Case 060Risk measurement and limitsHard

A fund holds Rs 80 crore of a mid-cap that trades Rs 8 crore a day, and its normal one-day VaR is Rs 3 crore. Add the cost of exiting at 20% of daily volume, with a 60 bps spread and square-root impact, and restate the risk.

The situation

Sunyata Quant holds Rs 80 crore of a mid-cap stock. The stock trades about Rs 8 crore a day, so the position is ten days of the whole market's volume. The risk system reports a one-day 99% value at risk of Rs 3 crore, computed from the stock's volatility as if the position could be sold at the mid price.

The execution desk says it can sell without dominating the stock at about 20% of daily volume. The quoted spread is 60 bps. The desk's impact model is the square-root rule: selling a quantity Q costs roughly the daily volatility times the square root of Q over daily volume. The chief risk officer asks you to restate the risk with the exit priced in.

Your task

Work out how long the exit takes, price the spread and impact, add the market risk carried during the exit, and give a liquidity-adjusted risk number with a view on what to do.

Quick check

At 20% of a Rs 8 crore daily volume, how long does selling Rs 80 crore take?

Worked solution

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

30-second answerThe answer to give first

Priced honestly, the risk is about Rs 16.7 crore, roughly 5.6 times the reported Rs 3 crore. Selling at 20% of volume takes 50 trading days, carrying market risk worth about Rs 12.4 crore at 99%. Half the spread costs Rs 0.24 crore and square-root impact about Rs 4.1 crore. The position is too large for its liquidity; a cap near three days of volume would bring it in line.

Step 1What does the reported VaR quietly assume?

Selling a flat in a hurry is different from knowing its market value: the valuation tells you what similar flats fetch, but if you must sell this month you take a lower price, and if you wait, the market can move while you search for a buyer. The Rs 3 crore VaR assumes Sunyata can sell all Rs 80 crore at the mid price within a day, which is impossible for a position worth ten days of the stock's entire volume. Backing the volatility out of the VaR, 3 / (2.326 x 80) gives a daily volatility of about 1.61%, which the rest of the calculation uses.

Step 2How long does the exit take, and what does that do to market risk?

At 20% participation the desk sells Rs 1.6 crore a day, so the exit takes 50 trading days. The fund is exposed to the stock's moves on a shrinking position for the whole of that time, and with the position falling in a straight line, the variance over the exit is the one-day variance times the sum of the squared fractions still held, about 17.2 days. The 99% market risk over the exit is therefore 3 x sqrt 17.2 = Rs 12.4 crore. Selling faster would shorten this but raise impact; the 20% pace is the desk's judgement of the balance.

Selling at 20% of volume takes 50 trading days, not one204060800VaR's assumption: out in one dayActual: Rs 1.6 crore a day for 50 daysmarket risk is carried on the shaded area01020304050Trading days into the exit. Vertical: position left, Rs crore
Selling at 20% of daily volume, Sunyata's Rs 80 crore position takes 50 trading days to run down, so the fund carries market risk on a shrinking position for about ten weeks, while the reported VaR assumes it is out in one day.
Step 3What do the spread and the impact cost?

Selling crosses half the spread: 30 bps of Rs 80 crore is Rs 0.24 crore. Impact is far larger: the square-root rule gives 1.61% x sqrt(80 / 8) = 5.10% of the position, about Rs 4.1 crore, because the fund's own selling pushes the price down as it goes. The rule depends on the total sold relative to daily volume, not much on the pace, so a slower exit does not buy much relief from it. Both costs are close to certain if the fund has to leave, which is why they are added to the risk rather than treated as a possibility.

The relationship
L-VaR=3.017.17⏟market, over exit+80×0.30%⏟half-spread+80×1.61%10⏟impact=12.43+0.24+4.08=16.75\text{L-VaR} = \underbrace{3.0\sqrt{17.17}}_{\text{market, over exit}} + \underbrace{80 \times 0.30\%}_{\text{half-spread}} + \underbrace{80 \times 1.61\% \sqrt{10}}_{\text{impact}} = 12.43 + 0.24 + 4.08 = 16.75
L-VaRliquidity-adjusted value at risk, Rs crore
\sqrt{17.17}scales one-day risk to a 50-day exit on a position falling in a straight line
\sqrt{10}square root of the position over daily volume, Rs 80 crore over Rs 8 crore
What it says in wordsThe risk of holding a position you cannot sell quickly is the market risk over the whole exit plus the near-certain cost of getting out: about Rs 16.7 crore here.
The risk number before and after you price the way out, Rs croreVaR 3.0Reported: 1-day VaR3.0market risk over exit: 12.4impact 4.08Liquidity-adjustedhalf-spread 0.2416.75.6 timesthe reported number
Adding the market risk carried over the 50-day exit, the half-spread and the square-root impact turns Sunyata's reported Rs 3 crore VaR into about Rs 16.7 crore of liquidity-adjusted risk, roughly 5.6 times the headline.
Step 4What should Sunyata do with the restated number?

Report it, and size the position to it. A position capped at three days of volume, Rs 24 crore, would carry about Rs 2.9 crore of liquidity-adjusted risk against Rs 16.7 crore for the current holding: 30% of the capital for 17% of the risk, because exit time and impact both grow faster than the position. Even if Sunyata keeps the holding, the risk limit should be set against the liquidity-adjusted figure, and the fund should know before a crisis that leaving takes ten weeks. Say the limitations: the square-root rule is a rough average, impact in a stressed market can be double, and volume tends to dry up exactly when everyone wants to sell.

Where candidates lose it

Candidates add the spread cost to the VaR and stop. On a position worth ten days of volume, the spread is the smallest piece; impact and the market risk carried during a long exit are each far larger.

The second miss is reading ten days of volume as a ten-day exit. The fund can take only a fraction of each day's trading, so the exit is five times longer, and the risk over the exit grows with it.

What the interviewer asks next

  • How would the liquidity-adjusted number change if the desk sold at 40% of volume?
  • Why does impact cost grow faster than the position, and what limit structure follows from that?
  • How would you stress the volume assumption itself?

Company names and figures are illustrative.