Technische Universiteit Delft Faculteit Elektrotechniek, Wiskunde en Informatica Delft Institute of Applied Mathematics

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1 Technische Universiteit Delft Faculteit Elektrotechniek, Wiskunde en Informatica Delft Institute of Applied Mathematics Het nauwkeurig bepalen van de verlieskans van een portfolio van risicovolle leningen (Engelse titel: Accurately determining the loss probability of a portfolio of risky loans) Verslag ten behoeve van het Delft Institute for Applied Mathematics als onderdeel ter verkrijging van de graad van BACHELOR OF SCIENCE in TECHNISCHE WISKUNDE door JELLE STRAATHOF Delft, Nederland November 2011 Copyright c 2011 door Jelle Straathof. Alle rechten voorbehouden.

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3 BSc verslag TECHNISCHE WISKUNDE Het nauwkeurig bepalen van de verlieskans van een portfolio van risicovolle leningen (Engelse titel: Accurately determining the loss probability of a portfolio of risky loans JELLE STRAATHOF Technische Universiteit Delft Begeleider Dr.ir. L.E. Meester Overige commissieleden Dr. J.G. Spandaw Prof. dr. ir. C.W. Oosterlee November, 2011 Delft

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5 Contents 1 Introduction Methodology Linear Model of Default Indicators Normal Copula Factor Model Simulation Monte Carlo Simulation Confidence Intervals Parameters of the Simulation Results of the Standard Monte Carlo Simulation Assessing the Variance Reduction and Batch Simulation Importance Sampling Method 1: Exponential Tilting Results of Exponential Tilting Method 2: Factor Twisting Results of the Factor Twisting Method 3: Zero-Variance Distribution Fitting the Zero-Variance Distribution Results of the zero-variance distribution Expected Shortfall for Importance Sampling 17 5 Variations in the model Reducing the default probability Reducing the loan size Increasing the parameters of the portfolio Conclusion 19 7 Appendix A 20 8 Appendix B 21 9 Appendix C Appendix C2 23 References 24 4

6 1 Introduction This project will evaluate different methods of importance sampling that can be used by banks to determine the loss probability of a portfolio of risky loans. Banks use portfolios to group together similar financial products, so that they can be more easily analyzed and traded. These portfolios can hold many different sorts of products, including loans. A loan is created when one party, named the lender, lends money to a second party, called an obligor. The obligor needs to repay it in a certain amount of time. It follows that an obligor may not be able to repay the loan: this is called a default. In this situation, the most basic problem... is determining the distribution of losses from default [Glasserman, 2004]. From this distribution of losses, a number of characteristics of a portfolio can be calculated. One of these is the loss probability, which is the probability that the loss of the portfolio is greater than a fixed number x: P(L > x), where L is the loss of the portfolio and x is called the default threshold. Another property is the expected shortfall, or E[L L > x]. This is the expected loss, given that the loss is greater than the default threshold. In order to determine the distribution of losses, mathematical models are used by banks. One of these is a linear model of default indicators. 1.1 Methodology To determine the loss probability of a portfolio of loans, a model of default indicators is used as described in the book by Glasserman [Glasserman, 2004]. Monte Carlo simulation is used to estimate the loss probability and the expected shortfall of a portfolio. The simulation is performed for select test cases so that results can be compared. A technique called importance sampling is then used to increase the accuracy of the results. This is done by drawing samples of the necessary variables from a new distribution in which there is a higher probability of obligors defaulting on a loan. The zero-variance distribution is the theoretical distribution that produces results with zero variance. This distribution is approximated and then used as the new distribution in the importance sampling to maximize the accuracy of the results. These variance reduction techniques are then applied to portfolios with different values for the default probability and the sizes of the loans. These techniques are used to find out how to maximize the variance reduction and so produce the most accurate results. The results presented will be the factor that each type of importance reduces the variance of the standard Monte Carlo simulation. This will make it simple to determine which method results in the greatest variance reduction. 1.2 Linear Model of Default Indicators A portfolio is a collection of loans held by a bank. Suppose the variable m represents the number of loans in the portfolio, and the variable c i is the value of the loan of the i th obligor, with i = 1,..., m. Each loan has an associated default probability, given by p i (for the i th obligor). The default indicator Y i takes the value 0 if obligor i has repaid the loan, or the value 1, if the obligor has not repaid the loan. The loss of the portfolio is then given by L = i Y ic i. In order to determine whether Y i is 1 or 0, each obligor has a corresponding variable X i that reflects the state of the loan. The default indicators are represented as: Y i = 1 {Xi >x i }, with x i the default threshold of obligor i. Thus, if the state variable X i is larger than the default threshold, the loan is not repaid. At this point, the obligors are independent of one another. In order to better reflect reality where there are often economic factors at play that affect all or most obligors, the model needs to be expanded so that the obligors are dependent. A factor 5

