CFA 2018 level 3 gostudy CFA l3 equation guide for 2017

www.gostudy.io GoStudy’s CFA Exam Level ® 2017 Equation Guide www.gostudy.io Powered by GoStudy™ Everything you need to pass & nothing you don’t www.gostudy.io Guided Notes for CFA® Level 2017 Copyright â 2015 by Go Study LLC.đ All Rights Reserved Published in 2015-2016 The “CFA® and Chartered Financial Analyst® are trademarks owned by CFA Institute CFA Institute does not endorse, promote, review, or warrant the accuracy of the products or services offered by www.gostudy.io Certain materials contained with this text are the copyrighted property of the CFA Institute The following is the copyright disclosure for those materials: “Copyright, 2015, CFA Institute Reproduced and republished from 2015 Learning Outcome Statements, Level III CFA® Program Materials, CFA Institute Standards of Professional Conduct, and CFA Institute’s Global Investment Performance Standards with permission from CFA Institute All rights reserved.” Disclaimer: These guided notes condense the original CFA Institute study material into 240 pages It is not designed to replace those notes, but to be used in conjunction with them While we believe we cover all of the core concepts accurately we cannot guarantee nor warrant that this is true Use of these notes is not a guarantee of exam success (although we think it will help a lot) and we cannot be held liable for your ultimate exam performance About www.gostudy.io Go Study offers in-depth exam strategies and subject review to help candidates pass the CFA exams In addition to this equation guide we offer full guided notes, a mobile app for on-the-go review with hundreds of notecards, and last-minute cram material such as equation lists and “week before” summary sheets We also highly recommend candidates subscribe to our free newsletter for exclusive offers, access to study tips, tricks, and in-depth discussions of the exam We also periodically provide bonus resources such as mock exams, practice problems, and more to our subscribers If you have any questions regarding this product, the exam, or the company’s future, please contact us via the website We strive to answer every question a candidate has www.gostudy.io GoStudy’s CFA Level – 2017 Exam Equation Sheet Contents Behavioral Finance Private Wealth Management Taxes & Private Wealth Estate Planning Institutional Investors Capital Market Expectations Capital Market Expectations – Financial Equilibrium Models Equity Market Valuation 10 Relative Valuation Models 11 Asset Allocation 12 Fixed Income 13 Bond Portfolio Management 14 Interest Rate Swaps & Options 15 International Bond Returns 15 Equity Portfolio Management 16 Alternatives 17 Risk Management 18 Currency Risk Management 19 Risk Management with Futures & Forwards 20 Options 21 Swaps 22 Portfolio Execution 22 Monitoring & Rebalancing 23 Return Calculations 23 Global Investment Performance Standards 25 www.gostudy.io Behavioral Finance Equation Formula 𝑟𝑒 = 𝑟𝑓 + 𝛽(𝑟𝑚 − 𝑟𝑓 ) CAPM Model Behavioral Asset Pricing Model (BAPM) Where: re = The required return on equity rf = Risk-free rate rm = The market return β = The stock market beta (rm-rf) = The Equity risk Premium (ERP) 𝑟𝑒 = 𝑟𝑓 + 𝛽(𝑟𝑚 − 𝑟𝑓 ) + 𝑠𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 The more analysts disagree, the wider the sentiment premium Private Wealth Management Equation     Solving for Required Return on IPS question   Formula List the client objectives Quantify their current assets This is their PV Calculate the time horizon This is n Calculate what they will need on an annual basis This is their PMT This is sometimes a predictable annual payment (like a mortgage) or the sum of their total living expenses (think time value) NET of their income (so total inflow or outflow) Make sure you apply nominal/real and pre/post tax to these inputs as well Calculate their FV This is often equal to the PV adjusted for inflation over the time horizon Calculate the % return needed This will be using your financial calculator      Comments Are you solving for after tax or before tax returns? If you are solving for nominal return (most likely) are you including expected inflation? Have you adjusted the PV (current investable assets) to account for immediate cash inflow/outflow? Have you adjusted next year’s required cash flows (CF) for inflation if needed? Are you using the correct (+/-) signs when adjusting for cash in and out? Have you adjusted the PV to account for any immediate cash flows in or out? www.gostudy.io Taxes & Private Wealth Key Principles:    The less you trade the more efficient your tax efficient your portfolio will be The longer you can defer paying taxes the more your overall portfolio will benefit (tax alpha) Relative tax rates matter when comparing future/terminal values Equation Accrual Taxes Formula 𝐹𝑉𝐼𝐹𝑖 = [1 + 𝑟(1 − 𝑇)]ⁿ Where FVIF = Future value interest factor 𝐹𝑉𝐼𝐹𝐶𝐺 = (1 + 𝑟)ⁿ − [(1 + 𝑟)ⁿ − 1]𝑡𝑐𝑔 Deferred Capital Gains Tax (with MV = Cost Basis) Deferred Capital Gains Tax (MV ≠ Cost Basis) Wealth Based Taxes Type of Tax Annual Accrual Deferred capital gains Wealth tax Or 