Value At Risk (VaR)
What you'll learn
- LOS 38.aExplain the use of value at risk (VaR) in measuring portfolio risk.
- LOS 38.bCompare the parametric (variance−covariance), historical simulation, and Monte Carlo simulation methods for estimating VaR.
- LOS 38.cEstimate and interpret VaR under the parametric, historical simulation, and Monte Carlo simulation methods.
Value at risk (VaR) measures downside risk of a portfolio. It has three components: the loss size, the probability (of a loss greater than or equal to the specified loss size), and a time frame. Consider the statement: “There is a 5% probability that the company will experience a loss of $25,000 or more in any given month.”
This is the same as stating that the monthly 5% VaR is $25,000. In the previous statement, the probability is 5%, the loss size is $25,000, and the time frame is one month.
Note that $25,000 is a minimum loss amount, so we can state, “5% of the time the minimum monthly loss that the company will experience is $25,000.”
VaR can also be expressed in percentage terms so that for a portfolio, we could state that the 5% monthly VaR is 3%, meaning that 5% of the time the monthly portfolio value will fall by at least 3%. We can also state VaR as a confidence level: we are 95% (i.e., 100%−5%) confident that the portfolio will experience a loss of no more than 3%.
To estimate a VaR, we must specify the time period and the size of the loss, so there is significant judgment involved in VaR estimation. If we choose the size of the loss, we will estimate the probability of losses of that size or larger; but, if we choose the probability of the loss, we will estimate the minimum size of the losses that will occur with that probability.
Figure 38.1 shows the 5% VaR for a given probability distribution of monthly returns. The 5% left hand tail of the distribution of possible monthly outcomes is bounded by the
5% VaR; VaR is the upper limit of the specified left tail.
Figure 38.1: Distribution of Monthly Returns
While any probability can be specified, VaR is typically expressed for 1%, 5%, or 16% (one standard deviation below the mean for a normal distribution) probability. The time frame specified also varies; we could estimate VaR for a day, a week, a month, or any other relevant period.
The first step in estimating the VaR for a portfolio is to identify the risk factors that enter into the determination of portfolio returns. These risk factors might include market risk, interest rate risk, or currency risk, among others.
One method of estimating VaR is the parametric or variance covariance method. Often we assume that the risk factors are distributed normally, but we could also assume other distributions. Assuming normality allows us to estimate the risk of the portfolio based only on the means, variances, and covariances (or correlations) of the various risk factors.
An assumption that risk factor probabilities are non normal would increase the complexity of the analysis and require that we estimate values for other parameters, such as skewness and kurtosis.
Assuming normality, we can use the portfolio variance formula to estimate the mean and variance of portfolio returns. Once we have estimated these parameters, we can identify portfolio VaR as the value bounding the left hand tail of the distribution, as we illustrated in Figure 38.1. To simplify the explanation of the parametric method, we will consider a case of only two risk factors, both of which are normally distributed.
Consider two securities, Asset A and Asset B. For a portfolio with portfolio fraction WA invested in Asset A and the remaining portfolio fraction WB invested in Asset B, portfolio variance is given by the formula:
The returns period that we use to estimate the mean and standard deviation of returns for each risk factor (each fund in our example) is called the lookback period. For estimating the variance of daily returns, we might use the last two years; but, for estimating the variance of annual returns, we would choose a longer lookback period. The important point is that the parameter estimates we use should be those we expect over the period for which we are estimating the VaR.
Estimates based on recent periods may be adjusted towards longer term averages.
- Imagine that we are provided the following information about two assets, Security A and Security B:
- How would we use this information to estimate the 5% annual VaR for a portfolio that is 60% invested in Security A and 40% invested in Security B?
- Answer:
Mean daily portfolio return = 0.6(0.0004) + 0.4(0.0003) = 0.00036
Variance of portfolio return = (0.6)2(0.0158)2 + (0.4)2(0.0112)2 + 2(0.4)(0.6) (0.000106) = 0.000161
Standard deviation of portfolio returns
- For a 5% VaR we want 5% in the left hand tail, so we calculate the value 1.65 standard deviations below the mean:
5% daily VaR = [0.00036 − 1.65(0.012682)] × (−1) = 0.0206
- Assuming the distribution of daily returns is constant over the year, that there are 250 trading days in one year, and that daily returns are independently distributed, we can calculate the annual mean return as 250(0.00036) = 0.09.
- The annual standard deviation can be calculated as
(0.012682) = 0.20052.
- Based on these estimates, the 5% annual VaR = [0.09 − 1.65(0.20052)] × (−1) = 0.2409.
- For a portfolio with a value of $10 million, the 5% daily and annual VaR are:
10 million (0.0206) = $206,000 and
10 million (0.2409) = $2,409,000.
The parametric method is relatively simple to apply under the assumption of normally distributed returns. Of course, its estimates will only be as good as the estimates of future mean returns and standard deviations. The calculated VaR is also very sensitive to the covariance estimate.
The length of the lookback period will affect the parameter estimates, and care must be taken to adjust estimates based on recent results when they may not reflect the future distribution of returns. In cases where normality cannot be reasonably assumed, such as when the portfolio contains options, the parametric method has limited usefulness.
The historical simulation method of estimating VaR is based on the actual periodic changes in risk factors over a lookback period. For a daily VaR, the change in the value of the current portfolio is calculated for each day of the lookback period, using the actual daily changes in portfolio value. By ordering the changes in portfolio value from most positive to most negative, we can find the largest 5% of losses.
The smallest of those losses is our estimate of the 5% VaR for the current portfolio.
Under the historical simulation method, no adjustments are made for the difference between the results for the lookback period and the results over a longer prior period.
One positive aspect of the historical simulation method is that we do not need the assumption of normality, or any other distributional assumption, to estimate VaR. Because the historical results for a portfolio containing options include the changes in option values, the historical simulation method can be used to estimate the VaR for portfolios that include options.
VaR estimates will depend on the lookback period and, as with any forecasts, will vary with the characteristics of the sample data used. VaR based on an unusually volatile lookback period will yield overestimates of VaR, just as VaR based on a lookback period with low volatility will likely underestimate the true VaR over subsequent periods.
A third method of VaR estimation is Monte Carlo simulation. Monte Carlo simulation is based on an assumed probability distribution for each risk factor. Additionally, an assumption must be made about the correlations between risk factors.
Computer software is used to generate random values for each risk factor, and pricing models are used calculate the change in portfolio value for that set of risk factor changes.
This procedure is repeated thousands of times. Then, just as with historical simulation, we can order the outcomes and identify the fifth percentile (i.e., a value for which 5% of the outcomes will be lower) to estimate the 5% VaR. As with the other methods, the data used and the assumptions about the distributions of the risk factors will have significant effects on the estimated VaR.
Assuming a large sample size, the Monte Carlo method will produce identical results as the parametric method if the distribution specified and the parameters are the same.