CAS Exam 8 Notes - Parts F, G, & H. Financial Risk Management Valuation International Securities

Transcription

1 CAS Exam 8 Notes - Parts F, G, & H Financial Risk Management Valuation International Securities

2

3 Part III Table of Contents F Financial Risk Management 1 Hull - Ch. 17: The Greek letters Hull - Ch. 20: Value at Risk (VaR) Hull - Ch. 22b: Credit risk Culp, Miller and Neves: Value at risk - Uses and abuses Stulz: Rethinking risk management Butsic: Solvency measurement for Property-Liability RBC applications Cummins: Allocation of capital in the insurance industry Goldfarb: Risk-adjusted performance measurement for P&C insurers BKM - Ch. 1: The investment environment BKM - Ch. 2: Asset classes and financial instruments Gorvett: Insurance securitization - The development of a new asset class G Valuation 83 BKM - Ch. 18: Equity valuation models Goldfarb: P&C insurance company valuation H International Securities 109 BKM - Ch. 25: International diversification Additional Notes 117

4

5 F Financial Risk Management

6

7 Hull - Ch. 17: The Greek letters Introduction Financial institution that sells an OTC option to a client is faced with problem of managing its risk When option tailored to client and standardized exchanges products, hedging is difficult Each Greek letter measures a different dimension to the risk in an option position and the aim of a trader is to manage the Greeks so that all risks are acceptable Illustration Financial institution sold for $300k a European call on 100,000 shares of non-dividend-paying stock Assume: Stock price = $49, strike price = $50, risk-free rate = 5%, stock volatility = 20% per annum, time to maturity = 20 weeks, and expected return = 13% S 0 = 49, K = 50, r = 0.05, σ = 0.20, T = , µ = 0.13 Black-Scholes price of option $240,000 Financial institution sold option for $60,000 more than its theoretical value, but it is faced with problem of hedging the risks Naked and covered positions Strategy open to financial institution: Do nothing (naked position) Very risky if call exercised As an alternative to a naked position, the financial institution can adopt a covered position: This involves buying 100,000 shares as soon as the option has been sold If option exercised, strategy works well, but can lead to significant loss in other circumstances E.g., if the stock price drops to $40, the financial institution loses $900,000 Put-call parity: Exposure from writing a covered call = exposure from writing a naked put Neither a naked position nor a covered position provides a good hedge: If the assumptions underlying the Black-Scholes formula hold, the cost to the financial institution should always be $240,000 on average for both approaches But on any one occasion the cost is liable to range from zero to over $1,000,000 A good hedge would ensure that the PV of the expected cost is always close to $240,000 A stop-loss strategy Involves buying one unit of stock as soon as its price > K and selling it as soon as price < K Objective: Hold naked position when stock price < K; Covered position when stock price > K The procedure is designed to ensure that at time T the institution owns the stock if the option closes in the money and does not own it if the option closes out of the money The strategy produces payoffs that are the same as the payoffs on the option It seems that the total cost Q of writing and hedging the option is the option s intrinsic value: Q = max(s 0 K, 0) (1) The cost of setting up the hedge initially is S 0 if S 0 > K and zero otherwise All purchases and sales subsequent to time 0 are made at price K If this were in fact correct, the hedging scheme would work perfectly in the absence of transactions costs Furthermore, the cost of hedging the option would always be less than its Black-Scholes price An investor could earn riskless profits by writing options and hedging them There are two basic reasons why Eq. (1) is incorrect: 1. The cash flows to the hedger occur at different times and must be discounted 2. Purchases and sales cannot be made at exactly the same price K Cannot legitimately assume that both purchases/sales are made at same price If markets efficient, cannot know if, when stock price = K, it will continue above/below K In practice, purchases must be made at price K + ε/sales must be made at price K ε Every purchase/subsequent sale involves a cost (apart from transaction costs) of 2ε As ε is made smaller, trades tend to occur more frequently The lower cost per trade is offset by the increased frequency of trading 3

8 Stop-loss strategy, although superficially attractive, does not work well as hedging scheme Consider out-of-the-money option: If stock price never reaches strike price K, hedging costs nothing If the path of stock price crosses strike price level many times, scheme is expensive Hedge performance measure: Ratio [std. dev. of cost of hedging] [Black-Scholes option price] A perfect hedge would have a hedge performance measure of zero With stop-loss strategy, using Monte Carlo simulations, it appears to be impossible to produce a hedge performance measure below 0.70 regardless of how small t is made Delta hedging Delta ( ) of an option Rate of change of the option price wrt. the price of the underlying asset It is the slope of the curve that relates the option price to the underlying asset price Suppose that the delta of a call option on a stock is 0.6: When stock price changes by small amount, option price changes by 60% of that amount Denoting c the price of the call option and S the stock price, = c/ S Terminology Delta neutral: A position with a delta of zero changes Investor s position remains hedged ( neutral) for only short period of time The hedge has to be adjusted periodically, aka rebalancing Delta-hedging is an example of dynamic hedging: It can be contrasted with static hedging, where the hedge is set up initially and never adjusted, aka hedge-and forget Delta hedging and Black-Scholes Black, Scholes, and Merton showed that it is possible to set up a riskless portfolio consisting of a position in an option on a stock and a position in the stock Expressed in terms of, the Black-Scholes portfolio is: 1: Option + : Shares of the stock BSM valuation: Set up a -neutral position and argue that return should be risk-free rate Delta of a European stock option European call on non-dividend-paying stock: (call) = N(d 1 ), where d 1 defined as in Eq. (13.15) Formula gives of long position in one call: Using hedging for long position in European call involves maintaining short position of N(d 1 ) shares for each option purchased of short position in one call option is N(d 1 ): Using hedging for short position in European call involves maintaining long position of N(d 1 ) for each option sold For European put option on non-dividend-paying stock: (put) = N(d 1 ) 1 Delta is negative Long position in put option hedged with long position in underlying stock Short position in put option hedged with short position in underlying stock 1.0 Delta of call 0.0 Delta of put K Stock price Delta of call In the money At the money Out of the money K Stock price Figure 1: Variation of with stock price Time to expiration Figure 2: Variation of with maturity for call 4