𝐹𝑉𝐼𝐹𝐶𝐺 = (1 + 𝑟)ⁿ(1 − 𝑡𝑐𝑔 ) + 𝑡𝑐𝑔 𝐹𝑉𝐼𝐹𝐶𝐺𝐵 = [(1 + 𝑟)ⁿ(1 − 𝑡𝑐𝑔 )] + 𝑡𝑐𝑔 𝐵 B = Cost Basis/Market Value 𝐹𝑉𝐼𝐹𝑊𝑇 = [(1 + 𝑟)(1 − 𝑇𝑊)]ⁿ Higher returns mean… Greater tax drag Less tax drag (greater impact of deferring payment) LOWER tax drag Comments -The percentage of total taxes paid is > than the stated tax rate (due to compounding) -As you increase the investment time horizon (N), the tax drag increases in both $ and % terms -As you increase the investment returns (R), the tax drag increases in both $ and % terms -The loss to deferred taxes is always a constant rate regardless of time or return -Tax rate = tax drag % -Thus the value of deferring taxes increases as the time or return increase -The lower the cost basis relative to current market value the more taxes you will pay -As with accrual taxes the longer the time period the greater the tax drag % -UNLIKE accrual taxes, the greater the returns the lower the tax drag % Long investment horizon means… Higher Tax Drag Less tax drag (greater impact of deferring payment) Higher Tax Drag www.gostudy.io Blended Tax Impacts PI = Proportion or weight of total return Equation Formula Realized Tax Rate realized tax rate = PITI+ PDTD + PCGTCG Return after realized taxes 𝑅𝐴𝑅𝑇 = 𝑅 ∗ [1 − 𝑃𝐼 𝑇𝐼 − 𝑃𝐷 𝑇𝐷 − 𝑃𝐶𝐺 𝑇𝐶𝐺 ] Effective Capital Gains Tax FVIF Equation tying together RART, TECG, and cost basis Accrual Equivalent Return Accrual Equivalent Tax Rate TECG = 𝑇𝐶𝐺 (1−𝑃𝐼 −𝑃𝐷 −𝑃𝐶𝐺 ) (1−𝑃𝐼 𝑇𝐼 −𝑃𝐷 𝑇𝐷 −𝑃𝐶𝐺𝑇𝐶𝐺 ) Comments PI = Proportion or weight of total return from each source (Interest, Dividends, Capital Gains) The Ts are the tax rates Just says realized return is the return x (1 – weighted average tax rate) -This ratio will always be less than one meaning the effective capital gains tax will always be less than the stated capital gains tax -Adjusts the capital gains tax rate gain to reflect previously taxed dividends, income and realized capital gains so that they will not be double taxed - Numerator is (1 – all individual proportions of returns) and the Denominator is (1 – realized tax rate) 𝐹𝑉𝐼𝐹𝑇 = [(1 + 𝑅𝐴𝑅𝑇 )ⁿ(1 − 𝑇𝐸𝐶𝐺 )] + 𝑇𝐸𝐶𝐺 − (1 − 𝐵)𝑇𝐶𝐺 𝑁 𝐹𝑉 𝑇 𝑅𝐴𝐸 = √ −1 𝑃𝑉 𝑇𝐴𝐸 𝑅𝐴𝐸 =1− 𝑅 -Return that produces the same terminal value as the taxable portfolio You can think of it as the effective annual return, or IRR, that connects PV and FV -Use your financial calculator to solve for I/Y by plugging in FV, PV, and N I/Y represents the geometric average return for the period in question -The tax rate that makes the pre-tax return equal to the accrual equivalent return www.gostudy.io Estate Planning Equation Formula 𝐹𝑉𝑇𝑎𝑥−𝑓𝑟𝑒𝑒 𝑔𝑖𝑓𝑡 [1 + 𝑟𝑔 (1 − 𝑡𝑖𝑔 )]ⁿ = 𝐹𝑉𝑏𝑒𝑞𝑢𝑒𝑠𝑡 [1 + 𝑟𝑒 (1 − 𝑡𝑖𝑒 )]ⁿ(1 − 𝑇𝑒 ) Relative value of tax free gift vs bequest rg = Pre-tax return if asset is gifted and held by the recipient tig = The recipient’s tax rate re = Pre-tax return if asset is held by the testator/estate (re is usually equal to rg) tie = The giftor (testator)’s tax rate Te = The estate tax rate 𝐹𝑉𝑇𝑎𝑥𝑎𝑏𝑙𝑒 𝑔𝑖𝑓𝑡 (1 − 𝑡𝑔 )[1 + 𝑟𝑔 (1 − 𝑡𝑖𝑔 )]ⁿ = 𝐹𝑉𝑏𝑒𝑞𝑢𝑒𝑠𝑡 [1 + 𝑟𝑒 (1 − 𝑡𝑖𝑒 )]ⁿ(1 − 𝑇𝑒 ) Relative value of taxable gift vs bequest Value of a Taxable Gift Relative to a Bequest IF the donor pays gift taxes Generation Skipping 𝐹𝑉𝑇𝑎𝑥𝑎𝑏𝑙𝑒 𝑔𝑖𝑓𝑡 (1 − 𝑇𝑔 + 𝑇𝑔 𝑇𝑒 )[1 + 𝑟𝑔 (1 − 𝑡𝑖𝑔 )]ⁿ = 𝐹𝑉𝑏𝑒𝑞𝑢𝑒𝑠𝑡 [1 + 𝑟𝑒 (1 − 𝑡𝑖𝑒 )]ⁿ(1 − 𝑇𝑒 ) Increases value of the estate by a factor of 1⁄ (1 − 𝑡) Calculating Core Capital Needs Core capital is defined as the amount of assets needed to meet liabilities plus a reserve for unexpected needs Excess capital is whatever is left over Core capital needs are calculated as: 𝑃𝑉 (𝑆𝑝𝑒𝑛𝑑𝑖𝑛𝑔 𝑛𝑒𝑒𝑑 ) = ∑𝑁 𝑗=1 𝑝(𝑆𝑢𝑟𝑣𝑖𝑣𝑎𝑙𝑗 ) 𝑥 𝑆𝑝𝑒𝑛𝑑𝑖𝑛𝑔𝑗 (1+𝑟)𝑗 Where 𝑝(𝑠𝑢𝑟𝑣𝑖𝑣𝑎𝑙) = 𝑝(𝐻𝑢𝑠𝑏𝑎𝑛𝑑 𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑠) + 𝑝(𝑊𝑖𝑓𝑒 𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑠) − [𝑝(𝐻𝑢𝑠𝑏𝑎𝑛𝑑 𝑆𝑢𝑟𝑣𝑖𝑣𝑒𝑠) 𝑥 𝑝(𝑊𝑖𝑓𝑒 𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑠] www.gostudy.io Institutional Investors Spending Rules Spending rules are used to determine how much money an endowment needs to spend in its next year which means they can come up as part of a return calculation The less variable the spending the more risk an endowment can take all else equal For all of the equations listed below S=the specified spending rate, R = smoothing rate (given) i=inflation, and MV = market value Equation Formula Simple Spending Rate 𝑆𝑝𝑒𝑛𝑑𝑖𝑛𝑔 = 𝑆(𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒)𝑡−1 Three Year Average Spending Rate 𝑆𝑝𝑒𝑛𝑑𝑖𝑛𝑔 = 𝑆 𝑥 [𝑀𝑉𝑡−1 + 𝑀𝑉𝑡−2 + 𝑀𝑉𝑡−3 ] 𝑆𝑝𝑒𝑛𝑑𝑖𝑛𝑔 = (𝑅)(𝑆𝑝𝑒𝑛𝑑𝑖𝑛𝑔𝑡−1 )(1 + 𝑖𝑡−1 ) + (1 − 𝑅)(𝑆)(𝑀𝑉𝑡−1 ) Geometric Spending Rule Capital Market Expectations Equation Multifactor Analysis Formula 𝑌 = 𝑎 + 𝛽1 𝑋1 + 𝛽2 𝑋2 + 𝛽3 𝑋3 … + 𝛽𝑁 𝑋𝑁 + 𝜀 Gordon Growth Model (DCF) 𝑃0 = 𝑅𝑖 = ( Grinold & Kroner Model 𝐷𝑖𝑣0 (1 + 𝑔) 𝑅𝑖 − 𝑔 𝐷𝑖𝑣1 𝑃 ) + 𝑖 + 𝑔 − ∆𝑆 + ∆( ) 𝑃0 𝐸 Nominal earnings growth: 𝑖 + 𝑔 𝑃 The re-pricing return: ∆( ) 𝐸 The expected income return from the current yield: 𝐷𝑖𝑣 ( ) − ∆𝑆 Comments Multifactor analysis forecasts total return based on the values of a set of return drivers or risk factors This is your standard econometrics Each factor (or beta) measures the change in E(R) given a unit change in that