J. P. Morgan developed the RiskMetrics methodology to VaR calculation; see Longerstaey and More (1995). In its simple form, RiskMetrics assumes that the continuously compounded daily return of a portfolio follows a conditional normal distribution. Denote the daily log return by rt and the information set available at time t − 1 by Ft−1. RiskMetrics assumes that 
, where μt is the conditional mean and 
is the conditional variance of rt. In addition, the method assumes that the two quantities evolve over time according to the simple model:
Therefore, the method assumes that the logarithm of the daily price, pt = ln(Pt), of the portfolio satisfies the difference equation pt − pt−1 = at, where at = σtϵt is an IGARCH(1,1) process without drift. The value of α is often in the interval (0.9, 1) with a typical value of 0.94.
A nice property of such a special random-walk IGARCH model is that the conditional distribution of a multiperiod return is easily available. Specifically, for a k-period horizon, the log return from time t + 1 to time t + k (inclusive) is rt[k] = rt+1 + ⋯ + rt+k−1 + rt+k. We use the square bracket [k] to denote a k-horizon return. Under the special IGARCH(1,1) model in Eq. (7.2), the conditional distribution rt[k]|Ft is normal with mean zero and variance 
, where 
can be computed using the forecasting method discussed in Chapter 3. Using the independence assumption of ϵt and model (7.2), we have
where Var(
can be obtained recursively. Using rt−1 = at−1 = σt−1ϵt−1, we can rewrite the volatility equation of the IGARCH(1,1) model in Eq. (7.2) as
In particular, we have
Since 
for i ≥ 2, the prior equation shows that
For the 1-step-ahead volatility forecast, Eq. (7.2) shows that 
. Therefore, Eq. (7.3) shows that Var(
for i ≥ 1 and, hence, 
. The results show that rt[k]|Ft 
. Consequently, under the special IGARCH(1,1) model in Eq. (7.2) the conditional variance of rt[k] is proportional to the time horizon k. The conditional standard deviation of a k-period horizon log return is then 
, which is 
times σt+1.
Given a tail probability, RiskMetrics uses the result 
to calculate VaR for the log return. If the tail probability is set to 5%, then VaR = 1.65σt+1 for the next trading day. This is the upper 5% quantile (or the 95th percentile) of a normal distribution with mean zero and standard deviation σt+1. For the next k trading days, 
, which is the 95th percentile of 
. Similarly, if the tail probability is 1%, then VaR = 2.326σt+1 for the next trading day and 
for the next k trading days.
Consider the case of 1% tail probability. The VaR for the portfolio under RiskMetrics is then
for the next trading day and that of a k-day horizon is
where the argument (k) of VaR is used to denote the time horizon and the portfolio value is measured in dollars. Consequently, under RiskMetrics, we have
This is referred to as the square root of time rule in VaR calculation under RiskMetrics.
If the log returns are in percentages, then the 1% VaR for the next trading day is VaR = Amount of position × 2.326σt+1/100, where σt+1 is the volatility of the percentage log returns.
Note that because RiskMetrics assumes log returns are normally distributed with mean zero, the loss function is symmetric and VaR are the same for long and short financial positions.
Example 7.1
The sample standard deviation of the continuously compounded daily return of the German mark/U.S. dollar exchange rate was about 0.53% in June 1997. Suppose that an investor was long in $10 million worth of mark/dollar exchange rate contract. Then the 5% VaR for a 1-day horizon of the investor is
The corresponding VaR for 10-day horizon is
Example 7.2
Consider the daily IBM log returns of Figure 7.1. As mentioned in Chapter 1, the sample mean of the returns is significantly different from zero. However, for demonstration of VaR calculation using RiskMetrics, we assume in this example that the conditional mean is zero and the volatility of the returns follows an IGARCH(1,1) model without drift. The fitted model is
7.4 
where {ϵt} is a standard Gaussian white noise series. As expected, this model is rejected by the Q statistics. For instance, we have a highly significant statistic Q(10) = 56.19 for the squared standardized residuals.
From the data and the fitted model, we have r9190 = − 0.0128 and 
. Therefore, the 1-step-ahead volatility forecast is 
. The 95% quantile of the conditional distribution r9191|F9190 is 
. Consequently, the 1-day horizon 5% VaR of a long position of $10 millions is
The 99% quantile is 
, and the corresponding 1% VaR for the same long position is $426, 500.