9 Dynamic aspects of Delta hedging The table below provides an example of the operation of delta hedging Stock Shares Cost of shares w/ Cumulative Interest Week price Delta purchased including interest ($000) cost ($000) ,200 2, , (6,400) (308.0) 2, (5,800) (274.7) 1, , , ,263.3 Initial for option being sold calculated as of short option position initially -52,200 When option is written, $2,557,800 borrowed to buy 52,200 shares at $49 Rate of interest = 5% Interest cost of $2,500 incurred in 1st week The stock price falls by the end of the first week to $48.12 Option declines to New position = -45,800 6,400 of initial shares sold to maintain the hedge Strategy realizes $308k cash, and cumulative borrowings at end of Week 1 = $2,252,300 Toward end of option life, clear that option will be exercised and 1.0 By Week 20, the hedger has a fully covered position Hedger receives $5M for stock held Total cost of writing option/hedging it = $263,300 The costs of hedging the option, when discounted to the beginning of the period, are close to but not exactly the same as the Black-Scholes price of $240,000 If the hedging scheme worked perfectly, the cost of hedging would, after discounting, be exactly equal to the Black Scholes price for every simulated stock price path Reason for variation in cost of hedging: The hedge is rebalanced only once a week As rebalancing takes place more frequently, the variation in the cost of hedging is reduced The example is idealized: Volatility is constant, there are no transaction costs Delta hedging is a great improvement over a stop-loss strategy: Unlike stop-loss strategy, -hedging gets better as hedge monitored more frequently -hedging keeps financial institution s position as close to unchanged as possible Where the cost comes from The delta-hedging procedure creates the equivalent of a long position in the option This neutralizes the short position the financial institution created by writing the option -hedging a short position generally involves selling stock just after price has gone down and buying stock just after price has gone up Buy-high, sell-low trading strategy! Cost of $240,000 comes from avg. difference between price paid for stock and price realized for it Delta of a portfolio of portfolio Π of options/derivatives dependent on single asset at price S is Π/ S The delta of the portfolio can be calculated from the deltas of individual options in portfolio If a portfolio consists of a quantity w i of option i (1 i n), the delta of the portfolio is: = n w i i i=1 where i is the delta of the i-th option Example - Portfolio with three positions in options A long position in 100,000 call options. The delta of each option is A short position in 200,000 call options. The delta of each option is A short position in 50,000 put options. The delta of each option is The delta of the whole portfolio is: 100, , , 000 ( 0.508) = 14, 900 The portfolio can be made delta neutral by buying 14,900 shares 5

10 Transactions costs Derivatives dealers usually rebalance their positions once a day to maintain delta neutrality When dealer has few options on an asset, this is liable to be prohibitively expensive For a large portfolio of options, it is more feasible: Only one trade in the underlying asset is necessary to zero out for whole portfolio The hedging transactions costs are absorbed by the profits on many different trades Theta The theta (Θ) of a portfolio of options is the rate of change of the value of the portfolio with respect to the passage of time with all else remaining the same, aka the time decay of the portfolio For a European call option on a non-dividend-paying stock [with d 1 and d 2 from Eq. (13.15)]: Θ(call) = S 0N (d 1 )σ 2 T rke rt N(d 2 ) where N (x) = 1 2π e x2 /2 (2) For a European put option on the stock: Θ(put) = S 0N (d 1 )σ 2 T + rke rt N( d 2 ), T in years Because N( d 2 ) = 1 N(d 2 ), Θ(put) > Θ(corresponding call) by rke rt Usually, Θ is quoted with T in days Θ = change in portfolio value when 1 day passes To obtain Θ per calendar day (trading day), formula for Θ must be divided by 365 (252) 0.0 Theta of call option K Stock price 0.0 Out of the money Time to maturity Theta of call option In the money At the money Figure 3: Θ vs. stock price for a call Figure 4: Θ vs. maturity for a call Theta is usually negative for an option Because, as time passes with all else equal, option tends to become less valuable An exception to this could be an in-the-money European put option on a non-dividend-paying stock or an in-the-money European call option on a currency with a very high interest rate The variation of Θ with stock price for a call option on a stock is shown in Fig. 3 When the stock price is very low, theta is close to zero For an at-the-money call option, theta is large and negative As the stock price becomes larger, theta tends to rke rt Fig. 4 shows typical patterns for the variation of Θ with the time to maturity Theta is not the same type of hedge parameter as delta There is uncertainty about future stock price, but no uncertainty about passage of time Makes sense to hedge against changes in price of underlying asset, but not against passage of time However, Θ = useful statistic because, in -neutral portfolio, Θ = proxy for Γ Gamma Γ is the rate of change of the portfolio s delta with respect to the price of the underlying asset It is the second partial derivative of the portfolio with respect to asset price: Γ = 2 Π/ S 2 6

