Risk Measures

VaR, Marginal VaR, and IVaR

Value-at-Risk (VaR) is one of the most important and widely used statistics that measure the potential risk of economic losses. VaR answers the question: What is the minimum amount that the company can expect to lose with a certain probability over a given horizon? In mathematical terms, VaR corresponds to a percentile of the distribution of portfolio P&L. For a given time horizon, the 100α% VaR, denoted VaR(α ) , is the size of loss that will be exceeded with probability (1 - α ). Suppose that the loss incurred by a portfolio during the specified period is given by the random variable L, having some (unknown) cumulative distribution function (cdf) F, so that Prob(L = y) = F ( y) . The portfolio’s 100α% VaR equals the αth population quantile of L, that is, VaR(α ) = F -1 (α ) .

The Marginal VaR of a position (business line, etc.) with respect to a portfolio (company) can be thought of as the amount of risk that the position is adding to the portfolio. In other words, Marginal VaR tells us how the VaR of our portfolio would change if we sold (ran off) or added a specific position. Marginal VaR can be formally defined as the difference between the VaR of the total portfolio and the VaR of the portfolio without the position. It can be easily shown that Marginal VaR is an increasing function of the correlation ρ between the position and the portfolio. When the VaR of the position is much smaller than the VaR of the portfolio, Marginal VaR will be positive when ρ > 0 , and negative when ρ < 0 .

Marginal VaR can be used to compute the amount of risk added by an entire position to the total risk of the portfolio. It is an appropriate risk measure in the context of run-off and acquisition decisions. However, we are also interested in the potential effect that buying or selling a relatively small portion of a position would have on the overall risk. For example, in the process of rebalancing a portfolio, we often wish to decrease our holdings by a small amount rather than liquidate the entire position. Since Marginal VaR can only consider the effect of selling the whole position, it would be an inappropriate measure of risk contribution for this example.

Incremental VaR (IVaR) is a statistic that provides information regarding the sensitivity of VaR to changes in the portfolio holdings. If we denote by IVaR i the Incremental VaR for the ith position in the portfolio, and by θi the percentage change in size of that position, we can approximate the change in VaR by

Δ VaR = Σ_i θ_i IVaR_i

An important difference between IVaR and Marginal VaR is that the IVaRs of the positions add up to the total VaR of the portfolio:

Σ_i IVaR_i = VaR

This additive property of IVaR has important applications in the allocation of risk to different units (sectors, countries), where the goal is to keep the sum of the risks equal to the total risk.

For a practical calculation of IVaR, we need a more rigorous definition. Let w_i be the amount of money invested in ith position. We define the Incremental VaR of position i as

IVaR_i = w_i (δ VaR / δ w _i)

To verify the additive property of IVaR we need to note that VaR is a homogeneous function of order one of the total amount invested. This means that if we double the investments on each position, the VaR of the new portfolio will be twice as large. That is,

VaR(tw1 , tw2 ,..., twn ) = t VaR(w1 , w2 ,..., wn ).

Then, by Euler’s homogeneous function theorem we have that

VaR = Σ _i w_i (δ VaR / δ w _i) ≡ Σ _i IVaR_i