Financial risk management course

Financial Risk Management 2010-11 Topics T1 Stock index futures Duration, Convexity, Immunization T2 Repo and reverse repo Futures on T-bills Futures on T-bonds Delta, Gamma, Vega hedging T3 Portfolio insurance Implied volatility and volatility smiles T4 Modelling stock prices using GBM Interest rate derivatives (Bond options, Caps, Floors, Swaptions) T5 Value at Risk T6 Value at Risk: statistical issues Monte Carlo Simulations Principal Component Analysis Other VaR measures T7 Parametric volatility models (GARCH type models) Non-parametric volatility models (Range and high frequency models) Multivariate volatility models (Dynamic Conditional Correlation DCC models) T8 Credit Risk Measures (credit metrics, KMV, Credit Risk Plus, CPV) T9 Credit derivatives (credit options, total return swaps, credit default swaps) Asset Backed Securitization Collateralized Debt Obligations (CDO) * This file provides you an indication of the range of topics that is planned to be covered in the module However, please note that the topic plans might be subject to change Topics Financial Risk Management Topic Managing risk using Futures Reading: CN(2001) chapter Futures Contract: Speculation, arbitrage, and hedging Stock Index Futures Contract: Hedging (minimum variance hedge ratio) Hedging market risks Futures Contract Agreement to buy or sell “something” in the future at a price agreed today (It provides Leverage.) Speculation with Futures: Buy low, sell high Futures (unlike Forwards) can be closed anytime by taking an opposite position Arbitrage with Futures: Spot and Futures are linked by actions of arbitragers So they move one for one Hedging with Futures: Example: In January, a farmer wants to lock in the sale price of his hogs which will be “fat and pretty” in September Sell live hog Futures contract in Jan with maturity in Sept Speculation with Futures Speculation with Futures Speculation with Futures Purchase at F0 = 100 Hope to sell at higher price later F1 = 110 Close-out position before delivery date Obtain Leverage (i.e initial margin is ‘low’) Profit/Loss per contract Long future $10 -$10 F1 = 90 F0 = 100 Example: Example: Nick Leeson: Feb 1995 Long 61,000 Nikkei-225 index futures (underlying value = $7bn) Nikkei fell and he lost money (lots of it) - he was supposed to be doing riskless ‘index arbitrage’ not speculating F1 = 110 Futures price Short future Speculation with Futures Profit payoff (direction vectors) F increase then profit increases Profit/Loss F increase then profit decrease Profit/Loss Arbitrage with Futures -1 +1 Underlying,S +1 or Futures, F -1 Long Futures Short Futures or, Long Spot or, Short Spot Arbitrage with Futures At expiry (T), FT = ST Else we can make riskless profit (Arbitrage) Forward price approaches spot price at maturity Forward price, F Forward price ‘at a premium’ when : F > S (contango) Stock price, St T At T, ST = FT Forward price ‘at a discount’, when : F < S (backwardation) Arbitrage with Futures General formula for non-income paying security: F0 = S0erT or F0 = S0(1+r)T Futures price = spot price + cost of carry For stock paying dividends, we reduce the ‘cost of carry’ by amount of dividend payments (d) F0 = S0e(r-d)T For commodity futures, storage costs (v or V) is negative income F0 = S0e(r+v)T or F0 = (S0+V)erT Arbitrage with Futures Arbitrage with Futures For currency futures, the ‘cost of carry’ will be reduced by the riskless rate of the foreign currency (rf) Arbitrage at t S0erT then buy the asset and short the futures contract If F0 < S0erT then short the asset and buy the futures contract Example of ‘Cash and Carry’ arbitrage: S=£100, r=4%p.a., F=£102 for delivery in months 0.04×0.25 = 101 £ We see Fɶ = 100 × e Since Futures is over priced, time = Now •Sell Futures contract at £102 time = in months •Pay loan back (£101) •Borrow £100 for months