variable holding all other variables constant Where: 𝑅𝑖 = expected return 𝐷𝑖𝑣0 = this year’s expected dividend 𝑃0 = this year’s share price 𝑔 = REAL earnings growth 𝑅𝑖 = expected return 𝐷𝑖𝑣1 = next year’s expected dividend 𝑃0 = this year’s share price 𝑔 = REAL earnings growth 𝑖 = inflation ∆𝑆 = Percentage Change in outstanding shares, negative (-∆S) when there are share repurchases whereas ∆S is positive when number of shares outstanding increases 𝑃 ∆( ) = change in P/E Ratio 𝐸 𝑃0 Risk Premium Approach 𝑅𝐵 = 𝑟𝑓 + 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 + 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 + 𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 + 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 + 𝑡𝑎𝑥 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 www.gostudy.io Capital Market Expectations – Financial Equilibrium Models Equation iCAPM Formula 𝑟𝑒 = 𝑟𝑓 + 𝛽(𝑟𝑔𝑖𝑚 − 𝑟𝑓 ) 𝛽= 𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑚 ) ⁄𝜎 𝑚 or Beta 𝛽= 𝜎𝑖 ∗ 𝜎𝑚 ∗ 𝜌𝑖,𝑚 ⁄𝜎 𝑚 𝐸𝑅𝑃𝑖 = 𝜌𝑖,𝑚 𝜎𝑖 ∗ Solving for ERP in ICAPM Comments Same as CAPM but Rgim is the return on the global investable market (gim) 𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑚 ) = the covariance between the asset and the global portfolio The covariance between any two assets = 𝛽1 ∗ 𝛽2 ∗ 𝜎𝑚 𝜎𝑚 = the global portfolio variance 𝜎𝑖 =the asset’s standard deviation 𝜎𝑚 = the global portfolio standard deviation 𝜌𝑖,𝑚 =the correlation between asset i and the market portfolio 𝐸𝑅𝑃𝐺𝐼𝑀 𝜎𝑚 Note that 𝐸𝑅𝑃𝐺𝐼𝑀 𝑅𝑀 − 𝑅𝑓 = = 𝑇ℎ𝑒 𝑔𝑙𝑜𝑏𝑎𝑙 𝑠ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 𝜎𝑚 𝜎𝑚 An Asset’s Risk Premium Asset Risk Premium = (Sharpe ratio of global portfolio) x (Asset’s own volatility) x (assets correlation with global portfolio) You may need to solve ERP = (𝑟𝑔𝑖𝑚 − 𝑟𝑓 ) Do this by substituting the Beta equation for beta and then simplifying This is solving from the above rows Calculate the ERP of the market assuming fully integrated with global portfolio (gim) 𝐸𝑅𝑃𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒𝑑 = 𝜌𝑖,𝑚 𝜎𝑖 ∗ 𝐸𝑅𝑃𝐺𝐼𝑀 𝜎𝑚 Calculate the ERP of the market assuming it is fully segmented from the global portfolio (i.e Solving the Singer & Terhaar Problem ignore the correlation with the global market) 𝐸𝑅𝑃𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒𝑑 = 𝜎𝑖 ∗ 𝐸𝑅𝑃𝐺𝐼𝑀 𝜎𝑚 Weight the results of Step and Step based on the actual level of market integration (which will be given on the exam) 𝐸𝑅𝑃𝑃𝑎𝑟𝑡𝑖𝑎𝑙 𝑆𝑒𝑔𝑚𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 = 𝑤𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒𝑑 𝐸𝑅𝑃𝑖𝑛𝑡 + (1 − 𝑤𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒𝑑 )𝐸𝑅𝑃𝑠𝑒𝑔 Add any illiquidity premium to the weighted average from Step This final number will be the expected return for the market in question www.gostudy.io 𝑟𝑡𝑎𝑟𝑔𝑒𝑡 = 𝑟𝑛𝑒𝑢𝑡𝑟𝑎𝑙 + [0.5(𝐺𝐷𝑃𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 − 𝐺𝐷𝑃𝑡𝑟𝑒𝑛𝑑 ) + 0.5(𝑖𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 − 𝑖𝑡𝑎𝑟𝑔𝑒𝑡 )] Taylor Rule  Where: 𝑟𝑡𝑎𝑟𝑔𝑒𝑡 = the target short-term interest rate 𝑟𝑛𝑒𝑢𝑡𝑟𝑎𝑙 = the short-term rate that would be targeted if GDP was on trend and inflation was on target 𝐺𝐷𝑃𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 = the forecasted GDP growth rate 𝐺𝐷𝑃𝑡𝑟𝑒𝑛𝑑 = the long term, observed GDP trend growth rate 𝑖𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 = the forecasted inflation rate 𝑖𝑡𝑎𝑟𝑔𝑒𝑡 = the target inflation rate  If expected GDP < GDPtrend then the central bank should cut rates to stimulate the economy If expected inflation > target inflation then the central bank would raise rates to minimize inflation Equity Market Valuation Equation Formula Comments α (1−α) 𝑌 = 𝐴𝐾 𝐿 α or 𝑌 = 𝐴𝐾 𝐾 𝑏 We can also get the percentage change: Cobb Douglas Function Total Factor Productivity Constant Growth DDM (same as previous) ∆𝑌 ∆𝐴 ∆𝐾 ∆𝐿 ≅ +α + (1 − α) 𝑌 𝐴 𝐾 𝐿 Y is GDP L is the quantity of labor K is the quantity of capital A is a positive constant representing technology/total factor productivity (Solow residual) α is the output elasticity of capital and is a number between and and (1-α) is the output elasticity of Labor Sometimes it is also called β Note that α + β = 𝑆𝑜𝑙𝑜𝑤 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = %∆𝑇𝐹𝑃 = %∆𝑌 − α(%∆𝐾) − (1 − α)%∆𝐿 𝑃0 = Always remember if you’re this year’s dividend, Div0 or next years, Div1 𝑉0 = The H Model 𝐷𝑖𝑣0 (1 + 𝑔) 𝑅𝑖 − 𝑔 𝐷0 𝑁 [(1 + 𝑔𝐿 ) + (𝑔𝑆 − 𝑔𝐿 )] 𝑟 − 𝑔𝐿 Where N = # years, gL = long term growth rate, gS = short term growth rate www.gostudy.io Asset Allocation Equation Multiplicative Return Calculation Formula 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 = [(1 + %𝑠𝑝𝑒𝑛𝑑𝑖𝑛𝑔)(1 + %𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛)(1 + %𝑚𝑔𝑚𝑛𝑡 𝑓𝑒𝑒)] − (generic to curriculum) 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑎𝑐𝑐𝑒𝑝𝑡𝑎𝑏𝑙𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑀𝐴𝑅 = 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 − (2 ∗ 𝑠𝑡𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛) Shortfall Risk (Shortfall risk is risk of exceeding a maximum acceptable dollar loss) 𝑅𝑆𝐹 = Roys Safety First Criteria 𝑅𝑃 − 𝑀𝐴𝑅 𝜎𝑃 