity Method* only time we use amortisation of excess Purchase Price +Acquirer share * Net Income - Acquirer share * dividends paid -Amortisation of excess (because BV ≠ MV) Ending BS Value US GAAP: No Reversals for Financial Assets Available for sale = Reversals are allowed but cannot exceed new cost basis on P/l instead any reversals beyond go into OCI The disappearance of an active market because an entity’s financial instruments are no longer publicly traded is not evidence of impairment Impaired Equites: Significant changes in the technological, market, economic, and or legal environments that adversely affect the investee and indicate that the initial cost of the equity investment may not be recovered A significant or prolonged decline in the fair value of an equity investment below its cost Business Model Test: To be measured at amortized cost, financial assets must meet two criteria:  A business model test: The financial assets are being held to collect contractual cash flows; and  A cash flow characteristic test: The contractual cash flows are solely payments of principal and interest on principal Recycling: Under AFS at sale of assets remove unrecognised P/L from OCI into Income Statement The shareholders’ equity section of the post-acquisition consolidated balance sheet will consist of the capital stock and retained earnings account of the parent and the non-controlling interest of the minority shareholders EMPLOYEE COMPENSATION: POST-EMPLOYMENT and SHARE-BASED Net Pension Liability = (Beginning PV of the Defined benefit obligation – Beginning FV of Plan Assets) Funded Status = (Fair value of the plan assets – PV of the Defined benefit obligation) = (Ending funded status – Employer contributions – Beginning funded status) Total Periodic Pension Cost = Economic Pension Expense = (Change in Funded Status - Contributions) = (Change in Assets - Change in Liability) - Contribution Periodic Pension Costs under IFRS and US GAAP IFRS Service Cost Past Service Cost Net Interest Income or Expense r x net pension liability / asset Re-measurements Actual Return - (r x Plan Assets) Actuarial gains and losses Assumption US GAAP Recognition Service Cost Income Statement Past Service Cost OCI and amortised over service life Interest expense on pension obligation Income Statement Expected return on plan assets will reduce the costs Income Statement Recognition Income Statement Income Statement OCI NOT Amortised Actual return – (Plan Assets × Expected return) Actuarial Gains / Losses Impact of Assumption on Balance Sheet OCI or Amortised using corridor method to P/L Impact of Assumption Higher discount rate ↓beginning PBO & ↓Service cost ↑ or↓ Interest cost Periodic pension costs will typically be lower because of lower opening obligation and lower service costs Higher rate of compensation increase Higher obligation Higher service costs Higher expected return on plan assets No effect, because fair value of return plan assets is used on balance sheet Not applicable for IFRS PBO Jan +Current Service Cost +Past Service Cost +/- Plan Amendments +Interest Costs -Benefits Paid +/-Actuarial P/L +Employee Contribution PBO Dec Plan Assets Jan +Actual Return of Plan Assets +Employer Contribution +Employee Contribution -Benefits Paid Plan Assets Dec Retroactive curtailment is a form of past service benefit for the frim hence will decrease pension cost Retroactive benefit i.e past service cost increases the pension expense Vested BO < Accumulated BO TPPE) = financing use of funds = equivalent to repayment of loan – use money from CFF to pay for something in CFO ↓CFF and ↑ CFO by the after tax figure Deficit = (Employer Contribution < TPPE) = financing source of funds = equivalent to getting a loan – hence take out money from CFO and increase CFF ↑CFF and ↓CFO by the after tax figure After-tax $ by which the firms contribution exceeds/below total pension cost Amount = (Company Contributions - Pension Cost) * – tax 10 2 Active Risk = (Rp-Rbm) = σ of Active Risk = Tracking error IR will be zero for an index that meets target Active risk squared = (active factor risk + active specific risk) active specific risk also called security selection risk Tracking error is a synonym for tracking risk or active risk Small - Big (SMB) > = Small Cap Tilt < = Large Cap Tilt High - Low (HML) > = Growth Tilt < = Value Tilt In a macroeconomic factor model, the factors are surprises in macroeconomic variables that significantly explain returns - develop time series of the factors (surprise) series first then sensitivities are estimated using regression In a fundamental factor model, the factors are attributes of stocks or companies that are important in explaining cross-sectional differences in stock prices- factors are returns - specify the factor sensitivities (attributes) first.Beta’s are standardized (x – μ)/ σX and specified first then returns are estimated using regression In a statistical factor model, statistical methods are applied to historical returns of a group of securities to extract factors that can explain the observed returns of securities in the group Factor sensitivity is a measure of the response of return to each unit of increase in a factor, holding all other factors constant (Factor sensitivities are sometimes called factor betas or factor loadings) The arbitrage pricing theory (APT) describes the expected return on an asset (or portfolio) as a linear function of the risk of the asset with respect to a set of factors Economics and Investment Markets Good times = low utility of consumption and Bad times = high utility of consumption Marginal utility of consumption (MUC) diminishes as people become wealthier Inter-temporal rate of substitution = ( 𝑀𝑈𝐶 tomorrow / Ratio of MUC today) Real Rate Return inverse relationship with future consumption relative to current consumption Period Rf return = (1 – inter temporal rate of sub) / inter temporal rate of sub To account for inflation risk, implied premium = (future pay off / price) – (1+ Rf+ average inflation + premiums) Output Gap = (Actual GDP – Potential GDP) Credit spread = Yield - BEI – Real Rf Breakeven Inflation Rate = (Yield on nominal indexed bonds - Yield on inflation indexed bonds) i.e (nominal yield – real yield) ↑ MUC = ↓required real rate ↓ MUC = ↑required real rate ↑Future Asset Price = ↓ MUC today ↓Future Asset Price = ↑MUC today Negative Covariance Upward Sloping Yield = future↑ interest rates, ↑ risk premium or (↓in rates + ↑in premium) or (↑inflation / (e) inflation) Assets with low or –ve correlation with bad times = Good hedge hence low risk premium Interest rates are positively related to GDP growth rate and to the expected volatility in GDP growth due to a higher risk premium For the required return to be less than the risk-free rate, the asset’s risk premium would need to be negative I.e investors want to pay a premium to hold risk Term spread (difference in yield between long dated government bonds and shortdated government bonds) Algorithmic Trading and High-Frequency Trading At its most basic, an algorithm is “a sequence of steps to achieve a goal,” and algorithmic trading is “using a computer to automate a trading strategy Recession: Short-term policy rates tend to be low Investor expectations about higher future GDP growth and inflation as the economy comes out of recession lead to higher longer-term rates This leads to positive slope of the yield curve Conversely, an inversely sloping yield curve is often considered a predictor of future recessions or lower interest rates Execution Algorithms - used to break down large orders and execute them over a period of time Volume-weighted average price (VWAP) - historical trading volume distribution over a day Implementation shortfall - dynamically adjusts the schedule of the trade in response to market conditions to minimize the difference Market participation algorithms - slices the order into segments intended to participate on a pro-rata basis The real default-free interest rate=yields on short-term inflation-protected government bonds High-Frequency Trading Algorithms - When and what to trade – focus on profit Equities are generally cyclical; they have higher values during good times and have poor consumption hedging properties +ve risk premium This risk premium will generally rise with the maturity of these bonds because longer-dated government bonds tend to be less negatively correlated with consumption, and therefore less useful as a consumption hedge Delta neutral strategies - trader can profit from the time decay of the option or from changes in volatility, irrespective of the direction in price of the underlying security Latency is the time difference between stimulus and response – the lower the better i.e quicker to process and trade Creating a new pairs trade is called instantiating a new instance of an algorithm Genetic tuning, in which many thousands of permutations of algorithms are run in parallel and fed with real market data but are not necessarily trading live in the market - Darwinian trading Keep +ve P/L and kill –ve P/L Complex event processing—a platform specifically designed for complex analysis and response to high-frequency data and low latency Painting the tape: A trader manipulates the top of the book to make the market price go in one direction before executing its own trade at the more favourable price – i.e to drive the price up and sell at higher price The term quote stuffing refers to one such practice in which large quantities of fictitious orders are rapidly entered into the market by an algorithm and then just as quickly cancelled Sharp increases in profit growth occur at the end of a period of recession, but in some cases, while recession conditions still persist Thus, corporate profitability can lead an economy out of recession as well as into it Real Estate cash flows have low sensitivity to the business cycle but capital appreciation has high sensitivity Discount rate applied to expected cash flows = (real default-free interest rate + expected inflation + other risk premia) On average, over time, according to the Taylor rule, a central bank’s policy rate should comprise the sum of an economy’s trend growth plus inflation expectation It is the aggregated opportunity cost of all investors that will determine the price of this asset today and its return over the investment horizon One interpretation of an upward-sloping yield curve is that short-dated bonds are less positively (or more negatively) correlated with bad times than are long-dated bonds Hence they are a better hedge for bad times, Long-dated bonds then must have a higher risk premium because they are not such a good hedge – in other words I would want more compensation for a less efficient hedge Liquidity aggregation refers to the process of monitoring a number of trading venues and then compiling the data into a "super book" that summarizes price and liquidity across these markets Growth stocks small caps = High PE (Buy during Good times) Value Stocks mature firms = Low PE (Buy during Bad times) Cyclical adjusted PE CAPE is derived in the same way as the P/E P = real (or inflation-adjusted) price of the equity market E= 10year moving average of the market’s real (or inflation-adjusted) earning Wash Trading: Investor buys and sells the same instrument simultaneously stimulates demand and boost trading volume Discount rate is positively related to inflation meaning high inflation = (high discount) = (low market price) Bad points: Acceleration and accentuation of market movements, Increased risk profile due to speed of execution The average level of real short-term interest rates is positively related to the trend rate of growth of the underlying economy and also to the volatility of economic growth in the economy 39 Analysis of Portfolio Management: Value added is defined as the difference between the return on the managed portfolio and the return on a passive benchmark portfolio This difference in returns might be positive or negative after the fact but would be expected to be positive before the fact or active management would generally not be justified Basic Fundamental Law: Basic fundamental law (un-constrained active weights): Expected Active Return = IC x √ BR x Active Risk -> There is no TC because it is unconstrained IR = IC x √BR using optimal weights hence Active Return = IR x ∆w* Value Added = Sum of (Active weights * Expected Return) affected by leverage ∆Risk Man pf = √sum of (∆w2 *∆vol2) Active Return = (Pf return – BM return) Active Return Decomposition = Asset allocation Security selection = [(Active weight * BM r of Equity) + (Active weight * BM r of Bonds)] + (Equity w * Active r + Bond w * Active r) The Full Fundamental Law: Full fundamental law (constrained): IR = (TC) * IC * √BR Expected Active Return = (TC) * IC * √BR * Active Risk (4 Terms) Security Selection = (Value Added - Asset Allocation) Expected Return = (IR x Pf σ) The Fundamental Law of Active Management separates the expected value added, or portfolio return relative to the benchmark return, into the basic elements of the strategy, which are identified correctly as: Alpha = Portfolio return – (BM return xβ) Sharpe ratio: Absolute reward-to-risk measure - unaffected by the addition of cash or leverage in a portfolio Hence if we want to reduce volatility we will increase cash which has zero volatility and hence will not affect Sharpe ratio Information Ratio: Unaffected by the aggressiveness of active weights but affected by cash and leverage Meaning we cannot adjust active weights by combining with cash Skill (information coefficient) Portfolio construction (transfer coefficient) Breadth (number of independent decisions per year) Aggressiveness (benchmark tracking risk) same as active risk IR for BM is always zero and Rf is also zer0 The Sharpe Ratio of the Benchmark is essentially the IR for the Benchmark The last three of these four elements may be beyond the control of the manager if they are specified by investment policy or constrained by regulation IR will drop if risk is added unless pf is UNCONSTRAINED then It wont Level of cash to keep in portfolio = 1- (target level of volatility/current volatility) Tracking error is a synonym for tracking risk or active risk A well-executed passive investment strategy would achieve the lowest amount of tracking error, whereas an aggressive active equity manager would be expected to have the highest tracking error IR = (Rp – Rbm) / σ (Rp – Rb) Optimal Portfolio Sharpe = Sqrt of (Sharpe Ratio Bm2 + IR2) Optimal level active risk = 𝑰𝑹 𝑩𝑴 𝑺𝒉𝒂𝒓𝒑𝒆 Total risk or Rp σ (Bm rσ2 + ∆rσ2) BM risk + Active risk * BMσ = ∆σ* To find the optimal weights to get highest Sharpe: Pf weight = (Pf active weight / Optimal active risk) which is the optimal portfolio BM weight = (1 – Pf weight) How to increase aggressiveness: Short benchmark or reduce exposure to it The Fundamental Law: µ= investors subjective expected return The idea is to have high correlation between: µ= investor’s subjective forecasted return Ra = Realised active return Ra = Realised active return ∆W = Active weights wi = Active weights In such a situation –ve forecasted returns will produce-ve realised returns and active weights will be –ve too IC = manager's skill for a market timer IC = 2(% correct) – = Risk weighted correlation between active returns and forecasted active returns i.e [µ and Ra] = Forecasting ability TC = degree of constraints on manager's active management (for unconstrained portfolio its 1) = Correlation * (µ/∆σ*), (∆w * ∆σ) = Correlation between actual active weights and optimal active weights i.e [µ and wi] = Cross sectional correlation between forecasted active returns and actual weights adjusted for risk BR = number of independent active bet N = number if decisions and r = correlation Hence BR = N / + (n-1) r Assuming individual decisions are correlated, not independent Investment Policy Statement: Investor's risk tolerance is included under objectives Constraints include:  Liquidity needs  Time horizon  Tax concerns  Legal and regulatory factors  Unique circumstances Investment objectives and constraints are identified and specified Investment strategies are developed Portfolio composition is decided in detail Portfolio decisions are initiated by portfolio managers and Implemented by traders Portfolio performance is measured and evaluated Investor and market conditions are monitored Any necessary rebalancing is implemented Liabilities and long-term wealth target are each direct determinants of an individual investor's ability to accept risk Market expectations will affect return achieved but is not a direct determinant of an investor's ability to accept risk In a passive investment strategy approach, portfolio composition does not react to changes in expectations; an example is indexing, which involves a fixed portfolio designed to replicate the returns on an index Performance measurement is the calculation of portfolio rates of return Performance attribution is the analysis of those rates of return to determine the factors that explain how the return was achieved Performance appraisal assesses how well the portfolio manager performed on a risk-adjusted basis, whether absolute or relative to a benchmark A tax concern is any issue arising from a tax structure that reduces the amount of the total return that can be used for current needs or reinvested for future growth Diversification reduces a portfolios active specific risk; therefore, the portfolio with the lowest active specific risk is likely to be the most diversified µ = [IC *σ * Score] = Standardised forecasts of return The indicates over performance and -1 would be under performance Zero would be a score of neutral Optimal Active Weight (∆w*) = (µ / ∆σ*2) * (∆σ* / IC* √BR) = (investor’s subjective forecasted return / optimal active risk 2) * (optimal active risk / IC * √BR) In summary, the information ratio is active return over active risk, in contrast to the excess return-to-risk measure known as the Sharpe ratio Information ratios help investors focus on the relative valued added by active management A long-term investor’s labor income may also be an asset sufficiently stable to support a higher level of portfolio risk Cash may be safe for a short-term investor but risky for a long-term investor who will be faced with continuously reinvesting These are the active weights we want to use so if bm w is 0.3 and we have ∆w* of 0.2 our Pf w will be 0.5 Which active w + bm w The information ratio will be unaffected by the reallocation from the active fund into the benchmark This change will reduce both the level of the active return and also the active risk such that the information ratio is unchanged 40 Economics Currency Exchange Rates: Determination and Forecasting Bid (sell) < Offer (buy) International Fisher: Interest rate differential = Expected inflation differential Large Notional, high credit risk, long forward contracts = wide spreads Price / Base = Foreign / Domestic Real interest rate parity: Real interest rates will converge to the same level across different markets if both uncovered interest rate parity and ex ante PPP hold Triangular Arbitrage: 1) Identify which source has the undervalued quote Compute Cross Rates only to identify if there is an arbitrage opportunity don’t use the cross in the triangle 2) Start with the currency stated in the question – during the transaction buy the undervalued currency at the quote of the source where it’s undervalued Sell the overvalued currency at quote of and source Make sure to arrive at the currency started with Forward Discount / Premium: Premium: if > id Discount: id>if 𝑺𝒑𝒐𝒕 𝒇/𝒅 = (rd − rf) > (Forward − Spot) / Spot: Borrow Foreign (rd − rf) < (Forward − Spot) / Spot: Borrow Domestic (𝒊𝒇 − 𝒊𝒅) ∗ 𝒕/𝑻 (𝟏 + 𝒊𝒅) ∗ 𝒕/𝑻 Assessing Equilibrium (IMF Approaches): The macroeconomic balance approach estimates how much exchange rates need to adjust in order to close the gap between the medium-term expectation for a country’s current account imbalance and that country’s normal (or sustainable) current account imbalance Focus on real flows The external sustainability approach focuses on stocks of outstanding assets or debt and calculates how much exchange rates would need to adjust to ensure that a country’s net foreign-asset/GDP ratio or net foreign-liability/GDP ratio stabilizes at some benchmark level (Deals with capital account focus is on financial assets) A reduced-form econometric model seeks to estimate the equilibrium path that a currency should take on the basis of the trends in several key macroeconomic variables, such as a country’s net foreign asset position, its terms of trade, and its relative productivity Hybrid approach using current / capital Statistics Mark to Market: Open contract and take opposite position to close If opened a year delivery, in month we sell months to close On valuation date, compute PV of (new fx rate – fx rate at initiation) * Notional The new