and buy stock •Deliver stock and get agreed price of £102 Hedging with Futures Hedging with Futures Hedging with Futures Simple Hedging Example: You long a stock and you fear falling prices over the next months, when you want to sell Today (say January), you observe S0=£100 and F0=£101 for April delivery so r is 4% Today: you sell one futures contract In March: say prices fell to £90 (S1=£90) So F1=S1e0.04x(1/12)=£90.3 You close out on Futures Profit on Futures: 101 – 90.3 = £10.7 Loss on stock value: 100 – 90 =£10 Net Position is +0.7 profit Value of hedged portfolio = S1+ (F0 - F1) = 90 + 10.7 = 100.7 F and S are positively correlated To hedge, we need a negative correlation So we long one and short the other Hedge = long underlying + short Futures Hedging with Futures F1 value would have been different if r had changed This is Basis Risk (b1 = S1 – F1) Final Value = S1 + (F0 - F1 ) = £100.7 = (S1 - F1 ) + F0 = b1 + F0 where “Final basis” b1 = S1 - F1 At maturity of the futures contract the basis is zero (since S1 = F1 ) In general, when contract is closed out prior to maturity b1 = S1 - F1 may not be zero However, b1 will usually be small in relation to F0 Stock Index Futures Contract Stock Index Futures contract can be used to eliminate market risk from a portfolio of stocks Stock Index Futures Contract Hedging with SIFs F0 = S0 × e( r − d )T If this equality does not hold then index arbitrage (program trading) would generate riskless profits Risk free rate is usually greater than dividend yield (r>d) so F>S Hedging with Stock Index Futures Hedging with Stock Index Futures Example: A portfolio manager wishes to hedge her portfolio of $1.4m held in diversified equity and S&P500 index Total value of spot position, TVS0=$1.4m S0 = 1400 index point Number of stocks, Ns = TVS0/S0 = $1.4m/1400 =1000 units We want to hedge Δ(TVSt)= Ns Δ(St) The required number of Stock Index Futures contract to short will be Use Stock Index Futures, F0=1500 index point, z= contract multiplier = $250 FVF0 = z F0 = $250 ( 1500 ) = $375,000  TVS   $1, 400, 000  NF = −  = −  = − 3.73  $375, 000   FVF0  In the above example, we have assumed that S and F have correlation +1 (i.e ∆ S = ∆ F ) In reality this is not the case and so we need minimum variance hedge ratio Hedging with Stock Index Futures Hedging with Stock Index Futures Minimum Variance Hedge Ratio To obtain minimum, we differentiate with respect to Nf (∂σ V / ∂N f = ) and set to zero ∆V = change in spot market position + change in Index Futures position + Nf (F1 - F0) z = Ns (S1-S0) Ns S0 ∆S /S0 TVS0 ∆S /S0 = = + + Nf F0 (∆ ∆F /F0) z Nf FVF0 (∆ ∆F /F0) where, z = contract multiple for futures ($250 for S&P 500 Futures); ∆S = S1 - S0, ∆F = F1 - F0 N f ( F V F0 ) σ ∆2 F / F N f = − 2 2 σ V = (TVS ) σ ∆S / S + ( N f ) ( FVF ) σ ∆F / F + 2N f TVS FVF0 σ ∆S / S , ∆F / F ∂σV / ∂N = f TVS β p Nf = − FVF0 (∆S ( σ ∆ S / S ,∆ F / F σ ∆F / F ) / S ) = α + β ∆S / ∆F ( ∆ F / F ) + ε Hedging with Stock Index Futures Application: Changing beta of your portfolio: “Market Timing Strategy” TVS implies  Value of Spot Position   = −  FaceValue of futures at t =   ⋅ F V F ⋅ σ ∆ S / S ,∆ F / F where Ns = TVS0/S0 and beta is regression coefficient of the regression Hedging with Stock Index Futures SUMMARY  TVS0  =−  β ∆ S / S ,∆ F / F  F V F0  The variance of the hedged portfolio is  TVS0     F V F0  = −TVS Nf = FVF0 ( β h − β p ) Example: βp (=say 0.8) is your current ‘spot/cash’ portfolio of stocks βp But • You are more optimistic about ‘bull market’ and desire a higher exposure of βh (=say, 1.3) • It’s ‘expensive’ to sell low-beta shares and purchase high-beta shares If correlation = 1, the beta will be and we just have TVS0 Nf = − FVF0 • Instead ‘go long’ more Nf Stock Index Futures contracts Note: If βh= 0, then Nf = - (TVS0 / FVF0) βp Hedging with Stock Index Futures Hedging with Stock Index Futures Application: Stock Picking and hedging market risk If you hold stock portfolio, selling futures will place a hedge and reduce the beta of your stock portfolio If you want to increase your portfolio beta, go long futures Example: Suppose β = 0.8 and Nf = -6 contracts would make β = If you short (-3) contracts instead, then β = 0.4 If you long (+3) contracts instead, then β = 0.8+0.4 = 1.2 You hold (or purchase) 1000 undervalued shares of Sven plc V(Sven) = $110 (e.g Using Gordon Growth model) P(Sven) = $100 (say) Sven plc are underpriced by 10% Therefore you believe Sven will rise 10% more than the market over the next months But you also think that the market as a whole may fall by 3% The beta of Sven plc (when regressed with the market return) is 2.0 Hedging with Stock Index Futures Hedging with Stock Index Futures Can you ‘protect’ yourself against the general fall in the market and hence any ‘knock on’ effect on Sven plc ? Application: Future stock purchase and hedging market risk Yes You want to purchase 1000 stocks of takeover target with βp = 2, in month’s time when you will have the cash Sell Nf index futures, using: N f = − TVS FVF β You fear a general rise in stock prices p If the market falls 3% then Sven plc will only change by about 10% - (2x3%) = +4% But the profit from the short position in Nf index futures, will give you an additional return of around 6%, making your total return around 10% Go long Stock Index Futures (SIF) contracts, so that gain on the futures will offset the higher cost of these particular shares in month’s time N f = TVS FVF β p SIF will protect you from market risk (ie General rise in prices) but not from specific risk For example if the information that you are trying to takeover the firm ‘leaks out’ , then price of ‘takeover target’ will move more than that given by its ‘beta’ (i.e the futures only hedges market risk) Topics Financial Risk Management Topic Managing interest rate risks Reference: Hull(2009), Luenberger (1997), and CN(2001) Duration, immunization, convexity Repo (Sale and Repurchase agreement) and Reverse Repo Hedging using interest rate Futures Futures on T-bills Futures on T-bonds Readings Books Hull(2009) chapters CN(2001) chapters 5, Luenberger (1997) chapters Journal Article Fooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis” Managerial Finance,Vol 25, no Hedging Interest rate risks: Duration Duration Duration (also called Macaulay Duration) Duration measures sensitivity of price changes (volatility) with changes in interest rates Lower the coupons for a given time to maturity, greater change in price to change in interest rates B = Greater the time to D = T T PB = ∑ C t t + ParValue T (1+ r ) t =1 (1+ r ) T T PB = ∑ C t t + ParValue T (1+ r ) t =1 (1+ r ) Duration of the bond is a measure that summarizes approximate response of bond prices to change in yields A better approximation could be convexity of the bond maturity with a given coupon, greater change in price to change in interest rates For a given percentage change in yield, the actual price increase is n ∑ c i e − y ti i =1 n ∑ i =1 weight t i ⋅ c i e − y ti B = n ∑ i =1  c e − y ti  ti  i   B  Duration is weighted average of the times when payments are made The weight is equal to proportion of bond’s total present value received in cash flow at time ti Duration is “how long” bondholder has to wait for cash flows greater than a price decrease Macaulay Duration For a small change in yields ∆ y / d y ∆B = dB ∆y dy Evaluating d B : n   d y ∆ B =  − ∑ t i c i e − y ti  ∆ y  i =1  = −B ⋅ D ⋅∆y ∆B = −D ⋅∆y B D measures sensitivity of percentage change in bond prices to (small) changes in yields Note negative relationship between Price (B) and yields (Y) Modified Duration and Dollar Duration For Macaulay Duration, y is expressed in continuous compounding When we have discrete compounding, we have