7 model is used to incorporate a dependence structure among the obligors, and a normal copula is used to transfer this structure to the default indicators. 1.3 Normal Copula Copulas are a mathematical mechanism of isolating the dependence structure between random variables. If the random vector X = (X 1,..., X d ) has the distribution function F with marginals F 1,..., F d, then the copula of F is the distribution function C of (F 1 (X 1 ),..., F d (X d )). According to Sklar s theorem, if the marginals F i of the distribution function F are continuous, then there is a unique function C : [0, 1] d [0, 1] with: F (X 1,..., X d ) = C(F 1 (X 1 ),..., F d (X d )) The vector (X 1,..., X m ) of correlated N(0,1) random variables can be used to create dependent uniform random variables U i = Φ 1 (X i ), where i = 1,..., m. Then, U 1,..., U m can be used to generate more dependent random variables. This copula is called a normal copula because of the use of the normal distribution function. The correlation between U i and U j, when i j, is called the copula correlation. 1.4 Factor Model Suppose there are m obligors, with associated random variables (X 1,..., X m ) N(0, I m ) and default thresholds x i = Φ 1 (1 p i ), where p i is the default probability of the obligor i. The next step is to determine the default indicators Y i = 1 {Xi >x i }, when X i are dependent. In order to introduce a dependence structure between the random variables X i, a 1-factor model is used. The following theory is based on Glasserman [Glasserman, 2004, Section 9.3]: let Z N(0, 1) be an independent common factor that affects all obligors, and let ɛ i N(0, 1) be the independent movement of an obligor. Represent this in the random variables X 1,..., X m with the form: X i = ρz + 1 ρ 2 ɛ i, i = 1,..., m Each ɛ i is independent of the others. The ρ is called the factor loading; in addition, the correlation between X i and X j, with i j, is ρ 2. The variance of X i gives the following: Var(X i ) = Var(ρZ + 1 ρ 2 ɛ i ) = ρ 2 Var(Z) + (1 ρ 2 )Var(ɛ i ) = ρ 2 + (1 ρ 2 ) = 1 The variance of the state variables, X i, is unchanged by the introduction of the factor model. Given Z = z, the conditional probability of default becomes: ( p i = P(Y i = 1 Z = z) = P(X i > x i Z = z) = P ρz + ( ) ) 1 ρ 2 x i ρz ɛ i > x i Z = z = 1 Φ 1 ρ 2 The last equals sign is correct because ɛ i N(0, 1). Given the common factor Z, the problem reduces to the independent case; when Z is fixed, the only variation in the state variables comes from the independent variables ɛ i. Conditionally, the portfolio loss L is a sum of m independent random variables. 6

8 2 Simulation 2.1 Monte Carlo Simulation Monte Carlo simulation is a technique used in financial mathematics to analyze complex instruments, investments and portfolios by simulating the uncertainty that affects their value. This is done by repeating the simulation thousands of times, and sampling from the distributions within the simulation each time. For example: α = 1 0 f(x)dx means that α = E[f(U)], where U U[0, 1]. If one were to draw realizations of this distribution, U 1,..., U n, then an estimate of α is given by ˆα n = 1 n n f(u i ) i=1 The law of large numbers implies that this value ˆα n converges to α as n with probability 1. Furthermore, set Var(f(U)) = σf 2 = 1 0 (f(x) α)2 dx. The error ˆα n α is approximately normally distributed with mean 0 and standard deviation σ f / n. 2.2 Confidence Intervals The confidence intervals that are presented are determined by simulating the loss probability 10,000 times. From these values, the 95% confidence interval is created: [ ] µ n 1.96 sn n µ n sn n where µ n is the arithmetic mean of n replications, and s n is the sample standard deviation. The standard error is given by s n / n. 2.3 Parameters of the Simulation To allow easy comparison between results, the portfolio is homogenized. The portfolio that is evaluated has m = 1000 obligors, all with the same loan, so the size of the loan is c i = 1 and the default probability is p i = for each obligor i. In addition, a number of test cases for the factor model are chosen. In order to compare the simulations of the test cases the desired loss probability needs to be fixed. The loss probability can be influenced by the default threshold of the portfolio, x. Large losses in the portfolio are events that do not occur frequently, and so rare-event simulation must be used to simulate these losses. The simulation will look for the 1% loss probability of the portfolio and its standard deviation; the corresponding default thresholds suggested by Glasserman [Glasserman, 2004] for each test case are as follows: ρ x These default thresholds give a loss probability that is very close to 1%. 2.4 Results of the Standard Monte Carlo Simulation The program used to simulate the model begins by randomly selecting a common factor for each replication. Then the default probability, p i, from Section 1.4, is calculated. The default indicators, Y i, are determined by creating a vector V with size m, the number of obligors, and 7

