Quantile-based risk measures: value at risk

Given the limitations of standard deviation and variance as risk measures, alternative ones have been proposed. To be specific, we will refer once more to a financial investment problem, where risk is related to portfolio loss. The most widely known such measure is value at risk [VaR; not to be confused with variance (Var)]. The VaR concept was introduced as an easy-to-understand measure of portfolio risk. In fact, measuring, monitoring, and managing risk are fundamental activities for any portfolio manager, but another fundamental side of the coin is the possibility of communicating about risk with top management, through a single number making sense for a broad class of assets. Bonds and stocks involve different forms of risk and derivatives, if used for speculation, may be even riskier. Basically, VaR aims at measuring the maximum portfolio loss one could suffer, over a given time horizon, within a given confidence level. Technically speaking, it is a quantile of the probability distribution of loss. Let L_T be a random variable representing the loss, in absolute value, of a portfolio over a holding period of length T. Then, VaR at confidence level 1 − α can be defined as the smallest number VaR_1−α such that

This definition is consistent with Definition 7.2 of a quantile of a discrete probability distribution. If L_T is a continuous random variable and its CDF is invertible, we may rewrite Eq. (13.14) as

In the following discussion, unless otherwise noted, we will assume for simplicity that loss is a continuous random variable. For instance, if we set α = 0.05, we obtain VaR at 95%. The probability that the loss exceeds VaR is α.

Actually, there are different definitions of VaR, which are best clarified by relating the distribution of loss to the distribution of return. Let W₀ be the initial portfolio wealth. If R_T is the random return over the holding period, the future wealth is

A loss occurs when the wealth increment

turns out to be negative. So, the absolute loss over the holding period is

Let us assume that the holding period return has a continuous distribution with density f_RT(r). Equation (13.15) implies that loss will exceed VaR with a low probability:

where we may indifferently write “≥” or “>,” since the distribution involved is continuous. This can be rewritten as follows:

where we have defined

The return r_α will be negative in most practical cases and is just the quantile at level α of the distribution of portfolio return:

The quantile r_α is obviously associated with a critical wealth ω_α, which is the wealth we end up with if our loss is exactly the VaR:

We may interpret r_α as the worst-case return with confidence level α; if α = 0.05, return will be worse than r_α only in 5% of the cases. By the same token, we will end up with a wealth lower than ω_α only with probability α.

Now we may define an absolute VaR as

Note once again that the critical return r_α is usually negative and VaR (absolute value of loss) is positive. We may also define a relative VaR, where the reference value to define loss is the expected value of future wealth. Let us denote the expected holding period return by μ = E[R_T]. Then

Relative VaR is defined as

The definitions in Eqs. (13.17) and (13.18) may yield approximately the same VaR over a short time horizon, say, a few days. In this case volatility dominates drift¹³ and E[W_T] ≈ W₀. This assumption is not unreasonable, as bank regulations require the use of a risk measure in order to set aside enough cash to be able to cover short-term losses.

Computing VaR is easy if we assume that return or, equivalently, loss are normally distributed. Let us assume that the holding period return R_T is normally distributed and that

Then, there is no difference between absolute and relative VaR. Since loss is L_T = −W₀R_T, we obtain . We should not overlook the fact that we are taking advantage of the symmetry of the normal distribution of return with respect to its expected value, which is 0. Hence, the critical return r_α is, in absolute value, equal to the quantile r_1−α. Then, to compute VaR_1−α, we may use the familiar standardization/destandardization drill for normal variables:

where Z is a standard normal variable. We have just to find the standard quantile z_1−α and set

Example 13.9 You have invested $100,000 in Quacko Corporation stock shares. Daily volatility is 2%. VaR at 95% level is

We are “95% sure” that we will not lose more than $3,289.71 in one day. VaR at 99% level is

Clearly, increasing the confidence level by 4% has a significant effect, since we are working on the tail of the distribution.

A commonly proposed scaling procedure allows us to compute VaR on multiple time periods. If we assume that daily returns are a stream of i.i.d. normal variables with standard deviation σ, over a time period spanning T days, the application of the square-root rule yields

Needless to say, this simple-minded approach hides a few dangers. To begin with, when we consider longer time periods, expected return does play a role, and we should clarify whether we are interested in the relative or absolute VaR. Furthermore, we are assuming that returns on consecutive days can be just summed, disregarding compounding effects; this allows us to consider return over T days as the sum of T normal variables, which is normal again. Last but not least, we are assuming that there is no correlation between the returns on consecutive days.¹⁴ The very assumption of normality of returns can be dangerous, as the normal distribution has a relatively low kurtosis; alternative distributions have been proposed, featuring fatter tails, in order to better account for tail-risk, which is what we are concerned about in risk management. Nevertheless, the calculation based on the normal distribution is so simple and appealing that it is tempting to use it even when we should rely on more realistic models. In practice, we are not interested in VaR for a single asset, but in VaR for the whole portfolio. Again, a normality assumption streamlines our task considerably.