Modified Duration (with these small modifications) If y is expressed as compounding m times a year, we divide D D by (1+y/m) ∆B = − B ⋅ ⋅ ∆y (1 + y / m) ∆B = − B ⋅ D* ⋅ ∆y Dollar Duration, D$ = B.D That is, D$ = Bond Price x Duration (Macaulay or Modified) ∆B = − D$ ⋅ ∆y So D$ is like Options Delta D$ = − ∆B ∆y McKinsey’s Credit Portfolio View, CPV Mark-to-market model with direct link to macro variables Explicitly model the link between the transition probability (e.g p(C to D)) and an index of macroeconomic activity, y pit = f(yt) where i = “C to D” etc y is assumed to depend on a set of macroeconomic variables Xit (e.g GDP, unemployment etc.) yt = g (Xit, vt) i = 1, 2, … n Xit depend on own past values plus other random errors εit.(say VAR(1)) It follows that: pit = k (Xi,t-1, vt, εit) Each transition probability depends on past values of the macrovariables Xit and the error terms vt, εit Clearly the pit are correlated CPV Monte Carlo simulation to adjust the empirical (or average) transition probabilities estimated from a sample of firms (e.g as in CreditMetrics) Consider one Monte Carlo ‘draw’ of the error terms vt, εit (which embody the correlations found in the estimated equations for yt and Xit above) This may give rise to a simulated probability pis = 0.25, whereas the historic (unconditional) transition probability might be pih = 0.20 This implies a ratio of ri = pis / pih = 1.25 Repeat the above for all initial credit rating states (i.e i = AAA, AA, … etc.) and obtain a set of r’s CPV Then take the (CreditMetrics type) historic x transition matrix Tt and multiply these historic probabilities by the appropriate ri so that we obtain a new ‘simulated’ transition probability matrix, T T , now embodies the impact of the macro variables and hence the correlations between credit migrations Then revalue our portfolio of bonds using new simulated probabilities which reflect one possible state of the economy This would complete the first Monte Carlo ‘draw’ and give us one new value for the bond portfolio Repeating this a large number of times (e.g 10,000), provides the whole distribution of gains and losses on the portfolio, from which we can ‘read off’ the portfolio value at the 1st percentile SUMMARY A COMPARISON OF CREDIT MODELS Characteristics J.P.Morgan CreditMetrics KMV Credit Monitor CSFP Credit Risk Plus Mark-to-Market (MTM) or Default Mode (DM) Source of Risk MTM MTM or DM DM McKinsey Credit Portfolio View MTM or DM Multivariate normal stock returns Multivariate normal stock returns Stochastic default rate (Poisson) Macroeconomic Variables Stock prices Transition probabilities Option prices Stock price volatility Correlation between mean default rates Correlation between macro factors Analytic or MCS Analytic Analytic MCS Correlations Solution Method Topic # 8b: Merton model and KMV Financial Risk Management 2010-11 March 3, 2011 FRM c Dennis PHILIP 2011 Merton (1974) model and KMV Assume …rm’s balance sheet looks like this: Consider a …rm with risky assets A; that follow a GBM Suppose …rm is …nanced by a simple capital structure, namely one debt obligation (D) and one type of equity (E) A0 = E0 + D0 where (Et )t is a GBM that describes evolution of equity of the …rm and (Dt )t is some process that describes market value of debt obligation of the …rm FRM c Dennis PHILIP 2011 Suppose debt holders pay capital D0 at time t = and get F (includes principal and interest) at time t = T: For debt holders (lending banks), credit risk increase iÔ P [AT < F ] > When default probability > 0, we can conclude that D0 < F e rt where r is the risk free rate That is, debt holders would need credit risk premium In other words, D0 is smaller, greater the riskiness of the …rm To hedge this credit risk, debt holders go Long a Put contract on A; with strike F and maturity T: This guarantees credit protection against default of