V A L U E-A T-R I S K (V A R) Value-at-Risk (VaR) The authors describe how to implement VaR, the risk measurement technique widely used in financial risk management. by Simon Benninga and Zvi Wiener n this article we discuss one of the modern risk- million dollars by year end (i.e., what is the probability measuring techniques Value-at-Risk (VaR). Mathemat- that the end-of-year value is less than $80 million)? I ica is used to demonstrate the basic methods for cal- 3. With 1% probability what is the maximum loss at the culation of VaR for a hypothetical portfolio of a stock and end of the year? This is the VaR at 1%. a foreign bond. We start by loading Mathematica ’s statistical package: VALUE-AT-RISK Needs "Statistics‘Master‘" Value-at-Risk (VaR) measures the worst expected loss un- [ ] Needs["Statistics‘MultiDescriptiveStatistics‘"] der normal market conditions over a specific time inter- val at a given confidence level. As one of our references We first want to know the distribution of the end-of- states: “VaR answers the question: how much can I lose year portfolio value: with x% probability over a pre-set horizon” (J.P. Mor- Plot[PDF[NormalDistribution[110,30],x],{x,0,200}]; gan, RiskMetrics–Technical Document). Another way of expressing this is that VaR is the lowest quantile of the potential losses that can occur within a given portfolio 0.012 during a specified time period. The basic time period T 0.01 and the confidence level (the quantile) q are the two ma- jor parameters that should be chosen in a way appropriate 0.008 to the overall goal of risk measurement. The time horizon 0.006 can differ from a few hours for an active trading desk 0.004 to a year for a pension fund. When the primary goal is 0.002 to satisfy external regulatory requirements, such as bank capital requirements, the quantile is typically very small 50 100 150 200 (for example, 1% of worst outcomes). However for an internal risk management model used by a company to The probability that the end-of-year portfolio value is control the risk exposure the typical number is around less than $80 is about 15.9%. 5% (visit the internet sites in references for more details). A general introduction to VaR can be found in Linsmeier, CDF[NormalDistribution[110.,30],80] [Pearson 1996] and [Jorion 1997]. 0.158655 In the jargon of VaR, suppose that a portfolio manager has a daily VaR equal to $1 million at 1%. This means that With a probability of 1% the end-of-year portfolio value there is only one chance in 100 that a daily loss bigger will be less than 40.2096; this means that the VaR of the than $1 million occurs under normal market conditions. distribution is 100 - 40.2096 = 59.7904. Quantile[NormalDistribution[110.,30],0.01] A REALLY SIMPLE EXAMPLE 40.2096 Suppose portfolio manager manages a portfolio which consists of a single asset. The return of the asset is nor- We can formalize this by defining a VaR function which mally distributed with annual mean return 10% and annual takes as its parameters the mean mu and standard devia- standard deviation 30%. The value of the portfolio today is tion sigma of the distribution as well as the VaR level x. $100 million. We want to answer various simple questions about the end-of-year distribution of portfolio value: ClearAll[VaR]; VaR[mu_,sigma_,x_]:= 1. What is the distribution of the end-of-year portfolio 100-Quantile[NormalDistribution[mu,sigma],x] value? VaR[110,30,0.01] 2. What is the probability of a loss of more than $20 59.7904 Vol. 7 No. 4 1998 Mathematica in Education and Research 1 V A L U E-A T-R I S K (V A R) LOGNORMAL DISTRIBUTIONS The daily VaR of the portfolio at 1% is $256,831. The As explained in our previous articles, the lognormal dis- probability that the firm will lose more than this amount tribution is a more reasonable distribution for many asset on its portfolio over the course of a single day is less than prices (which can not become negative) than the normal 1%. Similarly, the weekly and the monthly VaRs at 1% are distribution. This is not a problem: Suppose that the natu- $1.28 and $5.26 million. ral logarithm of the portfolio value is normally distributed with annual mean m and annual standard deviation s. De- A THREE-ASSET PROBLEM: THE IMPORTANCE OF THE noting the value of the portfolio by v it follows that the VARIANCE-COVARIANCE MATRIX logarithm of the portfolio value at time T , v , is normally t As can be seen from the above examples, VaR