fx rate = current spot +/- forward premium/discount) At this point we sell the currency we had bought at initiation hence difference will be in the other currency If we are closing a contract and using a swap to extend it, the mid-market rate is the new spot rate for closing and extension The above models have some power in predicting real exchange rate direction but are poor in predicting magnitude Parity Relations: (long run movements between fx, Interest rates and inflation) Covered Interest Rate Parity: A riskless arbitrage relationship in which an investment in a foreign money market instrument that is completely hedged against exchange rate risk should yield exactly the same return as an otherwise identical domestic money market investment Overvalued Currency & Trade Deficit: Encourage Depreciation because FX is too↑ so we need ↓FX and (X < M) so again we need↓FX to increase exports 𝑺𝒑𝒐𝒕 𝒇/𝒅 = (𝟏 + 𝒊𝒇) ∗ 𝒕/𝑻 (𝟏 + 𝒊𝒅) ∗ 𝒕/𝑻 Uncovered Interest Rate Parity: States that the current forward exchange rate is an unbiased predictor of the future spot rate High Yielding Currency will depreciate in order for UIRP to hold and there is no Arbitrage condition that forces UIRP to hold If – id = expected ∆ in S f/d The exchange rates must change so that the return on investments Interest rate differential = S f/d * (1+fr / 1+dr) – S f/d with identical risk will be the same in any currency F f/d = S f/d * 1+ (rf-rd) If Both CIRP and UIRP hold: Forward exchange rate = unbiased forecast of the future spot exchange rate and The real rate will be zero Absolute PPP: Equilibrium exchange rate between two countries is determined entirely by the ratio of their national price levels, after adjusting for price levels Assumes all goods are identical, tradable and have the same weight Relative PPP: The percentage change in the spot exchange rate will be completely (f/d) determined by the difference between the foreign and domestic inflation rates Costs are constant over time Holds in the long run ∆Spot = ∆ inflation = (inf f – inf d) Ex-Ante PPP focuses on expected changes in the spot exchange rate being entirely driven by expected differences in national inflation rates ∆Spot = ∆ expected inflation = e (inf f) – e (inf d) Future Spot = Spot (f/d)* (1+ e infl f - e inf d) Equilibrium Currency Rates According to PPP Overvalued Currency & Trade Surplus: Unclear because Currency because FX is too↑ so we need↓FX but (X > M) so we need↑FX to make exports more expensive &↑ imports Undervalued Currency & Trade Surplus: Encourage Appreciation because FX is too↓ and because we have (X>M) again encourage↑FX to bring it equilibrium Undervalued Currency & Trade Deficit: Unclear because Currency because FX is too↓ so we need↑FX but (X< M) so we need↓FX to bring it up to equilibrium Capital Flows: Push Factor: Pushes capital out of country due to low rates Pull Factor: Pulls capital into country due to high interest rates / low debt levels (pushes FX rates up and creates asset bubbles) Intervention & Controls: If there is no inflation threat, the authorities could engage in unsterilized intervention – excludes FX intervention If inflation is a concern, then this intervention would need to be a sterilized intervention operation - FX intervention If all of the above failed to stop the capital flow–induced upward pressure then capital controls might have to be considered as a final line of resistance in preventing capital flows from pushing currency values and asset prices to undesirable levels A Model That Includes Long-Term Equilibrium: Assumes Uncovered IRP holds and real exchange rate is expected to converge to its long-run equilibrium value Long run Equilibrium FX Rate = qf/d + [(rd−rf) – (Φd – Φf) - (φd−φf)] Φ = risk premia that reflects perceived sustainability of external balances Negative skew, more peaked around the mean and fat tails, indicating larger and more frequent losses – high probability of crash risk - due to leverage    Manage downside risk FX volatility filters Valuation filters using PPP will provide a high and low bound for exchange rates 41 Impact of Balance of Payment: BOP’s are monetary transaction – persistent current account deficit will lead to domestic currency depreciating Countries that run persistent current account surpluses often see their currencies appreciate over time During a deficit a currency appreciation is required to bring it back up to a surplus The Taylor Rule: Policy rate is a function of a central bank’s:  Neutral rate  Inflation  Output targets, and observed deviations from those targets Current account = real economy flows Capital Account = financing / investment flows (dominant factor in FX) Policy rate = neutral rate + inflation + α (inflation – target inflation) + β (output– potential output) How current account surpluses can affect exchange rates The Flow Supply/Demand Channel - focuses on purchases & sales of internationally traded goods and services require the exchange of domestic and foreign currencies in order to arrange payment The Portfolio Balance Channel - Current account imbalances shift financial wealth from deficit nations to surplus nations Over time, this may lead to shifts in global asset preferences, which impacts the path of exchange rates Surplus nations will build up on fx reserves Investors are assumed to hold a diversified portfolio of domestic and foreign assets, including bonds Their desired allocation is assumed to vary in response to changes in expected return and risk consideration In the long run, governments that run large budget deficits on a sustained basis could eventually see their currencies decline in value The Debt Sustainability Channel - According to this channel, there should be some upper limit on the ability of countries to run persistently wide current account deficits Rising debt levels at unsustainable levels = downward exchange move Mundell – Flemming: only Monetary and Fiscal policy are important Describes how changes in monetary and fiscal policy affect the level of interest rates and economic activity within a country, which in turn leads to changes in the direction and magnitude of trade and capital flows and ultimately to changes in the exchange rate Expansionary Monetary =↓Rates &↑Investment and consumption spending -> Capital outflows Restrictive Monetary=↑Rates -> Capital Inflows Expansionary Fiscal =↓Tax ↑Government Spending, leads to↑ Rates and capital inflow Large deficits will need to finance If a country follows an expansionary fiscal policy, government borrowing will increase leading to an increase in interest rates This increase in interest rates will attract capital inflows, leading to an appreciation of the currency Restrictive Fiscal = ↓Spending and ↑Taxes ↑ Rates Low capital mobility: The impact of monetary and fiscal policy changes on domestic interest rates will not induce major changes in capital flow Monetary and fiscal policy effects on exchange rates will operate primarily through trade flows Carry Arbitrage transaction T0: Borrow x in low yield currency Convert x into high yield currency at S f/d and invest it T1: Loan = (x * 1+r ly) in yield currency Proceeds = (x * 1+r hy) and convert into low yield at F f/d (Proceeds – Loan) = Arbitrage Profit Volatility Filters: Open carry trade when average fx volatility falls below threshold Close carry trade when average fx volatility is above threshold If the funding currency falls below its PPP-implied value, the likelihood of its appreciation increases At this point it becomes wise to exit the trade: buying the funding currency at this point and thus effectively closing out the position limits risk EM countries are better able to influence their exchange rates because their reserve levels as a ratio to average daily FX turnover are generally much greater than those of DM countries This means that EM central banks are in a better position to affect currency supply and demand than DM countries where the ratio is negligible EM policymakers use their foreign exchange reserves as a kind of insurance to defend their currencies, as needed Options: The level or trend in currency risk reversals can be used to correctly anticipate future exchange rate movements The evidence indicates that there exists a high, contemporaneous correlation between the trend in risk reversals and the trend in exchange rates, but no statistically significant relationship exists between lagged risk reversal data and future exchange rate movements Therefore, risk reversals are capable of confirming an exchange rate’s trend but cannot predict it FX Options and Deal Flow: A risk reversal is a currency option position that consists of the purchase of an out-of-the-money (25 delta) call and the simultaneous sale of an out-of-the-money (25 delta) put, both on the base currency in the P/B exchange rate quote and both with the same expiration date Restrictive fiscal & monetary policy leads to an improvement in the trade balance hence bullish for currency Both result in ↑IR Expansionary fiscal & monetary policy mix will be bearish for a currency because the trade balance under such conditions would deteriorate Fixed FX Rates Expansionary Monetary Policy: FX rates will depreciate but because the rates are fixed, the central Bank will have to buy its own currency in the market to support from weakening beyond the fixed rate and hence will deplete its FX reserves Hence limited to reserves the central bank holds This will tighten domestic credit because the central bank is buying and holding its currency hence it will offset the intended expansion policy Expansionary Fiscal: Central Bank has to sell currency and build FX reserves to remain the fixed rate This will expand the domestic money supply and hence reinforce the expansion policy Supports Intended action Sticky The Dornbusch Overshooting Model: Model of the exchange rate that assumes output prices exhibit limited flexibility in the short run but are fully flexible in the long run Equity Market Trends and Exchange Rates: The long-run correlation between the US equity market and the dollar is very close to zero, but over short- to medium-term periods, rolling correlations tend to swing from being highly positive to being highly negative It allows gauging whether FX market is attaching a higher probability to a currency appreciation or depreciation High contemporaneous probability but no statistically significant relationship between lagged risk reversal data In forex markets, dealer order books have