11 If Γ small, delta changes slowly Adjustments to keep portfolio -neutral infrequent However, if Γ large, highly sensitive to price of underlying asset Risky to leave a delta-neutral portfolio unchanged for any length of time Fig. 5 illustrates this point: When the stock price moves from S to S, delta hedging assumes that the option price moves from C to C, when in fact it moves from C to C The difference between C and C leads to a hedging error Size of error depends on curvature (measured by Γ) between option price/stock price Call price (a) Small > 0 (b) Large > 0 S S C'' C' (c) Small < 0 (d) Large < 0 C S S S S' Stock price Figure 5: Hedging error due to nonlinearity Figure 6: Π vs. S for -neutral portfolio S = price change of underlying asset during t and Π = corresponding price change in portfolio For -neutral portfolio, if terms of order higher than t are ignored: Π = Θ t Γ S2 [Relationship between Π and S, Fig. 6] (3) When Γ > 0, Θ tends to be negative: The portfolio declines in value if there is no change in S, but increases in value if there is a large positive or negative change in S When Γ < 0, Θ tends to be positive and reverse is true: The portfolio increases in value if no change in S but decreases if large positive/negative change in S As Γ, the sensitivity of the value of the portfolio to S increases Making a portfolio Gamma neutral Position in underlying asset has Γ = 0 and cannot be used to change portfolio s Γ What is required is position in instrument not linearly dependent on underlying asset Suppose that -neutral portfolio has Γ, and traded option has Γ T If w T traded options are added to portfolio, portfolio Γ is: w T Γ T + Γ Hence, position in traded option necessary to make portfolio Γ-neutral is Γ/Γ T Including the traded option is likely to change the delta of the portfolio, so the position in the underlying asset then has to be changed to maintain delta neutrality Portfolio gamma neutral only for short period of time: As time passes, Γ neutrality maintained only if position in traded option is adjusted so that it is always equal to Γ/Γ T Making a portfolio Γ-/ -neutral correction for hedging error in Fig. 5 -neutrality provides protection against small stock price moves between rebalancing Γ-neutrality provides protection against larger movements in stock price between rebalancing Assume portfolio -neutral and Γ = 3, 000. /Γ of particular call are 0.62/1.50 Portfolio can be Γ-neutral by including a long position of 3, 000/1.5 = 2, 000 in calls However, the delta of the portfolio will then change from zero to 2, = 1, 240 1,240 units of the underlying asset must be sold from the portfolio to keep it delta neutral Calculation of Gamma For a European call/put option on non-dividend-paying stock: Γ = N (d 1 ) / s 0 σ T Γ of long position always > 0 and varies with S 0 as indicated in Fig. 7 The variation of gamma with time to maturity is shown in Fig. 8 7

12 Gamma of a stock option Gamma for a stock option At the money Out of the money K Figure 7: Γ vs. stock price for an option Stock price 0.0 In the money Time to maturity Figure 8: Γ vs. maturity for an option For an at-the-money option, gamma increases as the time to maturity decreases Short-life at-the-money options have very high gammas The value of the option holder s position is highly sensitive to jumps in the stock price Relationship between delta, theta and gamma Price of single derivative dependent on non-dividend-paying stock must satisfy Eq. (13.13) The value of a portfolio Π of such derivatives must satisfy: Π t It follows that: Π + rs S σ2 S 2 2 Π Π = rπ, and since Θ = S2 t, = Π S, Γ = 2 Π S 2 Θ + rs σ2 S 2 Γ = rπ (4) For a delta-neutral portfolio, = 0 and: Θ σ2 S 2 Γ = rπ When Θ is large and positive, Γ tends to be large and negative, and vice versa Θ can to some extent be regarded as a proxy for Γ in a delta-neutral portfolio Vega In practice, volatilities change over time The value of a derivative is liable to change because of movements in volatility as well as because of changes in the asset price and the passage of time The vega of a portfolio of derivatives V is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset: V = Π/ σ If V high, the portfolio s value is very sensitive to small changes in volatility If V low, volatility changes have relatively little impact on the value of the portfolio Hedging for Vega A position in the underlying asset has zero vega However, the vega of a portfolio can be changed by adding a position in a traded option: If V is the vega of the portfolio and V T is the vega of a traded option, a position of V/V T in the traded option makes the portfolio instantaneously vega neutral Unfortunately, a portfolio that is Γ-neutral will not in general be V-neutral, and vice versa If hedger requires both Γ-/V-neutrality, at least two traded derivatives must be used Example Consider a portfolio that is delta neutral, with Γ = 5, 000 and V = 8, 000 Delta Gamma Vega Portfolio Option Option