is not–in distributed: 2 principle, at least–a very complicated concept. In the im- s plementation of VaR, however, there are two big practical Log[vT ] ~ Normal Log[v] + m - T , s T 2 problems (both problems are discussed in much greater s2 detail in the material available on the J.P. Morgan Web The term 2 T appears due to Ito’s Lemma (see [Hull 1997]). In our case, this means that v = 100, m = 10%, site): s = 30%. Thus the end-of-yearC log Kof the portfolioO valueG is 1. The first problem is the estimation of the parameters of distributed NormalDistribution[ Log[ 100 ] + asset return distributions. In “real world” applications (0.1-0.3ˆ2/2), 0.3] = NormalDistribution[ of VaR, it is necessary to estimate means, variances, 4.666017,0.3]. This means that the probability that and correlations of returns. This is a not-inconsiderable the end-of-year value of the portfolio is less than 80 is problem! In this section we illustrate the importance given by: of the correlations between asset returns. In the fol- CDF[NormalDistribution[Log[100]+ lowing section we give a highly-simplified example of (0.1-0.3ˆ2/2),0.3],Log[80]] the estimation of return distributions from market data. 0.176926 For example you can imagine that a long position in Similarly the VaR function has to be redefined: Deutschmarks and a short position in Dutch guldens is less risky than one leg only, because of a high prob- ClearAll[lognormalVaR]; ability that profits of one position will be mainly offset lognormalVaR mu_,sigma_,x_ [ ]:= by losses of another. 100-Exp[Quantile[NormalDistribution[ Log[100]+(mu-sigmaˆ2/2),sigma],x]]; lognormalVaR[0.10,0.30,0.01] 2. The second problem is the actual calculation of posi- tion sizes. A large financial institution may have thou- 47.4237 sands of loans outstanding. The data base of these Thus a portfolio whose initial value is $100 million and loans may not classify them by their riskiness, nor whose annual returns are lognormally distributed with even by their term to maturity. Or–to give a second parameters mu = 10% and sigma = 30%, has an annual example–a bank may have offsetting positions in for- VaR equal to $47.42 million at 1%. eign currencies at different branches in different loca- Most VaR calculations are not concerned with annual tions. A long position in Deutschmarks in New York value at risk. The main regulatory and management con- may be offset by a short position in Deutschmarks in cern is with loss of portfolio value over a much shorter Geneva; the bank’s risk–which we intend to measure time period (typically several days or perhaps weeks). by VaR–is based on the net position. It is clear that the distribution formula Log[vT ] ~ Normal[ 2 We start with the problem of correlations between asset Log[v ] + (m - s )T , sT ] can be used to calculate the VaR 2 returns. We continue the previous example, but assume over any horizon. Recall that T is measured in annual that there are three risky assets. As before the parameters terms; if there are 250 business days in a year, then the of the distributions of the asset returns are known: all daily VaR corresponds to T = 1/250 (for many fixed in- the means: m , m , m , as well as the variance covariance come instruments one should use 1/360, 1/365, or 1/365.25 1 2 3 matrix of the returns: depending on the market convention): ClearAll lognormalVaR [ ]; ÁÊ s11 s12 s13 ˜ˆ lognormalVaR[mu_,sigma_,x_,T_]:=100-Exp[ Á ˜ S = Á s21 s22 s23 ˜ Quantile[NormalDistribution[Log[100]+(mu- Á s s s ˜ sigmaˆ2/2)*T,sigma*T],x]]; Ë 31 32 33 ¯ lognormalVaR[0.10,0.30,0.01,1/250] lognormalVaR[0.10,0.30,0.01,5/250] The matrix S is of course symmetric; sij is the covariance lognormalVaR[0.10,0.30,0.01,21/250] of the returns of assets i and j (if i = j, s]ii is the variance 0.256831 of asset i’s return). 1.27758 Suppose that the total portfolio value today is $100 5.25717 million, with $30 million invested in asset 1, $25 million 2 Mathematica in Education and Research Vol. 7 No. 4 1998 V A L U E-A T-R I S K (V A R) in asset 2, and $45 million in asset 3. Then the return until the bond’s maturity determines its price in foreign distribution of the portfolio is given by: currency). mean return = x.m = x1m1 + x2m2 + x3m3 Í The time until the bond’s maturity; along with the for- variance of return = x.S.x T , eign market interest rate, this will determine the bond’s where x = {x1, x2, x3} = {0.3, 0.25, 0.45} is the vector of foreign currency price.