9 then determining whether or not V < p, where p is the vector of p i. The loss is then calculated with L = Y i c i, with L representing the loss. This simulation is repeated 10,000 times so that there exist 10,000 different instances of L. The value L > x shows when a portfolio has crossed the default threshold x, and so the loss probability is found by summing all these instances of default and dividing by the amount of repetitions. Simulating the portfolio for each test case produces the following: Loss Probability ρ Sample Mean Sample Standard Deviation Sample Standard Error It appears that the standard error is too large to give an accurate loss probability, given that it is between 20% and 33% of the mean of the loss probability. However, this can be remedied through the use of variance reduction techniques that are described in Chapter Assessing the Variance Reduction and Batch Simulation The use of Monte Carlo simulation brings with it the option of using a number of variance reduction techniques that can reduce the variance of the simulation, sometimes drastically. By reducing the variance, a choice can be made between accuracy and speed while simulating: if a specific accuracy is needed for the results, the variance can be reduced until the desired accuracy is achieved; if speed is desired, then the amount of replications can be lowered while maintaining the desired accuracy. One importance sampling technique uses an antithetic variable: this is a second identically distributed random variable that combines with the first to reduce the variance. Another technique uses a control variate, a similarly distributed random variable whose expectation is known; the variance is reduced through the correlation between the two variables. A third method called importance sampling uses a new distribution chosen specifically to increase the probability of the interesting event occurring, namely an obligor defaulting. This is the method that will be used and further explored in the next chapter. The variance reduction is determined by dividing the variance of the standard Monte Carlo simulation by the variance of the new simulation; this gives the factor by which the new variance is smaller than the variance of the standard Monte Carlo. The standard error of the simulated variance reduction is calculated by splitting the simulation into 201 batches, each of 1000 repetitions, and then determining the 95% confidence interval of the variance reduction. This works as follows: suppose the random variable X, with E[X] = µ and Var(X) = σ 2, is simulated repeatedly, producing n batches of k replications of X. Then the total number of replications is m = nk. Given that the result of replication j of batch i is X ij, with i = 1,..., n and j = 1,..., k, the mean of batch i is given by: X i = 1 k X ij k From this the overall mean is determined by: j=1 X = X X n n 8

10 The standard error of X is σ m. The random variable X i has an expected value of E[X i ] = µ and a variance of σb 2 = Var(X i) = σ2 k. By the Central Limit Theorem, if k is large then X i N(µ, σb 2). An unbiased estimator for σ2 b is given by: s 2 b = 1 n (X i X) 2 n 1 i=1 Then ˆσ 2 = s 2 b k is an unbiased estimator for σ2. If k is large enough to apply the Central Limit Theorem, then ˆσ 2 has a scaled χ 2 (n 1) distribution. Following from the definition of χ 2 distributions, if the random variables Y 1,..., Y m are independent identically distributed N(µ, σ 2 ), then for s 2 = ( 1 m 1 ) (Y i Y ) 2, where Y is the mean of Y 1,..., Y m, it is true that (m 1)s 2 χ 2 (m 1). σ 2 The variance of the estimator ˆσ 2 is given by: Var(ˆσ 2 ) = Var(s 2 b ( k) σ 2 = Var b k (n 1)s 2 ) b n 1 σb 2 ( ) k 2 ( (n 1)s 2 ) = n 1 σ2 b Var b σb 2 ( ) k 2 = n 1 σ2 b 2(n 1) = 2σ4 n 1 So the standard error for a simulation with n batches of k repetitions is given by s.e.(ˆσ 2 ) = 2 n 1 σ2. The relative standard error is the standard error divided by the true value of the parameter. Thus, the relative standard error is 2 n 1. If k is large, the Central Limit Theorem can be used to draw the above conclusions; in this simulation k = 1000 and n = 201, so that the relative standard error is 10%. 3 Importance Sampling Importance sampling can be a powerful technique in reducing the variance of a sample. It emphasizes important values that increase the probability of seeing a particular event, in this case, a default. The following theory is based on Glasserman [Glasserman, 2004, Section 9.3] and Glasserman & Li [Glasserman, 2005]. A biased distribution is chosen to sample from, and the samples of this distribution are then weighed to correct for the use of a bias. Suppose α = E[h(X)], with X a continuous random variable and h a function. Then E[h(X)] = h(x)f(x)dx with f the density function of X. Name g a density function such that g(x) > 0 when f(x) > 0, then α can be written as: α = h(x) f(x) g(x) g(x)dx The α is now found by sampling from a new density: α = use of the biased density g. The fraction f(x) g(x) f(x) Ẽ[h(X) g(x) ]. The tilde denotes the is called the likelihood ratio. If g is chosen well, 9

