VALUE AT RISK (VAR) (PART 3)

Value-at-Risk (VaR)

by Simon Benninga and Zvi Wiener

A THREE-ASSET PROBLEM: THE IMPORTANCE OF THE VARIANCE-COVARIANCEMATRIX
As can be seen from the above examples, VaR is not–in principle, at least–a very complicated concept. In the implementation of VaR, however, there are two big practical problems (both problems are discussed in much greater detail in the material available on the J.P. Morgan Web site):
1. The first problem is the estimation of the parameters of asset return distributions. In “real world” applications of VaR, it is necessary to estimate means, variances, and correlations of returns. This is a not-inconsiderable problem! In this section we illustrate the importance of the correlations between asset returns. In the following section we give a highly-simplified example of the estimation of return distributions from market data.
For example you can imagine that a long position in Deutschmarks and a short position in Dutch guldens
is less risky than one leg only, because of a high probability that profits of one position will be mainly offset by losses of another.
2. The second problem is the actual calculation of position sizes. A large financial institution may have thousands of loans outstanding. The data base of these loans may not classify them by their riskiness, nor
even by their term to maturity. Or–to give a second example–a bank may have offsetting positions in foreign currencies at different branches in different locations.
A long position in Deutschmarks in New York may be offset by a short position in Deutschmarks in
Geneva; the bank’s risk–which we intend to measure by VaR–is based on the net position.
We start with the problem of correlations between asset returns. We continue the previous example, but assume that there are three risky assets. As before the parameters of the distributions of the asset returns are known: all the means: m1, m2, m3, as well as the variance covariance matrix of the returns:
S =ÊÁÁÁÁÁÁË
s11
s21
s31
s12
s22
s32
s13
s23
s33
ˆ˜˜˜˜˜˜¯
The matrix S is of course symmetric; sij is the covariance of the returns of assets i and j (if i = j , s]ii is the variance of asset i’s return).
Suppose that the total portfolio value today is $100 million, with $30 million invested in asset 1, $25 Million 2 MathematicainEducationandResearch Vol. 7 No. 4 1998 VALUE-AT-RISK (VAR) in asset 2, and $45 million in asset 3. Then the return distribution of the portfolio is given by:
mean return = x.m = x1m1 + x2m2 + x3m3
variance of return = x.S.xT ,where x = {x1, x2, x3} = {0.3, 0.25, 0.45} is the vector of proportions invested in each of the three assets. Assuming that the returns are normally distributed (meaning that prices are lognormally distributed), we may calculate the VaR by:
Clear[x,S,means,portfolioMean,portfolioSigma];
X=Table[x[i],{i,3}];
S=Table[sigma[i,j],{i,3},{j,3}];
means=Table[mu[i],{i,3}];
portfolioMean[X_,means_,initial_]:=X.means*initial;
portfolioSigma[X_,s_,initial_]:=Sqrt[X.s.X]*initial;
To implement this:
Clear[VaR];
X={0.3,0.25,0.45};
means={0.1,0.12,0.13};
initial=100;
VaRlevel=0.01;
S={{0.1,0.04,0.03},{0.04,0.2,-0.04},
{0.03,-0.04,0.6}};
Print["The portfolio mean = ",
portfolioMean[X,1+means,initial] ]
Print["The portfolio sigma = ",
portfolioSigma[X,S,initial]]
VaR[X_,s_,initial_,level_]:=initial-Quantile[NormalDistribution[portfolioMean[X,means,initial],
portfolioSigma[X,s,initial]],VaRlevel]
Print["The portfolio VaR at the ",
VaRlevel*100, "% level is ",
VaR[X,S,initial,VaRlevel] ]
The portfolio mean = 111.85
The portfolio sigma = 38.4838
The portfolio VaR at the 1. % level is 177.677
USING EMPIRICAL DATA TO CALCULATE THE VAR
In this section we use market data to do some VaR calculations. Our data consists of 40 business days of Data for a market index, foreign interest rates and foreign exchange rates. We use this data to do three kinds of VaR calculations: Historic simulations, variance-covariance calculations, and Monte Carlo simulations.
Data Description and Preliminary Calculations We consider the VaR of a portfolio manager who has invested in only two assets: A domestic stock index and a foreign bond. For simplicity we assume that the value of this portfolio is wholly determined by only several parameters:
Í The current price of the stock index
Í The foreign market interest rate (the bond is assumed to have a zero coupon, so that only the interest rate
until the bond’s maturity determines its price in foreign currency).
Í The time until the bond’s maturity; along with the foreign market interest rate, this will determine the bond’s foreign currency price.
Í The exchange rate between the foreign and the domestic currency.
We use historic price and return data to do our VaR calculations. It is convenient to translate all dates to a numerical format. We set January 1, 1997 to be day number 0 and then any day is given by a number of days since the initial date (we assume that all days are business days for simplicity).
Needs["Miscellaneous‘Calendar‘"];
Needs["Statistics‘Master‘"]
Needs["Statistics‘MultiDescriptiveStatistics‘"]
Day0 ={1997, 1, 1};
dayN[ day_]:=DaysBetween[ Day0, day];
Thus January 13, 1997 is the day number 12 and so on.
dayN[{1997, 1,13}]12
Consider February 10, 1997. Suppose that on February 10, 1997, the stock index value is 293., the foreign interest rate is 5.3%, and the exchange rate is 3.4; February 10, 1997 is the 40th business day since January 1, 1997. Thus we write the data for this day as a list: {40, 293., 5.3,3.4}. Obviously any real market data must contain much more information, however will use this simple example to illustrate the VaR approach.
The whole data set with which we will be working is the following (in a real-world situation, we would obviously use much more data about many more assets):
dataVAR ={
{1, 282., 5.28, 3.5},{2, 283., 5.26, 3.47},
{3, 285., 5.23, 3.46},{4, 280., 5.24, 3.45},
{5, 282., 5.25, 3.45},{6, 281., 5.24, 3.46},
{7, 282., 5.24, 3.45},{8, 286., 5.25, 3.43},
{9, 285., 5.25, 3.47},{10, 286., 5.26, 3.443},
{11, 288., 5.27, 3.42},{12, 289., 5.28, 3.42},
{13, 290., 5.28, 3.41},{14, 289., 5.28, 3.42},
{15, 291., 5.29, 3.46},{16, 293., 5.31, 3.41},
{17, 294., 5.32, 3.40},{18, 290., 5.34, 3.49},
{19, 287., 5.35, 3.47},{20, 288., 5.34, 3.48},
{21, 289., 5.35, 3.46},{22, 281., 5.36, 3.44},
{23, 283., 5.23, 3.45},{24, 285., 5.24, 3.42},
{25, 288., 5.25, 3.41},{26, 289., 5.26, 3.41},
{27, 287., 5.26, 3.43},{28, 285., 5.28, 3.42},
{29, 290., 5.27, 3.44},{30, 291., 5.27, 3.42},
{31, 289., 5.27, 3.37},{32, 288., 5.29, 3.39},
{33, 290., 5.28, 3.41},{34, 293., 5.31, 3.44},
{35, 292., 5.32, 3.41},{36, 293., 5.28, 3.42},
{37, 293., 5.30, 3.42},{38, 293., 5.31, 3.44},
{39, 292., 5.32, 3.41},{40, 293., 5.30, 3.4}};
Suppose that the financial institution holds two shares of the stock index portfolio and a short position in a zerocoupon foreign bond having face value 100 rubles with maturity May 8, 2000. We represent this portfolio as a list:
Vol. 7 No. 4 1998 MathematicainEducationandResearch 3
VALUE-AT-RISK (VAR)
portfolio = { {STOCK,{}, 2},
{FBOND,{dayN[{2000, 5, 8}]}, -1}}
Note that the empty paramters list in the stock part means that there are no parameters and we always keep
the same portfolio of stocks. The list of parameters of the bond part consists of one number - day of maturity. In general both lists can have many parameters describing a specific portfolio of stocks or bonds.