Financial Risk Measurement/Management

550.446 Financial Risk Measurement/Management Week of September 23, 2013 Interest Rate Risk & Value at Risk (VaR) 3.1 Where we are Last week: Introduction continued; Insurance company and Investment company business models and risks; Financial Instrument Recap & Hedging (Chapters 4, 5,7) This week: Finish some Hedging Ideas (Chapter 7, Linear vs. Nonlinear) Interest Rate Risk and an Introduction to Value at Risk (VaR) (Chapters 8-9) Next week+: Volatility, Correlation and the Copula model (Chapters 10-11) 3.2 Assignment Assignment For September 23 rd (Next Week) Read: Hull Chapters 8-9 (Interest Rate Risk & VAR) Problems (Due September 23 rd, today) Chapter 5: 3, 8, 10, 13, 21 Chapter 7: 1, 3, 6, 14; 16 Problems (Due September 30 th ) Chapter 8: 6, 7, 10, 11; 16 Chapter 9: 1, 3, 4, 5; 12 For September 30 th (Next Week) Read: Hull Chapters 10-11 (Volatility, Correlation and the Copula model) Problems (Due September 30 th ) Chapter 8: 6, 7, 10, 11; 16 Chapter 9: 1, 3, 4, 5; 12 Problems (Due Oct 7 th + ) Chapter 10 (Oct 7): 1, 2, 5, 7, 8, 9, 11, 15, 16; 19, 21 Chapter 11 ( Oct 7): 1, 2, 5, 6, 8, 10, 14; 19 3.3 3.4 1

Assignment Midterm: October 30, 2013 Final Exam Wednesday, December 18 th ; 9am - Noon Shaffer 202 A Trader s Gold Portfolio How Should Trading Risks Be Hedged? Position Value ($) Spot Gold 180,000 Forward Contracts 60,000 Futures Contracts 2,000 Swaps 80,000 Options 110,000 Exotics 25,000 Total 117,000 3.5 3.6 Hedging on the Trading Desk using Delta Linear vs. Nonlinear Products - Hedging Delta of a portfolio is the partial derivative of a portfolio with respect to the price of the underlying asset (gold in this case) Suppose that a $0.1 increase in the price of gold leads to the gold portfolio increasing in value by $100 The delta of the portfolio is 1000 The portfolio could be hedged against short-term changes in the price of gold by selling 1000 ounces of gold 10 futures. This is known as making the portfolio delta neutral When the price of a product is linearly dependent on the price of an underlying, a hedge and forget strategy can be used Minimum variance hedge ratios work best for futures S or P F A P is the value of the portfolio, is its beta, and A is the value of the assets underlying one futures contract Non-linear products require the hedge to be rebalanced to preserve delta neutrality Or you use a non-linear hedge Options are an example of a non-linear hedge vehicle non-constant delta 3.7 3.8 2

Delta of the Option Example Short Call B Option price A Slope = Stock price A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend paying stock S 0 = 49, K = 50, r = 5%, = 20%, T = 20 weeks, = 13% The Black-Scholes value of the option is $240,000 (DerivaGem) How does the bank hedge its risk to lock in a $60,000 profit? 3.9 3.10 Delta Hedging a Short Call Example Short Call (Expires In-the-Money) Initially the delta of the option is 0.522 This means that 52,200 shares of stock are purchased to create a delta neutral position But, if a week later delta falls to 0.458, 6,400 shares must be sold to maintain delta neutrality The option is non-linear so the delta changes with price Tables 7.2 and 7.3 (pages 142 and 143) provide examples of how delta hedging might work for the option. 3.11 Offset by the $300K option premium 3.12 3

Example Short Call (Expires Worthless) Gamma Gamma ( ) is the rate of change of delta ( ) with respect to the price of the underlying asset Gamma is greatest for options that are close to the money Offset by the $300K option premium 3.13 3.14 Gamma Measures the Delta Hedging Errors Caused By Curvature (Figure 7.4, page 145) C'' C' C Call price S S' Stock price 3.15 Gamma Suppose a delta-neutral PF has gamma and a traded option has gamma the total gamma of the PF with option is T T Where T / T is the number of options to make the PF gamma-neutral Since this is likely to change the delta of the PF with option, the underlying asset position will need rebalancing 3.16 4