11 the second moment, and therefore the variance, can be reduced: [ ( Ẽ h(x) f(x) ) ] 2 ] [ = [h(x) g(x) Ẽ 2 f(x)2 g(x) 2 = E h(x) 2 f(x) ] g(x) This last value can be greater or smaller than E[h(X) 2 ] depending on g. When simulating a portfolio of loans, the density function g can be chosen using different methods of importance sampling. Three methods will be used and analyzed as part of this project: exponential tilting, factor twisting with a normal distribution, and factor twisting with the zerovariance distribution. These will be called Method 1, 2, and 3, respectively. The results of the variance reduction are presented without standard errors; however, the number of batches used in the simulation translates into a standard error of approximately 10% of the variance reduction. For more detailed results, consult Appendix A. 3.1 Method 1: Exponential Tilting Exponential tilting is a specific type of importance sampling. In order to understand how to use it, some other terms must be explained. Define the cumulant generating function of the distribution function F as follows: ψ(θ) = log( eθx df (x)); this is the logarithm of the moment generating function of F ; it is defined for all θ for which ψ(θ) remains finite. It can also be written as: ψ(θ) = log E[e θx ] The exponential tilting of F is defined as: F θ (x) = x eθu ψ(θ) df (u). Then each F θ is a probability distribution and F = F 0. If F has a density f, then F θ has density function f θ (x) = e θx ψ(θ) f(x) [Glasserman, 2004, Section 4.6.2]. Given independent, identically distributed random variables X 1,..., X n, with probability distribution F = F 0, the likelihood ratio for changing the distribution to F θ is: n i=1 df 0 (X i ) n df θ (X i ) = i=1 It follows from the definition of ψ(θ) that f(x i ) n e θxi ψ(θ) f(x i ) = exp( θ X i + nψ(θ)) ψ (θ) = d ) (E dθ log 0 [e θx ] = E 0[Xe θx ] E 0 [e θx = E 0 [Xe θx ]e ψ(θ) = E 0 [Xe θx ψ(θ) ] = E θ [X] ] and ψ (θ) = Var θ (X) following a similar calculation. E θ [X] is the expectation of X when tilted, and Var θ (X) is the variance of X when tilted. As stated in the description of the dependent model, conditioning on Z reduces the problem to the independent case. Therefore, the calculations of this section can be used on the dependent model when conditional on Z. Write φ as follows: φ L Z (θ) = log(e[e θl Z]). Recall that L = c i Y i and that Y i = 1 with probability p i and Y i = 0 with probability 1 p i. This means that [ φ L Z (θ) = log(e[e θ m ] m Y i c i Z]) = log(e e θy ic i Z ) = log( p i e ciθ + (1 p i )) i=1 The optimal θ for reducing the variance is determined by solving φ L Z ( θ x ) = x for a particular x, which follows from the calculations in Glasserman [Glasserman, 2004, Section 9.2.2]. Name i=1 i=1 10

