SOLUTION NOTES FOR EXERCISES ON FUNCTIONS
4. Code for Standard Deviation of Cash Flows
7.5 VALUE-AT-RISK
In this section, we examine the concept of Value-at-Risk (VaR), implemented for equities as an analytic calculation derived from the assumed lognormal distribution of returns.
Empirical observation of share returns shows that they typically demonstrate some skewness. However, rescaling returns by taking their logs (usually ‘natural’ logs) gives a more symmetrical spread of values. So when compared with returns, log returns are usually more symmetric and tend to follow the normal distribution. When log returns are normally distributed, the distribution of returns is said to be lognormal. Although it is frequently possible to overlook the distinction between the lognormal and the normal
distribution in some instances (such as dealing with daily returns or with assets of low volatility), in accurate work the difference cannot be ignored. It is important to at least understand the difference between the two cases. One broad distinction is that academic users prefer to use log returns in their analyses whilst commercial software typically uses ordinary (raw) returns and assumes these are normally distributed. Another variation in what is termed ‘returns data’ is that excess returns (the return after differencing off the risk-free rate) are preferred as the basis for analysis. Essentially, the crucial thing is to know what variant of the underlying share price returns data is being assumed in any analysis.
Strictly speaking, some check should be made on ‘returns’ data to ensure that it is approximately normally distributed. An easy approach is to produce a normal probability plot, for example for ShareA log returns (column B) of the Beta sheet. The methodology of normal probability plots is described in Chapter 3 (section 3.6.2). The resulting plot of ShareA log returns is shown in Chart1 of the EQUITY2.xls workbook.
Using the normal distribution of log returns, we can define a lower-tail value for a portfolio of assets below which there is a known chance that the asset value will fall.
This lower tail value is called the Value-at-Risk or VaR. Essentially the measure is a translation from the volatility of equity returns into the appropriate percentile of the normal distribution function.
As well as the mean and variance of the monthly log returns, which we denote asMand V, we also need to find the mean of the returns data which has a lognormal distribution (denoted by M1). Since the moments of the returns distribution are required in this and the following section, they have been summarised in Figure 7.10 later, the crucial linking formula being thatM1DexpMC0.5V.
If the distribution of the monthly log returns on an asset is normal, with mean Mand variance V, then over time υt the distribution is also normal, but with mean Mυt and variance Vυt. From the standard normal distribution, we know there is a 5% chance of being 1.64 standard deviations or more below the mean, that is of being:
[Mυt1.64Vυt]
or lower. Similarly, there is a 2.5% chance of being 1.96 standard deviations below and so forth. The ‘z-value’ (herezD1.645) determines the area in the lower tail of the distribution (here 5%). If the asset starts with value S, then afterυt months there will be a 5% chance that its value will be as low as:
S[Mυt1.64Vυt]
which is the VaR for the asset. Strictly speaking this is the absolute VaR. If only the volatility is captured (and the expected return SMυt, is ignored), the VaR measure is calledrelative.
VaR is an attempt to estimate the downside risk of an asset or portfolio with a single number, where the VaR represents the maximum loss expected of the asset given an appropriate percentile (say 5%) and time period (say one month). In the real world, VaR is used by the major investment banks as an aggregate measure of the daily downside risk of their trading portfolios. These portfolios are likely to contain the whole gamut of financial assets such as options, bonds and futures as well as equities. Of necessity, the illustration here is chosen with equity portfolios in mind and the complications that arise when calculating VaR for other assets are not addressed.
Asset Pricing 133 Calculation of VaR is illustrated for the portfolio of eight Swiss shares, whose returns are displayed in the VCV sheet. We use the monthly log excess returns for the eight shares, as shown in Figure 7.7. In rows 8 and 10 the mean and variance of the monthly log returns are calculated for each asset from the 60 log returns readings in column C through to column J. For example for UBS, the mean [evaluated from formula DAVERAGE(C13:C72)] is in cell W8 and the variance [formulaDVARP(C13:C72)] is in cell W10. The formula for the relative VaR uses the asset value for UBS (from cell W6), the UBS variance (from cell W10), in conjunction with the selected time interval (one month from cell W15) and the number of standard deviations or ‘z-value’ (in cell W19).
The relative VaR estimate (in cell W21) can be interpreted as saying that there is only a 5% probability that a holding of UBS shares with a current value of 1000 will lose 105.59 or more in value over the course of the next month. The relative VaR assumes that the asset has an expected return over the period that is small enough to be ignored (as will typically be the case when considering daily VaRs).
The absolute VaR incorporates the expected return on the asset (here 1.21% over the month for UBS from the calculation of M1 in cell W12) and subtracts this typi- cally positive amount for long positions from the relative VaR. The expected return (M1 in cell W12) uses the formula linking the moments of lognormal and normal returns explained in section 7.7. In our example the absolute VaR of 93.50 in cell W24 is thus smaller than the relative VaR. The share with the lowest VaR estimates of the group is Roche, which of course given our simple formula will have had the smallest variance.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
V W X Y Z AA AB AC AD
Estimating VaR using Lognormal Distribution
UBS CS Zurich Ins Winterthur Roche Sandoz Ciba-Geigy Swissair Asset value (S) 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 Return (M) lnS 0.0100 0.0047 0.0007 -0.0036 0.0200 0.0071 0.0002 -0.0001
Variance (V) 0.0041 0.0056 0.0050 0.0053 0.0034 0.0041 0.0044 0.0073
Exp ret (M1) S 1.21% 0.75% 0.32% -0.09% 2.20% 0.92% 0.24% 0.36%
Time months (δt) 1 Conf level (c) 95.00%
’z-value’ 1.64
VaR Asset Rel 105.59 122.97 115.83 119.77 96.29 104.80 108.80 140.68
via fn 105.59 122.97 115.83 119.77 96.29 104.80 108.80 140.68
VaR Asset Abs 93.50 115.43 112.63 120.68 74.31 95.64 106.40 137.10
via fn 93.50 115.43 112.63 120.68 74.31 95.64 106.40 137.10
Estimating Portfolio VaR using Lognormal Distribution Var Port Rel 702.70
via fn 702.70
Var Port Abs 658.67
via fn 658.67
Figure 7.7 Calculations for Value-at-Risk for the eight Swiss shares in sheet VCV
It is straightforward to estimate the VaR for a portfolio (here assumed to have equal amounts invested in the eight shares) from the VaRs for the individual shares, in conjunc- tion with the correlation matrix of the individual returns. The relative VaR for the portfolio in cell W30 is 702.70, compared to the sum of the individual VaRs totalling 914.73, thus demonstrating the diversification benefits of holding assets in portfolios. As indicated in Figure 7.7, we provide user-defined functions to calculate the various VaR measures.