valuable inside information not available to general market participants Technical analysis using FX dealer order books focuses on this market imperfection Macroeconomic variable on Recessions  In the period leading up to a crisis, the real exchange rate is substantially higher than its mean level during tranquil periods  The trade balance displays no significant difference between its behaviour in pre-crisis periods and in tranquil periods  Foreign exchange reserves tend to decline precipitously as the crisis approaches  On average, there is some deterioration in the terms of trade in the months leading up to a crisis  Inflation tends to be significantly higher in pre-crisis periods compared with tranquil periods  The ratio of M2, a measure of money supply, to bank reserves tends to rise in the 24-month period leading up to a crisis and then plummets sharply in the months immediately following a crisis  Broad money growth in nominal and real terms tends to rise sharply in the two years leading up to a currency crisis, peaking around 18 months before a crisis hits  Nominal private credit growth also tends to rise sharply in the period leading up to a crisis  Currency crises are often preceded by a boom–bust cycle in financial asset (equity) prices  Real economic activity displays no distinctive pattern ahead of a crisis but falls sharply in the aftermath 42 Economic Growth and the Investment Decision: Why potential growth matters: The idea is that potential GDP is the maximum amount of output an economy can sustainably produce without inducing an increase in the inflation rate High output gap or slack means there is downward pressure on inflation Central banks may need to reduce short term rates to spur economic activity which in turn will increase bond prices TFP = Function of Technology and (Labour x capital) -> implies a proportionate increase in output for any combination of inputs Marginal product of capital The Cobb–Douglas production function exhibits diminishing marginal productivity with respect to each individual input Marginal productivity is the extra output produced from a one-unit increase in an input keeping the other inputs unchanged A value of α close to zero means diminishing marginal returns to capital are very significant and the extra output made possible by additional capital declines quickly as capital increases Profit Maximisation: Firms will add capital until marginal product of capital (rental price of capital) = (marginal product of labour) Real interest rate = Real wage rate (rate of steady state) Real wage rate (MPL) = (1 - αL) / Y Real Interest Rate (MPK) = αL * Y / Capital Constant return to scale: (if you double input, output will double too) Diminishing Marginal Productivity: (each extra unit of output produced with additional input will decline as we increase L or K) Growth Accounting: To calculate (Y) potential output Production function expressed as change in rates: Y/Y = ∆/TFP + (α* ∆K) + (1-α)*∆L α = the elasticity of output in respect to K (change in Y for 1% change in K) Potential output per capita as function of Labour: y = TFP + F (k, 1) k = K/L = capital / labour = L/L y = Y/L= output / labour Labour productivity Method Growth rate in potential output = (Long term growth rate of labour force + Long term growth rate in Labour productivity) Capital Deepening: An increase in the capital to labour ratio and reflects a move along the per capita production function Once capital to labour ratios reaches high levels – increase in output per worker from increase in capital to labour ratio becomes insignificant DM Countries have high capital-labour ratios hence gain little from capital deepening – rely on TFP EM Counties have low capital – labour ratio hence gain a lot from capital deepening and TFP Capital: ICT: Impact of IT on economic growth – leads to network externalities which increase economic growth Non ICT: Results in capital deepening – temporary impact on economic growth High correlation between physical capital and GDP can be explained by capital deepening in the short run and TFP in the long run Labour quality / Human Capital  Accumulated knowledge and skills acquired from education and training and life expectancy  Public Infrastructure increases productivity of private investments Theories of Growth Classical (Malthusian): Growth in real GDP per capita only grows until subsistence level then falls due to population explosion In the long run technology results in larger population not richer population Contends that there is a subsistence real wage, defined as the minimum real wage necessary to support life Neoclassical (Solow): Equilibrium growth occurs when the economy grows at steady state rate of growth (indicates capital deepening is occurring but has no impact on growth rate) Both Labour and Capital are variable factors and suffer from diminishing marginal productivity but show constant marginal product of capital Only TFP change will impact growth rate of output per worker Steady State Growth Rate: Growth rate of output per capita (y) = (TFP g/ – α) or simply put (TFP g / labour share of GDP) Growth rate output (Y) = (TFP g / – α) + ∆L/L or simply put (TFP g / labour share of GDP) + growth in labour force Absolute convergence: Output of emerging countries will converge to those of developed countries regardless of characteristics Conditional: Depends on EM country having same savings rate, population growth and production function as DM country Club: Poor members of the club will grow to level of rich members assuming same (economic policy, monetary union and techonology) Because of the capital inflows, the physical capital stock in the developing countries should grow more rapidly than in rich countries even if the saving rate is low in the poorer countries Faster capital growth will result in higher productivity growth, causing per capita incomes to converge Endogenous Growth Theory: Focuses on explaining TFP- includes human capital and R&D into Capital Human capitals have positive externalities and spill over effects Hence R&D results in growth for entire economy which prevents diminishing marginal returns to capital Savings and Investment will generate self-sustaining growth permanently No income convergence possible due to permanent growth Private firms often fail to consider external social benefits and hence will not invest in the optimal level of R&D for the economy as a whole A more open trade policy will permanently raise the rate of economic growth Convergence can happen (based on conditional requirements) Empirical evidence shows poor countries are diverging Growth rate of per capital income Savings r - (Y/K) – labour force growth Trade Barriers: Removal of barriers: Savings / Investments↑ which makes countries more efficient Resources shift to comparative advantage countries Allows import of technologies Resource curse: First, countries rich in natural resources may fail to develop the economic institutions necessary for growth The Dutch disease: Where currency appreciation driven by strong export demand for resources makes other segments of the economy, in particular manufacturing, globally uncompetitive Although population growth may increase the growth rate of the overall economy, it has no impact on the rate of increase in per capita GDP Increase in participation rate however will increase growth per capita GDP ↑Labour force participation growth =↑output and Output per capita ↑Population growth =↑ increases output ≠↑ Output per capita ↑Net Migration growth = ↑Potential output (if country is running out gap, net migration will make it worse) 43 Economics of Regulation: Regulatory bodies include government agencies and independent regulators granted authority by a government or governmental agency Some independent regulators may be self-regulating organizations Typically, legislative bodies enact broad laws or statutes Regulatory bodies issue administrative regulations, often implementing statutes Courts interpret statutes and administrative regulations and these interpretations may result in judicial law SRO – regulating bodies with regulatory authority given by the government (some are independent / some are not) If independent, they are immune from political pressure – most likely to face conflict of interest with members They typically regulate behaviours and provide public goods in the form of standards (do not set price mechanism) Reasons for relying on Self-Regulation: Information frictions Externalities Moral Hazard Reasons for decreased reliance on self-regulation: Privatization of securities exchanges Intense competition Uncertainty regarding the effectiveness of self-regulation Internationalization Strengthening of government regulators Trend toward consolidation of financial regulators Cooperative regulation Pressure to increase efficiency and lower cost Regulatory tools available to regulators: -Price mechanisms (such as taxes and subsidies) -Mandates and restrictions on behaviours (cap ratio banks) -Provision of public goods -Public financing of private projects Regulatory capture – regulators end up serving interests of the regulated firms (regulation can create demand for sector) Regulatory competition – Regulators compete to attract specific entities Regulatory Arbitrage – Firms look for loopholes and in other locations with regulations for them to best exploit Bank supervisors (whether as a function of the central bank, another entity, or combination of entities) focus on prudential supervision—regulation and monitoring of the safety and soundness of financial institutions in order to promote financial stability, reduce system-wide risks, and protect customers of financial institutions Regulatory burden refers to the costs of regulation for the regulated entity; this cost is sometimes viewed as the private costs of regulation or government burden Net regulatory burden is the private costs of regulation less the private benefits of regulation (covers direct & indirect) Regulators view some costs associated with regulation as ―unintended, two types of such costs are implementation costs that were unanticipated, and indirect costs because of unintended consequences Regulation of financial institutions Focus on protecting investors and price stability Ensuring safety and soundness of financial intuitions Smooth operations of payments and maintain availability of credit Tackle issues on unemployment and economic growth Maintaining integrity of markets and