12 φ i (θ) = log( p i e c iθ +(1 p i )). It follows from E θ [L] = x that E θ [Y i c i ] = φ i ( θ x ) [Glasserman, 2004, Section 9.2.2]. Then the tilted default probability for each obligor is: p i ( θ x ) = φ i ( θ x ) c i = p i e θ xc i p i e θ xc i + 1 pi In order to transform the results from the simulation, they need to be multiplied by the likelihood ratio of the exponential twist. The estimator of the loss probability becomes e θ xl+φ L Z ( θ x) 1 {L>x} Results of Exponential Tilting The exponential tilting is implemented in the simulation by solving φ L Z ( θ x ) = x in order to calculate θ x, and φ L Z is then determined using θ x. The tilted default probability, p i ( θ x ), can then be found using the formula in Section 3.1. The indicators Y i and the portfolio loss L are determined in the same manner as the standard Monte Carlo simulation, but the estimator 1 {L>x} is multiplied by the likelihood ratio e θ xl+φ L Z ( θ x) to account for the exponential tilting. The variance of the estimator from the standard Monte Carlo simulation is then divided by the variance of the estimator with exponential tilting to determine the variance reduction. The simulation with the help of exponential tilting produces the following variance reductions: ρ Variance Reduction In the first test case where ρ = 0.05, there is a reasonably high variance reduction, but the other values are not of the same magnitude. Thus, another method of importance sampling is needed: a technique called factor twisting is chosen. 3.2 Method 2: Factor Twisting It is now possible to apply importance sampling to the normally distributed common factor Z. From the law of total variance, also known as the variance decomposition formula, it follows that the variance of the estimator given above in Section 3.1 is equal to Var(e θ xl+φ L Z ( θ x) 1 {L>x} ) = E[Var(e θ xl+φ L Z ( θ x) 1{L > x} Z)] + Var(P(L > x Z)) This shows that tilting Y i conditional on Z makes the first term small but has no effect on the second. Using importance sampling on Z may reduce this second term. Introduce the mean µ so that Z is realized from a N(µ,I m ) distribution instead of a standard normal distribution; the likelihood ratio for this change is e µz+ 1 2 µ2. Multiplying this likelihood ratio with the estimator of the estimator of the loss probability for exponential tilting, the complete estimator for the tilted and twisted distribution becomes e µz+ 1 2 µ2 θ xl+φ L Z ( θ x) 1 {L>x} 11

13 The implementation of this step means that the simulation of the model can be checked: a graph in Glasserman & Li [Glasserman, 2004] can be reproduced to confirm that the simulation works correctly. Figure 1 shows the variance reduction of the test cases, with the common factor having a standard normal distribution with different means as produced by Glasserman. The reproduction in Figure 2 confirms that the simulations were performing correctly. The reproduction also confirms the results of the exponential tilting, as those are the same variance reductions shown in the graph at µ = 0. Figure 1: Variance reduction using change of mean from Glasserman Figure 2: Variance reduction using change of mean reproduction by Jelle Straathof Results of the Factor Twisting Using a similar approach as with the exponential tilting, the factor twisting adjusts the common factor Z with the mean µ so that it samples from a N(µ, 1) distribution. The new default probability p i ( θ x ) is calculated, after which the default indicators and the losses are determined as described in Section The estimator from Section 3.2 is calculated and its variance is determined. Once again the variance of the estimator of the loss probability from the standard Monte Carlo simulation is divided by the variance of the estimator of the loss probability from the tilted and twisted simulation to determine the variance reduction. In Figures 1 and 2, the maximum variance reduction is as follows: Variance Reduction ρ µ Method 1 & 2 Method The optimal µ is determined by calculating the variance reduction for each µ between 0 and 2 for the test case ρ = 0.05, between 0 and 2.5 for the test case ρ = 0.1 and between 0 and 4 for the test cases ρ = 0.25, 0.5, 0.8. These variance reductions are between 2 and 40 times 12

