Case 087Risk measurement and limitsWarm up

Aviratam holds two positions of Rs 50 crore each, with annual volatilities of 20% and 25%. What are the portfolio volatility and one-day 99% VaR at the modelled correlation of -0.3, and at a crisis correlation of +0.8?

The situation

Aviratam Capital holds two positions of Rs 50 crore each: a long position in crude oil futures with annual volatility of 20%, and a long basket of invented airline and travel stocks with annual volatility of 25%. Over the last three years the two have moved with a correlation of -0.3, because dearer fuel hurts airline profits, and the risk model uses that figure.

The chief risk officer asks for the portfolio's volatility and one-day 99% value at risk under the model, and again under a scenario in which a demand collapse drives oil and airlines down together, with correlation +0.8. Use 252 trading days a year and 2.326 for the 99% point of the normal distribution.

Your task

Compute the portfolio volatility and one-day 99% VaR at both correlations, and say what the comparison means for how the book is limited.

Quick check

Moving the correlation from -0.3 to +0.8 raises the portfolio VaR by roughly:

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Portfolio volatility is Rs 13.46 crore a year at -0.3 and Rs 21.36 crore at +0.8, so one-day 99% VaR rises from Rs 1.97 crore to Rs 3.13 crore, about 59% more. The variance more than doubles. At +0.8 the book is close to the Rs 3.30 crore it would carry with no diversification at all, which is why limits should be set on a stressed correlation, not the calm one.

Step 1How does correlation enter the portfolio's risk?

Two friends commute by different routes. If one route jams when the other is clear, at least one of them is usually on time; if both routes jam in the same rain, they are late together. Portfolio variance is each position's variance plus a cross term that depends on how the two move together, and that cross term is where diversification lives. In Rs crore a year, the positions have standard deviations of 50 x 20% = 10 and 50 x 25% = 12.5. Their variances are 100 and 156.25. The cross term is 2 x rho x 10 x 12.5, which is -75 at rho = -0.3 and +200 at rho = +0.8.

The relationship
σp=σ12+σ22+2ρσ1σ2100+156.25−75=13.46,100+156.25+200=21.36\sigma_p = \sqrt{\sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2} \qquad \sqrt{100 + 156.25 - 75} = 13.46, \quad \sqrt{100 + 156.25 + 200} = 21.36
sigma 1, sigma 2annual standard deviations of the positions in Rs crore, 10 and 12.5
rhocorrelation between the positions
sigma pportfolio standard deviation, Rs crore a year
What it says in wordsThe portfolio's standard deviation is the square root of the two variances plus twice the correlation times the two standard deviations.
Step 2What are the VaR figures?

Convert to one day by dividing by the square root of 252, then multiply by 2.326. At -0.3, daily volatility is Rs 0.848 crore and VaR is Rs 1.97 crore; at +0.8 they are Rs 1.346 crore and Rs 3.13 crore. The two standalone VaRs are Rs 1.47 crore and Rs 1.83 crore, which add to Rs 3.30 crore, the VaR at perfect correlation. Under the model, diversification removes 40% of the risk; under the crisis correlation it removes only 5%.

Same positions, same volatilities: the correlation sets the risk12301.97Model: -0.33.13Crisis: +0.83.30No diversification: +1.0x 1.59Rs crore, one day at 99%. Standalone VaRs 1.47 and 1.83; their sum is the last bar.
Aviratam's one-day 99% VaR is Rs 1.97 crore at the modelled correlation of -0.3 but Rs 3.13 crore at a crisis correlation of +0.8, close to the Rs 3.30 crore of no diversification at all.
Step 3What should the risk officer take from it?

The model's diversification benefit rests on one number estimated in calm years. Correlations between risky positions tend to rise in a sell-off, when investors cut everything at once, so diversification tends to vanish exactly when it is needed. For limits, run the book at the stressed correlation as well and size against the worse figure, or hold capital for the gap: here Rs 1.16 crore of one-day VaR that the model does not show. The limitation to say: VaR at 99% says nothing about the size of losses beyond it, and a demand collapse is a fat-tailed event, so a scenario loss, for example oil down 30% and airlines down 25% together, belongs beside the VaR number. That scenario costs Rs 27.5 crore.

Where candidates lose it

The common error is adding the correlation linearly, as if VaR scaled with rho, or forgetting the factor of 2 in the cross term. Write the variance out in full; the answer then follows mechanically.

The judgement error is presenting the -0.3 figure as the risk. A correlation estimated in a calm period is the least reliable number in the model, and the question is really asking whether you know that.

What the interviewer asks next

  • What correlation would make the portfolio VaR equal to the larger position's standalone VaR?
  • How would you hedge the book so that its risk depends less on the correlation?
  • Why does VaR at 99% understate the oil-and-airlines crash scenario?
  • How would you estimate a stressed correlation from data?

Company names and figures are illustrative.