Introduction to Risk Management

Transcription

1 Introduction to Risk Management ACPM Certified Portfolio Management Program c 2010 by Martin Haugh Introduction to Risk Management We introduce some of the basic concepts and techniques of risk management in these lecture notes. Much of the content and notation is drawn from McNeil, Frey and Embrechts ( Risk Factors and Loss Distributions Let be a fixed period of time such as 1 day, 1 week or 6 months. This will be the horizon of interest when it comes to measuring risk and calculating loss distributions. Let V t be the value of a portfolio at time t so that the portfolio loss between times t and (t + 1 is given by L t+1 := (V t+1 V t. (1 Note that we treat a loss as a positive quantity so, for example, a negative value of L t+1 denotes a profit. The time 1 t value of the portfolio depends of course on the time t value of the securities in the portfolio. More generally, however, we may wish to define a set of d risk factors, Z t := (Z t,1,..., Z t,d so that V t is a function of t and Z t. That is V t = f(t, Z t. for some function f. In a stock portfolio, for example, we might take the stock prices or some function of the stock prices as our risk factors. In an options portfolio, however, Z t might contain stock factors together with implied volatility and interest rate factors. Now let X t := Z t Z t 1 denote the change in the values of the risk factors between times t and t + 1. Then we have L t+1 (X t+1 = (f(t + 1, Z t + X t+1 f(t, Z t (2 and given the value of Z t, the distribution of L t+1 then depends only on the distribution of X t+1. Linear Approximations to the Loss Function Assuming f(, is differentiable, we can use a first order Taylor expansion to approximate L t+1 with ( d L t+1 (X t+1 := f t (t, Z t + f zi (t, Z t X t+1,i (3 where the f-subscripts denote partial derivatives and time is measured in units. If time is measured in years then we would replace f t (t, Z t in (3 with f t (t, Z t. The first order approximation is commonly used when X t+1 is likely to be small. This is often the case when is small, e.g. 1 day, and the market is not too volatile. Second and higher order approximations also based on Taylor s Theorem can also be used. It is important to note, however, that if X t+1 is likely to be very large then Taylor approximations of any order are likely to work poorly. 1 When we say time t we typically have in mind time t.

2 Introduction to Risk Management 2 Conditional and Unconditional Loss Distributions When we discuss the distribution of L t+1 it is important to clarify exactly what we mean. In particular, we need to distinguish between the conditional and unconditional loss distributions. Consider the series X t of risk factor changes and assume that they form a stationary 2 time series with stationary distribution, F X. We also let I t denote all information available in the system at time t including {X s : s t} in particular. We then have the following two definitions. Definition 1 The unconditional loss distribution is the distribution of L t+1 given the time t composition of the portfolio and assuming the CDF of X t+1 is given by F X. Definition 2 The conditional loss distribution is the distribution of L t+1 given the time t composition of the portfolio and conditional on the information in I t. It is clear that if the X t s are IID then the conditional and unconditional distributions coincide. For long time horizons, e.g. = 6 months, we might be more inclined to use the unconditional loss distribution. However, for short horizons, e.g. 1 day or 10 days, then the conditional loss distribution is clearly the appropriate distribution. This would be particularly true in times of high market volatility when the unconditional distribution would bear little resemblance to the true conditional distribution. Portfolio Examples Example 1 (A Stock Portfolio Consider a portfolio of d stocks with S t,i denoting the time t price of the i th stock and λ i denoting the number of units of this stock in the portfolio. If we take log stock prices as our factors then we obtain X t+1,i = ln S t+1,i ln S t,i and The linear approximation satisfies L t+1 d ( = λ i S t,i e X t+1,i 1. L t+1 d d = λ i S t,i X t+1,i = V t ω t,i X t+1,i where ω t,i := λ i S t,i /V t is the ] i th portfolio weight. If X t+1,i has mean vector µ and variance-covariance matrix Σ then we obtain E t [ Lt+1 = V t ω µ and Var t ( Lt+1 = Vt 2 ω Σω. Note that µ and Σ could refer to the first two moments of either the conditional or unconditional loss distribution. Example 2 (An Options Portfolio The Black-Scholes formula for the time t price of a European call option with strike K and maturity T on a non-dividend paying stock satisfies C(S t, t, σ = S t Φ(d 1 e r(t t KΦ(d 2 where d 1 = log ( S t K + (r + σ 2 /2(T t σ, d 2 = d 1 σ T t T t 2 A time series, X t, is strongly stationary if (X t1,..., X tn is equal in distribution to (X t1 +k,..., X tn +k for all t 1,..., t n, k Z. Most risk factors are assumed to be stationary.