payment: FRM c Dennis PHILIP 2011 Debt holder’s portfolio consist on a loan and Put contract At t = 0; D0 + P0 (A0 ; A ; F; T; r) At t = T; portfolio value is F Using non-arbitrage principal, at t = D0 + P0 (A0 ; A ; F; T; r) = Fe rt So the present value of D0 is D0 = F e FRM rt P0 (A0 ; A ; F; T; r) (1) c Dennis PHILIP 2011 The …gure below summarizes this Next, we can think about the value of equity E0 in terms of a Call option Equity holders of the …rm have the right to liquidate the rm (i.e paying oÔ the debt and taking over remaining assets) Suppose liquidation happens at maturity date t = T: Two Scenarios: FRM c Dennis PHILIP 2011 AT < F : In this case, there is default and there is not enough to pay the debt holders Moreover, equity holders have a payoÔ of zero AT F : In this case, there is a net prot for equity holders after paying oÔ the debt (AT F ) So the total payoÔ to equity holders is max(AT F; 0) which is the payoÔ of an European call option on A with strike F and maturity T: The present value of equity is therefore E0 = C0 (A0 ; A ; F; T; r) (2) Combining equation and 2, we can obtain A0 = E0 + D0 = C0 (A0 ; A ; F; T; r) + F e P0 (A0 ; A ; F; T; r) A0 + P0 (A0 ; A ; F; T; r) = C0 (A0 ; A ; F; T; r) + F e which is nothing but the put-call parity FRM c Dennis PHILIP 2011 rt rt The above also shows that equity and debt holders have contrary risk preferences By choosing risky investment in some asset A, with higher volatility A ; equity holders will increase the option premium for both call and put This is good for equity holders as they are long a call and this is bad for debt holders as they have a short position Equity holders have limited downside risk but unlimited upside potential The main application of this Merton’s option pricing framework is to estimate default probabilities This is implemented in the KMV credit monitor system KMV applies Black-Scholes formula in reverse Conventionally, we observe price of the underlying, strike price, etc and we calculate FRM c Dennis PHILIP 2011 the value of the derivative In this applcation, we begin with value of the derivative (value of equity E0 as a call option) and given strike F ; and calculate the unobserved value of A0 : Another complication: we need the value of A but since we not observe value of A, we have to infer A from volatility of returns on equity E : The Black-Scholes gives the value of equity today as E0 = A0 N (d1 ) Fe rT N (d2 ) (3) where d1 = ln(A0 =F ) + (r + p A T and d2 = d1 A p A =2)T T The value of debt today is D0 = A0 FRM E0 c Dennis PHILIP 2011 To calculate above we need A0 and knowns) A (un- Using Ito lemma, we yield the relationship between the volatilities at t = E E0 = @E @A A A0 (4) Here @E=@A is the delta of the equity So @E=@A = N (d1 ) So we have two equations (3 and 4) and two unknowns (A0 and A ) We can use Excel Solver to obtain the solution to both these equations Now we can calculate the default probabilities (called Expected Default frequency, EDF) as pi = P [AT < F ] This can be shown to be pi = P [Zi < FRM DD] c Dennis PHILIP 2011 10 where Zi N (0; 1) and DD is the ‘distance to default’ (i.e number of standard deviations the …rm’s assets are away from default) DD = ln(A0 =F ) + ( A p A T A =2)T where A is the mean return/growth rate of the assets FRM c Dennis PHILIP 2011 11 FRM c Dennis PHILIP 2011 ... that given by its ‘beta’ (i.e the futures only hedges market risk) Topics Financial Risk Management Topic Managing interest rate risks Reference: Hull(2009), Luenberger (1997), and CN(2001) Duration,... factor for the cheapest to deliver bond Financial Risk Management Topic 3a Managing risk using Options Readings: CN(2001) chapters 9, 13; Hull Chapter 17 Topics Financial Engineering with Options...Topics Financial Risk Management Topic Managing risk using Futures Reading: CN(2001) chapter Futures Contract: Speculation,