Case 032Market risk limits and VaRCore

A desk holds a small-cap position worth eight days of trading volume. Its one-day VaR fits the limit, but can it actually get out? Rescale the risk to the exit horizon, add the exit cost and test the limit.

The situation

Brisanth Securities holds Rs 60 crore of a single small-cap stock. The position is equal to eight days of the stock's average trading volume. The desk's one-day 99% value at risk (VaR) on the position is Rs 3 crore, measured at mid prices, and its limit for the position is Rs 8 crore.

The stock's bid-ask spread is 1.5%. The desk's liquidity policy measures an illiquid position over the number of days of volume it represents, and treats daily returns as independent.

Your task

What is the VaR over the time it takes to exit, what does crossing the spread add, does the position fit its limit, and what size would?

Quick check

Scaled to the eight-day exit and with the exit cost added, where does the position sit against the Rs 8 crore limit?

Worked solution

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

30-second answerThe answer to give first

Measured over its eight-day exit, the position carries about Rs 8.94 crore of risk, over the Rs 8 crore limit. One-day VaR of Rs 3 crore scales by the square root of 8 to Rs 8.49 crore, and selling at the bid rather than the mid costs another Rs 0.45 crore. The one-day number hid the breach. Cutting the position to about Rs 56 crore brings it inside, because a smaller position is both less at risk and quicker to sell.

Step 1Why is one-day VaR the wrong horizon here?

One-day VaR asks what you could lose before you get out tomorrow. For a large-cap stock that is fair: you can sell in minutes. For a stock that trades an eighth of your holding each day, you are stuck in it for about eight days, and the price keeps moving while you sell. An illiquid position's risk is measured over the time it takes to leave, not over one day. It is the difference between a flat you can sell tomorrow and a flat that takes months to find a buyer: the price risk is the same per day, but you carry it for longer.

Step 2How do you rescale the VaR and add the exit cost?

With independent daily returns, variance grows with time, so VaR grows with the square root of time. Rs 3 crore times the square root of 8, about 2.83, is Rs 8.49 crore. VaR is also measured at mid prices, but a seller receives the bid, half the spread below mid. That exit costThe cost of turning a position into cash at the price buyers actually pay, usually half the bid-ask spread for a sale, more if the sale itself moves the price. is 0.75% of Rs 60 crore, Rs 0.45 crore, and it is certain rather than probable, so it is added on top.

The relationship
LVaR=VaR1dh+12 s P=3.08+0.5×1.5%×60=8.49+0.45=8.94\text{LVaR} = \text{VaR}_{1d}\sqrt{h} + \tfrac{1}{2}\,s\,P = 3.0\sqrt{8} + 0.5\times1.5\%\times60 = 8.49 + 0.45 = 8.94
\text{VaR}_{1d}one-day 99% VaR at mid prices, Rs 3 crore
hthe exit horizon in days, 8
sthe bid-ask spread, 1.5%
Pthe position value, Rs 60 crore
What it says in wordsLiquidity-adjusted VaR is the one-day VaR scaled to the exit horizon, plus the certain cost of selling at the bid.
Measured over the time it takes to leave, the position breaks its limit3.00One-day VaR8.49Over 8 days8.49exit cost +0.458.94Plus exit costlimit Rs 8 croreone day x square root of 8, about 2.83Rs crore
Brisanth's one-day VaR of Rs 3.0 crore becomes Rs 8.49 crore over an eight-day exit and Rs 8.94 crore with the Rs 0.45 crore exit cost, over the Rs 8 crore limit.
Step 3What size would fit, and why is the curve bent?

Halving a liquid position halves its VaR. Halving an illiquid one does more, because the exit horizon shrinks too. Risk here grows faster than the position: size times the square root of the days it takes to sell. Solving for the limit gives a position of about Rs 55.7 crore, sold in about 7.4 days. So Brisanth needs to sell only about Rs 4 crore to comply, and should do it over the coming days rather than dump it in one session and pay far more than the spread.

Cutting the position helps twice: less at risk, and a faster exit4812020406080Position size, Rs croreLiquidity-adjusted VaR, Rs crorelimit 8held 60: 8.94fits at 55.7
Liquidity-adjusted VaR rises faster than position size because a bigger holding also takes longer to sell; it crosses the Rs 8 crore limit at about Rs 55.7 crore, below Brisanth's Rs 60 crore.
Step 4Which assumptions could move the answer most?

The horizon, in both directions. Selling evenly over eight days means the average holding is smaller than Rs 60 crore; that version gives about Rs 5.8 crore. But selling a full day's volume every day would move the price hard. If the desk can realistically sell only a quarter of daily volume, the exit takes 32 days and the same method gives about Rs 17.4 crore. The spread in those answers is the real lesson: for an illiquid position, the liquidation assumption matters more than the VaR model, and a risk manager should see it written down and approved.

Where candidates lose it

The common error is to scale by 8 instead of the square root of 8, giving Rs 24 crore and a dramatic breach. Risk over several independent days grows with the square root of time.

The second is to forget the spread, or to charge the full spread. VaR is measured at mid, so selling costs half the spread; leaving it out hides Rs 45 lakh that will certainly be paid.

What the interviewer asks next

  • How would you set the exit horizon for a stock whose volume halves in a sell-off?
  • Why might the square-root-of-time rule understate risk for a small-cap stock?
  • Should the limit framework use the liquidity-adjusted number for every position, or only above a threshold?

Company names and figures are illustrative.