acting as a referee for its fairness Regulations of Securities Market Protecting retail investors Increase Market confidence Encourages capital formation Discount window borrowers (write checks) – firms borrow from central banks at discounts and it’s kept confidential 44 Quantitative Methods Correlation = Cov (x,y) / sdx * sdy Covariance: n-1 denominator Significance of correlation: df = n-2 Fail to reject H0: -ve critical T < t-stat < +ve critical t value p – Value > alpha Reject H0: critical t value < t-stat (absolute number) p – Value < alpha Level of significance = alpha = (1 – confidence level) Assumptions of the Linear Regression Model I The relationship between the dependent variable, Y, and the independent variable, X is linear in the parameters b0 and b1 II The independent variable, X, is not random III The expected value of the error term is 0: E(ε) = zero IV The variance of the error term is the same for all observations (if not we have heteroskedasticity) V The error term is uncorrelated across observations hence expected error term = VI The error term is normally distributed Standard Error of Estimate: df = n – Sum of squared errors = unexplained variation= standard error or error Hypothesis Testing of slope and intercept: df = n – Confidence interval = intercept –/+ (critical t x sb1) Two-tailed: Alt Hypothesis ≠ zero MSR = RSS/k MSE= SSE/n-k-1 The F-statistic tests whether all the slope coefficients in a linear regression are equal to zero The F-statistic measures how well the regression equation explains the variation in the dependent variable Single regression F-Test = square of t-test Multiple Regressions: When predicting the dependent variable using a linear regression model, we encounter two types of uncertainty: uncertainty in the regression model itself, as reflected in the standard error of estimate, and uncertainty about the estimates of the regression model’s parameters SEE = √ (SSE / (n-k-1)) with df of (n / n-k-1) F-Test H0: All slopes = Ha: at least one slope ≠0 2 R always positive but Adjusted R can be negative Heteroskedasticity - the standard errors will be underestimated and the t-statistics will be inflated Conditional heteroskedasticity Error variance is correlated with (conditional on) the values of the independent variables in the regression Corrected using generalised least squares or robust standard errors Serial correlation: Use Hansen method to correct standard errors – will also correct conditional heteroskedasticity When regression errors are correlated across observations, we say that they are serially correlated (or auto correlated) Positive serial correlation is serial correlation in which a positive error for one observation increases the chance of a positive error for another observation Negative error would be more likely to produce other negative errors First, the F-statistic to test for overall significance of the regression may be inflated because the (MSE) will tend to be underestimated Second, positive serial correlation typically shows small error values hence inflated t-stats Multicollinearity occurs when two or more of the independent variables, or linear combinations of independent variables, may be highly correlated with each other In a classic effect of multicollinearity, the R2 is high and the F-statistic is significant, but the t-statistics on the individual slope coefficients are insignificant Slope coefficient is interpreted as the change in the dependent variable for the case when the dummy variable is one The best potential remedy is to attempt to eliminate highly correlated variables The probit model, which is based on the normal distribution, estimates the probability that Y = (a condition is fulfilled) given the value of the independent variable X The logit model is identical, except that it is based on the logistic distribution rather than the normal distribution Both models must be estimated using maximum likelihood methods The intercept of the regression measures the average value of the dependent variable of the omitted category, and the coefficient on each dummy variable measures the average incremental effect of that dummy variable on the dependent variable Discriminant analysis yields a linear function, similar to a regression equation, which can then be used to create an overall score Based on the score, an observation can be classified into the bankrupt or not bankrupt category.A parameter is a measure that describes the entire population, whereas a statistic describes a sample of the population Issue Test Conditional heteroskedasticity Breusch–Pagan nR^2 No Conditional Heteroskedasticity Reject H0 if BP stat > critical value (will show low r2 if no CK) Df = k Serial correlation Durbin Watson (1-r) No Serial Correlation Reject H0: if DW stat < dw low (there is positive serial correlation) Reject H0: if DW stat > dw high (there is +ve or -ve serial correlation) Inconclusive: lower bound high bound Unit Root H0 Dickey Fuller g1 = Unit root is present (random walk) ARCH Regress r^2 Error on r^2 t-1 Error H0: No ARCH Autocorrelation in AR Time Series Autocorrelation / (1 /√T) H0:No Autocorrelation Conclusion Reject H0 if b1-1 = i.e g=0 where (b1-1 = g) nd Test the slope “a1” coefficient of the regression is= Reject H0: if test is significant and move on to AR2 model (1 /√T) = Standard Error Probabilistic Approaches: Steps Determine “probabilistic” variable Define probability distributions for these variables Check for correlation across variables: When there is correlation between inputs, we can A) pick only one of the inputs to vary; it makes sense to focus on the input that has the bigger impact on value B) we can build the correlation explicitly into the simulation As with the distribution, the correlations can be estimated by looking at the past Run the simulation Define probability distributions for these variables: This is a key and the most difficult step in the analysis.3 different approaches:  Historical data if available: Implicit in this approach is the assumption that there have been no structural shifts in the market that will render the historical data unreliable  Cross sectional data  Statistical distribution and parameters (if histor &structural are insufficient or unreliable) Pros of Simulations It yields a distribution for expected value rather than a point estimate Provides Better input estimation Allow us to not only quantify the likelihood of distress but also build in the cost of indirect bankruptcy costs into valuation A proper simulation provides us with more than just an expected value for an asset or investment Simulations with Constraints Regulatory Capital Restrictions Negative Book Value for Equity Earnings and Cash Flow Constraints Market Value Constraints We can use simulations to both assess the likelihood that these constraints will be violated and to examine the effect of risk hedging products on this likelihood All approaches’ use expected rather than risk-adjusted cash flows and the discount rate that is used should be a risk-adjusted discount rate: the risk-free rate cannot be used to discount expected cash flows Comparing the Approaches: With simulations, we use probability distributions to capture all possible outcomes Put in terms of probability, the sum of the probabilities of the scenarios we examine in scenario analysis can be less than one, whereas the sum of the probabilities of outcomes in decision trees and simulations has to equal one Exception: When using standard deviation in values from simulations as risk measure Here using risk-adjusted rate will be double counting With Scenarios, we will not have a complete assessment of all possible outcomes from risky investments or assets because we only consider best / worst case and ignore all others Simulations are a key component of Value at Risk and other risk management tools used, especially in firms that have to deal with risk in financial assets In decision trees, we try to accomplish this by converting continuous risk into a manageable set of possible outcomes 45 Alternative Investments Real Estate: Time Series Analysis: Unlike in multiple regressions, serial correlation in the error term causes estimates of the intercept (b0) and slope coefficient (b1) to be inconsistent Cannot use Durbin Watson to test autocorrelation in AR1 Model time series An autoregressive model (AR) a time series regressed on its own past values Mean Reversion = b0 / (1-b1) If a trend is persistently above or below the value of the time series, however, the residuals (the difference between the time series and the trend) are serially correlated Assumptions:  First, the expected value of the time series must be constant and finite in all periods  Second, the variance of the time series must be constant and finite in all periods  Third, the covariance of the time series with itself for a fixed number of periods in the past or future must be constant and finite in all periods We can test autocorrelation by looking at the autocorrelations of the residuals We can estimate an autoregressive model using ordinary least squares if the time series is covariance stationary and the errors are uncorrelated The standard error of the residual correlation, which is equal to (1/ √ T) (where T is the number of observations in the time series) If there is autocorrelation – add more lags of DV and IV T-stat = Residual of autocorrelations / standard error A random walk: A time series in which the value of the series in one period is the value of the series in the previous period plus an unpredictable random error If data has a random walk – we have to transform it using is first-differencing = Δxt = (xt − xt−1) = New DV Can’t use AR models or t-stats Once differencing is conducted – check if autocorrelation is present if not we can use the model Typically, for a stationary time series, either autocorrelations at all lags are statistically indistinguishable from zero, or the autocorrelations drop off rapidly to zero as the number of lags becomes large A random walk with drift has b0 ≠ compared to a simple random walk, which has b0 = zero… hence best predictor of the time series in the next period is its current value Random walk equation is a special case of an AR(1) model with b0 = and b1 = Hence best predictor of future rate is this rate If the lag coefficient is equal to 1.0, the time series has a unit root Hence on coefficients < are covariance stationary The autocorrelations of most AR time series start large