3 Introduction to Risk Management 3 Φ( is the CDF of the standard normal distribution, S t is the time t price of the underlying security and r is the risk-free interest rate. While the Black-Scholes model assumes a constant volatility, σ, in practice an implied volatility, σ(k, T, t, that depends on the strike, maturity and current time, t, is observed in the market. Consider now a portfolio of European options all on the same underlying security. The portfolio may also contain a position in the underlying security itself. If the portfolio contains d different options with a position of λ i in the i th option, then d = λ 0 (S t+1 S t λ i (C(S t+1, t + 1, σ(k i, T i, t + 1 C(S t, t, σ(k i, T i, t. (4 L t+1 where λ 0 is the position in the underlying security. Note that by put-call parity we can assume that all options are call options. We can also use the linear approximation technique to approximate L t+1 in (4. This would result in a delta-vega-theta approximation. For derivatives portfolios, the linear approximation technique based on the first-order Greeks is often inadequate and second order approximations involving gamma and possibly higher-order Greeks are often employed. For risk factors, we can again take the log stock prices but it is not clear how to handle the implied volatilities. There are several possibilities: 1. One possibility is to assume that the σ(k, T, t s simply do not change. This is not very satisfactory but is commonly assumed when historical simulation 3 is used to approximate the loss distribution and historical data on the changes in implied volatilities are not available. 2. Let each σ(k, T, t be a separate factor. In addition to approximately doubling the number of factors, a particular problem with this approach is that the implied volatilities are not free to move around independently. In fact the assumption of no-arbitrage imposes strong restrictions on how the volatility surface may move. It is therefore important to choose factors in such a way that those restrictions are easily imposed when we estimate the loss distribution. 3. In light of the previous point, it may be a good idea to parameterize the volatility surface with just a few parameters and assume that only those parameters can move from one period to the next. The parameterization should be such that the no-arbitrage restrictions are easy to enforce. Example 3 (A Bond Portfolio Consider a portfolio containing quantities of d different default-free zero-coupon 4 bonds where the i th bond has price P t,i, maturity T i and face value equal 5 to 1. Let S t,ti denote the continuously compounded spot interest rate for maturity T i so that P t,i = e S t,t i (T i t. If there are λ i units of the i th bond in the portfolio, then the total portfolio value is given by d V t = λ i e S t,t i (T i t. Assume now that we only consider the possibility of parallel 6 changes in the spot rate curve. Then if the spot curve moves by δ the portfolio loss satisfies d L t+1 = λ i (e (S t+,t i +δ(t i t e S t,t i (T i t (5 d λ i (S t,ti (T i t (S t+,ti + δ(t i t. (6 3 See Section 4. 4 There is no loss of generality here since we can decompose a default-free coupon bond into a series of zero-coupon bonds. 5 There is also no loss of generality in assuming a face value of 1 since we can compensate by adjusting the quantity of each bond in the portfolio. 6 While unrealistic, this is a common assumption in practice and leads to the ideas of duration and convexity.

4 Introduction to Risk Management 4 We therefore have a single risk factor, δ, that we can use to study the distribution of the portfolio loss. 2 Risk Measurement Approaches to Risk Measurement We now outline several approaches to the problem of risk measurement. Notional Amount Approach This approach to risk management defines the risk of a portfolio as the sum of the notional amounts of the individual positions in the portfolio. Each notional amount may be weighted by a factor representing the perceived riskiness of the position. While it is simple, it has many weaknesses. It does not reflect the benefits of diversification, does not allow for any netting and does not distinguish between long and short positions. Moreover, it is not always clear what the notional amount of a derivative is and so the notional approach can be difficult to apply when such derivatives are present in the portfolio. Factor Sensitivity Measures A factor sensitivity measure gives the change in the value of the portfolio for a given change in the factor. Commonly used examples include the Greeks of an option portfolio or the duration and convexity of a bond portfolio. These measures are often used to set position limits on trading desks and portfolios. They are generally not used for capital adequacy decisions as it is often difficult to aggregate these measures across different risk factors and markets. Scenario or Stress Approach The scenario approach defines a number of scenarios where in each scenario the various risk factors are assumed to have moved by some fixed amounts. For example, a scenario might assume that all stock prices have fallen by 10% and all implied volatilities have increased by 5 percentage points. Another scenario might assume the same movements but an additional steepening of the volatility surface. A scenario for a credit portfolio might assume that all credit spreads have increased by some fixed absolute, e.g. 100 bps, or relative amount. The risk of a portfolio could then be defined as the maximum loss over all of the scenarios that were considered. A particular advantage of this approach is that it does not depend on probability distributions that are difficult to estimate. Therefore the scenario approach to risk management is more robust and should play a vital role in any financial risk management operation. In order to construct good scenarios it is very important to: 1. Understand the dynamics of risk-factors in extreme environments. For example, suppose we want to stress-test an options portfolio by creating a scenario where implied volatilities increase dramatically. Should we increase the implied volatilities of all maturities by the same amount? In general the answer is no. This is because in times of market stress the absolute change in implied volatility typically decreases with approximately the square-root-of-time. Can you guess why this might be the case? 2. Understand the composition of the portfolio and in particular, what risk factors to which the the portfolio is exposed. Measures Based on Loss Distribution Many risk measures such as value-at-risk (VaR or conditional value-at-risk (CVaR are based on the loss distribution of the portfolio. Working with loss distributions makes sense as the distribution contains all the information you could possibly wish to know about possible losses. A loss distribution implicitly reflects the benefits of netting and diversification. Moreover it is easy to compare the loss distribution of a derivatives

5 Introduction to Risk Management 5 portfolio with that of a bond or credit portfolio, at least when the same time horizon is under consideration. However, it must be noted that it may be very difficult to estimate the loss distribution. This may be the case for a number of reasons including a lack of historical data, non-stationarity of risk-factors and poor model choice among others. 3 Value-at-Risk and Conditional Value-at-Risk Value-at-Risk Value-at-Risk (VaR is the most widely used risk measure in the financial industry. Despite the many weaknesses of VaR, financial institutions are required to use it under the Basel II capital-adequacy framework. In addition, many institutions routinely report their VaR numbers to shareholders, investors or regulatory authorities. VaR is a risk measure based on the loss distribution and our discussion will not depend on whether we are dealing with the conditional or unconditional loss distribution. Nor will it depend on whether we are using the true loss distribution or some approximation to it. We will assume that the horizon has been fixed and that the random variable L represents the loss on the portfolio under consideration over the time interval. We will use F L ( to denote the cumulative distribution function (CDF of L. We first define the quantiles of a CDF. Definition 3 Let F : R [0, 1] be an arbitrary CDF. Then for α (0, 1 the α-quantile of F is defined by q α (F := inf{x R : F (x α}. Note that if F is continuous and strictly increasing, then q α (F = F 1 (α. For a random variable L with CDF F L (, we will often write q α (L instead of q α (F L. Definition 4 Let α (0, 1 be some fixed confidence level. Then the VaR of the portfolio loss at the confidence interval, α, is given by VaR α := q α (L, the α-quantile of the loss distribution. Example 4 Normal and t Distributions Because the normal and t CDFs are both continuous and strictly increasing, it is straightforward to calculate their VaR α. If L N(µ, σ 2 then it is straightforward to show that VaR α = µ + σφ 1 (α where Φ is the standard normal CDF. (7 If L t(ν, µ, σ 2 so that (L µ/σ has a standard t distribution with ν > 2 degrees-of-freedom, then VaR α = µ + σt 1 ν (α where t ν is the CDF for the t distribution with ν degrees-of-freedom. Note that in this case we have E[L] = µ and Var(L = νσ 2 /(ν 2. VaR has several weaknesses: 1. VaR attempts to describe the entire loss distribution with a single number and so significant information is not captured in VaR. This criticism does of course apply to all scalar risk measures. One way around this is to report VaR α for several different values of α. 2. There is significant model risk attached to VaR. If the loss distribution is heavy-tailed, for example, but a normal distribution is assumed, then VaR α will be severely underestimated as α approaches 1. A fundamental problem with VaR and other risk measures based on the loss distribution is that it can be very difficult to estimate the loss distribution. Even when there is sufficient historical data available there is no guarantee that the historical data-generating process will remain unchanged in the future. For example, in the buildup to the sub-prime housing crisis it was clear that the creation and selling of vast quantities of structured credit products could change the price dynamics of these and related securities. And of course, that is precisely what happened.

6 Introduction to Risk Management 6 3. VaR is not a sub-additive risk measure so that it doesn t lend itself to aggregation. For example, let L = L 1 + L 2 be the total loss associated with two portfolios, each with respective losses, L 1 and L 2. Then q α (F L > q α (F L1 + q α (F L1 is possible. (8 In the risk literature this is viewed as being a very undesirable property as we would expect some diversification benefits when we combine two portfolios together. Such a benefit would be reflected by the combined portfolio having a smaller risk measure than the sum of the two individual risk measures. We now give an example of how VaR fails to be sub-additive and how severe this problem could potentially be. Example 5 (VaR for a Portfolio of Defaultable Bonds 7 Consider a portfolio of n = 100 defaultable corporate bonds where the probability of a default over the next year is identical for all bonds and is equal to 2%. We assume that defaults of different bonds are independent from one another. The current price of each bond is 100 and if there is no default, a bond will pay 105 one year from now. If the bond defaults then there is no repayment. This means we can define L i, the loss on the i th bond, as L i := 105Y i 5 where Y i = 1 if the bond defaults over the next year and Y i = 0 otherwise. By assumption we also see that P (L i = 5 =.98 and P (L i = 100 =.02. Consider now the following two portfolios: A: A fully concentrated portfolio consisting of 100 units of bond 1. B: A completely diversified portfolio consisting of 1 unit of each of the 100 bonds. We can compute the 95% VaR for each portfolio as follows: Portfolio A: The loss on portfolio A is given by L A = 100L 1 so that VaR.95 (L A = 100VaR.95 (L 1. Note that P (L 1 5 =.98 >.95 and P (L 1 l = 0 <.95 for l <.5. We therefore obtain VaR.95 (L 1 = 5 and so VaR.95 (L A = 500. So the 95% VaR for portfolio A corresponds to a gain(! of 500. Portfolio B: The loss on portfolio B is given by L B = L i = 105 Y i 500 and so VaR.95 (L B = 105 VaR.95 ( 100 Y i 500. Note that M := 100 Y i Binomial(100,.02 and by inspection we see that P (M >.95 and P (M <.95. Therefore VaR.95 (M = 5 and so VaR.95 (L B = = 25. So according to VaR.95, portfolio B is riskier than portfolio A. This is clearly nonsensical. Note that we have shown that ( VaR.95 L i 100 VaR.95 (L 1 = VaR.95 (L i demonstrating again that VaR is not subadditive. An advantage of VaR is that it is generally easier 8 to estimate. This is true when it comes to quantile estimation in general as quantiles are not very sensitive to outliers. This is not true of other risk measures such as CVaR which we discuss below. Despite this fact, it becomes progressively more difficult to estimate VaR α as α gets closer to 1. 7 This example is taken from McNeil, Frei and Embrechts ( This assumes of course that we have correctly specified the appropriate probability model so that the second weakness above is not an issue. This assumption is often not justified!