13 The portfolio can be made vega neutral by including a long position in 4,000 of Option 1 increases to 2,400 and requires that 2,400 units of asset be sold to maintain -neutrality The gamma of the portfolio changes from -5,000 to -3,000 To make the portfolio gamma and vega neutral, both Option 1 and Option 2 can be used If w 1 /w 2 are the quantities of Option 1/2 that are added to the portfolio, we require: 5, w w 2 = 0 8, w w 2 = 0 } w 1 = 400, w 2 = 6, 000 Portfolio can be made Γ-/V-neutral by including 400 of Option 1 and 6,000 of Option 2 after addition of both traded options = , = 3, 240 3,240 units of asset needed to be sold to maintain -neutrality For European call/put option on non-dividend-paying stock, vega is given by: V = S 0 T N (d 1 ) The vega of a long position in a European or American option is always positive The general way in which vega varies with S 0 is shown in Fig. 9 Vega of a stock option K Stock price Figure 9: Variation of V with stock price for an option The vega calculated from a stochastic volatility model is very similar to the Black-Scholes vega, so the practice of calculating vega from a model in which volatility is constant works reasonably well Gamma neutrality protects against large changes in the price of the underlying asset between hedge rebalancing. Vega neutrality protects for a variable σ Whether it is best to use an available traded option for vega or gamma hedging depends on the time between hedge rebalancing and the volatility of the volatility When volatilities change, the implied volatilities of short-dated options tend to change by more than the implied volatilities of long-dated options The vega of a portfolio is often calculated by changing the volatilities of long-dated options by less than that of short-dated options Rho Rho: Rate of change of the value of the portfolio with respect to the interest rate: Π/ r Measures sensitivity of portfolio to a change in the interest rate when all else remains the same For European call option on non-dividend-paying stock: ρ(call) = KT e rt N(d 2 ) For European put option: ρ(put) = KT e rt N( d2) The realities of hedging In an ideal world, traders working for financial institutions would be able to rebalance their portfolios very frequently in order to maintain all Greeks equal to zero In practice, this is not possible: When managing a large portfolio dependent on a single underlying asset, traders usually make delta zero, or close to zero, at least once a day by trading the underlying asset Unfortunately, Γ = 0 and V = 0 are less easy to achieve because it is difficult to find options/other nonlinear derivatives that can be traded in volume required at competitive prices 9

14 There are big economies of scale in trading derivatives: Maintaining delta neutrality for a small number of options on an asset by trading daily is usually not economically feasible. The trading costs per option being hedged is high But when a derivatives dealer maintains delta neutrality for a large portfolio of options on an asset, the trading costs per option hedged are likely to be much more reasonable Scenario analysis In addition to monitoring risks such as, Γ, and V, traders also carry out a scenario analysis: Calculate gain/loss over specified period under various scenarios The time period chosen is likely to depend on the liquidity of the instruments The scenarios can be either chosen by management or generated by a model Extension of formulas Formulas for, Θ, Γ, V, and ρ change when stock pays continuous dividend yield q Greek Call option Put option e qt N(d 1 ) e qt [N(d 1 ) 1] Γ Θ S0N (d 1)σe qt 2 T N (d 1)e qt S 0σ T + qs 0 N(d 1 )e qt rke rt N(d 2 ) S0N (d 1)σe qt 2 T N (d 1)e qt S 0σ T qs 0 N( d 1 )e qt + rke rt N( d 2 ) V S 0 T N (d 1 )e qt S 0 T N (d 1 )e qt ρ KT e rt N(d 2 ) KT e rt N( d 2 ) The expressions for d 1 and d 2 are as for Eqs. (15.4) and (15.5) With q = dividend yield on an index, we obtain Greek letters for European options on indices With q = foreign risk-free rate, we obtain Greek letters for European options on currency With q = r, we obtain Greek letters for European options on a futures contract An exception lies in the calculation of rho for European options on a futures contract: ρ for call futures option is ct and ρ for European put futures option is pt In the case of currency options, there are two rhos corresponding to the two interest rates: ρ corresponding to domestic interest rate: Given by formula in table above ρ for foreign interest rate for European call on currency: ρ = T e r f T S 0 N(d 1 ) For European put: rho = T e r f T S 0 N( d 1 ) Delta of forward contracts The concept of delta can be applied to financial instruments other than options Consider a forward contract on a non-dividend-paying stock: The value of a forward contract is S 0 Ke rt. When the price of the stock changes by S, with all else equal, the value of a forward contract on the stock also changes by S The delta of a long forward contract on one share of the stock is always 1.0 A long forward contract on one share can be hedged by shorting one share; A short forward contract on one share can be hedged by purchasing one share (hedge-and-forget schemes) For an asset providing a dividend yield at rate q, the forward contract s delta is e qt For of forward contract on stock index, q dividend yield on the index For of forward contract on currency, q foreign risk-free rate r f Delta of a futures contract The futures price for a contract on a non-dividend-paying stock is S 0 e rt When the price of the stock changes by S, with all else equal, the futures price changes by Se rt Since futures contracts are marked to market daily, the holder of a long futures position makes an almost immediate gain of this amount The delta of a futures contract is e rt For a futures position on an asset providing a dividend yield q, similarly, = e (r q)t Marking to market makes the deltas of futures and forward contracts slightly different Sometimes a futures contract is used to achieve a delta-neutral position. Define: T : Maturity of futures contract H A : Required position in asset for delta hedging H F : Alternative required position in futures contracts for delta hedging 10