Example 13.10 Suppose that we hold a portfolio of two assets. The portfolio weights are  and , respectively. We also assume that the returns of the two assets have a jointly normal distribution; the two daily volatilities are σ₁ = 2% and σ₂ = 1% respectively, and the correlation is ρ = 0.7. Let the time horizon be T = 10 days; despite this, we assume again that expected holding period return is zero. To obtain portfolio risk, we first compute the variance of the holding period return:

Hence, σ_p = 0.05011. If the overall portfolio value is $10 million, and the required confidence level is 99%, we obtain

Once again, we stress that the calculations in Example 13.10 are quite simple (maybe too simple) since they rely on a few rather critical assumptions. Alternative ways of estimating VaR have been proposed:

• Monte Carlo simulation. In the previous examples, we have taken advantage of the analytical tractability of the normal distribution, and the fact the return of stock shares was the only risk factor involved. Other risk factors may be involved, such as inflation and interest rates, and the portfolio can include derivatives, whose value is a complicated function of underlying asset values. Even if we assume that the underlying risk factors are normally distributed, the portfolio value may be a nonlinear function of them, and the analytical tractability of the normal distribution is lost. In this case, we may resort to Monte Carlo simulation,¹⁵ which is a remarkably flexible tool, even though not necessarily the most efficient one.Fig. 13.8 Value at risk can be the same in quite different situations.
• Historical VaR. What we have illustrated so far is a parametric approach, since it relies on a theoretical probability distribution, not necessarily normal. One advantage of the normal distribution is that it simplifies the task of characterizing the joint distribution of returns, since we need only a correlation matrix. However, we know that correlations need not capture dependence between random variables. Alternative distributions can be used, possibly requiring sophisticated numerical methods or Monte Carlo simulation, but it is generally difficult to capture dependence. Rather than assuming a specific joint distribution, we may rely on a nonparametric approach based on historical data. The advantage of historical data is that they should naturally capture dependence. Hence, we may combine them, according to bootstrapping procedures, to generate future scenarios and estimate VaR by historical simulation.

Whatever approach we use to compute VaR, it is not free from some fundamental flaws, which depend on its definition as a quantile, and we should be well aware of them. For instance, a quantile cannot distinguish between different tail shapes. Consider the two densities in Fig. 13.8. In Fig. 13.8(a) we see a normal loss distribution and its 95% VaR, which is just its quantile at probability level 95%; the area of the right tail is 5%. In Fig. 13.8(b) we see a sort of truncated distribution obtained by appending a uniform tail to a normal density, which accounts for 5% of the total probability. By construction, VaR is the same in both cases, since the areas of the right tails are identical. However, we should not associate the same measure of risk with the two distributions. In the case of the normal distribution there is no upper bound to loss; in the second case, there is a clearly defined worst-case loss. Hence, the risk for density (a) should be larger than with density (b), but VaR does not indicate any difference between them. In order to discriminate the two cases, we may consider the expected value of loss conditional on being on the right (unfortunate) tail of the loss distribution. This conditional expectation yields the midpoint of the uniform tail in the truncated density; conditional expected value is larger in the normal case, because of its unbounded support. This observation has led to the definition of alternative risk measures, such as conditional value at risk (CVaR), which is the expected value of loss, conditional on being to the right of VaR.

Risk measures like VaR or CVaR could also be used in portfolio optimization, by solving mathematical programs with the same structure as problem (13.13), where variance is replaced by such measures. The resulting optimization problem can be rather difficult. In particular, it may lack the convexity properties that are so important in optimization. It turns out that minimizing VaR, when uncertainty is modeled by a finite set of scenarios (which may be useful to capture complex distributions and dependencies among asset prices), is a nasty nonconvex problem, whereas minimizing CVaR is (numerically) easier as it yields a convex optimization problem.¹⁶

There is one last issue with VaR that deserves mention. Intuitively, risk is reduced by diversification. This should be reflected by any risk measure ρ(·) we consider. A little more formally, we should require a subadditivity condition like

where A and B are two portfolio positions. The following counterexample is often used to show that VaR lacks this property.

Example 13.11 (VaR is not subadditive) Let us consider two corporate bonds, A and B, whose issuers may default with probability 4%. Say that, in the case of default, we lose the full face value, $100 (in practice, we might partially recover the face value of the bond). Let us compute the VaR of each bond with confidence level 95%. Since loss has a discrete distribution in this example, we should use the more general definition of VaR provided by (13.14). The probability of default probability is 4%, and 1 − 0.04 = 0.96 > 0.95; therefore, we find

Now what happens if we hold both bonds and assume independent defaults? We will suffer:

• A loss of $0, with probability 0.96² = 0.9216
• A loss of $100, with probability 2 × 0.96 × 0.04 = 0.0768
• A loss of $200, with probability 0.04² = 0.0016

Note that the probability of losing $0 is smaller than 95%, but

Hence, with that confidence level, VaR(A + B) = 100 > VaR(A) + VaR(B), which means that diversification increases risk, if we measure it by VaR.

Subadditivity is one of the properties that any sensible risk measure should enjoy. The term coherent risk measure has been introduced to specify a risk measure that meets a set of sensible requirements. VaR is not a coherent risk measure, whereas it can be shown that CVaR is.