7 Introduction to Risk Management 7 Conditional Value-at-Risk (CVaR We now define conditional value-at-risk which is also commonly referred to as expected shortfall. Definition 5 For a portfolio loss, L, satisfying E[ L ] < the CVaR at confidence level α (0, 1 is given by CVaR α := 1 1 α 1 α VaR u (L du. It is therefore clear that CVaR α (L VaR α (L. A more well known representation of CVaR α (L holds when F L is continuous. In particular, if F L is a continuous CDF then CVaR α = E [L L VaR α ] (9 so that CVaR α can be interpreted as the expected loss conditional on the loss exceeding the corresponding VaR α. In other words, the CVaR tells you how bad you can expect the loss to be given that you are going to incur a large loss. Example 6 (CVaR for a Normal Distribution We can use (9 to compute the CVaR of an N(µ, σ 2 random variable. In particular we can check that CVaR α = µ + σ φ ( Φ 1 (α where φ( is the PDF of the standard normal distribution. 1 α (10 Example 7 (CVaR for a t Distribution Let L t(ν, µ, σ 2 so that L := (L µ/σ has a standard t distribution with ν > 2 degrees-of-freedom. Then as in the previous example it can be shown that CVaR α (L = µ + σcvar α ( L. It is straightforward using direct integration to check that CVaR α ( L = g ν ( t 1 ν (α 1 α ( ν + (t 1 ν (α 2 ν 1 where t ν ( and g ν ( are the CDF and probability density function (PDF, respectively, of the standard t distribution with ν degrees-of-freedom. (11 Remark 1 The t distribution has been found to be a much better model of stock (and other asset returns than the normal model. In empirical studies, values of ν around 5 or 6 are often found to fit best. Remark 2 It is well known that CVaR is a sub-additive risk measure. Returning to Example 5 we therefore obtain ( CVaR L i CVaR(L i = 100CVaR(L 1 and so CVaR would correctly classify portfolio A as being riskier than portfolio B. The CVaR-to-VaR Ratio One method of comparing VaR α and CVaR α is to consider their ratio as α 1. It is not too difficult to see that in the case of the normal distribution, CVaR α /VaR α 1 as α 1. However, CVaR α VaR α ν/(ν 1 > 1 in the case of the t distribution with ν > 1 degrees-of-freedom.

8 Introduction to Risk Management 8 4 Standard Techniques for Risk Measurement We now discuss some of the principal techniques for estimating the loss distribution. Historical Simulation Instead of using some probabilistic model to estimate the distribution of L t+1 (X t+1, we could estimate the distribution using a historical simulation. In particular, if we know the values of X t i+1 for i = 1,..., n, then we can use this data to create a set of historical losses: { L i := L t+1 (X t i+1 : i = 1,..., n}. L i is the loss on the portfolio that would occur if the changes in the risk factors on date t i + 1 were to recur. In order to calculate the value of a given risk measure we simply assume that the distribution of L t+1 (X t+1 is discrete and takes on each of the values L i with probability 1/n for i = 1,..., n. That is, we use the empirical distribution of the X t s to determine the loss distribution. If we wish to compute VaR, then we can do so using the appropriate quantiles of the L i s. For example suppose we have ordered the L i s by L n,n L 1,n. Then a possible estimator of VaR α (L t+1 is L [n(1 α],n where [n(1 α] is the largest integer not exceeding n(1 α. The associated CVaR could be estimated by averaging L n,n,..., L [n(1 α],n. The historical simulation approach is generally difficult to apply for derivative portfolios as it is often the case that historical data on at least some of the associated risk factors is not available. If the risk factor data is available, however, then the method is easy to apply as it does not require any statistical estimation of multivariate distributions. It is also worth emphasizing that the method estimates the unconditional loss distribution and not the conditional loss distribution. As a result, it is likely to be very inaccurate in times of market stress. Unfortunately these are precisely the times when you are most concerned with obtaining accurate risk measures. Monte-Carlo Simulation The Monte-Carlo approach is similar to the historical simulation approach except now we use some parametric distribution of the change in risk factors to generate sample portfolio losses. The distribution (conditional or unconditional of the risk factors is estimated and, assuming we know how to simulate from this distribution, we can generate a number of samples, m say, of portfolio losses. We are free to make m as large as possible, subject to constraints on computational time. Variance reduction methods are often employed to obtain improved estimates of the required risk measures. While Monte-Carlo is an excellent tool, it is only as good as the model used to generate the data: if the distribution of X t+1 that is used to generate the samples is poor, then the Monte-Carlo samples will be of little value. Variance-Covariance Approximations In the variance-covariance approach we assume that X t+1 has a multivariate normal distribution so that X t+1 MVN (µ, Σ. We also assume that the linear approximation in (3 is sufficiently accurate. Writing L t+1 (X t+1 = (c t + b t X t+1 for a constant scalar, c t, and constant vector, b t, we therefore obtain that L t+1 (X t+1 N ( c t b t µ, b t Σb t.