15 If the underlying asset is a non-dividend-paying stock: H F = e rt H A (5) When the underlying asset pays a dividend yield q (or for a stock index): H F = e (r q)t H A (6) For a currency, we set q foreign risk-free rate r f, so that: H F = e (r r f )T H A (7) Portfolio insurance A portfolio manager is often interested in acquiring a put option on his or her portfolio Protection against market declines while preserving potential for gain if market does well One approach is to buy put options on a market index such as the S&P 500 An alternative is to create the options synthetically Creating an option synthetically involves maintaining a position in the underlying asset (or futures on the underlying asset) so that of position = of required option Position necessary to create an option synthetically = reverse of that necessary to hedge it Procedure for hedging an option involves creation of an equal and opposite option synthetically Two reasons why it may be more attractive to create option synthetically than to buy it in the market: 1. Options markets do not always have liquidity to absorb large trades 2. Managers require strike prices/exercise dates unavailable in exchange-traded markets The synthetic option can be created from trading the portfolio or index futures contracts: 1. Creation of a put option by trading the portfolio From the table above, the delta of a European put on the portfolio is: = e qt [N(d 1 ) 1] with d 1 = ln(s 0/K) + (r q + σ 2 /2)T σ T (8) The volatility of the portfolio can usually be assumed to be its beta times the volatility of a well-diversified market index 2. Creation of a put option synthetically The fund manager should ensure that at any given time a proportion e qt [1 N(d 1 )] of the stocks in the original portfolio has been sold and the proceeds invested in riskless assets As the value of the original portfolio declines, the delta of the put given by Eq. (8) becomes more negative and the proportion of the original portfolio sold must be increased As value of original portfolio increases, of put becomes less negative and proportion of original portfolio sold must be decreased (i.e., repurchase some of original portfolio) Using the second strategy to create portfolio insurance means that at any given time funds are divided between the stock portfolio on which insurance is required and riskless assets As value of portfolio increases, riskless assets are sold and position in stock portfolio is increased As value of portfolio declines, position in stock portfolio is decreased and riskless assets are bought The cost of the insurance arises from the fact that the portfolio manager is always selling after a decline in the market and buying after a rise in the market Use of index futures Using index futures to create options synthetically can be preferable to using the underlying stocks because the transaction costs associated with trades in index futures are generally lower Dollar amount of futures contracts shorted as proportion of portfolio should be [Eqs. (6)/(8)]: e qt e (r q)t [1 N(d 1 )] = e q(t T ) e rt [1 N(d 1 )] with T = maturity of futures contract 11

16 If the portfolio is worth A 1 times the index and each index futures contract is on A 2 times the index, the number of futures contracts shorted at any given time should be: e q(t T ) e rt [1 N(d 1 )]A 1 /A 2 As time passes and the index changes, the position in futures contracts must be adjusted This analysis assumes that portfolio mirrors index. When not the case, it is necessary to: Calculate the portfolio s beta Find the position in options on the index that gives the required protection Choose a position in index futures to create the options synthetically Options strike price should = expected level of index when portfolio = insured value # of options required = β [# required if portfolio had β = 1.0] Stock market volatility Portfolio insurance strategies have the potential to increase volatility When the market declines, they cause portfolio managers either to sell stock or to sell index futures contracts. Either action may accentuate the decline The sale of stock is liable to drive down the market index further in a direct way The sale of index futures contracts is liable to drive down futures prices. This creates selling pressure on stocks via the mechanism of index arbitrage Similarly, when the market rises, the portfolio insurance strategies cause portfolio managers either to buy stock or to buy futures contracts. This may accentuate the rise Also, many investors consciously or subconsciously follow portfolio insurance rules of their own Whether portfolio insurance trading strategies (formal or informal) affect volatility depends on how easily the market can absorb the trades that are generated by portfolio insurance If portfolio insurance trades are a very small fraction of all trades, there is likely to be no effect As portfolio insurance becomes more popular, it is liable to have a destabilizing effect on the market 12

17 Hull - Ch. 20: Value at Risk (VaR) Introduction -Γ-V analysis does not measure total risk to which financial institution is exposed VaR tries to provide a single number summarizing total risk in portfolio of financial assets The VaR measure When using the VaR measure, an analyst is interested in making a statement of the form: I am X% certain there will not be a loss of more than V dollars in the next N days The variable V is the VaR of the portfolio. It is a function of two parameters: (i) The time horizon (N days) and, (ii) The confidence level (X%) It is the loss level over N days that has a probability of only (100 X)% of being exceeded Calculation of VaR Calculation of VaR Same VaR, larger potential loss (100 { X)% (100 { X)% VaR loss Gain (loss) over N days Figure 1: Calculation of VaR VaR loss Gain (loss) over N days Figure 2: Same VaR - Alternative situation When N days = time horizon and X% = confidence level, VaR = loss corresponding to (100 X)-th percentile of the distribution of the change in the value of the portfolio over next N days (for the probability distribution of the change in value, gains are positive, losses are negative) How bank regulators use VaR: Bank regulators require banks to calculate VaR for market risk with N = 10 and X = 99 Capital required to be held is k [VaR measure], with adjustment for specific risks The multiplier k is chosen on a bank-by-bank basis by the regulators and must be at least 3.0 For a bank with excellent well-tested VaR estimation procedures, it is likely that k will be set equal to the minimum value of 3.0. For other banks it may be higher VaR is an attractive measure because easy to understand. It asks: How bad can things get? An interesting question is whether VaR is the best alternative: Some researchers have argued that VaR may tempt traders to choose a portfolio with a return distribution similar to that in Fig. 2: Portfolios in Figs. 1 and 2 have same VaR, but portfolio in Fig. 2 is much riskier because potential losses are much larger Expected shortfall Measure that deals with the VAR problem above Expected shortfall asks: If things do get bad, how much can the company expect to lose? Expected shortfall is the expected loss during an N-day period conditional that an outcome in the (100 X)% left tail of the distribution occurs E.g., with X = 99 and N = 10, the expected shortfall is the average amount the company loses over a 10-day period when the loss is in the 1% tail of the distribution The time horizon In practice, analysts almost invariably set N = 1 in the first instance because there is not enough data to estimate directly the behavior of market variables over periods of time > 1 day The usual assumption is: N-day VaR = 1-day VaR N Formula exactly true when changes in the value of the portfolio on successive days have independent identical normal distributions with mean zero. In other cases it is an approximation Minimum capital set by regulators (10-day VaR) for banks is 3 10 = 9.49 [1-day 99% VaR] 13