The higher the ratio the better Utility Adjusted Return Adding a new investment to portfolio? (Sharpe Ratio Comparison) Solving a Corner Portfolio 𝑈𝑡𝑖𝑙𝑖𝑡𝑦 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 (𝑈𝑃 ) = 𝑅𝑃 − 0.005(𝐴)(𝜎 ) A = Risk aversion score which will be given on exam Make sure not to confuse variance and standard deviation in the equation and make sure you express the variance in nondecimal terms (i.e 50 not 0.50) Add if: 𝐸 (𝑅𝑛𝑒𝑤 ) − 𝑅𝑓 𝐸 (𝑅𝑃 ) − 𝑅𝑓 ] 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑅𝑁𝑒𝑤 , 𝑅𝑃 ) >[ 𝜎𝑛𝑒𝑤 𝜎𝑃 Solve for required return for a portfolio You will then have a table with a list of other portfolios on the efficient frontier You’ll need to determine which two portfolios bracket your portfolio based on expected return (i.e one will be higher and one will be lower than your required return) Then solve for the weighted average of each bracketed portfolio you will hold Check that maximum acceptable standard deviation is not exceeded From there you can use those two portfolio weights to solve for each individual asset class weight in your portfolio or to determine your standard deviation Note: The Tangency portfolio is one with highest Sharpe ratio www.gostudy.io Fixed Income Equation Formula 𝑛 𝐷𝑃 = ∑ 𝑤𝑖 𝐷𝑖 = 𝑤1 𝐷1 + 𝑤2 𝐷2 + 𝑤3 𝐷3 … + 𝑤𝑛 𝐷𝑛 𝑖=1 Effective Duration Effective Duration measures the percent change in a bond’s price for a 1% change in its yield to maturity Where: 𝐷𝑃 = the effective duration of the portfolio wi = the weight of bond i in the portfolio 𝐷𝑖 = the effective duration of bond i % Change in Price for Given Change in Interest Rates %∆𝑃 ≈ −𝐷𝑢𝑟𝑒𝑓𝑓 ∗ ∆𝑦 Where: %∆𝑃 = the percentage change in the bonds price 𝐷𝑢𝑟𝑒𝑓𝑓 = bond′ s effective duration ∆𝑦 = the change in the bond’s YTM (yield to maturity) Take the present value of the given period’s cash flows and divide it by the present value of all cash flows This gives you the percentage of market value attributed to cash flows from each period In other words, it gives us the weight of a period’s 𝑃𝑉 𝑜𝑓 𝑃𝑒𝑟𝑖𝑜𝑑𝑠 𝐶𝐹 cash flow = 𝑀𝑉 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 Present Value Distribution of Cash Flows Now that we have the weight of that period’s contribution to the portfolio’s value we need to convert that to its duration contribution To get the duration contribution we simply multiply the duration of a given period by its weight Duration Contribution = Duration ∗ 𝑃𝑉 𝑜𝑓 𝑃𝑒𝑟𝑖𝑜𝑑𝑠 𝐶𝐹 𝑀𝑉 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 Divide that duration contribution by the overall index duration to get PVD Dollar Duration 𝐷𝑜𝑙𝑙𝑎𝑟 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = −(𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛)(0.01)(𝑃𝑟𝑖𝑐𝑒) Dollar Duration (Portfolio Basis) 𝐷𝑃 = ∑ 𝐷𝐷𝑖 = 𝐷𝐷1 + 𝐷𝐷2 + 𝐷𝐷3 … + 𝐷𝐷𝑛 Rebalancing Ratio Adjusting Dollar Duration Callable Bond 𝑛 = 𝑂𝑙𝑑 𝐷𝐷 𝑖=1 𝑁𝑒𝑤 𝐷𝐷 Calculate the new DD of the portfolio Calculate the rebalancing ratio to determine the required percentage change (cash needed) to restore the value of the portfolio If needed, multiply the % change needed (rebalancing ratio – 1) by the market value of the portfolio to calculate the necessary increase/decrease in dollar value 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎 𝑐𝑎𝑙𝑙𝑎𝑏𝑙𝑒 𝑏𝑜𝑛𝑑 = −(𝐶𝑎𝑙𝑙 𝑜𝑝𝑡𝑖𝑜𝑛) + 𝐵𝑜𝑛𝑑 𝑃𝑟𝑖𝑐𝑒 www.gostudy.io Bond Portfolio Management Equation Return on a Leveraged Investment Formula Where: Rp = return on portfolio Ri = return on invested assets B = amount of leverage E = amount of equity invested C = cost of borrowed funds 𝐷𝑃 = Leverage & Duration Repurchase Agreement 𝑅𝑝 = 𝑅𝑖 + [(𝐵⁄𝐸 ) ∗ (𝑅𝑖 − 𝑐) 𝐷𝑖 𝐼 − 𝐷𝐵 𝐵 𝐸 Where: Dp = duration of portfolio Di = duration of invested assets DB = duration of borrowed assets I = amount of invested funds B= amount of leverage E = amount of equity invested 𝐷𝑜𝑙𝑙𝑎𝑟 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 = 𝑎𝑚𝑜𝑢𝑛𝑡 𝑏𝑜𝑟𝑟𝑜𝑤𝑒𝑑 ∗ 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 (𝑇𝑖𝑚𝑒⁄360) Know the factors influencing the repo rate (+/-) # 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 𝑡𝑜 𝑏𝑢𝑦 𝑜𝑟 𝑠𝑒𝑙𝑙 = (𝑦𝑖𝑒𝑙𝑑 𝑏𝑒𝑡𝑎)( Bond Futures & Duration MDt = target modified duration, MDp = Portfolio MD, MDF = futures MDf)  If we want to increase DD we BUY FUTURES  If we want to decrease DD we SELL FUTURES ℎ𝑒𝑑𝑔𝑒 𝑟𝑎𝑡𝑖𝑜 = Hedge Ratio 𝑀𝐷𝑇 − 𝑀𝐷𝑃 𝑉𝑃 )( ) 𝑀𝐷𝑓 (𝑃𝑓 ∗ 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟) 𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒 𝑜𝑓 𝑏𝑜𝑛𝑑 𝑡𝑜 𝑟𝑖𝑠𝑘 𝑓𝑎𝑐𝑡𝑜𝑟 𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒 𝑜𝑓 𝑓𝑢𝑡𝑢𝑟𝑒𝑠 𝑡𝑜 𝑟𝑖𝑠𝑘 𝑓𝑎𝑐𝑡𝑜𝑟 If the bond has greater risk exposure than the futures, you would need more futures contracts to fully cover the risk 𝑦𝑖𝑒𝑙𝑑 = 𝛼 + 𝛽(𝑦𝑖𝑒𝑙𝑑 𝑜𝑛 𝐶𝑇𝐷) + 𝜖 Yield Beta Yield Beta measures the sensitivity to interest rate changes of the CTD bond versus the original bond You’ll use the value when using futures to modify a portfolio’s duration www.gostudy.io Interest Rate Swaps & Options Equation Formula Binary Credit Options 𝑂𝑝𝑡𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑒 (𝑂𝑉) = [(𝑠𝑡𝑟𝑖𝑘𝑒 − 𝑣𝑎𝑙𝑢𝑒), 0] Credit Spread Option 𝑂𝑉 = max[(𝑎𝑐𝑡𝑢𝑎𝑙 𝑠𝑝𝑟𝑒𝑎𝑑 − 𝑠𝑡𝑟𝑖𝑘𝑒 𝑠𝑟𝑒𝑎𝑑) ∗ 