9 Introduction to Risk Management 9 We can now use our knowledge of the normal distribution to calculate any risk measures of interest. Note that this technique can be either conditional or unconditional, depending on how µ and Σ are estimated. The strength of this approach is that it provides a straightforward analytically tractable method of determining the loss distribution. It has several weaknesses, however. Risk factor distributions are often 9 fat- or heavy-tailed but the normal distribution is light-tailed. As a result, we are likely to underestimate the frequency of extreme movements which in turn can lead to seriously underestimating the risk in the portfolio. This problem is generally easy to overcome as there are other multivariate distributions that are also closed under linear operations. In particular, if we assume that X t+1 has a multivariate t distribution so that X t+1 t (ν, µ, Σ then we obtain L t+1 (X t+1 t ( ν, c t b t µ, b t Σb t. where ν is the degrees-of-freedom of the t-distribution. A more serious problem with this approach is the assumption that the linear approximation will work well. This is generally not true for portfolios of derivative securities or other portfolios where the portfolio value is a non-linear function of the risk-factors. It is also problematic when the time horizon is large. 5 Evaluating Risk Measures An important task of any risk manger is to constantly evaluate the risk measures that are being reported. For example, if the daily 95% VaR is reported then we should see the daily losses exceeding the reported VaR approximately 95% of the time. Indeed, suppose the reported VaR numbers are calculated correctly and let Y i be the indicator function denoting whether or not the portfolio loss in period i exceeds VaR i, the corresponding VaR for that period. That is { 1, Li VaR Y i = i ; 0, otherwise. If we assume the Y i s are IID, then n Y i should be Binomial(n,.05. We can use standard statistical tests to see if this is indeed the case. Similar tests can be constructed for CVaR and other risk measures. More generally, it is important to check that the true reported portfolio losses are what you would expect given the composition of the portfolio and the realized change in the risk factors over the previous period. For example, if you know the delta and vega of your portfolio at time t i 1 and you observe the change in the underlying and volatility factors between t i 1 and t i, then the actual realized gains or losses should be consistent with the delta and vega numbers. If they are not, then further investigation is required. It may be that (i the moves required second order terms such as gamma, vanna or volga or (ii other risk factors influencing the portfolio value also changed or (iii the risk factor moves were very large and cannot be captured by any Taylor series approximation or (iv there s a bug in the risk management system! 9 We will define light- and heavy-tailed distributions formally when we discuss multivariate distributions.