Gamma - Example Suppose a delta-neutral PF has gamma -3,000 And the delta-gamma of a traded call option is 0.62 and 1.50, respectively The PF can be made gamma-neutral with a long position of T / T 3000 /1.5 2,000 options The delta of the PF is now 2,000 x 0.62 = 1,240 Therefore, 1,240 of the underlying asset must be sold to keep the total PF delta-neutral as well as gamma-neutral Vega Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility P Vega tends to be greatest for options that are close to the money 3.17 3.18 Vega As with gamma, a delta-neutral PF can be made (both gamma and) Vega neutral A delta-neutral PF has gamma-vega of -5,000 and -8,000, respectively Suppose there are traded options as shown DELTA GAMMA VEGA Portfolio 0.0-5,000-8,000 Option 1 0.6 0.5 2.0 Option 2 0.5 0.8 1.2 If 1, 2 are the quantities of Options 1&2, then for a gamma-vega neutral PF we must have 3.19 Vega Simultaneously For gamma-neutral: 5,000 0.5 1 0.8 2 0 and For Vega-neutral: 8,000 2.0 1 1.2 2 0 So 1 400 & 2 6,000 But now the delta of the PF after adding these 2 options is 400 0.6 6, 000 0.5 3, 240 So to restore delta neutrality to the combined PF with Options 1&2, we need to sell 3,240 units of the asset 3.20 5

Gamma and Vega Limits In practice a trader responsible for all trading involving a particular asset must keep Gamma and Vega within limits set by risk management Theta Theta ( ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option declines 3.21 3.22 Hedging in Practice Traders usually ensure that their portfolios are delta-neutral at least once a day Whenever the opportunity arises, they improve gamma and Vega As portfolio becomes larger hedging becomes less expensive Sources of Interest Rate Risk More complicated and difficult than the risk of exposure to other market factors Many Interest Rates and the Term Structure Types of IR risk include Interest Rate Margin for Banks P/L in the value of the asset / liability PF We look at the process to control these risks Liquidity & Solvency 3.23 3.24 6

Management of Net Interest Income Suppose that the market s best guess is that future short term rates will equal today s rates What would happen if a bank posted the following rates? Maturity (yrs.) Deposit Rate Mortgage Rate 1 3% 6% 5 3% 6% How can the bank manage its risks? Management of Net Interest Income The following might make more sense Maturity (yrs) Deposit Rate Mortgage Rate 1 3% 6% 5 4% 7% A source of liquidity preference in higher rates w/longer maturities And why the yield curve rises (even when rates are expected to remain the same) 1.25 1.26 LIBOR Rates LIBOR rates are 1-, 3-, 6-, and 12-month borrowing rates for financial companies that have a AA-rating To extend the LIBOR zero curve we can Create a zero curve to represent the rates at which AA-rates companies can borrow for longer periods of time Create a zero curve to represent the future short term borrowing rates for AA-rated companies In practice we do the second 3.27 Swap Rates A bank can Lend to a series AA-rated borrowers for ten successive six month periods Swap the LIBOR interest received to the fiveyear swap rate This shows that swap rates can be used to extend the LIBOR zero curve in the preferred way To avoid ambiguity we refer to it as the LIBOR/swap zero curve 3.28 7