Remark
To implement RiskMetrics in S-Plus, one can use ewma1 (exponentially weighted moving average of order 1) under the mgarch (multivariate GARCH) command to obtain the estimate of 1 − α. Then, use the command predict to obtain volatility forecasts. For the IBM data used, the estimate of α is 1 − 0.036 = 0.964 and the 1-step-ahead volatility forecast is 
. Please see the demonstration below. This leads to VaR = $10, 000, 000 × (1.65 × 0.01888) = $311, 520 and VaR = $439, 187 for p = 0.05 and 0.01, respectively. These two values are slightly higher than those of Example 7.2, which are based on estimates of the RATS package. □
7.2.1.1 S-Plus Demonstration
The following output has been simplified:
> ibm.risk=mgarch(ibm˜−1, ˜ewma1)
> ibm.risk
ALPHA 0.036
> predict(ibm.risk,2)
$sigma.pred 0.01888
7.2.2 Discussion
An advantage of RiskMetrics is simplicity. It is easy to understand and apply. Another advantage is that it makes risk more transparent in the financial markets. However, as security returns tend to have heavy tails (or fat tails), the normality assumption used often results in underestimation of VaR. Other approaches to VaR calculation avoid making such an assumption.
The square root of time rule is a consequence of the special model used by RiskMetrics. If either the zero mean assumption or the special IGARCH(1,1) model assumption of the log returns fails, then the rule is invalid. Consider the simple model
where {ϵt} is a standard Gaussian white noise series. The assumption that μ ≠ 0 holds for returns of many heavily traded stocks on the NYSE; see Chapter 1. For this simple model, the distribution of rt+1 given Ft is 
. The 95% quantile used to calculate the 1-period horizon VaR becomes μ + 1.65σt+1. For a k-period horizon, the distribution of rt[k] given Ft is 
, where as before rt[k] = rt+1 + ⋯ + rt+k. The 95% quantile used in the k-period horizon VaR calculation is 
. Consequently, 
when the mean return is not zero. It is also easy to show that the rule fails when the volatility model of the return is not an IGARCH(1,1) model without drift.
7.2.3 Multiple Positions
In some applications, an investor may hold multiple positions and needs to compute the overall VaR of the positions. RiskMetrics adopts a simple approach for doing such a calculation under the assumption that daily log returns of each position follow a random-walk IGARCH(1,1) model. The additional quantities needed are the cross-correlation coefficients between the returns. Consider the case of two positions. Let VaR1 and VaR2 be the VaR for the two positions and ρ12 be the cross-correlation coefficient between the two returns—that is, ρ12 = Cov(r1t, r2t)/[Var(r1t)Var(r2t)]0.5. Then the overall VaR of the investor is
The generalization of VaR to a position consisting of m instruments is straightforward as
where ρij is the cross-correlation coefficient between returns of the ith and jth instruments and VaRi is the VaR of the ith instrument.
The prior formula is obtained using the assumption that the joint distribution of the log returns of assets involved in the portfolio is multivariate normal with mean zero and covariance matrix 
. Under such an assumption, the log return of the portfolio is normal with mean zero and finite variance; see Appendix B of Chapter 8 for properties of multivariate normal variables.
7.2.4 Expected Shortfall
Given a tail probability p, VaR is simply the (1 − p)th quantile of the loss function. In practice, the actual loss, if it occurs, can be greater than VaR. In this sense, VaR may underestimate the actual loss. To have a better assessment of the potential loss, one can consider the expected value of the loss function if the VaR is exceeded. This consideration leads to the concept of expected shortfall (ES). Under RiskMetrics, the loss function is normally distributed so that the conditional distribution of the loss function given that a VaR is exceeded is a truncated (from below) normal distribution. Properties such as mean and variance of a truncated normal distribution have been well-studied in the statistical literature. We can use the mean of the distribution to calculate expected shortfall. Specifically, consider the standard normal distribution X ∼ N(0, 1). For a given upper tail probability p, let q = 1 − p and VaRq be the associated VaR, that is, VaRq is the qth quantile of X. Then the expectation of X given X > VaRq is E(X|X > VaRq) = f(VaRq)/p, where 
is the pdf of X. The expected shortfall for a log return rt with conditional distribution 
is then
For example, if p = 0.05, then VaR0.95 ≈ 1.645 and f(VaRq)/p = f(1.645)/0.05 = 2.0627 so that the expected shortfall under RiskMetrics is ES0.95 = 2.0627σt. If p = 0.01, then ES0.99 = 2.6652σt.