and decline gradually The autocorrelations of an MA(q) time series suddenly drop to zero after the first q autocorrelations Seasonality: if present add lag of IV So change from Y= b0 +b1 (xt1- xt-1) to Y = b0 +b1 (xt1-xt-1) + b2x (t4-xt-5) assuming quarterly time series Repeat until there is no more autocorrelation Highest and Best Use = property with highest implied land value Commercial Real Estate: Office - demand for office properties depends heavily on employment growth Industrial - demand is heavily dependent on the overall strength of the economy and economic growth Warehouse - demand is dependent on the overall strength of economy & import and export activity in the economy Retail- demand depends on trends in consumer spending In turn, depends on the health of the economy, job growth, population growth, and savings rate Multi-family (apartments) - demand depends on population growth Net Lease = tenant covers operating expenses (Going-in cap rate < Terminal cap rate) = investor is willing to paying more Income approach - GOOD FOR UNSUAL PROPERTIES Annual Rental income at full occupancy + Other income (such as parking) = Potential gross income (PGI) – Vacancy and collection loss = Effective gross income (EGI) – Operating expenses (OE) Net operating income (NOI) Value = (NOI t1 / Cap Rate) where cap rate = r – g Market value = Rent/All Risk Yield - based on 1st year NOI  gross multiplier = sales price / gross income 1st year (not good) r = cap + g Property Value = PV of Resale Value of (NOI / terminal cap rate) PMT = current NOI, i/y = r, n = years until property is sold FV = resale value => for terminal value Implied going in cap = current NOI / new property value The terminal cap rate to be higher than the going-in cap rate because it is being applied to income that is more uncertain Cost Approach: Market value of the land (from comparable) +Replacement cost, including constructor’s profit +Developer’s profit Leveraged IRR: N = when Property is sold PMT = NOI – Interest expense PV = Equity FV = Sale Price at time n Compute I/Y to get leveraged IRR Sales comparison approach Pro: Sets an upper limit on the value Cons: Difficult to estimate the depreciation for a property that is older and/or has much obsolescence Depreciation adjustment: (age of subject property – age of comparable) x (depreciation of subject property) Commercial uses with higher management involvement, such as restaurants, hotels, shopping centers, also have higher operational risks One way to check this given the specifics in this case is to look at management fees as a percentage of effective gross income for the three properties Time series are COINTEGRATED if a long-term financial or economic relationship exists between them such that they not diverge from each other without bound in the long run Use (Engle–Granger) Dickey–Fuller for the COINTEGRATION test If the time series has an exponential trend: Take the natural log of the time series Then first-difference it ARCH: If a large error occurs in one period, the variance of the error in the next period will be even larger - intercept>0 H0: No Arch Test if error term is significant by computing t-test of error terms To test for ARCH Regress the R^2 residuals on the R^2 form the previous period If the estimate of the slope of the regression of the squared residuals on the lagged one period squared residuals is statistically significantly different from 0, the time series is ARCH For Log-linear values: Y1 =Y-1 + (log1 – log-1) where: (log1 – log-1) = b0 + b1 (log4 – log3) + b1 (log1 – log4) apply exponential function Use whenever the rate of change is constant Physical Deterioration -Curable = value of loss -Incurable = (eff life / total life) * (Replacement cost - Curable Deterioration) Functional Obsolescence -Curable = value of loss -Incurable = (annual loss of NOI / cap rate) Locational Obsolescence: (Total loss - loss in land value) in form of a reduction in NOI or loss value Economic obsolescence: -Value of loss (applies when replacement cost > value using current NOI) A Hedonic index requires only one sale of a property and thus can usually include more properties than a repeat sales index, but it must control for “hedonic” characteristics of the property, such as its size and location Appraisal-based index will tend to have less volatility and low correlation and lag a transaction-based index – unsmooth to mitigate issue Repeats Sales Index – uses a time series regression Transaction based Index – Leads appraisal based index and includes a random element error term DSCR = NOIt1 / (interest + principal payment) NOI – (Interest + principal pay) = cash flow Cash flow / equity = equity dividend Maximum debt service = Year NOI / DSCR Hence loan amount = Max Deb Service / mortgage interest rate Value with Renovation = (Stabilise NOI) / cap rate – (PV of loss due to renovation) Stabilise NOI = NOI once renovation is complete NOI = Rental Income + Other Income – Property Management Fee 46 REIT: Distributes 75% of Income Private Equity: Returns have a J curve UPREITs: the REIT has a controlling interest in and serves as the general partner (with responsibility for operations) of a partnership that owns and operates all or most of the properties DOWNREIT structure owns more than one partnership and may own properties at both the REIT level and the partnership level  Relatively low volatility of income / NAV pricing is more volatile & low rate of reinvestment for future growth  Access to superior quality and range of properties – landmark and trophy buildings  High income payout ratios and yield / up & down structure can cause conflict of interest Term Structure: Effective structuring of investments terms results in a balance of rights and obligations between the private equity firm and the management team Co-investment - LPs generally have a first right of co-investing along with the GP REOC – have more operating flexibility – not just limited to income producing property Short remaining lease terms provide mark-to-market opportunities on rents- especially in booming economic periods Shopping center/Retail – largest type of REITS – typically net lease Industrial property and industrial REITs are less cyclical than hotel, health care, and storage The Hotel sector is cyclical because it is not protected by long-term leases and is thus exposed to business-cycle driven short-term changes in regional, national, and international business and leisure travel Net Asset Value: Under IFRS, companies are allowed to value investment properties using either a cost model or a fair value model Cash and Equivalents GAV *Cash NOI = x 1+g Cap Rate + FFO NOI = GAV - Total Debt - Other Liabilities = NOI - SGA Expenses - Interest Expenses = EBITDA – Interest Expense = NI + Depreciation and Amortisation -/+ (P/L from property disposal ) = Gross Rental Revenue+ Other Income -Vacancy and collection loss -Insurance -Taxes -Utilities, Repairs and Maintenance Expenses = Gross Rental Revenue – Operating expenses Reserved matters: some domains of strategic importance (such as changes in the business plan, acquisitions, or divestitures) are subject to approval or veto by the private equity firm Earn-outs: (mostly in venture capital): mechanism linking the acquisition price paid by the private equity firm to the company’s future financial performance over a predetermined time horizon, generally not exceeding to years Ratchet is a mechanism that determines the allocation of equity between shareholders and the management team of the private equity controlled company It enables the management team to increase its equity allocation depending on the company’s actual performance and the return achieved by the private equity firm Ratchet will lower PE returns The LBO model is not a valuation methodology per se It is a negotiation tool that helps develop a range of acceptable prices to conclude the transaction Source of Value = earnings growth, expansion, debt reduction Discount rate = PE firm IRR Land for Development Accounts Receivable and Prepaid Assets NAV How is Value created? 1) Ability to re-engineer the private firm to generate superior returns 2) The ability to access credit markets on favourable terms 3) A better alignment of interests between private equity firm owners and the managers of the firms they control *Cash NOI = NOI – Non cash rents + Impact of acquisitions FFO -Non Cash Rent -Maintenance type capital expenditures -leasing commissions AFFO EBITDA = NOI - SGA Expenses AFFO = (FFO – NCR – Recurring Cap Ex) ↓cap rate = ↓FFO AFFO is superior to FFO as a measure of economic income because it takes into account the capital expenditures necessary to maintain the economic income of a property portfolio Neither FFO nor AFFO take into account differences in leverage Straight-line rent is the average contractual rent over a lease term and this figure is recognized as revenue under IFRS and US GAAP Intangibles and Goodwill = Soft assets FFO and AFFO may not capture the intrinsic value of all real estate assets held by the REIT or REOC – empty buildings which will produce income in the future are not factored in P/FFO does not adjust for the impact of recurring capital expenditures needed to keep properties operating smoothly – due variations in calculation The conventional argument is that the risk premium associated with REITs and REOCs should be lower than stock because the underlying business of owning income-producing real estate is less volatile due to contractual revenue streams from leases Private equity real estate portfolios are less risky than stock portfolios and have lower expected returns Private equity real estate has bond-like characteristics because of the stream of lease payments and at the same time has stock-like characteristics because of the dependency on the strength of the overall economy when leases are renewed Distribution waterfall: A distribution waterfall is a mechanism providing an order of distributions to LPs first before the GP receives carried interest Deal by deal or total return A crossover co-investment occurs when a subsequent fund launched by the same GP invests in a portfolio company that has received funding from a previous fund No-fault divorce A GP may be removed without cause, provided that a super majority (above 75%) of LPs approve that removal Tag-along: Minority shareholders have their shares bought at the same term as majority Drag-along: Majority shareholders can force minority shareholder to join