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙 ∗ 𝑟𝑖𝑠𝑘 𝑓𝑎𝑐𝑡𝑜𝑟, 0]) Credit Spread Forward 𝐹𝑉 = (𝑠𝑝𝑟𝑒𝑎𝑑 𝑎𝑡 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 − 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡 𝑠𝑝𝑟𝑒𝑎𝑑) ∗ 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙 ∗ 𝑟𝑖𝑠𝑘 𝑓𝑎𝑐𝑡𝑜𝑟 International Bond Returns Equation Formula Foreign Yield Calc ∆𝑦𝑖𝑒𝑙𝑑𝑓𝑜𝑟𝑒𝑖𝑔𝑛 = 𝛽(∆𝑦𝑖𝑒𝑙𝑑𝑑𝑜𝑚𝑒𝑠𝑡𝑖𝑐 ) + 𝜖 Foreign Yield Change given Domestic Change Credit Spread Forward Equation Breakeven Analysis %∆𝑦𝑖𝑒𝑙𝑑𝑓𝑜𝑟𝑒𝑖𝑔𝑛 = −𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 ∗ (∆𝑦𝑖𝑒𝑙𝑑𝑑𝑜𝑚𝑒𝑠𝑡𝑖𝑐 ) ∗ 𝛽 𝐹𝑉 = (𝑠𝑝𝑟𝑒𝑎𝑑 𝑎𝑡 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 − 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡 𝑠𝑝𝑟𝑒𝑎𝑑) ∗ 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙 ∗ 𝑟𝑖𝑠𝑘 𝑓𝑎𝑐𝑡𝑜𝑟 Formula ∆𝑝𝑟𝑖𝑐𝑒 %∆𝑦 = −𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 Always divide the yield advantage by the number of periods in question The spreads of the two bonds will always have to widen www.gostudy.io Equity Portfolio Management Equation Formula 𝑅𝑃 = 𝑏0 + 𝑏1 𝑆𝐶𝐺 + 𝑏2 𝐿𝐶𝐺 + 𝑏3 𝑆𝐶𝑉 + 𝑏4 𝐿𝐶𝑉 Returns-based Style Analysis Where: Rp = returns on manager’s portfolio SCG = returns on a small-cap growth index LCG = returns on a large cap growth index SCV = returns on a small cap value index LCV = returns on a large cap value index The weight of each coefficient (beta) is non-negative and together they sum to R2 is the coefficient of determination, and it shows the percentage of an investor’s returns explained by the style indices Here, 1-R2 is the amount of the returns unexplained by style In other words, 1-R2 shows us the percent of returns due to the manager’s ability to select securities 𝐼𝑅 ≈ 𝐼𝐶 ∗ √𝐼𝐵 Fundamental Law of Active Management Where: IR = the information ratio IC = the information coefficient IB = investor breadth On the exam, the information coefficient will be given, while the investor breadth is equal to the number of independent investment decisions a manager is making 𝑡𝑜𝑡𝑎𝑙 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑖𝑠𝑘 = √(𝑡𝑟𝑢𝑒 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑖𝑠𝑘)2 + (𝑚𝑖𝑠𝑓𝑖𝑡 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑖𝑠𝑘)2 Total Active Risk Information Ratio True Active Return = Managers total return – Manager’s normal benchmark portfolio Misfit Active Return = Manager’s normal benchmark portfolio return – investor benchmark return 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 = 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑅𝑃 − 𝑅𝐵 = 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑖𝑠𝑘 𝜎(𝑃−𝐵) www.gostudy.io Alternatives Note the grayed out equations are now OPTIONAL & are presented to help facilitate your understanding Equation Commodity Futures Return Formula 𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 = 𝑠𝑝𝑜𝑡 𝑟𝑒𝑡𝑢𝑟𝑛 + 𝑐𝑜𝑙𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 + 𝑟𝑜𝑙𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 The spot return is the price return on the underlying commodity Collateral return is the risk free rate of return Roll return is the return that comes from rolling a futures contract forward at expiration 𝑟𝑜𝑙𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 = ∆𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑝𝑟𝑖𝑐𝑒 − ∆𝑆𝑝𝑜𝑡 𝑝𝑟𝑖𝑐𝑒 Roll Return Roll return is negative when markets are in contango & positive when in backwardation 𝐹0,𝑇 = 𝑆0 𝑒 (𝑅𝑓−𝛿)𝑇 Commodity Futures Price with lease rates Commodity Futures Price with Storage Costs 𝐹𝑜,𝑇 = Futures price 𝑆𝑜 = Spot Price 𝑅𝑓 = Risk free rate 𝛿 = Lease rate T = Time 𝐹0,𝑇 = 𝑆0 𝑒 (𝑅𝑓 +λ)𝑇 𝐹0,𝑇 = 𝑆0 𝑒 (𝑅𝑓 )𝑇 + λ0,𝑇 No Arbitrage Range for Commodity Futures (continuous compounding) Where λ = storage cost (discrete cost) where λ𝑜,𝑇 = FV storage costs 𝐹0,𝑇 = 𝑆0 𝑒 (𝑅𝑓 +λ−c)𝑇 Commodity Futures Price with Convenience Yield Commodity Futures Price Relationship with Return factors Forward markets will be in contango if 𝛿1 < 𝑅𝐹 Forward markets will be in backwardation if 𝛿1 > 𝑅𝐹     where c = the convenience yield The lease rate is the convenience yield – storage costs: 𝛿 = 𝑐 − λ 𝑅𝑓 ↑; 𝐹𝑃 ↑ as financing costs are avoided λ ↑; FP ↑ as storage costs are avoided 𝛿 ↑; 𝐹𝑃 ↓ as you lose the opportunity to earn the lease rate 𝑐 ↑ 𝐹𝑃 ↓ as it impacts ability to engage in a reverse cash & carry strategy 𝑆𝑜 𝑒 (𝑅𝑓+λ−c)𝑇 ≤ 𝐹𝑜,𝑇 ≤ 𝑆𝑜 𝑒 (𝑅𝑓+λ)𝑇 𝑆ℎ𝑎𝑟𝑝𝑒𝐻𝐹 𝐴𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 = Sharpe Ratio for Hedge Funds Swap Price (generic) 𝐴𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 − 𝐴𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑟𝑓 𝑎𝑛𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 σ The Sharpe ratio is larger for longer holding periods by the square root of time So we use monthly returns, we get the annualized returns by multiplying by 12, but the annual standard deviation is multiplied by √12 𝑆𝑤𝑎𝑝 𝑃𝑟𝑖𝑐𝑒 = ∑𝑇1 𝐹𝐷𝐹𝑡 𝑥 𝐹𝑃𝐶𝑜𝑚𝑚𝑜𝑑𝑖𝑡𝑦/𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 ∑𝑇1 𝐹𝐷𝐹𝑡 Where FP = Futures Price and FDF = Future Discount Factors, which are equal to 1⁄ (1 + 𝑟)𝑇 www.gostudy.io Risk Management Equation Formula 𝑉𝐴𝑅 = [Ȓ𝑝 − (𝑧)( 𝛿)] 𝑉𝑝 VAR – Analytical