10 Introduction to Risk Management 10 6 Principal Components Analysis and Factor Analysis We now consider some dimension reduction techniques that are often useful for identifying the key components that drive the risk or randomness in a system. We begin with principal components analysis (PCA. Principal Components Analysis Let Y = (Y 1,..., Y n T denote an n-dimensional random vector with variance-covariance matrix, Σ. In the context of risk management, we take this vector to represent the (normalized changes, over some appropriately chosen time horizon, of an n-dimensional vector of risk factors. These risk factors could represent security price returns, returns on futures contracts of varying maturities, or changes in spot interest rates, again of varying maturities. The goal of PCA is to construct linear combinations P i = n w ij Y j for i = 1,..., n j=1 in such a way that: (1 the P i s are orthogonal so that E[P i P j ] = 0 for i j and (2 the P i s are ordered so that: (i P 1 explains the largest percentage of the total variability in the system and (ii each P i explains the largest percentage of the total variability in the system that has not already been explained by P 1,..., P i 1. In practice it is common to apply PCA to normalized 10 random variables that satisfy E[Y i ] = 0 and Var(Y i = 1. This is achieved by subtracting the means from the original random variables and dividing by their standard deviations. This is done to ensure that no one component of Y can influence the analysis by virtue of that component s measurement units. We will therefore assume that the Y i s have already been normalized. The key tool of PCA is the Spectral Decomposition of linear algebra which states that any symmetric matrix, A R n n can be written as A = Γ Γ T (12 where: (i is a diagonal matrix, diag(λ 1,..., λ n, of the eigen-values of A which, without loss of generality, are ordered so that λ 1 λ 2 λ n and (ii Γ is an orthogonal matrix with the i th column of Γ containing the i th standardized 11 eigen-vector, γ i, of A. The orthogonality of A implies Γ Γ T = Γ T Γ = I n. Since Σ is symmetric we can take A = Σ in (12 and the positive semi-definiteness of Σ implies λ i 0 for all i = 1,..., n. The principal components of Y are then given by P = (P 1,..., P n satisfying Note that: (a E[P ] = 0 since E[Y ] = 0 and P = Γ T Y. (13 (b Cov(P = Γ T Σ Γ = Γ T (Γ Γ T Γ = which is a diagonal matrix. The components of P are therefore uncorrelated and Var(P i = λ i. This is consistent with (1 above. 10 Working with normalized random variables is equivalent to working with the correlation matrix of the un-normalized variables. 11 By standardized we mean γi T γ i = 1.

11 Introduction to Risk Management 11 The matrix Γ T is called the matrix of factor loadings. Note that we can invert (13 to obtain Y = Γ P. (14 We can measure the ability of the first few principal components to explain the total variability in the system. We see from (b that n n n Var(P i = λ i = trace(σ = Var(Y i (15 where we have used the fact that the trace of a matrix, i.e. the sum of its diagonal elements, is also equal to the sum of its eigen-values. If we take n Var(P i = n Var(Y i to measure the total variability, then by (16 we may interpret the ratio k λ i n λ i as measuring the percentage of the total variability that is explained by the first k principal components. This is consistent with (2 above since the λ i s are non-increasing. In particular, it is possible to show that the first principal component, P 1 = γ T 1 Y, satisfies Var(γ T 1 Y = max { Var(a T Y : a T a = 1 }. Moreover, it is also possible to show that each successive principal component, P i = γi T Y, satisfies the same optimization problem but with the added constraint that it be orthogonal, i.e. uncorrelated, to P 1,..., P i 1. In financial applications, it is often the case that just two or three principal components are sufficient to explain anywhere from 60% to 95% or more of the total variability. Moreover, it is often possible to interpret the first two or three components. For example, if Y represents (normalized changes in the spot interest rate for n different maturities, then the first principal component can usually be interpreted as an (approximate parallel shift in the spot rate curve, whereas the second component represents a flattening or steepening of the curve. In equity applications, the first component often represents a systematic market factor that impacts all of the stocks whereas the second (and possibly other components may be identified with industry specific factors. Empirical PCA In practice we do not know the true variance-covariance matrix but it may be estimated using historical data. Suppose then that we have the multivariate observations, X 1,... X m, where X t = (X t1,..., X tn T, represents the date t sample observation. It is important that these observations represent a stationary time series such as asset returns or yield changes. However X t should not represent a vector of price levels, for example, as the latter generally constitute non-stationary time series. If µ j and σ j for j = 1,..., n, are the sample mean and standard deviation, respectively, of {X tj : t = 1,..., m}, then we can normalize the data by setting Y tj = X tj µ j σ j for t = 1,..., m and j = 1,..., n. Let Σ be the sample 12 variance-covariance matrix and let us assume that the Y i s come from some stationary time-series. Using the same notation as before, we see that the sample covariance matrix of Y is then given by Σ = 1 m m Y t Yt T. t=1 The principal components are then computed using this covariance matrix. From (14, we see that the original data is obtained from the principal components as X t = diag(σ 1,..., σ n Y t + µ 12 Usually we would write Σ for a sample covariance but we will stick with Σ here. = diag(σ 1,..., σ n Γ P t + µ (16