14 better than the ones achieved with only the exponential tilting. However, with the change in distribution for the common factor, there may be a better type of distribution than the shifted standard normal distribution. The zero-variance distribution provides an option of looking for such a distribution. 3.3 Method 3: Zero-Variance Distribution The change in distribution as considered in the previous section comes from the theory of the zero-variance distribution, mentioned in Glasserman & Li [Glasserman, 2005]. This states that there is a distribution of the common factor that results in an estimator of the loss probability with a variance of zero. With the use of the importance sampling theory, where: h(x)f(x) I = E[h(X)] = g(x)dx g(x) so that if the random variable Y has the distribution function g: I = E[h(Y )LR(Y )] where LR(Y ) is the likelihood ratio for using Y, and thus: Var(h(Y )LR(Y )) = ( h(x) f(x) ) 2 g(x) I g(x)dx In addition, with h non-negative, the product h(x)f(x) is non-negative. This means that it can be normalized to a probability density. Supposing that g is this density, and g(x) is proportional to h(x)f(x), then the variance is equal to zero. In this case, the constant of proportionality is 1 I. Applying this to the situation of the loan portfolio, name h(y) the loss probability P(L > x Z = z) and f(y) is the original density of the common factor, which is the standard normal density. Then the zero-variance distribution of the common factor should have a density proportional to: z P(L > x Z = z) exp( z 2 /2) The goal is to simulate the loss probability P(L > x Z = z), and it is needed to determine the zero-variance distribution. Therefore, the loss probability needs to be approximated so that the zero-variance distribution can also be approximated. In this case, the normal approximation for the loss probability is used, so P(L > x Z = z) is replaced: ( ) x E[L Z = z] P(L > x Z = z) 1 Φ Var(L Z = z) noting that E[L Z = z] = k p k(z)c k and Var(L Z = z) = k c2 k p k(z)(1 p k (z)). The accuracy of this approximation can be seen in Figures 3 through 7, where the loss probability and its 95% confidence interval is plotted against the approximation of the loss probability. The normal approximation is in all cases larger than the simulated loss probability. However, as rho increases, the approximation approaches the simulated loss probability. Additionally, Glasserman recommends fitting a normal distribution with a mean equal to the mode of the optimal density [Glasserman, 1999]. This step suggests that using a different distribution that fits the calculated density better might increase the variance reduction. 13

15 Figure 3: Approximation of the loss probability for ρ = 0.05 Figure 4: Approximation of the loss probability for ρ = 0.1 Figure 5: Approximation of the loss probability for ρ = 0.25 Figure 6: Approximation of the loss probability for ρ = 0.5 After determining the zero-variance distribution, it became clear that a shifted lognormal distribution would fit well. In order to determine the parameters of both the normal and lognormal fits of the zero-variance distribution, the moment-method was used. This method uses the results of the simulation as a sample from which to determine the mean and standard deviation of the normal and lognormal distributions that fit the results the best. Therefore, the first parameter of the normal distribution, µ, is the mean of the simulated values, and the second parameter, σ, is the sample standard deviation of the simulated values. The lognormal parameters were determined by first using the moment-method to calculate the mean, standard deviation and skewness of the results, and then changing these to the parameters of the lognormal distribution. The exact transformations can be seen in Appendix B. 14

16 Figure 7: Approximation of the loss probability for ρ = Fitting the Zero-Variance Distribution The estimator of the loss probabilities, when the common factor is lognormally distributed, is very close to the loss probabilities shown in Glasserman [Glasserman, 2004]: Loss Probability ρ Glasserman Sample Mean ± Sample Standard Error ± ± ± ± ± This means that the lognormal distribution of the common factors can be used for reducing the variance of the estimator of the loss probability. As seen in Figures 8 through 12, when ρ is small, there is very little difference between the simulated distribution and the fitted normal and lognormal distributions. However, at higher values of ρ, the simulated distribution becomes more skewed to the left, and as such, the lognormal distribution becomes a better fit Results of the zero-variance distribution The zero-variance distribution is close to a shifted lognormal distribution. The simulation for the estimator of the loss probability and its variance with the zero-variance distribution is the same as the simulation for the factor twisting, except that the common factor takes on the shifted lognormal dsitribution instead of the shifted normal distribution. The variance reduction achieved with a lognormally distributed common factor is as follows: Variance Reduction ρ Method 1 & 2 Method 1 &

17 Figure 8: Normal and lognormal fits of ρ = 0.05 distribution Figure 9: Normal and lognormal fits of ρ = 0.1 distribution Figure 10: Normal and lognormal fits of ρ = 0.25 distribution Figure 11: Normal and lognormal fits of ρ = 0.5 distribution This suggests that shifted lognormally distributed common factor should be used because the variance is only slightly smaller at low ρ, when compared to the shifted normal distribution. When ρ is low the approximation of the conditional loss probability used to determine the new distribution of the common factor is not accurate, as seen in Figures 3 through 7 With increasing ρ the lognormal distribution shows increased variance reduction compared to the variance reduction achieved by the shifted normal distribution. More detailed results can be found in Appendix A, as well as the relative variance reduction for the different methods. 16