Risk-Free Rate In practice traders and risk managers assume that the LIBOR/swap zero curve is the risk-free zero curve The Treasury curve is about 30 basis points below the LIBOR/swap zero curve Treasury rates are considered to be artificially low for a variety of regulatory and tax reasons 3.29 Duration A measure of PF exposure to yield curve movements Suppose y is a bond s yield and B is its corresponding price n (where cash flow c i is at time t i ), then: yti B ce i B Duration, D, is given by D i y 1 B, the percent change in price to a small change in yield n yt ce i i This leads to the explicit relationship: D ti i 1 B Dollar Duration, D $ = D B, is the product of duration & price B D$ y the dollar change due to small change in yield Or using continuous notation db (like the delta measure) D$ dy 3.30 Calculation of Duration 3-year bond paying a coupon 10%; Bond yield=12%; B = 94.213 & D = 2.653 (Table 8.3, page 166) Time (yrs) Cash Flow ($) PV ($) Weight =PV($)/B Time Weight 0.5 5 4.709 0.050 0.025 1.0 5 4.435 0.047 0.047 1.5 5 4.176 0.044 0.066 2.0 5 3.933 0.042 0.083 2.5 5 3.704 0.039 0.098 3.0 105 73.256 0.778 2.333 Total 130 94.213 1.000 2.653 3.31 Duration (Continued) When the yield y is expressed with periodic compounding m times per year BD y B 1 y m The expression D 1 y m is referred to as the modified duration Modified for periodic compounding vs. continuous compounding (McCauley s Duration) as before 3.32 8

Convexity The convexity of a bond is defined as n 2 yti 2 ct i ie 1 B i 1 C B y 2 B so that B 1 D y C ( y ) B 2 2 Portfolios Duration and convexity can be defined similarly for portfolios of bonds and other interest-rate dependent securities The duration of a portfolio is the weighted average of the durations, D i, of each of the i components of the portfolio weighted by the PF percent of the i th component s value, X n i X i 1 X i D Di where Di i 1 P Xi y Similarly for Convexity 3.33 3.34 Portfolios Non-Parallel YC Moves Starting Zero Curve (Figure 8.3, page 173) Duration represents the sensitivity of the PF to a parallel shift in the (spot) yield curve Very often traders want protection for non-parallel shifts Therefore, a measure of PF sensitivity to non-parallel shifts is desired The most common measure is Partial Duration Where P is value of PF 1 Pi ΔP i is the sensitivity in PF value to Δy i Di P yi D i is the partial duration for y i For the several (spot) yield terms, i (compare to last slide) The sum of the partial durations equals the usual duration measure 3.35 3.36 9

Calculating a Partial Duration (Figure 8.4, page 173) Portfolios Non-Parallel YC Moves From the definition of duration, we have for the sensitivity of the PF value to each discrete yield change Pi DP i yi This is the dollar duration, or Delta, of the portfolio So for a $1 million PF and a 1 bp increase 3.37 3.38 Portfolios Non-Parallel YC Moves For non-parallel shifts, we can use partial durations Define a particular non-parallel rotation (Fig 8.5) amount -3e -2e -e 0 e 3e 6e term (yrs) 1 2 3 4 5 7 10 Where we let e = 1 bp (.0001) for example From the partial durations in Table 8.5 & $1mm PF The %PF change to the 1-yr move is -0.2 x (-3e) = 0.6e And for the 2-yr move: -0.6 x (-2e) = 1.2e Yrs 1 2 3 4 5 7 10 Tot D$ 60 120 90 0-200 630 1800 2500 Compare to result in Table 8.6 3.39 Combining Partial Durations to Create Rotation in the Yield Curve (Figure 8.5, page 174) Zero Rate (%) 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 Maturity (yrs) 3.40 10

IR Deltas in Practice In practice a number of approaches are used to calculate IR Deltas Define Delta as Dollar Duration Similarly, the DV01 (dollar value of a basis point) DV01 = Dollar Duration x.0001 Moreover, traders like to capture YC shape change Partial Duration (as before) Rather than moving one point w/linear interpolation, consider moving a bucket: Preferred approach in ALM (see next slide) Vary hedge instruments used to determine Zero Curve Principal Component Analysis 3.42 Change When One Bucket Is Shifted (Figure 8.6, page 149) Zero Rate (%) 6 5 4 3 2 1 0 0 2 4 6 8 10 12 Maturity (yrs) 3.43 Principal Components Analysis Attempts to identify standard shifts (or factors) for the yield curve so that most of the movements that are observed in practice are combinations of the standard shifts 3.44 Principal Components Analysis Common approach for representing correlated data through factors that capture the essence For IR we use daily changes of the several term yields Find Variance-Covariance matrix: determine eigenvectors and eigenvalues Eigenvectors are Factor Loadings Eigenvalues are variance of the Factor Scores which represent each days change Factor Scores are uncorrelated across the data The following comes from 2,780 observations of swap rates between 2000 and 2011 3.45 11