12 Introduction to Risk Management 12 where P t := (P t1,..., P tn = Γ T Y t is the t th sample principal component vector. Applications of PCA in Finance There are many applications of PCA in finance and risk management. They include: 1. Scenario Generation It is easy to generate scenarios using PCA. Suppose today is date t and we want to generate scenarios over the period [t, t + 1]. Then (16 evaluated at date t + 1 states X t+1 = diag(σ 1,..., σ n Γ P t+1 + µ. We can then apply stresses to the first few principal components, either singly or jointly, to generate loss scenarios. Moreover, we know that Var(P i = λ i and so we can easily control the severity of the stresses. 2. Building Factor Models If we believe the first k principal components explain a sufficiently large amount of the total variability then we may partition the n n matrix Γ according to Γ = [Γ 1 Γ 2 ] where Γ 1 is n k and Γ 2 is n (n k. Similarly we can write P t = [P (1 t P (2 t ] T where P (1 t is k 1 and P (2 t is (n k 1. We may then use (16 to write X t+1 = µ + diag(σ 1,..., σ n Γ 1 P (1 t+1 + ɛ t+1 (17 where ɛ t+1 := diag(σ 1,..., σ n Γ 2 P (2 t+1 now represents an error term. We can interpret (17 as a k-factor model for the changes in risk factors, X t+1. Note, however, that the components of ɛ t+1 are not independent which would be the case in a typical factor model. 3. Estimating VaR and CVaR We can use the model (17 and Monte-Carlo to simulate portfolio returns. This could be done by: (i ignoring the error term, ɛ t+1, which we know is small relative to the uncertainty in P (1 t+1 and (ii estimating the joint distribution of the first k principal components. Since they are uncorrelated by construction and we know their variances we could, for example, assume P (1 t+1 MVN k(0, diag(λ 1,..., λ k. Otherwise, we could use standard statistical techniques to estimate the distribution of P (1 t Portfolio Immunization It is also possible to hedge or immunize a portfolio against moves in the principal components. For example, suppose we wish to hedge the value of a portfolio against movements in the first k principal components. Let V t be the time t value of the portfolio and assume that our hedge will consist of positions, φ i, in the securities with time t prices, S ti, for i = 1,..., k. As usual let Z (t+1j be the date t + 1 level of the j th risk factor so that Z (t+1j = X (t+1j. If the change in value of the hedged portfolio between dates t and t + 1 is denoted by Vt+1, then we have ( V t+1 = n V t Z tj + ( n V t Z tj + j=1 j=1 k S ti φ i Z tj k φ i S ti Z tj Z (t+1j X (t+1j (18

13 Introduction to Risk Management 13 = (( n V t k + Z j=1 tj ( n V t k + Z j=1 tj ( k n + l=1 j=1 φ i V t Z tj + S ti φ i Z tj S ti Z tj k φ i ( k µ j + σ j Γ jl P l l=1 µ j (19 S ti Z tj σ j Γ jl P l (20 where, ignoring ɛ, we used the factor model representation in (17 in going from (18 to (19. We can now use (20 to hedge the risk associated with the first k principal components: we simply solve for the φ i s so that the coefficients of the P l s in (20 are zero. This is a system of k linear equations in k unknowns and so it is easily solved. If we include an additional hedging asset then we could also, for example, ensure that the total value of the hedged portfolio is equal to the value of the original un-hedged portfolio. Factor Models Factor models play an important 13 role in finance, particularly in the equity space where they are often used to build low-dimensional models of stock returns. In this context they are often used for both portfolio construction and portfolio hedging. Our discussion is very brief 14 and we begin with the definition 15 of a k-factor model. Definition 6 We say the random vector X = (X 1,..., X n T follows a linear k-factor model if it satisfies where X = a + B F + ɛ (21 (i F = (F 1,..., F k T is a random vector of common factors with k < n and with a positive-definite covariance matrix; (ii ɛ = (ɛ 1,..., ɛ n is a random vector of idiosyncratic error terms which are uncorrelated and have mean zero; (iii B is an n k constant matrix of factor loadings, and a is an n 1 vector of constants; (iv Cov(F i, ɛ j = 0 for all i, j. If X is multivariate normally distributed and follows (21 then it is possible to find a version of the model where F and ɛ are also multivariate normally distributed. In this case the error terms, ɛ, are independent. If Ω is the covariance matrix of F then the covariance matrix, Σ, of X must satisfy (why? Σ = B Ω B T + Υ where Υ is a diagonal matrix of the variances of ɛ. Exercise 1 Show that if (21 holds then there is also a representation X = µ + B F + ɛ (22 where E[X] = µ and Cov(F = I k so that Σ = B (B T + Υ. 13 There are many vendors of factor models in the financial services industry. It is arguable as to how much value these models actually provide when it comes to portfolio construction. 14 See Section 3.4 of MFE for further details and in particular, references to more complete sources. 15 This is Definition 3.3 in MFE.

