How much you may hold: VaR and position limits
Every share held overnight is a bet the desk did not choose. Limits are set in dollars of value at risk, and a position you cannot sell in a day is exposed for as long as it takes to sell, so risk grows faster than size.
- Compute a one-day 95% value at risk from the position, the price and the daily volatility
- Stretch it over the days an exit takes, using the square root of time
- Turn a dollar VaR limit into a maximum position, with and without the liquidity charge
- Explain why halving a large, illiquid position removes far more than half its risk
The intuition
You are long 400,000 shares of a $50 stock: $20 million. If the stock moves 2% on a normal day, one standard deviation is $400,000, and the move you would expect to be beaten only one day in twenty is 1.645 of those: $658,000. That is the one-day 95% value at risk, and the desk limit is written in those units. Because the limit is in dollars, the shares you may hold shrink when the price rises, when volatility rises, or when the limit is cut.
The subtler point is liquidity. If the stock trades a million shares a day and you will not be more than a fifth of the volume, you can sell 200,000 a day: two days to get out. Over two days the risk is √2 times bigger, about $930,600. And the horizon itself grows with the position, so risk rises with size to the power 1.5: double the shares and you double the exposure and take √2 times longer to shed it. Halve a large position and its liquidity-adjusted VaR falls by almost two thirds, not a half.
1-day VaR = 1.645 × daily vol × position. Days to unwind = shares ÷ (ADV × participation). Liquidity-adjusted VaR = 1-day VaR × √days. Max shares on a one-day limit = limit ÷ (1.645 × vol × price). With the liquidity charge, VaR ∝ q^1.5, so the maximum is (limit × √capacity ÷ (1.645 × vol × price))^(2/3).
Why it works
- The conventions here: a 95% one-sided value at risk on normal returns with zero mean: 1.645 standard deviations. Days are independent, so risk scales with the square root of the number of days. Days to unwind = shares ÷ (average daily volume × participation rate), not rounded; the liquidity-adjusted VaR uses that as its horizon. The recipe's own draws keep the unwind at a day or more.
- Dollars, not shares. The limit is a dollar loss, so the position it allows is limit ÷ (1.645 × vol × price). Any of the three rising cuts the shares you may hold in proportion.
- Why the square root. Independent daily moves add in variance, not in standard deviation. Over d days the standard deviation is √d times the daily one, so VaR is too.
- Why the power 1.5. VaR = 1.645 × vol × price × q × √(q ÷ capacity): one q from the size and a √q from the horizon. Set that equal to the limit and q^1.5 falls out, so the maximum is the limit-based term raised to two thirds.
- Is √days fair? Slightly harsh: you hold only about half the position on average as you sell, so a straight-line unwind carries roughly √(1/3) of the risk of holding it all. It also ignores the price impact of your own selling, which pushes the other way. Desks keep the simple rule and add a separate impact charge.
- The systemic problem. Every desk running the same limit is forced to sell at the same moment: right after volatility jumps, which is usually right after prices fall. Market makers pulling back when they are most needed is how liquidity disappears in a stress.
| Position: 400,000 × $50 | $20,000,000 |
| One standard deviation: $20,000,000 × 2% | $400,000 |
| One-day 95% VaR: 1.645 × $400,000 | $658,000 |
| Days to unwind: 400,000 ÷ (1,000,000 × 20%) | 2 days |
| Liquidity-adjusted VaR: $658,000 × √2 | $930,553 |
| Maximum shares on a $500,000 one-day limit: 500,000 ÷ (1.645 × 2% × $50) | 303,951 |
| Sell half: VaR × ½^1.5 = × 0.3536 | $329,000, a fall of 64.6% |
The formulas
The loss beaten only one day in twenty, zero mean.
How long the exit takes at the participation you will accept.
Independent days add in variance, so risk scales with the square root of time.
A dollar limit turned into shares.
Because VaR grows with size to the power 1.5.
Worked example
Shares you can sell a day, days to exit, the one-day VaR, then stretch it by the square root of the days. The follow-up asks whether √days is generous or harsh.
See it move
Same stock and book. Change the position, the volatility, how much the stock trades, how much of the volume you will be, and the limit.
- Add shares. The position, the one-day VaR, the days to exit and the adjusted VaR all rise, and the adjusted VaR rises faster than in proportion. Neither maximum moves: they belong to the limit, not to the book.
- Raise the volatility. Both VaRs rise and both maximums fall; the one-day maximum falls in inverse proportion, the liquidity-adjusted one more gently, because only the size part of its formula carries the volatility.
- Raise the daily volume or your participation. The days to exit and the adjusted VaR fall and the liquidity-adjusted maximum rises; the one-day figures do not move, because liquidity only enters through the horizon.
- Raise the limit. Both maximums rise; the VaR of the book you hold is unchanged.
Run it backwards
The desk's one-day VaR limit, the price and the volatility are given. What is the largest position you may carry?
Set 1.645 × vol × price × shares equal to the limit and solve for the shares. The VaR per share is 1.645 × vol × price; the limit divided by it is the answer.
With the liquidity charge the same move gives q^1.5 on the left, so the maximum is the limit-based term raised to two thirds, and it is smaller: a position that size takes longer than a day to exit.
Traps
Say it in the interview
“How do you set a position limit for a market-making book?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- 1-day VaR = 1.645 × daily vol × position, in dollars; the limit is a dollar figure.
- Days to exit = shares ÷ (ADV × participation); VaR scales with √days.
- Liquidity-adjusted VaR grows with size to the power 1.5, so halving a large position cuts risk by about 65%.
- Maximum shares = limit ÷ (1.645 × vol × price); the liquidity version raises the limit term to two thirds.