Risk Management. Exercises

Transcription

1 Risk Management Exercises

2 Exercise Value at Risk calculations

3 Problem Consider a stock S valued at $1 today, which after one period can be worth S T : $2 or $0.50. Consider also a convertible bond B, which after one period will be worth max(1, S T ). Determine which is the following three portfolios has lower VaR: 1. B 2. B-S 3. B+S

4 FRM exam 1999 The VaR of one asset is 300, and another one is 500. If the correlation between changes in asset prices is 1/15, what is the combined VaR?

5 Exercise Hedging

6 Problem Consider a stock S valued at $1 today, which after one period can be worth S T : $2 or $0.50. Consider also a convertible bond B, which after one period will be worth max(1, S T ). Determine the optimal trading strategy adding a stock portfolio to the bond.

7 Exercise Credit VaR

8 Problem Consider a stock S valued at $1 today, which after one period can be worth S T : $2 or $0.50. Consider also a convertible bond B, which after one period will be worth max(1, S T ). Assume the stock can default, after which event S T =0. Determine which is the following three portfolios has lower Credit-VaR: 1. B 2. B-S 3. B+S

9 Exercise The Merton Model

10 The Merton Model Consider a firm with total asset worth $100, and asset volatility equal to 20%. The risk free rate is 10% with continuous compounding. Time horizon is 1 year. Leverage is 90% (i.e., debt-to-equity ratio 900%) Find: The value of the credit spread. The risk neutral probability of default Calculate the PV of the expected loss.

11 Credit Spread A leverage of 0.9 implies that which says that K= Using Black-Scholes, we get that the call option is worth S=$ The bond price is then for a yield of B = V S Ke 0.1 / V or a credit spread of 4.07%. = 0.9 = $ 100 $13.59 = $86.41 ln( K / B) = ln(99.46 / 86.41) = 14.07%

12 Option Calculation Underlying Data Graph Results Underlying Type: Time Dividend Vertical Axis: Equity 1 Theta Horizontal Axis: Stock Price: Volatility (% per year): 20.00% Asset price Risk-Free Rate (% per year): 10.00% Minimum X value 80 Maximum X value 120 Calculate Draw Graph Option Type: Analytic: European Option Data Time to Exercise: Exercise Price: Imply Volatility Put Call Theta Price: Delta (per $): Gamma (per $ per $): Vega (per %): Theta (per day): Rho (per %): Asset Price

13 The risk neutral probability of default Given by N ( 2 d ) = , EDF= 1 N( d ) = %

14 Expected loss It is given by $ $100 $ ) ( ) ( ) ( = = = d N d N V Ke d N ECL rτ

15 Additional considerations Variations on the same problem: If debt-to-equity ratio is 233%, the spread is 0.36% If debt-to-equity ratio is 100%, the spread is about 0. In other words, the model fails to reproduce realistic, observed credit spreads.

16 Exercise Calibrating the asset volatility

17 The Goodrich Corporation From company s financials Debt/equity ratio: 2.27 Shares out: 117,540,000. Expected dividend: $0.20/share. From NYSE, ticker symbol GR Stock volatility: 49.59% Real rate of return (3 years): 0.06% Share price: $17.76 (May 2003) From interest rate market Annual risk free rate: 3.17% S =117,540,000*$17.76 S = $2,087B. V=S+B=3.27S=$6.826B Current debt = $4.759B Future debt (Strike price) K = $4.759 e = $4.912 Dividend = $0.20*117,540,000 = 23,508,

18 Bootstrapping asset volatility Underlying Data Graph Results Underlying Type: Time Dividend Vertical Axis: Equity Theta Horizontal Axis: Stock Price: 6.83 Volatility (% per year): 49.59% Asset price Risk-Free Rate (% per year): 3.17% Minimum X value 80 Maximum X value 120 Calculate Draw Graph Option Type: Analytic: European Option Data Imply Volatility Slightly high Time to Exercise: Exercise Price: 4.91 Price: Delta (per $): Gamma (per $ per $): Vega (per %): Theta (per day): Rho (per %): Put Call Theta σ V = σ S V V S Asset Price S

19 Bootstrapping asset volatility (iterative process) Underlying Data Graph Results Underlying Type: Time Dividend Vertical Axis: Equity Theta Horizontal Axis: Stock Price: 6.83 Volatility (% per year): 14.77% Asset price Risk-Free Rate (% per year): 3.17% Minimum X value 80 Maximum X value 120 Calculate Draw Graph Option Type: Analytic: European Option Data Time to Exercise: Exercise Price: 4.91 Price: Delta (per $): Gamma (per $ per $): Vega (per %): Theta (per day): Rho (per %): Same as Imply Volatility Put Call Theta S V σ V = σ S V S = / 6.83 = 14.77% Asset Price

20 Goodrich-Morgan swap Pricing a credit instrument

21 G-RB CreditMetrics analysis: setup The leg to consider for Credit Risk is the one between JPMorgan and BF Goodrich Cashflows of the leg (in million USD): upfront 5.5 per yr, during 8 years Assume: constant spread h = 180 bpi 2 state transition probabilities matrix