18 Historical simulation Historical simulation: Popular way of estimating VaR involving using past data The first step is to identify the market variables affecting the portfolio Exchange rates, equity prices, interest rates,... Data is then collected on movements in these market variables over most recent (e.g., 501) days Provides 500 alternative scenarios for what can happen between today and tomorrow Scenario 1 is where the % changes in values of all variables are the same as they were between Day 0 and Day 1, scenario 2 is where they are the same as between Day 1 and Day 2,... Define v i as the value of a market variable on Day i and suppose that today is Day m v The ith scenario assumes that the value of the market variable tomorrow will be: v i m v i 1 For each scenario, dollar change in portfolio value between today and tomorrow is calculated: This defines a probability distribution for daily changes in the value of the portfolio The fifth-worst daily change is the first percentile of the distribution The estimate of VaR is the loss at this first percentile point Assuming that the last 501 days are a good guide to what could happen during the next day, the company is 99% certain that it will not take a loss greater than the VaR estimate Alternatively, we can use extreme value theory to smooth numbers in left tail of distribution to obtain a more accurate estimate of the 1% point of the distribution The N-day VaR for a 99% confidence level is calculated as N times the 1-day VaR Each day the VaR estimate would be updated using the most recent 501 days of data Model-building approach The main alternative to historical simulation is the model-building approach Daily volatilities When using the model-building approach to calculate VaR, time is usually measured in days and the volatility of an asset is usually quoted as a volatility per day Assuming 252 trading days/year, σ day = σ year / 252 Daily volatility 6% annual volatility σ day std. dev. of % change in asset price in one day For the purposes of calculating VaR we assume exact equality Single asset case Simple situation where portfolio = position in a single stock: $10M in shares of Microsoft Assume that the volatility of Microsoft is 2% per day (about 32% per year) Std. dev. of daily changes in the position is 2% of $10 million, or $200,000 Customary to assume that expected change in market variable over time period = zero Not strictly true, but it is a reasonable assumption The expected change in the price of a market variable over a short time period is generally small when compared with the standard deviation of the change For an expected return of µ per annum, the expected return over a 1-day period is µ/252 For an annual standard deviation of σ, the std. dev. over a 1-day period is σ/ 252 Assume that change in portfolio value over a 1-day period is normally distributed There is a 1% probability that a normally distributed variable will decrease in value by more than 2.33 std. dev. The 1-day 99% VaR = , 000 = $466, 000 The 10-day 99% VaR for Microsoft is: 466, = $1, 473, 621 Two-asset case Suppose that returns on two shares have bivariate normal distribution with correlation of 0.3 If X/Y have standard deviation σ X /σ Y with correlation between them equal of ρ, the standard deviation of X + Y is: σ X+Y = σx 2 + σ2 Y + 2ρσ Xσ Y X change in value of position in Microsoft over a 1-day period and Y change in value of position in AT&T over a 1-day period, so that: σ X = 200, 000 and σ Y = 50, 000 Std. dev. of change in value of portfolio consisting of both stocks over 1-day period is: 200, , , , 000 = 220,

19 The mean change is assumed to be zero and the change is normally distributed The 1-day 99% VaR is: 220, = $513, 129 The 10-day 99% VaR is 10 times this, or $1,622,657 The benefits of diversification In our example: 1. The 10-day 99% VaR for the portfolio of Microsoft shares is $1,473, The 10-day 99% VaR for the portfolio of AT&T shares is $368, The 10-day 99% VaR for the portfolio of both Microsoft and AT&T shares is $1,622,657 The amount (1, 473, , 405) 1, 622, 657 = $219, 369 represents benefits of diversification If Microsoft and AT&T were perfectly correlated, the VaR for the portfolio of both Microsoft and AT&T would equal the VaR for the Microsoft portfolio plus the VaR for the AT&T portfolio Less than perfect correlation leads to some of the risk being diversified away Linear model Suppose P consists of n assets with α i invested in asset i (1 i n) and x i = return on i in 1 day The dollar change in the value of the investment in asset i in 1 day is α i x i The dollar change P in the value of the whole portfolio in 1 day is: P = n α i x i i=1 (1) If we assume that the x i in Eq. (1) are multivariate normal, P is normally distributed To calculate VaR, we only need to calculate the mean and standard deviation of P We assume that the expected value of each x i is zero The mean of P is zero σ i = daily volatility of the i-th asset (i.e. the standard deviation of x i ) ρ ij = correlation between returns on asset i and j (between x i and x j ). Variance σ 2 p of P : σ 2 p = n n ρ ij α i α j σ i σ j = i=1 j=1 n n n αi 2 σi ρ ij α i α j σ i σ j (2) i=1 i=1 j<i Std. dev. of change over N days = σ P N. 99%-VaR for N-day horizon is 2.33σP N Handling interest rates Some simplifications are necessary when the model-building approach is used One possibility is to assume that only parallel shifts in the yield curve occur It is then necessary to define only one market variable: The size of the parallel shift Changes in portfolio value is then calculated using the duration relationship: P = DP y where D = modified duration, and y = parallel shift in 1 day This approach does not usually give enough accuracy The procedure usually followed is to choose as market variables the prices of zero-coupon bonds with standard maturities from 1 month to 30 years and use cash-flow mapping Cash-flow mapping For calculating VaR, the CFs from instruments in the portfolio are mapped into cash flows occurring on the standard maturity dates Consider $1M position in T-bond lasting 1.2 years that pays coupon of 6% semiannually Coupons are paid in 0.2, 0.7, and 1.2 years, and the principal is paid in 1.2 years Bond $30,000 position in 0.2-year zero-coupon bond, plus $30,000 position in 0.7-year zerocoupon bond, plus $1.03M position in 1.2-year zero-coupon bond The position in the 0.2-year bond is then replaced by an equivalent position in 1-month and 3-month zero-coupon bonds, and so on Result: Position in 1.2-year coupon-bearing bond is for VaR purposes regarded as position in zero-coupon bonds having maturities of 1 month, 3 months, 6 months, 1 year, and 2 years Cash-flow mapping is not necessary when the historical simulation approach is used because the complete term structure of interest rates can be calculated for each of the scenarios considered 15