Principal Components Analysis Example Principal Components Analysis Example The first factor is a roughly parallel shift (90.1% of variation explained) 2 (17.55) / (338.8) 90.9% The second factor is a twist (explanation of 1 st 2 two factors) (17.55 4.77) / (338.8) 97.7% The third factor is a bowing (a further 1.3% of variation explained) 3.46 3.47 Principal Components Analysis Example Gamma for Interest Rates Gamma has the form 2 x x i j where x i and x j are yield curve shifts considered for delta To avoid too many numbers being produced one possibility is consider only i = j (ignore cross gammas) Another is to consider only a single gamma measure: the 2 nd partial derivative of the PF value w/r to a parallel shift of the zero curve Another is to consider the first two or three types of shifts given by a principal components analysis (better!) 3.48 3.49 12

Vega for Interest Rates One possibility is to make the same change to all interest rate implied volatilities. (However implied volatilities for long-dated options change by less than those for short-dated options.) Another is to do a principal component analysis on implied volatility changes More on this when we look closely at volatility next week 3.50 The End for Interest Rates We ve looked at Risk on the Trading Desk Cash prices and Interest Rate sensitive instruments Now we look across the enterprise and the myriad of risk variables rolled up to measures of total/extreme risk 3.51 The Value at Risk (VaR) Measure VaR is an attempt to provide a single number to characterize, in a summary way, the risk for loss in a portfolio total risk It complements trader risk measures for presentation to senior management in assessing risk to capital and/or capital adequacy We are X percent certain that we will not loose more than V dollars in the next N days JP Morgan: RiskMetrics 1990s The Value at Risk (VaR) Measure The variable V is the VaR of the portfolio It is a function of two parameters The time horizon (N days, for example), and The confidence level (X %) The loss level, V, over N days that we are X % certain will not be exceeded LOSS DISTRIBUTION 3.52 3.53 13

The Value at Risk (VaR) Measure When N days is the time horizon and X % the confidence level, VaR is the loss corresponding to the (100-X)th percentile of the distribution of the change in the value of the portfolio over the next N days The Value at Risk (VaR) Measure VaR is attractive because, in some sense, it answers the question how bad can things get? Though, not entirely; it answers the question: X% of the time things won t get worse than V Doesn t really tell us how bad things can get P/L DISTRIBUTION 3.54 3.55 The Value at Risk (VaR) Measure To get a closer representation of how bad can things get? We need to answer, If things are as bad as they get only (100-X)% of the time, how much can we expect to lose? This is Expected Shortfall or the Conditional VaR (C-VaR) measure The expected loss during an N day period conditioned on the event that we are in the (100-X)% left tail P/L DISTRIBUTION VaR vs. Expected Shortfall VaR is the loss level that will not be exceeded with a specified probability Expected shortfall is the expected loss given that the loss is greater than the VaR level (also called C-VaR and Tail Loss) Two portfolios with the same VaR can have very different expected shortfalls 3.56 8.57 14

Distributions with the Same VaR but Different Expected Shortfalls VaR and Regulatory Capital Bank regulators require banks to calculate VaR with N=10 and X=99% for their Trading Book The market-risk capital is k times the 10-day 99% VaR where k is at least 3.0 Under Basel II capital for Credit risk and Operational risk is based on a one-year 99.9% VaR 3.58 3.59 VaR and Regulatory Capital Suppose VaR of a PF for a confidence level of 99.9% and time horizon of 1-year is $50million In extreme circumstances (once every 1,000 years) the institution is expected to lose more than $50million If it keeps $50million in capital, it will have a 99.9% probability of not running out of capital in the course of one year If we are trying to define a risk measure that will equate to the amount of capital an institution should hold, is VaR the best to do this? This question has been studied 3.60 Coherent Risk Measure A number of properties have been proposed as desirable for a risk measure if it is to be used to set capital requirements A risk measure that satisfies all these properties is called a coherent risk measure These properties follow 3.61 15