a sale Gross IRR relates cash flows between the private equity fund and its portfolio companies and is often considered a good measure of the investment management team’s track record in creating value Excludes management fee and carried interest Net IRR relates cash flows between the private equity fund and LPs – includes management fee and carried interest Paid in Capital = Called capital / committed capital DPI = Distributed Capital / Paid in Capital = cash on cash return RVPI = Residual Value / Paid in Capital = unrealised return TVPI = Total Value / Paid in Capital = (RVPI + DPI) = (unrealised + distributed capital as a % of total distributed capital) Carried interest = (%) (Increase in NAV before distributions)*** NAV before distribution = NAV after distribution t-1 + called capital – management fee + operating results NAV after distributions = NAV before Distribution – Carried Interest – Distributions Management fee is based on paid in capital 1st Round POST money valuation = V/ (1 + r)t PRE money valuation = Post money – Investment Fractional Ownership = Investment / Post money N shares Investor = % x (FO / 1- FO) Price per Share = Investment / N Shares Investor 2nd Round Value / (1+R2) POST2 − Investment2 Investment2 / Post money (%inv1 + %inv2) x (FO2 / 1- FO2) Investment / N shares Inv Adjusted Discount rate = (1+r / 1-risk of failure) – Hurdle Rate = Min IRR Risks: Illiquidity Taxation risk – due to carried interest being taxed when it’s paid Market risk reliance on management of portfolio companies Loss of Capital 47 Commodity and Commodity Derivatives: An Introduction: Show an equity-like performance over the long run and highest real returns during inflation – exhibit positive skewness Commodities are a diverse asset class comprised of various sectors: energy, grains, industrial (base) metals, livestock, precious metals, and softs (cash crops) Each of these sectors has a number of characteristics that are important in determining the supply and demand for each commodity, including ease of storage, geo-politics, and weather The life cycle of commodities varies considerably depending on the economic, technical and structural (i.e., industry, value chain) profile of each commodity as well as the sector A short life cycle allows for relatively rapid adjustment to outside events, whereas a long life cycle generally limits the ability of the market to react The theory of storage emphasizes the role of inventories, and conceptually links inventories with commodity futures prices The difference between futures prices and spot prices can be explained by storage costs and the so-called convenience yield of holding specific commodities in inventory This theory predicts an inverse relationship between the level of inventories and the convenience yield—the lower the inventories, the higher the convenience yield Future price = sport + storage costs – convenience yield (hence high convenience yield = low future price) Low density = Light Crude (cheaper to extract and process) Low Energy Costs = High growth in economy The key differentiating characteristics of commodity indexes are  The breadth and selection methodology of coverage (number of commodities and sectors) included in each index, noting that some commodities have multiple reference contracts  The relative weightings assigned to each component/commodity, and the related methodology for how these weights are determined  The methodology and frequency for rolling the individual futures contracts  The methodology and frequency for rebalancing the weights of the individual commodities and sectors  The governance that determines which commodities are selected Total return = Spot return + Roll return + Collateral return + Rebalancing return Total index return = Spot return + Roll return + Collateral return + Rebalancing return + Futures Return Excess return = Futures return = (Spot return + Roll return) Arbitrageurs have the ability to hold inventory of physical commodities and can capitalize on mispricing between the commodity (along with storage and financing cost) versus the futures price by purchasing the commodity in the spot market and holding it in storage until a future date The term “sweet” refers to low sulfur content, a contaminant that destroys metal piping because of its form as sulphuric acid     Theoretically, the spot return is the component of the commodity futures return that is most strongly correlated with unexpected inflation The roll return results from the extension of the futures contract and the shape of the term structure curve If the commodity market is in backwardation (contango), the rolling from the maturing to the next shortest futures contract generates positive (negative) income Also impacted by sector diversification or concentration o Roll Return has small impact in the short term but large impact in the longer run Collateral return is based on the assumption that the whole futures position is collateralized by cash Interest is thus paid on this capital at the US Treasury bill rate, which is explicitly considered in the total return index o Collateral % * risk free rate Rebalancing: Rebalancing to original fixed weights entails selling contracts that have appreciated in value and purchasing contracts that have lost value In volatile markets, contracts that have risen in value will be sold and contracts that have lost value will be purchased This results in a significant positive return in markets that are flat in the long term and volatile in the short term Speculators cannot take inventory – only bet on price movements Under the Insurance Theory, the shape of the futures price curve can be explained by producers of a commodity (i.e market participants that are long the physical good) selling the commodity for future delivery in order to hedge their exposure to price risk The Hedging Pressure Hypothesis extends the insurance perspective to include consumers who hedge long positions, not solely producers with short positions The spot return index is a general indicator of existing price trends in commodity markets, and cannot be used as a performance measure or for comparison with other financial asset returns In contrast to the excess return indexes, the total return index is based on a fully cash-collateralized commodity investment Hence, in the long run, tremendous return differences can arise between the total and the excess return indexes The roll return is effectively the weighted accounting difference (in percentage terms) between the near-term commodity futures contract price and the farther-term commodity futures contract price The insurance perspective argues that risk premiums are available if hedgers use commodity futures to avoid commodity price risk Hedgers (producers) hold commodities in stock, and therefore must have a short position in commodity futures To attract speculators, hedgers must offer an insurance premium Assumes normal backwardation The hedging pressure hypothesis argues that investors will receive a risk premium that is a positive excess return for going short in a “normal contangoed” commodity futures market Hedging pressure theory – curve is influence by demand and supply of producer’s vs users Producers are selling and users are buying hence: If (producers > users) = downward slope = Backwardation If (users > producers) = upwards slope = Contango 48 Portfolio Man - VAR The use of VaR as a risk measure has the following advantages: Simple concept VaR is relatively easy to understand Although the methodology is fairly technical, the concept itself is not very difficult, so decision makers without technical backgrounds should be able to grasp the likelihood of possible losses that might endanger the organization Reporting that a daily 5% VaR is, for example, €2.2 million allows the user to assess the risk in the context of the capital deployed If a portfolio is expected to incur losses of a minimum of €2.2 million on 5% of the trading days, about once a month, this information is valuable in the context of the size of the portfolio Easily communicated concept VaR captures a considerable amount of information into a single number If the recipient of the information fully understands the meaning and limitations of VaR, it can be a very significant and practical piece of information Provides a basis for risk comparison VaR can be useful in comparing risks across asset classes, portfolios, and trading units, giving the risk manager a better picture of which constituents are contributing the least and the most to the overall risk As such, the risk manager can be better informed as he looks for potential hot spots in the organization This point will be discussed further in Section 2.4 Facilitates capital allocation decisions The ability to compare VaR across trading units or portfolio positions provides management with a benchmark that can be used in capital allocation decisions A proprietary trading firm, for example, can find that its VaR in equity trading is $20 million, and its VaR in fixed-income trading is $10 million If its equity trading portfolio is not expected to take more risk than its fixed-income trading portfolio, then the equity trading activities are taking too much risk or there is too much capital allocated to equity trading The firm should either make adjustments to realign its VaR or allocate capital in proportion to the relative risks If a firm is looking to add a position to a portfolio or change the weights of existing portfolio positions, certain extensions of VaR allow the manager to assess the risk of these changes This topic will be covered in more detail in Section 2.4 Can be used for performance evaluation Risk-adjusted performance measurement requires that return or profit be adjusted by the level of risk taken VaR can serve as the basis for risk adjustment Without this adjustment, more profitable units could be perceived as more successful but, when adjusted by VaR, a less profitable unit that poses less risk of loss may be judged more desirable Reliability can be verified VaR is easily capable of being verified, a process known as backtesting For example, if the daily VaR is $5 million at 5%, we would expect that on 5% of trading days, a loss of at least $5 million would be incurred To determine whether a VaR estimate is reliable, one can determine over a historical period of time whether losses of at least $5 million were incurred on 5% of trading days, subject to reasonable statistical variation