18 Figure 12: Normal and lognormal fits of ρ = 0.8 distribution 4 Expected Shortfall for Importance Sampling The expected shortfall has also been calculated for the test cases, with the corresponding variance reduction compared to standard Monte Carlo simulation. All the following results are with 10,000 replications. The standard Monte Carlo gives: Expected Shortfall ρ Sample Mean Sample Standard Error The variance reduction of the expected shortfall for the different methods of importance sampling is as follows: Variance Reduction ρ Method 1 Method 1 & 2 Method 1 & Variance reduction for the expected shortfall follows a similar pattern as seen with the loss probability in variance reduction in Section The change-of-mean of the normal distribution provides less variance reduction than the lognormal distribution of the common factor. 17

19 5 Variations in the model The results described in Chapters 3 & 4 were all based on the assumption that the portfolio was homogeneous with a loss probability p i = and loans worth c i = 1. To confirm the robustness of the model these values can be changed. Appendix C1 shows the results for default probabilities of p i = 0.001, 0.002, Appendix C2 shows the results for the loan sizes c i = 0.5, 1, Reducing the default probability Reducing the default probability of the obligors, so that p i = and c i = 1, results in a standard Monte Carlo simulation with very few defaults occuring. This in turn causes the importance sampling to greatly increase the number of defaults and so increases the variance reduction compared to a portfolio with p i = and c i = 1, for both the loss probability and the expected shortfall. For example, when ρ = 0.05, the variance reduction for the loss probability is for Methods 1 & 2 and for Methods 1 & 3, which is 33 and 43 times better than when p i = Although the loss probability still achieves the greatest variance reduction by using lognormally distributed common factor, the variance reduction for the expected shortfall is greater when using the shifted normal distribution for the common factor. 5.2 Reducing the loan size Using importance sampling on a portfolio with p i = and c i = 0.5 also results in greater variance reduction in a number of test cases compared to a portfolio with larger loan sizes. In this case however, the variance reduction for the test case ρ = 0.05 results in a variance reduction of 0. This is because no defaults occur in the standard Monte Carlo simulation and so there is a variance of 0 for both the loss probability and the expected shortfall. In addition, when using the lognormally distributed common factor, the variance reduction for ρ = 0.5, 0.8 is lower than the variance reduction achieved with a portfolio with p i = and c i = 1, by a factor 2 and 2.3 respectively, for the loss probability, and a factor 2.2 and 1.4, respectively, for the expected shortfall. As such, the best distribution to use for the common factor is the shifted normal distribution. 5.3 Increasing the parameters of the portfolio Greater p i and c i results in smaller variance reduction for the importance sampling methods used on both the loss probability and the expected shortfall. The variance reduction increases for increasing ρ both the loss probability and the expected shortfall. The smaller variance reduction is because more defaults occur in the standard Monte Carlo simulation, and so the importance sampling cannot reduce the variance as much as when the portfolio has the parameters p i = and c i = 1. For the portfolios with p i = 0.004, c i = 1 and p i = 0.002, c i = 2, using the shifted normal distribution results in greater variance reduction than using the lognormal distribution for the common factor. These results also suggest that both methods of changing the distribution of the common factor should be used so that the most accurate results can be achieved. However, the zero-variance 18

20 distribution used to determine the variance reductions for the different portfolios was optimized for p i = and c i = 1. Determining the zero-variance distribution for each of the different portfolios may result in greater variance reductions. 6 Conclusion Loan portfolios are important financial instruments for banks. They allow them to easily group similar products and to determine the outcomes of many products. It is necessary to provide accurate information regarding the losses that can be suffered by the bank. This can be done by modeling a portfolio and using Monte Carlo simulation to calculate the results, such as the loss probability and the expected shortfall. These results are only useful if the bank can determine whether or not they are accurate. It appears that Monte Carlo simulations combined with importance sampling yields accurate results for the loss probability of a portfolio of loans. Using the zero-variance distribution increases the reliability of the simulation by reducing the variance the most. The results are optimized for reducing the variance of the loss probability, so using a zero-variance distribution optimized for the expected shortfall could produce greater variance reduction. Further research can be done into using a better approximation of the loss probability than the normal approximation used here. Other areas that can be investigated further are the results of the variance reduction on other characteristics of the portfolio, such as the second moment of (L x) given that L > x, as well the use of other variance reduction techniques on such a portfolio. 19