Coherent Risk Measures Properties of coherent risk measures If one portfolio always produces a worse outcome than another its risk measure should be greater (monotonicity) If we add an amount of cash K to a portfolio its risk measure should go down by K (translation invariance) buffer against loss Changing the size of a portfolio by should result in the risk measure being multiplied by (homogeneity) 2x the PF; 2x loss The risk measures for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were merged (subadditivity) allows for diversification to reduce risk 3.62 VaR vs Expected Shortfall VaR satisfies the first three conditions but not always does it satisfy the fourth one Expected shortfall (C-VaR) satisfies all four conditions. 3.63 VaR vs Expected Shortfall VaR vs Expected Shortfall Example 9.6 : Two $10 million one-year loans each of which has a 1.25% chance of defaulting. All recoveries between 0 and 100% are equally likely. If there is no default the loan leads to a profit of $0.2 million. If one loan defaults it is certain that the other one will not default. 3.64 Example 9.6 (Continued): Single Loan 1-yr 99% VaR is $2m 1.25% chance of loss; if loss, then 80% chance it is > $2m (uniform) Unconditional prob of loss > $2m is 80% of 1.25% = 1% (99% VaR) Two Loan PF 1-yr 99% VaR is $5.8m Default occurs 1.25% of time, but never together Pr a default occurs is 2.5% (1.0 -.975 = probability of a default) If there is a loss, then 40% chance it is > $6m Unconditional prob of loss > $6m is 40% of 2.5% = 1% (99% VaR) A profit of $.2m on other loan plus loss of $6m => $5.8 Two Loans Separately = 2 + 2 = $4m Two Loans Together = $5.8m > Separately => No Subadd. 3.65 16

VaR vs Expected Shortfall Example 9.8 : Consider same situation again, but for C-VaR C-VaR (Expected Shortfall) from 1 loan for 1-year and 99% confidence level is Expected Loss conditional on loss > $2m = 99% 1-yr VaR W/uniform loss [0,$10m]; expected loss, conditioned on > $2m,is halfway along the interval [$2m,$10m] = $6m VaR for PF of 2 loans = $5.8m C-VaR is expected loss on PF conditional on loss > $5.8m When 1 loan defaults, other doesn t; outcomes = uniform [+$.2,-$9.8m] Expected loss given we are in the part of the distribution [$5.8m,$9.8m] is $7.8m Note: $6m + $6m > $7.8m, so C-VaR is subadditive 3.66 Spectral Risk Measures Risk measures are characterized by the weights assigned to quantiles of the loss distribution VaR assigns all weight to Xth quantile Expected shortfall assigns equal weight to all quantiles greater than the Xth quantile and zero to all below the Xth quantile We now consider alternatives 3.67 Spectral Risk Measures A spectral risk measure may be defined by making other assumptions about weights assigned to quantiles of the loss distribution A spectral risk measure is coherent, that is, It satisfies the Subadditivity condition, IF The weights are a non-decreasing function of the quantiles such as with Conditional VaR Another possibility for a risk measure is to make the weight assigned to q-th quantile proportional (1 to q )/ e where γ is a constant This is called the exponential spectral risk measure 3.68 Spectral Risk Measures 3.69 17

Normal Distribution Assumption The simplest assumption is that daily gains/losses are normally distributed and independent It is then easy to calculate 99%, 1-Day VaR from the standard deviation of the 1-day loss distribution (1-day VaR=2.33 ) Since N( 2.33)=0.01 or N(2.33)=0.99 The N-day VaR equals N times the one-day VaR Regulators allow banks to calculate the 10 day VaR as 10 times the one-day VaR 3.70 Choice of VaR Parameters Time horizon should depend on how quickly portfolio can be unwound. Regulators in effect use 1-day for bank market risk and 1-year for credit/operational risk. Fund managers often use one month Confidence level depends on objectives. Regulators use 99% for market risk and 99.9% for credit/operational risk. A bank wanting to maintain a AA credit rating will often use 99.97% for internal calculations. (VaR for high confidence levels cannot be observed directly from data and must be inferred in some way.) 3.71 18