20 Applications of the linear model Simplest application: Portfolio with no derivatives consisting of stocks, bonds, FX, and commodities In this case, the change in the value of the portfolio is linearly dependent on the % changes in the prices of the assets comprising the portfolio For purposes of VaR, all asset prices are measured in the domestic currency Derivative handled by the linear model: Forward contract to buy a foreign currency: Suppose the contract matures at time T. It can be regarded as the exchange of a foreign zero-coupon bond maturing at time T for a domestic zero-coupon bond maturing at time T For the purposes of calculating VaR, the forward contract is therefore treated as a long position in the foreign bond combined with a short position in the domestic bond Each bond can be handled using a cash-flow mapping procedure Consider an interest rate swap: This can be regarded as the exchange of a floating-rate bond for a fixed-rate bond The fixed-rate bond is a regular coupon-bearing bond The floating-rate bond is worth par just after the next payment date. It can be regarded as a zero-coupon bond with a maturity date equal to the next payment date The interest rate swap therefore reduces to a portfolio of long and short positions in bonds and can be handled using a cash-flow mapping procedure The linear model and options Consider first a portfolio consisting of options on a single stock whose current price is S Suppose that the delta of the position is δ. δ = rate of change of portfolio value with S: δ P S P δ S (3) Define x as the % change in stock price in 1 day, so that: x = S/S P = Sδ x When we have a position in several underlying market variables that includes options, we can derive an approximate linear relationship between P and the x i similarly: P = n S i δ i x i i=1 (4) This is Eq. (1) with α i = S i δ i Eq. (2) can be used to calculate the std. dev. of P Quadratic model When portfolio includes options, linear model = approximation: Does not take Γ into account Probability distribution for value of portfolio Value of short call Normal distribution (a) Positive - E.g., long call (b) Negative - E.g., short call Lighter tail than Normal Dist. Heavier tail than Normal Dist. Negative distribution Underlying asset (a) Figure 3: Probability distribution of portfolio value (b) Figure 4: Translation of Normal distribution Fig. 3 shows the impact of a nonzero gamma on the probability distribution of the value of the portfolio: When gamma is positive, the probability distribution tends to be positively skewed Positive gamma portfolio has a less heavy left tail than Normal distribution If the distribution of P is normal, the calculated VaR tends to be too high 16

21 When gamma is negative, it tends to be negatively skewed Negative gamma portfolio has a heavier left tail than Normal distribution If the distribution of P is normal, the calculated VaR tends to be too low For a more accurate estimate of VaR than that given by the linear model, both delta and gamma measures can be used to relate P to the x i : Suppose δ and γ are the delta and gamma of the portfolio. Then, with x = S/S: P = δ S γ( S)2 = Sδ x S2 γ( x) 2 (5) More generally for a portfolio with n underlying market variables, with each instrument in the portfolio being dependent on only one of the market variables, Eq. (5) becomes: P = n S i δ i x i + i=1 n i=1 1 2 S2 i γ i ( x i ) 2 When individual instruments may be dependent on more than one market variable: P = n S i δ i x i + i=1 n n i=1 j=1 1 2 S is j γ ij x i x j (6) Where γ ij is a cross gamma defined as: γ ij = 2 P/ S i S j Monte Carlo simulation Model-building can be implemented using Monte Carlo simulation to generate distribution for P Suppose we wish to calculate a 1-day VaR for a portfolio. The procedure is: 1. Value the portfolio today in the usual way using the current values of market variables 2. Sample once from the multivariate normal probability distribution of the x i 3. Use sampled x i to determine the value of each market variable at the end of one day 4. Revalue the portfolio at the end of the day in the usual way 5. Subtract the value calculated in Step 1 from the value in Step 4 to determine a sample P 6. Repeat Steps 2 to 5 many times to build up a probability distribution for P The VaR is calculated as the appropriate percentile of the probability distribution of P Suppose that we calculate 5,000 different sample values of P as described The 1-day 99% VaR is the value of P for the 50th worst outcome The N-day VaR is usually assumed to be the 1-day VaR multiplied by N Partial simulation approach Drawback of MC simulation: Slow because complete portfolio has to be revalued many times One way of speeding things up: Assume Eq. (6) describes relationship between P and the x i We can then jump straight from Step 2 to Step 5 in the Monte Carlo simulation and avoid the need for a complete revaluation of the portfolio Partial simulation approach Comparison of approaches Historical approach Advantages Historical data determine the joint probability distribution of the market variables Avoids the need for cash-flow mapping Disadvantage Computationally slow Does not easily allow volatility updating schemes to be used Model-building approach Advantages Results can be produced very quickly Can easily be used in conjunction with volatility updating schemes 17