22 G-RB CreditMetrics: expected cashflows Since Expected[cashflows] Then E[cashflows] = = ($cashflows) * Prob{non_default} Sum( 5.5 * each year}) But at the same time 8 E[cashflow] = (5.5)(exp( ( ri + h) ti) i= 1

23 G-RB CreditMetrics: probability of default Under our assumptions (see class notes) : P {non-default} = exp(-h) = exp(-.018) = constant for each year The 2 state matrix: BBB D BBB D 0 1

24 G-RB CreditMetrics: compute cashflows Notice P{default of BBB corp.} = 1.8% is very HIGH Rates in these days were around 10% Using a gvmnt zero curve for August 1983, the rates we ll use are: r = (.08850,.09297,.09656, ,.10550, ,.11770, ) for years (1,2,3,4,5,6,7,8)

25 G-RB CreditMetrics: cashflows (cont) E[cashflows] 8 = (5.5) exp( ( r t)) (.9822) = i= 1 i i i Non-Risk Cashflows 8 = (5.5) exp( ( r t)) = i= 1 i i

26 G-RB CreditMetrics: Expected losses Therefore E[loss] = 1 ( E[cashflows] / Non-Risk Cashflow) = i.e. the proportional expected loss is around 6.58% of USD million Or roughly E[loss] = (USD million)

27 The full swap If we consider the full swap, we need to consider the default process b and the interest rate process r. The random variable that describes losses is given by Loss = 50 8 years (11 libor ) If we assume the credit process and the market process are independent, we get 8 years This will overstimate the risk in the case that the default process and the market process are negatively correlated. t + e t r t b [ ] r t E(11 libor ) e t [Eb ] ECL = 50 t + t t

28 The MonteCarlo approach Correlation on market variables drive correlations of default events: ρ( Libor,GR) = Then, and ρ( Libor,b ) = years is calculated with Monte-Carlo techniques. t t [ ] r t E(11 libor ) e t [Eb ] ECL = 50 t + t

29 The CreditMetrics Approach Assume a 1 year time horizon, and that we wish to calculate the loss statistics for that time horizon. Assume credit ratings with transition probabilities from BBB given by AAA 25 AA 40 and spreads given by A 100 BBB 180 BB 250 B 320 CCC 500 Default

30 The loss statistics (1 year forward) The loss statistics can be summarized as follows Credit event Mtm Change in $K Spread (bpi) prob default AAA AA A BBB BB B CCC Default Average Std

31 Loss stats over the life of the asset Expected exposures, and exposure quantiles (in the case of this swap) will generally decrease over the life of the asset. They are pure market variables, which can be calculated with monte carlo methods. Probability of default, and the probability of other credit downgrades, increase over the live of the asset. They are calculated, either with transition probability matrices, or with default probability estimations (Merton s model, for instance) Discount factors will also decrease with time, and are given by the discount curve. t [ ] PV - ECL = E CL t PV t

32 Pricing the deal Assume the ECL=$50,000, and UCL=$200,000. GR swap bps $K Capital at Risk (UL, or CVAR) 200 Cost of capital is (15-8=7%) Required net income (8 years) 112 Tax (40%) 75 Pretax net income 187 Operating costs 100 Credit Provision (ECL) 50 Hedging costs Required revenue

33 Goodrich Calculating credit exposure

34 Credit VaR Credit Exposure How much one can loose due to counterparty default max( Swap Value t, 0 )

35 Credit VaR 99% Credit VaR Sort losses and take the 99 th percentile

36 Expected Shortfall Expected Loss given 99% VaR Take the average of the exposure greater than 99% percentile.

37 Simulation Monte Carlo simulation 10,000 simulations Simulate Interest Rates Credit Spreads

38 Interest Rates Black-Karasinski Model d ln r θ = a ln a r dt + σ r dw Tenor Init. IR Mean Vol 0.5yrs 8.18% 7.99% 5.98% 10yrs 10.56% 8.93% 5.64% Est. from Bonds

39 Spreads Vasicek Model ds = θ a a s dt + σ s dw Tenor Init. IR Mean Vol 5yrs 2.4% 2.546% 0.535%

40 Algorithm IR-Spread Choleski Decomposition Sample from Normal distribution Interest Rate-Spread Corr Corr of 6mo and 10yr rate Est. from Bond Data Corr of spread to 5yr IR Est. from New Car Sales and Bond rates 71-83

41 Algorithm Iterate the Black-Karasinski Calculate the Value of the Swap as the difference of the values of Non-Defaultable Fixed and Floating Bonds After 10,000 calculate the credit VaR and the expected shortfall

42 Simulation: Credit Exposure Credit Exposure $ $ $ Exposure $ $ Series1 $ $ $ Time/Month

43 Simulation: Expected Shortfall Expected Shortfall $ $ $ $ Shortfall $ $ $ $ $ $ Series Time/Month