Risk Management Exercises
Exercise Value at Risk calculations
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
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? 1. 525 2. 775 3. 600 4. 700
Exercise Hedging
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.
Exercise Credit VaR
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
Exercise The Merton Model
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.
Credit Spread A leverage of 0.9 implies that which says that K=99.46. Using Black-Scholes, we get that the call option is worth S=$13.59. 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%
Option Calculation Underlying Data Graph Results Underlying Type: Time Dividend Vertical Axis: Equity 1 Theta Horizontal Axis: Stock Price: 100.00 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: 1.0000 Exercise Price: 99.46 Imply Volatility Put Call 0.012 0.01 0.008 0.006 Theta 0.004 Price: 13.5923574 Delta (per $): 0.73469442 Gamma (per $ per $): 0.01638677 Vega (per %): 0.32773533 Theta (per day): -0.0253837 Rho (per %): 0.59877085 0.002 0 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00-0.002-0.004 Asset Price
The risk neutral probability of default Given by N ( 2 d ) = 0.6653, EDF= 1 N( d ) = 2 33.47%
Expected loss It is given by $3.96 0.3347 0.2653 $100 $90 0.3347 ) ( ) ( ) ( 2 1 2 = = = d N d N V Ke d N ECL rτ
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.
Exercise Calibrating the asset volatility
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 = 23,508,000. 0.0317
Bootstrapping asset volatility Underlying Data Graph Results Underlying Type: Time Dividend Vertical Axis: Equity 1 0.023 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 0.012 0.01 Slightly high Time to Exercise: 1.0000 Exercise Price: 4.91 Price: 2.42732841 Delta (per $): 0.83368681 Gamma (per $ per $): 0.07395122 Vega (per %): 0.01697587 Theta (per day): -0.001435 Rho (per %): 0.03244841 Put Call Theta 0.008 0.006 0.004 0.002 σ V = σ S V V S 0 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00-0.002-0.004 Asset Price S
Bootstrapping asset volatility (iterative process) Underlying Data Graph Results Underlying Type: Time Dividend Vertical Axis: Equity 1 0.023 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: 1.0000 Exercise Price: 4.91 Price: 2.04713078 Delta (per $): 0.99368819 Gamma (per $ per $): 0.01769645 Vega (per %): 0.00120993 Theta (per day): -0.0004339 Rho (per %): 0.04713643 Same as Imply Volatility Put Call Theta 0.012 0.01 0.008 0.006 0.004 0.002 S V σ V = σ S V S = 0.4959 2.087 0.9936 / 6.83 = 14.77% 0 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00-0.002-0.004 Asset Price
Goodrich-Morgan swap Pricing a credit instrument
G-RB CreditMetrics analysis: setup The leg to consider for Credit Risk is the one between JPMorgan and BF Goodrich Cashflows of the leg (in million USD): 0.125 upfront 5.5 per yr, during 8 years Assume: constant spread h = 180 bpi 2 state transition probabilities matrix
G-RB CreditMetrics: expected cashflows Since Expected[cashflows] Then E[cashflows] = = ($cashflows) * Prob{non_default}.125 + Sum( 5.5 * P{nondefault @ each year}) But at the same time 8 E[cashflow] =.125 + (5.5)(exp( ( ri + h) ti) i= 1
G-RB CreditMetrics: probability of default Under our assumptions (see class notes) : P {non-default} = exp(-h) = exp(-.018) =.98216 constant for each year The 2 state matrix: BBB D BBB.9822.0178 D 0 1
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,.0987855,.10550,.104355,.11770,.118676) for years (1,2,3,4,5,6,7,8)
G-RB CreditMetrics: cashflows (cont) E[cashflows] 8 =.125 + (5.5) exp( ( r t)) (.9822) = 23.0527 i= 1 i i i Non-Risk Cashflows 8 =.125 + (5.5) exp( ( r t)) = 24.67581 i= 1 i i
G-RB CreditMetrics: Expected losses Therefore E[loss] = 1 ( E[cashflows] / Non-Risk Cashflow) =.065776 i.e. the proportional expected loss is around 6.58% of USD 24.67581 million Or roughly E[loss] = 1.623 (USD million)
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
The MonteCarlo approach Correlation on market variables drive correlations of default events: ρ( Libor,GR) = -0.47 Then, and ρ( Libor,b ) = 0.47 8 years is calculated with Monte-Carlo techniques. t t [ ] r t E(11 libor ) e t [Eb ] ECL = 50 t + t
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
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 155 25 0.02 31 AA 140 40 0.33 462 A 80 100 2.95 2360 BBB 0 180 86.93 0 BB -70 250 5.3-3710 B -140 320 1.17-1638 CCC -320 500 0.12-384 Default -10000 0.18-18000 Average -2609.88 Std 6457.41
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
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 0.50 0.50 327
Goodrich Calculating credit exposure
Credit VaR Credit Exposure How much one can loose due to counterparty default max( Swap Value t, 0 )
Credit VaR 99% Credit VaR Sort losses and take the 99 th percentile
Expected Shortfall Expected Loss given 99% VaR Take the average of the exposure greater than 99% percentile.
Simulation Monte Carlo simulation 10,000 simulations Simulate Interest Rates Credit Spreads
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
Spreads Vasicek Model ds = θ a a s dt + σ s dw Tenor Init. IR Mean Vol 5yrs 2.4% 2.546% 0.535%
Algorithm IR-Spread Choleski Decomposition Sample from Normal distribution Interest Rate-Spread Corr Corr of 6mo and 10yr rate Est. from Bond Data 1 0.9458 0.53 1 0.53 1 Corr of spread to 5yr IR Est. from New Car Sales and Bond rates 71-83
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
Simulation: Credit Exposure Credit Exposure $14.0000 $12.0000 $10.0000 Exposure $8.0000 $6.0000 Series1 $4.0000 $2.0000 $0.0000 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 Time/Month
Simulation: Expected Shortfall Expected Shortfall $18.00000 $16.00000 $14.00000 $12.00000 Shortfall $10.00000 $8.00000 $6.00000 $4.00000 $2.00000 $0.00000 Series1 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 Time/Month