Despite its many advantages, users of VaR must also understand its limitations The primary limitations of VaR are the following: Subjectivity In spite of the apparent scientific objectivity on which it is based, VaR is actually a rather subjective method As we saw in the descriptions of the three methods of estimating VaR, there are many decisions to make At the fundamental level, a decision must be made as to the desired VaR cutoff (5%, 1%, or some other cutoff); over what time horizon the VaR will be measured; and finally, which estimation method will be used As we have seen here, for each estimation method, there are numerous other discretionary choices to make about inputs, source of data, and so on Underestimating the frequency of extreme events In particular, use of the normal distribution in the parametric method and sometimes in the Monte Carlo method commonly underestimates the likelihood of extreme events that occur in the left tail of the distribution In other words, there are often more extreme adverse events, called “left tail events,” than would be expected under a normal distribution As mentioned previously, there is no particular requirement that one use the normal distribution Although the historical simulation method uses whatever distribution the data produce, we chose to illustrate the Monte Carlo method with a normal distribution, and it is virtually always used in the parametric method Nonetheless, the tendency to favor the normal distribution and other simple and symmetrical distributions often leads to an understatement of the frequency of left tail events Failure to take into account liquidity If some assets in a portfolio are relatively illiquid, VaR could be understated, even under normal market conditions Additionally, liquidity squeezes are frequently associated with tail events and major market downturns, thereby exacerbating the risk Although illiquidity in times of stress is a very general problem that affects virtually all of the financial decisions of a firm, reliance on VaR in non-normal market conditions will lead the user to underestimate the magnitude of potential losses Sensitivity to correlation risk Correlation risk is the risk that during times of extreme market stress, correlations among all assets tend to rise significantly Thus, markets that provide a reasonable degree of diversification under normal conditions tend to decline together under stressed market conditions, thereby no longer providing diversification Vulnerability to trending or volatility regimes A portfolio might remain under its VaR limit every day but lose an amount approaching this limit each day Under such circumstances, the portfolio could accumulate substantial losses without technically breaching the VaR constraint Also, during periods of low volatility, VaR will appear quite low, underestimating the losses that could occur when the environment returns to a normal level of volatility Misunderstanding the meaning of VaR VaR is not a worst-case scenario Losses can and will exceed VaR Widely accepted by regulators In the United States, the SEC requires that the risk of derivatives positions be disclosed either in the form of a summary table, by sensitivity analysis (a topic we cover later), or by VaR Thus, VaRs are frequently found in annual reports of financial firms Global banking regulators also encourage banks to use VaR These regulations require or encourage the use of VaR, but they not prescribe how it is implemented, which estimation method is used, or the maximum acceptable VaR PM: High Frequency Trading Execution: Implementation shortfall – algorithm will balance market impact vs market drift Market Participation - exploit periods in the market when volumes is high or low Oversimplification Although we noted that VaR is an easily communicated concept, it can also oversimplify the picture And although VaR does indeed consolidate a considerable amount of information into a single number, that number should be interpreted with caution, with an awareness of the other limitations, and supported by additional risk measures Disregard of right-tail events VaR focuses so heavily on the left tail (the losses) that the right tail (potential gains) are often ignored By examining both tails of the distribution, the user can get a better appreciation of the overall risk–reward trade-off, which is often missed by concentrating only on VaR HFT: Quote event – algorithm spots a new bid/offer of a certain quantity Regulatory oversight: Fictitious order – entering orders and cancelling to distract other algos Pros: Minimised market impact of large trades Lower cost of execution Improved market efficiencies More open and competitive trading markets Negatives: Fear of unfair advantage Spoofing Increased Risk profile Excessive order repositioning Increased difficulty if controlling the market (trading venues not publish liquidity) 49 A Term and Reversion Valuation – Real Estate Assuming lease has n more years remaining at 400k, then 450 in n period Term Reversion: Find PV of 400 over n years with discount rate r = Term Value After the n period Resale Value = 450 / new r Discount Resale Value by n periods by new r to get PV of Resale Value Add PV of Resale Value + Term Value = Capital Value Layer Method: Term Value = 400/r 450 – 400 = 50 = top slice i.e difference in rent Find perpetuity value of top slice = 50/r Discount perpetuity value of top slice by n Private Equity 50 51 Under IFRS, factors and (dealing with revenues and sales prices) rank higher than factors and (dealing with operating costs) when determining functional currency Under IFRS when the functional and financing Step 1: Given the assumptions about benchmark interest rates, interest rate volatility, and a call and/or put rule, calculate the OAS for the issue, using the binomial model Disclosures about pensions and post-employment benefits including actuarial assumptions are normally disclosed in the Notes to the Financial Statements Not in Management discussion and analysis Step 2: Impose a small parallel shift to the interest rates used in the problem by an amount equal to +Di Binary Logistic Regression is a special type of regression where binary response variable is related to a set of explanatory variables, which can be discrete and/or continuous Step 3: Build a new binomial tree using the new yield curve Step 4: Add the OAS to each of the 1-year forward rates in the interest rate tree to get a "modified" tree (We assume that the OAS does not change when the interest rates change.) Step 5: Compute the new value for V+ using this modified interest rate tree Step 6: Repeat steps through using a parallel shift of -DI to obtain the value for V- Step 7: Use the formula duration = (V- + V+) / 2V0(DI) 52 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Current ratio = Current assets ÷ Current liabilities Quick ratio = (Cash + Short-term marketable investments + Receivables) ÷ Current liabilities excludes inventor Cash ratio = (Cash + Short-term marketable investments) ÷ Current liabilities excludes accounts receivable Defensive interval ratio = (Cash + Short-term marketable investments + Receivables) ÷ Daily cash expenditures Receivables turnover ratio = Total revenue ÷ Average receivables Days of sales outstanding (DSO) = Number of days in period ÷ Receivables turnover ratio Inventory turnover ratio = Cost of goods sold ÷ Average inventory Days of inventory on hand (DOH) = Number of days in period ÷ Inventory turnover ratio Payables turnover ratio = Purchases ÷ Average trade payables Number of days of payables = Number of days in period ÷ Payables turnover ratio Cash conversion cycle (net operating cycle) = DOH + DSO – Number of days of payables Working capital turnover ratio = Total revenue ÷ Average working capital Fixed asset turnover ratio = Total revenue ÷ Average net fixed assets Total asset turnover ratio = Total revenue ÷ Average total assets Gross profit margin = Gross profit ÷ Total revenue Operating profit margin = Operating profit ÷ Total revenue Pretax margin = Earnings before tax but after interest ÷ Total revenue Net profit margin = Net income ÷ Total revenue Operating return on assets = Operating income ÷ Average total assets Return on assets = Net income ÷ Average total assets Return on equity = Net income ÷ Average shareholders’ equity Return on total capital = Earnings before interest and taxes ÷ (Interest bearing debt + Shareholders’ equity) Return on common equity = (Net income – Preferred dividends) ÷ Average common shareholders’ equity Tax burden = Net income ÷ Earnings before taxes Interest burden = Earnings before taxes ÷ Earnings before interest and taxes EBIT margin = Earnings before interest and taxes ÷ Total revenue Financial leverage ratio (equity multiplier) = Average total assets ÷ Average shareholders’ equity Total debt = The total of interest-bearing short-term and long-term debt, excluding liabilities such as accrued expenses and accounts payable Debt-to-assets ratio = Total debt ÷ Total assets Debt-to-equity ratio = Total debt ÷ Total shareholders’ equity Debt-to-capital ratio = Total debt ÷ (Total debt + Total shareholders’ equity) Interest coverage ratio = Earnings before interest and taxes ÷ Interest payments Fixed charge coverage ratio = (Earnings before interest and taxes + Lease payments) ÷ (Interest payments + Lease payments) Dividend Payout ratio = Common share dividends ÷ Net income attributable to common shares Retention rate = (Net income attributable to common shares – Common share dividends) ÷ Net income attributable to common shares =( – Payout ratio) Sustainable growth rate = Retention rate × Return on equity Earnings per share = (Net income – Preferred dividends) ÷ Weighted average number of ordinary shares outstanding Book value per share = Common stockholders’ equity ÷ Total number of common shares outstanding 53 ... flow 0.998004 400.00 399 .20 CF2 0.98 522 2 400.00 394.09 Notional 0.98 522 2 20 ,000.00 19,704.44 Total 20 ,497.73 Equity Return = (1+ % return * Notional) Equity Return = (1.03 * 20 ,000,000) Quoted Price... Price per Share = Investment / N Shares Investor 2nd Round Value / (1+R2) POST2 − Investment2 Investment2 / Post money (%inv1 + %inv2) x (FO2 / 1- FO2) Investment / N shares Inv Adjusted Discount... Arbitrage Opportunity: Value 1L = (½) [(V2U + C) / (1 + r1L)] + [(V2L + C) / (1 + r1L)] T0: T1: Bond A 100 105 Bond B 20 0 22 0 Bond B is dominant as I can sell (2* bond) to finance Bond B which will