14 Introduction to Risk Management 14 Example 8 (Factor Models Based on Principal Components The factor model of (17 may be interpreted as a k-factor model with F = P (1 and B = diag(σ 1,..., σ n Γ 1. Note that as constructed, the covariance of the error term, ɛ, in (17 is not diagonal and so it does not satisfy part (ii of our definition above. Nonetheless, it is quite common to construct factor models in this manner and to then make the assumption that ɛ is indeed a vector of uncorrelated error terms. Calibration Approaches There are two different types of factor models and the calibration approach depends on the type: 1. Observable Factor Models are models where the factors have been identified in advance and are observable. The factors in these models typically have a fundamental economic interpretation. A classic example would be a 1-factor model where the market index plays the role of the single factor. Other potential factors include macro-economic and other financial variables. These models are usually calibrated and tested for goodness-of-fit using multivariate 16 regression techniques. 2. Latent Factor Models are models where the factors have not been identified in advance. They therefore need to be estimated as part of the overall statistical analysis. Two standard methods for building such models are factor analysis and principal components analysis which we have already seen. Factor Models in Risk Management It is straightforward to use a factor model such as (21 to manage risk. For a given portfolio composition and fixed matrix, B, of factor loadings, the sensitivity of the total portfolio value to each factor, F i for i = 1,..., k, is easily computed. The portfolio composition can then be adjusted if necessary in order to achieve the desired overall factor sensitivity. Obviously this process is easier to understand and justify when the factors are easy to interpret. When this is not the case then the model is purely statistical. This tends to occur when statistical methods such as factor analysis or PCA are used to build the factor model. Of course, as we have seen in the case of PCA, it is still possible even then for the identified factors to have an economic interpretation. 7 Other Topics Model Risk Model risk can arise when calculating risk measures or estimating loss distributions. For example, if we are considering a portfolio of European options then there should be no problem in using the Black-Scholes formula to price these options as long as we use appropriate volatility risk factors with appropriate distributions. But suppose instead that the portfolio contains barrier options and that we use the corresponding Black-Scholes model for pricing these options. Then we are likely to run into several difficulties as the model is notoriously bad. It is possible, for example, that reported risk measures such as delta or vega will have the wrong sign! Moreover, the range of possible barrier option prices is limited by the assumption of constant volatility. A more sophisticated model would allow for a greater range of prices, and therefore losses. A more obvious example of model risk is using a light-tailed distribution to model risk factors when in fact a heavy-tailed distribution should be used. Data Risk When using historical data to either estimate probability distributions or run a historical simulation analysis, for example, it is important to realize that the data may be biased in several ways. For example, survivorship bias is a common problem: the stocks that are still around today are more likely to have done well than the stocks that 16 A multivariate regression refers to a regression where there is more than one dependent variable.

15 Introduction to Risk Management 15 are no longer around. So when we run a historical simulation to evaluate the risk of a long stock portfolio, say, we are implicitly omitting the worst returns from our analysis and therefore introducing a bias into our estimates of the various risk measures. We can get around this problem by somehow including the returns of stocks that no longer exist in our analysis. We also need to be mindful of biases that arise due to data mining or data snooping whereby people trawl through data-sets looking for spurious data patterns to justify some investment strategy. Multi-Period Risk Measures and Scaling It is often necessary to report risk measures with a particular investment horizon in mind. For example, many hedge funds are required by investors to report their 10-day 95% VaR and the question arises as to how this should be calculated. They could of course try to estimate the 10-day VaR directly. However, as hedge funds typically calculate a 1-day 95% VaR anyway, it would be convenient if they could somehow scale the 1-day VaR in order to estimate the 10-day VaR. Before considering this problem, it is worth emphasizing one of the implicit assumptions that we make when calculating our risk measures: The Constant Portfolio Composition Assumption: The risk measures are calculated by assuming that the portfolio composition does not change over the horizon [t, (t + 1 ]. Indeed regulatory requirements typically require that VaR be calculated under this assumption. While this is usually ok for small values of, e.g. 1 day, it makes little sense for larger values of, e.g. 10 days or 6 months, during which the portfolio composition is very likely to change. In fact in some circumstance the portfolio composition must change. Consider an options portfolio where some of the options are due to expire before the interval,, elapses. If the options are not cash settled, then any in-the-money options will be exercised resulting automatically in new positions in the underlying securities. Regardless of how the options settle, the returns on the portfolio around expiration will definitely not be IID. It can therefore be very problematic scaling a 1-day VaR into a 10-day VaR in these circumstances. More generally, it may not even make sense to consider a 10-day VaR when you know the portfolio composition is likely to change dramatically. When portfolio returns are IID and factor changes are normally distributed then a square-root scaling rule can be justified. In this case, for example, we could take the 10-day VaR equal to 10 times the 1-day VaR. But this kind of scaling can lead to very misleading results when portfolio returns are not IID normal, even when the portfolio composition does not change. Data Aggregation A particularly important topic is the issue of data aggregation. In particular, how do we aggregate risk measures from several trading desks or several units within a firm into one single aggregate measure of risk? This issue arises when a firm is attempting to understand its overall risk profile and determining its capital adequacy requirements as mandated by regulations. One solution to this problem is to use sub-additive risk measures that can be added together to give a more conservative measure of aggregate risk. Another solution is to simply ignore the risk measures from the individual units or desks. Instead we could try to directly calculate a firm-wide risk number. This might be achieved by specifying scenarios that encompass the full range of risk-factors to which the firm is exposed or using the variance-covariance approach with many risk factors. The resulting mean vector, µ and variance-covariance matrix, Σ may be very high-dimensional, however. For a small horizon,, it is generally ok to set µ = 0 but it will often be very difficult to estimate Σ due to its high dimensionality. References McNeil, A.J., R. Frey and P. Embrechts Quantitative Risk Management, Princeton University Press, New Jersey.