Method VAR – Historical Method Credit Risk of a Forward Contract to Long Party Where: Ȓ𝑝 = Expected return of portfolio z = z-value at desired level of significance (will be given on exam OR…5% VAR = 1.65, 1% VAR = 2.33)1 𝛿 = standard deviation of returns 𝑉𝑝 = Value of portfolio Rank historical returns from highest to lowest Calculate the lowest 5% of returns and use the highest value of that lowest 5% to set the 5% VAR for that time period (usually daily) So if you have 40 observations, the lowest 5% of observations would be the lowest (0.05*40) returns We take the higher of those observations and multiply it by the portfolio value to get the dollar VAR = 𝑠𝑝𝑜𝑡 − 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 (1+𝑑)𝑡 𝜎𝑝 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 (1+𝑑)𝑡 where 𝑅𝑝 = 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑒𝑡𝑢𝑟𝑛, 𝑅𝑓 = 𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑟𝑎𝑡𝑒, 𝜎𝑝 = 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑠𝑡𝑑 𝑑𝑒𝑣 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 = 𝑅𝑝 −𝑀𝐴𝑅 Return over maximum drawdown (ROMAD) − Risk Adjusted Performance Measures Formula 𝑅𝑝 − 𝑅𝑓 Sortino Ratio (1+𝑓)𝑡 𝑐𝑟𝑒𝑑𝑖𝑡 𝑟𝑖𝑠𝑘 = 𝑃𝑉(𝑅𝑒𝑐𝑒𝑖𝑣𝑒𝑑) − 𝑃𝑉(𝑃𝑎𝑖𝑑) Equation Information Ratio 𝑠𝑝𝑜𝑡 Where: f = the foreign interest rate & d = the domestic interest rate If the long value is positive, that position bears the credit risk If it is negative, the short position bears the risk Swap Credit Risk Sharpe Ratio or for currency: = 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑅𝑃 − 𝑅𝐵 = 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑖𝑠𝑘 𝜎(𝑃−𝐵) where MAR = Minimum Acceptable return 𝐷𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑅𝑝 𝑀𝑎𝑥 𝐷𝑟𝑎𝑤𝑑𝑜𝑤𝑛 Reflecting the number of standard deviations away from the mean www.gostudy.io Currency Risk Management Returns on an international investment have two components: the return on the assets in its local currency (RFC) and the currency return (RFX) that results as a function of the relative performance between an investor’s domestic currency (DC) and the foreign currency (FC) Because currency movements can also be tied to underlying GDP or economic activity, there is also an interaction between RFC and RFX that we need to account for Equation Formula 𝑅𝐷𝐶 = (1 + 𝑅𝐹𝐶 )(1 + 𝑅𝐹𝑋 ) − Return (in DC) or alternatively as 𝑅𝐷𝐶 = 𝑅𝐹𝐶 + 𝑅𝐹𝑋 + (𝑅𝐹𝐶 )(𝑅𝐹𝑋 ) Variance of Returns (in DC) 𝜎 (𝑅𝐷𝐶 ) ≈ 𝜎 (𝑅𝐹𝐶 ) + 𝜎 (𝑅𝐹𝑋 ) + 2𝜎(𝑅𝐹𝐶 )𝜎(𝑅𝐹𝑋 )𝜌(𝑅𝐹𝐶 , 𝑅𝐹𝑋 ) If 𝑅𝐹𝐶 is the risk free return it has no risk (no variance) and no correlation with 𝑅𝐹𝑋 𝑓= Covered Interest Rate Parity 𝐹−𝑆0 𝑆0 ≈ 𝑖𝑑 − 𝑖𝑓 Where: 𝑓 = Forward exchange premium/discount F = Forward exchange rate S0 = Spot exchange rate Id = Domestic interest rate If = Foreign interest rate Under CIRP a currency with a higher interest rate will trade at a forward discount (F < S0) whereas the one with a lower interest rate will trade at a forward premium 𝑅𝐷,𝑈 = ∝ +h(𝑅𝐹𝑢𝑡 ) Hedge Ratio (1) Where: RD,U = The unhedged domestic return h = the optimal hedge ratio ℎ =1+ 𝐶𝑜𝑣(𝑅𝐿, , 𝑅𝐶𝑢𝑟𝑟𝑒𝑛𝑐𝑦 ) 𝜎 (𝑅𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 ) Hedge Ratio (2) Where: RL = The asset return in local currency RCurrency = The currency return 𝜎 (𝑅𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 ) = The variance of currency returns www.gostudy.io Risk Management with Futures & Forwards Equation Formula 𝐵𝑇 − 𝐵𝑃 𝑉𝑃 # 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 𝑡𝑜 𝑏𝑢𝑦 𝑜𝑟 𝑠𝑒𝑙𝑙 = ( )( 𝐵𝑓 (𝑃𝑓 ∗ 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟) Adjusting Equity Portfolio Duration Adjusting Bond Portfolio Duration   To INCREASE a portfolio’s beta BUY futures contracts To DECREASE a portfolio’s beta SELL futures contracts Where: 𝐵𝑇 = Target or desired beta 𝑉𝑃 = Current value of portfolio 𝐵𝑃 = Portfolio beta 𝑃𝑓 = Futures price 𝐵𝑓 = Beta of futures contract multiplier = # indicating index value per contract 𝑀𝐷𝑇 − 𝑀𝐷𝑃 𝑉𝑃 # 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 𝑡𝑜 𝑏𝑢𝑦 𝑜𝑟 𝑠𝑒𝑙𝑙 = (𝑦𝑖𝑒𝑙𝑑 𝑏𝑒𝑡𝑎)( )( ) 𝑀𝐷𝑓 (𝑃𝑓 ∗ 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟) Determine the # of futures contracts necessary Note the numerator is the FV of the T-bill position…its saying you will go long more equity futures contracts than you would if your money wasn't growing at the risk-free rate (𝑇𝐻𝑒𝑙𝑑 )(1 + 𝑅𝑓 )𝑡 # futures necessary = (𝑃𝑓 )(𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟) Build a Synthetic Equity Position Round the value to the nearest whole number Next calculate the PV of the cash you will need to settle your futures contract when it comes due: 𝑇𝐸𝑞𝑢𝑖𝑡𝑖𝑧𝑒𝑑 = (# 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠𝑅𝑜𝑢𝑛𝑑𝑒𝑑 )(𝑃𝑓 )(𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟) (1 + 𝑅𝑓 )𝑡 Then calculate the effective # of equity units purchased (if needed) = Build a Synthetic Cash Position (# 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠𝑅𝑜𝑢𝑛𝑑𝑒𝑑 )(𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟) (1 + 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑) # 𝑒𝑞𝑢𝑖𝑡𝑦 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 𝑡𝑜 𝑠𝑒𝑙𝑙 = −[ 𝑉𝑃 (1+𝑅𝑓 )𝑇 (𝑃𝑓 )𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟) ] where VP = the value of the portfolio So to reallocate from bonds to equities: Adjusting Portfolio Between Equity & Debt Remove all duration (MD=0) by shorting bond futures Add systemic risk to the position (𝛽 > 0) by buying stock index futures And to reallocate from equities to bonds: Remove all systematic risk (𝛽 = 0) by shorting the stock index futures Go long/buy bond futures (MD > 0) to add the correct level of duration www.gostudy.io Options Note: We not include payoff equations for the different option strategies Equation Delta Formula ∆𝑜𝑝𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒 𝑂𝑝𝑡𝑖𝑜𝑛 𝐷𝑒𝑙𝑡𝑎 = ⁄∆𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒 Sell calls = short underlying = buy shares to hedge Sell puts = long underlying = sell shares to hedge 𝐺𝑎𝑚𝑚𝑎 = Gamma ∆𝐷𝑒𝑙𝑡𝑎 ∆𝑆𝑡𝑜𝑐𝑘 𝐶𝑎𝑙𝑙 𝑃𝑎𝑦𝑜𝑓𝑓 = 𝑁𝑃 ∗ [max(0, 𝐿𝐼𝐵𝑂𝑅 − 𝑋 )] ( Interest Rate Calls Interest Rate Put 𝐷 ) 360 Where: NP = Notional Principal, or the amount you wish to borrow LIBOR = the reference rate on which the contract is based X = the strike price D = days in the underlying rate, or the time until the option expires (and the loan begins) 𝑃𝑢𝑡 𝑃𝑎𝑦𝑜𝑓𝑓 = 𝑁𝑃 ∗ [max(0, 𝑋 − 𝐿𝐼𝐵𝑂𝑅 )] ( 𝐷 ) 360 1) Calculate the FV of the premium…e.g you buy a call starting in 31 days, maturity =31: 𝐹𝑉 = ($𝑃𝑟𝑒𝑚𝑖𝑢𝑚)(1 + 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑙𝑖𝑏𝑜𝑟 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦⁄ + 𝑠𝑝𝑟𝑒𝑎𝑑)( 360) 2) Calculate the effective interest rate cost = 𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙(𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑙𝑖𝑏𝑜𝑟 + 𝑠𝑝𝑟𝑒𝑎𝑑)(𝐷⁄360) − (𝑐𝑎𝑙𝑙 𝑝𝑎𝑦𝑜𝑓𝑓) Interest Rate Call Payoff 3) Calculate the effective annual rate of borrowing (EAR)  𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙+𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 $ 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑐𝑜𝑠𝑡 365 ]𝐷 𝑁𝑒𝑡 𝐴𝑚𝑜𝑢𝑛𝑡 𝐸𝐴𝑅 = [ −1 Note that the net amount is the Principal Borrowed minus the premium paid for the call The only difference in the calculation for a put is that we add the premium received for the put to the net amount Note also that when we annualize our interest rate we use 365 days instead of 360 which is the convention with LIBOR www.gostudy.io Swaps 𝐷𝑆𝑤𝑎𝑝 = 𝐷𝑎𝑠𝑠𝑒𝑡 − 𝐷𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝐷𝑝𝑎𝑦 𝑓𝑙𝑜𝑎𝑡𝑖𝑛𝑔 = 𝐷𝑓𝑖𝑥𝑒𝑑 − 𝐷𝑓𝑙𝑜𝑎𝑡𝑖𝑛𝑔 𝐷𝑝𝑎𝑦 𝑓𝑖𝑥𝑒𝑑 = 𝐷𝑓𝑙𝑜𝑎𝑡𝑖𝑛𝑔 − 𝐷𝑓𝑖𝑥𝑒𝑑 Duration of Swap Position Pay-floating position Pay-fixed position Will be a positive number Will be a negative number Note the duration of the floating position will usually be 0.25 It is just (Time to PMT/2) Equation Formula Changing Duration of a Swap 𝑁𝑃 = 𝑉𝑃 ( 𝑀𝐷𝑡 − 𝑀𝐷𝑝 ) 𝑀𝐷𝑠𝑤𝑎𝑝 1) Divide foreign cash flow by the foreign interest rate to obtain the foreign NP 2) 3) 4) Convert the foreign NP into the domestic NP at the current exchange rate Enter a swap with the domestic NP Pay the foreign CF, receive the domestic payment (calculated using the domestic interest rate and NP…again make sure that the annual interest rate is divided by the # of periods) (MAKE SURE to divide the annual interest rate given by the # of periods first) Converting Foreign CF into Domestic CF Portfolio Execution Equation Effective Spread Converting Foreign CF into Domestic CF Implementation Shortfall Components of Implementation Shortfall Formula 𝐵𝑢𝑦 𝑜𝑟𝑑𝑒𝑟 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑠𝑝𝑟𝑒𝑎𝑑 = (𝑒𝑥𝑒𝑐𝑢𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒 − 𝑚𝑖𝑑𝑞𝑢𝑜𝑡𝑒) ∗ 𝑆𝑒𝑙𝑙 𝑜𝑟𝑑𝑒𝑟 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑠𝑝𝑟𝑒𝑎𝑑 = (𝑚𝑖𝑑𝑞𝑢𝑜𝑡𝑒 − 𝑒𝑥𝑒𝑐𝑢𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒) ∗ 1) Divide the foreign cash flow by the foreign interest rate to obtain the foreign NP  MAKE SURE to divide the annual interest rate given in the problem by the # of periods first 2) Convert the foreign NP into the domestic NP at the current exchange rate 3) Enter a swap with the domestic NP 4) Pay the foreign CF, receive the domestic payment (calculated using the domestic interest rate and NP…again make sure that the annual interest rate is divided by the # of periods) 𝑇𝑜𝑡𝑎𝑙 𝑖𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑠ℎ𝑜𝑟𝑡𝑓𝑎𝑙𝑙 (%) = 𝑃𝑎𝑝𝑒𝑟 𝑔𝑎𝑖𝑛 − 𝑅𝑒𝑎𝑙 𝑔𝑎𝑖𝑛 𝑃𝑎𝑝𝑒𝑟 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝐸𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝑐𝑜𝑠𝑡𝑠 = 𝑐𝑜𝑚𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑝𝑎𝑝𝑒𝑟 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑙𝑜𝑠𝑠 = 𝑒𝑥𝑒𝑐𝑢𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒 − 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑑𝑎𝑦 𝑐𝑙𝑜𝑠𝑒 𝑠ℎ𝑎𝑟𝑒𝑠 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑑 ∗ 𝑏𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 𝑝𝑟𝑖𝑐𝑒 𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝐷𝑒𝑙𝑎𝑦 𝑐𝑜𝑠𝑡𝑠 = 𝑀𝑇𝑂𝐶 = 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑑𝑎𝑦 𝑐𝑙𝑜𝑠𝑒 − 𝑏𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 𝑝𝑟𝑖𝑐𝑒 𝑠ℎ𝑎𝑟𝑒𝑠 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑑 ∗ 𝑏𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 𝑝𝑟𝑖𝑐𝑒 𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑐𝑎𝑛𝑐𝑒𝑙𝑙𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒 − 𝑏𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 𝑝𝑟𝑖𝑐𝑒 𝑠ℎ𝑎𝑟𝑒𝑠 𝑁𝑂𝑇 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑑 ∗ 𝑏𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 𝑝𝑟𝑖𝑐𝑒 𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 The Benchmark price (BP) is the price the manager decides to buy/sell at (on the exam this will be the price at the end of day 1) Decision price (DP) is the closing price on the day before any part of the order gets filled www.gostudy.io Monitoring & Rebalancing Equation Tolerance Bands (for % of Portfolio Rebalancing) CPPI Rebalancing Formula 𝑇𝑜𝑙𝑒𝑟𝑎𝑛𝑐𝑒 𝐵𝑎𝑛𝑑𝑠 = 𝑇 ± (𝑃 ∗ 𝑇) Where T = target allocation and P = the %∆ maximum allowable deviation 𝑉𝑎𝑙𝑢𝑒 𝑖𝑛 𝑆𝑡𝑜𝑐𝑘𝑠 $ = 𝑚(𝐴𝑠𝑠𝑒𝑡𝑠 − 𝐹𝑙𝑜𝑜𝑟) Where m is the multiplier and the value in parenthesis is your cushion Return Calculations Equation Return with External Cash Flows at T=0 Return with External Cash Flows at end Time Weighted Rate of Return (TWRR) Money Weighted Rate of Return (MWRR) Don’t memorize… Formula 𝑟𝑡 = 𝑟𝑡 = 𝑀𝑉1 −(𝑀𝑉0 +𝐶𝐹) 𝑀𝑉0 + 𝐶𝐹 (𝑀𝑉1 − 𝐶𝐹) − 𝑀𝑉0 𝑀𝑉0 𝑇𝑊𝑅𝑅 = (1 + 𝑟1 )(1 + 𝑟2 )(1 + 𝑟𝑖 ) − TWRR is the compound growth rate in an account over a period of time We account for cash flows that occur into or out of the portfolio over this time period by chaining together discrete sub-periods that not have a cash flow 𝑀𝑊𝑅𝑅 = 𝑀𝑉1 − 𝑀𝑉0 (1 + 𝑅)𝑀 + ∑ 𝐶𝐹𝑖 (1 + 𝑅)𝐿(𝑖) Where L(i) = # days of cash is in the portfolio 𝑃 =𝑀+𝑆+𝐴 M = market, S = style, A = active management (and P = portfolio return) Portfolio Return Components Specifically: P = Portfolio return M = Return on market index S = B- M = Excess return to style, or the difference between a manager’s benchmark style and market return A = P – B = active return, or Portfolio return – Benchmark www.gostudy.io Equation Net Contributions Risk Free Asset Asset Categories Macro Performance Evaluation Formula Net sum of external cash flows Risk Free Return 𝑅𝐴𝐶 = ∑(𝑤𝑖 )(𝑅𝑖 − 𝑅𝑓 ) 𝐴 𝑀 𝑅𝐵 = ∑ ∑(𝑤𝑖 )(𝑤𝑖,𝑗 )(𝑅𝑏,𝑖,𝑗 − 𝑅𝑖 ) 𝑖=1 𝑗=1 Benchmarks Where wi = weight of benchmark, wi,j = weight assigned to manager j in asset category i, A= the number of asset categories and M = the number of managers in a given asset category 𝐴 Investment Managers Allocation Effects Equation Pure Sector Allocation 𝑀 𝑅𝐼𝑀 = ∑ ∑(𝑤𝑖 )(𝑤𝑖,𝑗 )(𝑅𝐴,𝑖,𝑗 − 𝑅𝐵,𝑖,𝑗 ) 𝑖=1 𝑗=1 Plug value Micro Performance Evaluation Formula 𝑠 ∑(𝑤𝑃,𝑗 − 𝑤𝐵,𝑗 )(𝑅𝐵,𝑗 − 𝑅𝐵 ) 𝑗=1 𝑠 Allocation Selection Interaction ∑(𝑤𝑃,𝑗 − 𝑤𝐵,𝑗 )(𝑅𝑃,𝑗 − 𝑅𝐵,𝑗 ) 𝑗=1 𝑠 Within Sector Security Selection ∑(𝑤𝐵,𝑗 )(𝑅𝑝,𝑗 − 𝑅𝐵,𝐽 ) 𝑗=1 𝑠 𝑅𝑉 = ∑(𝑤𝑃,𝑗 − 𝑤𝐵,𝑗 )(𝑅𝐵,𝑗 − 𝑅𝐵 ) 𝑗=1 Micro Performance Evaluation 𝑠 𝑠 + ∑(𝑤𝑃,𝑗 − 𝑤𝐵,𝑗 )(𝑅𝑃,𝑗 − 𝑅𝐵,𝑗 ) + ∑(𝑤𝐵,𝑗 )(𝑅𝑝,𝑗 − 𝑅𝐵,𝐽 ) 𝑗=1 Where: Rv = value added return wp,j = manager’s weight to sector j wB,j = benchmark weight to sector j 𝑗=1 www.gostudy.io Summary of Risk Adjusted Performance Metrics Measure Formula Notes Jensen’s ex-post alpha ∝𝑃 = 𝑅𝑎𝑐𝑡𝑢𝑎𝑙 − 𝑅𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 Systematic risk measure Positive alpha = plot above SML Excess return over systematic risk Will have same relative ranking as Jensen’s ex-post alpha Alpha compared against market Measures excess returns to total risk Appropriate if normally distributed Alpha if SP > SM Measures the value-add or lost relative to the market if the portfolio has the same TOTAL risk as the market Excess return per unit of total excess risk Similar to Sharpe ratio, but compares excess risk/return Positive alpha if IR is positive number 𝑇𝑟𝑒𝑦𝑛𝑜𝑟 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 = Treynor Measure Sharpe Ratio M2 𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 = 𝑅𝑃 − 𝑅𝐹 𝛽𝑃 𝑅𝑝 − 𝑟𝑓 𝜎𝑃 𝑅𝑝 − 𝑟𝑓 𝑀2 = 𝑅𝐹 + ( )𝜎𝑀 𝜎𝑃 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 = Information Ratio = 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑎𝑐𝑡𝑖𝑣𝑒 𝑟𝑖𝑠𝑘 𝑅𝑃 − 𝑅𝐵 𝜎(𝑃−𝐵) Global Investment Performance Standards Equation Original Dietz (until 2005) Formula 𝐸𝑀𝑉 − 𝐵𝑀𝑉 − 𝐶𝐹 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐷𝑖𝑒𝑡𝑧 = 𝐵𝑀𝑉 + 0.5𝐶𝐹 Where BMV/EMV = Beginning/Ending market value Modified Dietz (2005-2010) 𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝐷𝑖𝑒𝑡𝑧 = where the weight, 𝑊𝑖 = 𝐸𝑀𝑉 − 𝐵𝑀𝑉 − 𝐶𝐹 𝐵𝑀𝑉 + ∑(𝑊𝑖 ∗ 𝐶𝐹) 𝐷𝑎𝑦𝑠 𝑖𝑛 𝑃𝑒𝑟𝑖𝑜𝑑 (𝐶𝐷)−𝐷𝑎𝑦 𝐶𝐹 𝑟𝑒𝑐𝑖𝑒𝑣𝑒𝑑 (𝐷𝑖 ) 𝐷𝑎𝑦𝑠 𝑖𝑛 𝑃𝑒𝑟𝑖𝑜𝑑 (𝐶𝐷) www.gostudy.io Real Estate GIPS Calculations Formula Equation Capital Employed 𝑛 𝐶𝐸 = 𝐶0 + ∑(𝐶𝐹𝑖 ∗ 𝑊𝑖 ) 𝑖=1 𝑅𝐶 = Capital Return 𝑉1 − 𝑉0 − 𝐸𝐶 + 𝑆 𝐶𝐸 The Capital Return (RC) gives us the change in market value of a property (V) AFTER considering any capital improvement expenses (EC) and sale proceeds (S) 𝑅𝐼 = Investment Return 𝑌𝐴 − 𝐸𝑅 − 𝐼𝐷 − 𝑇𝑃 𝐶𝐸 Where: 𝑌𝐴 = gross investment income 𝐸𝑅 = nonrecoverable expenses 𝐼𝐷 = Debt or interest payments 𝑇𝑃 = property taxes 𝐶𝐸 = capital employed Access full notes, a web and study app, and dozens or in-depth walkthroughs of common exam problems at www.gostudy.io ...www .gostudy. io Guided Notes for CFA Level – 2017 Copyright © 2015 by Go Study LLC.® All Rights Reserved Published in 2015-2016 The CFA and Chartered Financial Analyst®... PMT/2) Equation Formula Changing Duration of a Swap