22 Disadvantage Assumes that the market variables have a multivariate normal distribution. In practice, daily changes in market variables often have distributions that are quite different from Normal Tends to give poor results for low-delta portfolios Stress testing and back testing Stress testing Involves estimating how a company s portfolio would have performed under some of the most extreme market moves seen in the last 10 to 20 years E.g., to test the impact of an extreme movement in US equity prices, a company might set the percentage changes in all market variables equal to those on October 19, 1987 The scenarios used in stress testing are also sometimes generated by senior management Senior management may meet periodically and brainstorm to develop extreme scenarios that might occur given current economic environment/global uncertainties Stress testing taking into account extreme events that do occur from time to time but are virtually impossible according to probability distributions assumed for market variables Back testing Whatever the method used for calculating VaR, an important reality check is back testing It involves testing how well the VaR estimates would have performed in the past Suppose that we are calculating a 1-day 99% VaR: Back testing would involve looking at how often the loss in a day exceeded the 1-day 99% VaR that would have been calculated for that day If this happened 1% of the days, we can feel reasonably comfortable with methodology for VaR Principal components analysis Principal components analysis: Handle risk arising from highly correlated market variables Takes historical data and attempts to find components that explain movements in market variables Example Consider US Treasury rates with maturities between 3 months and 30 years using 10 factors: P C1 P C2 P C3... P C10 3m m m y The interest rate move for a particular factor is known as factor loading Because there are 10 rates and 10 factors, the interest rate changes observed on any given day can always be expressed as a linear sum of the factors by solving a set of 10 simultaneous equations Quantity of a factor in the rate changes on a day is the factor score for that day The importance of a factor is measured by the standard deviation of its factor score P C1 P C2 P C3... P C The numbers in the table above are measured in basis points A quantity of the 1st factor equal to one std. dev. corresponds to the 3-month rate moving by = 3.67 bp, the 6-month rate moving by = 4.55 bp The factors are chosen so that the factor scores are uncorrelated 1st factor score (parallel shift) uncorrelated with 2nd factor score (twist) across period Variances of factor scores add up to the total variance of the data Using principal components analysis to calculate VaR Consider a portfolio with the following exposures to interest rate moves: Change in portfolio value for a 1-basis-point rate move ($M) 1-yr rate 2-yr rate 3-yr rate 4-yr rate 5-yr rate

23 A 1-basis-point change in the 1-year rate causes the portfolio value to increase by $10 million Suppose the first two factors are used to model rate moves The exposure to 1st factor (measured in $M per factor score basis point) is = 0.08 The exposure to the second factor is: 10 ( 0.32) + 4 ( 0.10) = 4.40 Suppose that f 1 and f 2 are the factor scores (in bp). The change in the portfolio value is: P = 0.08f f 2 The factor scores are uncorrelated Std. dev. of P = = Hence, the 1-day 99% VaR is = A PC analysis can in theory be used for market variables other than interest rates Effectiveness of PC analysis depends on how closely correlated market variables are VaR usually calculated by relating actual changes in portfolio to % changes in market variables ( x i ) For VaR calculation, appropriate to carry out PC analysis on % changes in market variables rather than actual changes Appendix - Cash-flow mapping Consider portfolio: Long position in single T-bond with a principal of $1M maturing in 0.8 years Suppose that the bond provides a coupon of 10% per annum payable semiannually The Treasury bond can be regarded as a position in a 0.3-year zero-coupon bond with a principal of $50,000 and a position in a 0.8-year zero-coupon bond with a principal of $1,050,000 Position in 0.3-year zero mapped into equivalent position in 3m and 6m zeros Position in 0.8-year zero mapped into equivalent position in 6m and 1-yr zeros The position in the 0.8-year coupon-bearing bond is, for VaR purposes, regarded as a position in zero-coupon bonds having maturities of 3 months, 6 months, and 1 year The mapping procedure Suppose that zero rates, daily bond price volatilities, and correlations between bond returns are: Maturity: 3-month 6-month 1-year Zero rate (% with annual compounding): Bond price volatility (% per day): Correlation between daily returns 3m bond 6m bond 1yr bond 3-month bond month bond year bond The first stage is to interpolate between the 6-month rate of 6.0% and the 1-year rate of 7.0% to obtain a 0.8-year rate of 6.6% (Annual compounding is assumed for all rates) PV of $1,050,000 CF to be received in 0.8 years is: 1, 050, 000/ = 997, 662 We also interpolate between the 0.1% volatility for the 6-month bond and the 0.2% volatility for the 1-year bond to get a 0.16% volatility for the 0.8-year bond Allocate α of PV to 6m bond and 1 α of PV to 1-yr bond. Using Eq. (2) and matching variances: = α (1 α) α(1 α) Quadratic equation α = The 0.8-year bond worth $997,662 is replaced by a 6-month bond worth 997, = $319, 589 and a 1-year bond worth 997, = $678, 074 This CF mapping scheme preserves both the value and the variance of the cash flow Also, the weights assigned to the two adjacent zero-coupon bonds are always positive 19