21 7 Appendix A Presented here are the variance reductions generated by simulating with 201 batches, which should give a standard error of 10%. Due to the nature of simulation, the standard errors will not be exactly 10%. The variance reduction achieved by each method of importance sampling is shown. In the cases of Methods 2 and 3, the variance reduction was calculated by determining the difference between the variance reduction achieved with Method 1 and with Methods 1 and 2 or Methods 1 and 3, respectively. Variance Reduction of the Loss Probability ± Standard Error ρ Method 1 Methods 1 & 2 Method 2 Method 1 Method 1 & 3 Method 3 Method ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Variance Reduction of the Expected Shortfall ± Standard Error ρ Method 1 Methods 1 & 2 Methods 1 & ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± From these results it can be concluded that the best method of importance sampling for the loss probability as well as the expected shortfall is the combination of Methods 1 and 3, because the variance reduction achieved is the greatest compared to the other methods. 20

22 8 Appendix B The moment-method uses the moments to determine the parameters of a distribution. The first step is then to determine the mean, standard deviation and skewness of the data. The first moment is the expectation. The simulated distribution is made of discrete data points, so the expectation can be determined as follows: x = i P(X = x i) x i, where x i are the discrete data points. The standard deviation is determined as follows: s = P(X = x i )(x i x) 2 The skewness is determined as follows: [ (X ) ] x 3 γ = E = s i i P(X = x i)(x i x) 3 i (P(X = x i)(x i x) 2 ) 1.5 The three parameters for the shifted lognormal distribution, Y ln(n(µ, σ, δ)), where µ is the mean, σ is the standard deviation, and δ is the x-axis shift, are determined by solving for µ, σ and δ in the following equations: µ+ σ2 x = δ + e 2 ( ) s 2 = e σ2 1 e 2µ+σ2 ( ) γ = e σ2 + 2 e σ2 1 Solving these equations for the required variables gives: σ = log( 1 2 (8 + 4γ γ 2 + γ 4 ) (8 + 4γ γ 2 + γ 4 ) 1 3 µ = 1 2 σ2 + 1 ( ) s 2 2 ln e σ2 1 δ = x e (µ+ 1 2 σ2 ) 1) With these parameters, the fitted distributions are plotted as seen in Section

23 9 Appendix C1 These are the variance reductions generated by changing the default probability p i of the portfolio. p i = 0.001, c i = 1 Loss Probability Expected Shortfall ρ Method 1 & 2 Method 1 & 3 Method 1 & 2 Method 1 & ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± p i = 0.002, c i = 1 Loss Probability Expected Shortfall ρ Method 1 & 2 Method 1 & 3 Method 1 & 2 Method 1 & ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± p i = 0.004, c i = 1 Loss Probability Expected Shortfall ρ Method 1 & 2 Method 1 & 3 Method 1 & 2 Method 1 & ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± As seen in the tables, reducing the default probability of the portfolio increases the variance reduction that can be achieved for both the loss porbability and the expected shortfall, whereas increasing the default probability decreases the variance reduction. 22

24 10 Appendix C2 These are the variance reductions generated by changing the loan size c i of the portfolio. p i = 0.002, c i = 0.5 Loss Probability Expected Shortfall ρ Method 1 & 2 Method 1 & 3 Method 1 & 2 Method 1 & ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± p i = 0.002, c i = 1 Loss Probability Expected Shortfall ρ Method 1 & 2 Method 1 & 3 Method 1 & 2 Method 1 & ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± p i = 0.002, c i = 2 Loss Probability Expected Shortfall ρ Method 1 & 2 Method 1 & 3 Method 1 & 2 Method 1 & ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± On the whole, decreasing the size of the loan of the homogeneous portfolio increases the variance reduction that can be achieved in comparison to the case of c i = 1. Conversely, increasing the size of the loan decreases the achievable variance reduction. 23

25 References [Glasserman, 1999] Glasserman, Paul, Heidelberger, Philip and Shahabuddin, Perwez, Asymptotically optimal importance sampling and stratification for pricing path-dependent options. Mathematical Finance, vol. 9, p , [Glasserman, 2004] Glasserman, Paul, Monte Carlo in Financial Engineering, first edition. New York, Springer-Verlag, [Glasserman, 2005] Glasserman, Paul and Li, Jinyi, Importance Sampling for Portfolio Credit Risk. Management Science, vol. 51, no. 11, November [Hull] Hull, John C., Risk Management and Financial Institutions, second edition. Pearson Education,