Case 094Market risk limits and VaRHard

An FX desk is long USD 50 million and short EUR 30 million against the rupee. With daily volatilities of 0.35% and 0.5% and a correlation of 0.6, compute the one-day 99% VaR and each position's component VaR, and decide which to reduce.

The situation

Mirovex Bank's FX desk is long USD 50 million at Rs 84, a position of Rs 420 crore, and short EUR 30 million at Rs 91, Rs 273 crore. The daily volatility of the dollar-rupee rate is 0.35% and of the euro-rupee rate 0.5%, and the two rates have a correlation of 0.6, because both move when the rupee itself moves.

The desk's one-day 99% VaR limit is Rs 2.70 crore, measured with the variance-covariance method and a 99% multiplier of 2.33. Risk has asked the desk to come back inside the limit by reducing one position.

Your task

What is the book's VaR, how much does each position contribute, and which position should be reduced?

Quick check

What is the book's one-day 99% VaR?

Worked solution

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

30-second answerThe answer to give first

The book's one-day 99% VaR is about Rs 2.96 crore, above the Rs 2.70 crore limit, and the dollar long contributes 59% of it. Standalone, the legs carry Rs 3.42 crore and Rs 3.17 crore, but the short euro cancels much of the dollar's rupee risk. Cutting the dollar long by Rs 100 crore takes VaR to Rs 2.63 crore, inside the limit; no cut to the euro short can get there. Reduce the dollar.

Step 1Why is the book's VaR not the sum of the two?

Because the two positions face the same risk from opposite sides. If you owe a friend in Mumbai and are owed by another friend in Pune, a rise in train fares hurts one trip and not the other. When the rupee weakens, the dollar long gains and the euro short loses, so with a correlation of 0.6 much of each leg's swing cancels the other's. One-day swings are 0.35% of Rs 420 crore, Rs 1.47 crore, and 0.5% of Rs 273 crore, Rs 1.365 crore. Combined with the correlation and opposite signs, the book moves Rs 1.27 crore a day, and 2.33 times that is Rs 2.96 crore.

The relationship
σp=1.4702+1.3652−2(0.6)(1.470)(1.365)=1.271VaR99=2.33×1.271≈2.96\sigma_p = \sqrt{1.470^2 + 1.365^2 - 2(0.6)(1.470)(1.365)} = 1.271 \qquad \text{VaR}_{99} = 2.33 \times 1.271 \approx 2.96
1.470one-day volatility of the dollar long, Rs crore
1.365one-day volatility of the euro short, Rs crore
0.6correlation of the two rupee rates; the minus sign comes from the short
What it says in wordsThe short euro offsets the long dollar through their correlation, so the book swings far less than either leg alone.
Step 2How much does each position contribute?

Component VaRThe share of a portfolio VaR that belongs to one position, allowing for its correlation with the rest; the components add up exactly to the total. splits the book's VaR so the pieces add up. The dollar long contributes Rs 1.75 crore, 59%, and the euro short Rs 1.21 crore, 41%; the euro's contribution is far below its standalone Rs 3.17 crore because correlation cancels most of it. The offsetting leg earns its place. It is slightly larger than the size that would cancel the most risk, about Rs 176 crore, which is why its component is still positive.

What each leg risks alone, and what it adds to the bookLong USD 50malone 3.42in the book 1.75Short EUR 30malone 3.17in the book 1.21Sum of standalone VaRs 6.59Book VaR 2.96USD long is 59% of book VaRCorrelation cancels most of the euro's risk
Alone, the dollar long and euro short carry VaRs of Rs 3.42 crore and Rs 3.17 crore, but inside the book they contribute Rs 1.75 crore and Rs 1.21 crore, a total VaR of Rs 2.96 crore against a Rs 6.59 crore sum of standalones.
Step 3Which position should come down?

Test a cut of each and watch the total. Cutting the dollar long by Rs 100 crore takes VaR from Rs 2.96 crore to Rs 2.63 crore, inside the limit; cutting the euro short by Rs 100 crore only reaches Rs 2.74 crore, and no euro cut of any size gets below about Rs 2.74 crore, because the euro short is the dollar's hedge: past about Rs 176 crore of short, cutting it adds risk back. Reduce the dollar long, which is where most of the risk sits, and leave the offset in place.

Book VaR as each leg is cut, Rs crore2.52.72.9050100150200Rs crore cut from the positiontoday 2.96limit 2.70USD cut by 100: 2.63, inside the limitEUR cut by 100: 2.74, still above iteuro cutdollar cut
Cutting Mirovex's dollar long by Rs 100 crore lowers book VaR from Rs 2.96 crore to Rs 2.63 crore, while the same cut to the euro short only reaches Rs 2.74 crore and larger euro cuts push VaR back up.

Close with the model's limits. The answer leans on a correlation of 0.6 that can fall in a stress, when the dollar and euro sometimes move apart against the rupee; if it drops to zero, the euro short stops hedging and becomes a second outright risk. Run the same book at a stressed correlation before signing off the reduction.

Where candidates lose it

The usual error is adding the two standalone VaRs to get Rs 6.6 crore, or ignoring that one leg is short and treating the correlation as adding risk. The sign of the position flips the sign of the correlation term.

The second is cutting the smaller-looking or riskier-looking euro position because its volatility is higher. Its standalone risk is mostly cancelled inside the book; component VaR, not standalone VaR, says where to cut.

What the interviewer asks next

  • The correlation falls to 0.2 in a stress. What is the VaR now, and does the decision change?
  • What euro short would minimise VaR for the current dollar long?
  • How would a historical simulation VaR differ from this answer?

Company names and figures are illustrative.