Using VaR

VaR, as a measure of portfolio risk, has many benefits but suffers from the same limitations as many other forward looking estimates of portfolio risk.

Advantages of VaR

• The concept of VaR is simple and easy to explain, although the details of the methodology can be complex. VaR allows the risk of different portfolios, asset classes, or trading operations to be compared to gain a sense of relative riskiness. VaR can be used for performance evaluation (i.e., returns generated vs. risk taken).
• Rather than evaluating a trading group’s performance based only on returns, VaR allows calculation of the ratio of trading income to VaR. When allocating capital to various trading units, a firm’s risk managers can also look at the allocation of VaR and optimize the allocation of capital given the firm’s determination of the maximum VaR that the organization should be exposed to (sometimes referred to as risk budgeting). In the same manner, managers can estimate risk adjusted performance of trading units or profits per dollar of VaR.
• For example, the equity trading desk may be assigned a maximum daily VaR of $5 million while the more profitable currency trading desk may be assigned a daily VaR of $15 million. Global banking regulators accept VaR as a measure of financial risk, although they do not prescribe estimation methods or impose a maximum VaR. Reliability of VaR as a measure of risk can be verified by backtesting.

Limitations of VaR

• VaR estimation requires many choices (loss percentage, lookback period, distribution assumptions, and parameter estimates) and can be very significantly affected by these choices. An unscrupulous analyst can choose assumptions that lead to a low estimate of VaR. The assumption of normality leads to underestimates of downside (tail) risk because actual returns distributions frequently have “fatter tails” than a normal distribution.
• When this is the case, VaRs based on an assumption of normality tend to underestimate the probability of extreme outcomes. Although the assumption of normality is not a requirement of VaR, it is almost always used, especially with the parametric method.
• Liquidity often falls significantly when asset prices fall. A VaR which does not account for this will understate the actual losses incurred when liquidating positions that are under extreme price pressure. It is well known that correlations increase, or spike, during periods of financial stress.
• Increasing correlations mean that VaR measures based on normal levels of correlation will overestimate diversification benefits and underestimate the magnitude of potential losses. While VaR is a single number that can be used to quantify risk, as with any summary measure, many aspects of risk are not quantified or included. Users of VaR must understand the limitations of VaR as a measure of risk in order to use it appropriately.
• VaR focuses only on downside risk and extreme negative outcomes. Including consideration of right hand tail values will give a better understanding of the risk- return trade off.

Another measure based on VaR is the conditional VaR (CVaR). The CVaR is the expected loss, given that the loss is equal to or greater than the VaR. For this reason, the CVaR is also referred to as the expected tail loss or expected shortfall.

The CVaR is expected loss given that the loss is in the left hand tail past the VaR.

When the VaR is estimated using the historical simulation method or Monte Carlo simulation, we have all the losses greater than the VaR loss, so it is straightforward to take the average of these to get the CVaR. With the parametric method, we don’t know the magnitude of losses greater than the VaR, so calculating the expected loss in the left hand tail is mathematically complex.

Incremental VaR (IVaR) is the change in VaR from a change in the portfolio allocation to a security. If a 2% increase in the weight of a security in the portfolio increases the portfolio’s VaR from $1,345,600 to $1,562,400, the IVaR for the 2% increase in the portfolio weight of the security is 1,562,400 − 1,345,600 = $216,800.

A related measure is the marginal VaR (MVaR). The MVaR is estimated as the slope of a curve that plots VaR as a function of a security’s weight in the portfolio. The MVaR is calculated at the point on the curve corresponding to the security’s current weight, so we can interpret it as the change in VaR for a 1% increase in the security’s weight.

This is not precisely correct because the MVaR is the slope at a point on the curve, not the slope for a 1% change in weight. It is, however, a reasonable approximation of the sensitivity of VaR to a 1% change in weight of a security. Thus, both the MVaR and IVaR can be used to estimate the change in VaR that will result from a change in the weight of a single security.

Ex ante tracking error, also referred to as relative VaR, measures the VaR of the difference between the return on a portfolio and the return on its manager’s benchmark portfolio. A 5% monthly relative VaR of 2.5% implies that 5% of time, the portfolio’s relative underperformance will be at least 2.5%. The relative VaR can be calculated as the VaR of a combination of a long position in the subject portfolio and a short position in the benchmark portfolio.

Given the limitations of VaR as a risk measure, analysts should use other risk measures that complement VaR.

Risk assessment using sensitivity analysis focuses on the effect on portfolio value given a small change in one risk factor. By examining the sensitivity of a portfolio’s value to several risk factors, portfolio risk can be better understood and more effectively managed. Sensitivity analysis complements VaR in understanding portfolio risk, but, unlike VaR, it does not involve any prediction of the probability of losses of any specific amount.

While sensitivity analysis provides an estimate of the change in portfolio value due to a small change in a single risk factor, scenario analysis provides an estimate of the effect on portfolio value of a set of changes of significant magnitude in multiple risk factors. The changes in risk factors used in scenario analysis are often a set of changes that are expected to result in a significant decline in portfolio value, although a scenario of changes in risk factors that would increase portfolio value may also be considered.

A historical scenario approach uses a set of changes in risk factors that have actually occurred in the past, especially changes during a period of financial disruption and stress such as the subprime mortgage crisis of 2008 or the equity market crash of 1987.

With a hypothetical scenario approach, any set of changes in risk factors can be used, not just one that has happened in the past. A hypothetical scenario could have more extreme changes in risk factors than those that have occurred in the past, but that have some non zero probability of occurring in the future.

Stress tests examine the effect on value (or solvency) of a scenario of extreme risk- factor changes.

The risk factors used to measure the risks of equities, fixed income securities, and options are all different. For equities, the most often used risk factor is beta. Beta is a measure of how the returns of a security or a portfolio are expected to be affected by overall market returns.

The capital asset pricing model (CAPM) is based on market risk as measured by beta and concludes that the expected return on an asset is equal to the risk free rate plus beta times the market risk premium:

E(Ri) = Rf + Betai [E(RMKT) − Rf]

For fixed income securities and portfolios, duration provides an estimate of how market values are affected by changes in interest rates (yields to maturity). For larger changes in interest rates, including the effects of convexity on fixed income security values improves estimates of the sensitivity of the values of fixed income securities to changes in interest rates. Together, duration and convexity are used to estimate the sensitivity of the values of fixed income securities (and portfolios) to changes in interest rates.

An estimate of the percentage change in value of a fixed income security or portfolio in response to a change in YTM (∆Y) is given by:

change in price = −duration (∆Y) + 1⁄2 convexity (∆Y)2

Several risk factors affect the values of options positions. Delta is an estimate of the sensitivity of options values to changes in the value of the underlying asset. Delta is the ratio of the change in an option’s value to a change in the price of the underlying security.

A call delta of 0.6 means that for every $1 increase in the price of the underlying asset, the call value increases by $0.60. A put delta of −0.5 means that for every $1 increase the value of the underlying asset, the put value will decrease by $0.50.

Gamma is an estimate of how delta changes as the price of the underlying asset changes and is calculated as the ratio of the change in delta to a change in the price of the underlying asset. Just as convexity improves estimates of the impact of interest rate changes captured by delta, gamma improves estimates of the impact of a change in the price of the underlying asset on option values. Both convexity and gamma are considered second order effects, while duration and delta measure first order effects of risk factor changes.

Vega is a measure of the sensitivity of option values to changes in the expected volatility of the price of the underlying asset. We can incorporate all three of these option risk measures in the following equation:

where ∆S is the change in the price of the underlying asset and ∆V is the change in future volatility.