Aug 01

Value at Risk (VaR): 2007 FRM. Part 2

by David Harper, CFA, FRM, CIPM

2007 FRM Learning Outcomes

7.2 â€¦ calculate VaR for a single asset on both a dollar and percentage basis
7.3 Convert a daily VaR measure into a weekly, monthly, or annual VaR
7.5 Explain why it is best to use continuously compounded rates of return when calculating VaR.
7.6 Calculate portfolio VaR and describe the primary factors that affect portfolio risk.

Value at Risk (VaR) for a single asset (dollar and percentage)

Please note: these initial learning outcomes refer to conventional parametric value at risk (VaR) where volatility is the parameter. (There are other ways to approach VaR. As Jorion says, "the conventional VaR measure is the quantile of the distribution measured in dollars. This single number is a convenient summary, but its very simplicity may be dangerous.)

Percentage VaR for a single asset is simply the critical z-value (sometimes called the reliability factor) multiplied by volatility (standard deviation):

If the volatility is 1% and the critical value is 1.645, then the percentage VaR is 1.645%:

We typically select confidence levels of 95% and 99% (as one-tailed confidence levels, their critical values correspond to two-tailed confidence intervals of, respectively, 97.5% and 99.5%), such that the corresponding critical values are 1.645 and 2.33:

Now if our portfolio value given by W, we simply multiply the above formula by W to get the dollar VaR (technically, the relative dollar VaR). So, the dollar VaR is simply the product of the portfolio value, the critical value (that corresponds to a desired confidence level), and the volatility:

To continue the example, if the portfolio value is $100, the dollar VaR is $1.645:

Convert Daily VaR into weekly, monthly, or annual VaR

To convert daily VaR, we employ the "square root of time rule:" volatility scales with the square root of time (Why? Because variances scales with time). To convert daily VaR then, we do the following (the annual conversion is based on 250 trading days).

But you are better to recognize the generic format, which is the following:

To use this, for example, consider you want to scale a 5-day VaR into a 10-day VaR. In this case, you multiply the 5-day VaR by SQRT(10/5) = SQRT (2) = about 1.4. You are scaling by the proportional change in time. So, we could view the weekly VaR conversion above as: Weekly VaR = Daily VaR x SQRT(5/1). The one (1) is implicitly in the denominator!

Note: keep in mind the square-root-of-time rule assumes, and requires, that returns are i.i.d. If returns are not independent, the rule will over- or -understate true VaR.

Why prefer continuously compounded?

We have three choices for compounding periodic returns:

Absolute change (today's price â€“ yesterday's price): violates the stationarity requirement.
Simple change ([today price â€“ yesterday's price] Â¸ [yesterday's price]): satisfies stationarity requirement, but does not comply with time consistency requirement.
Continuous compounded return is best because it satisfies the time consistency requirement: the two-period return is the sum of two single period returns. The sum of two random variables that are jointly distributed is itself (i.e., the sum) normally distributed.

As Allen says, continuous compounding is best because it satisfied time consistency.

Portfolio VaR

We start with a simple two-asset portfolio. Here, we use the same VaR as above but we use portfolio portfolio volatility. Portfolio volatility is the square root of portfolio variance: