Sharpe ratio and leverage
Return per unit of risk, and the dial that trades one for the other. Free borrowing would leave the Sharpe untouched; real borrowing wears it down.
- Calculate a Sharpe ratio
- Lever a strategy and find its return, volatility and Sharpe
- Find the leverage for a target volatility, and what it delivers
- Say why allocators compare Sharpe ratios before returns
The intuition
Two drivers finish a rally. One averaged 150 km an hour while swerving all over the road; the other averaged 120 without a wobble. Put the steady driver in a faster car and they could beat the first at the same level of danger. Speed for the risk taken is what matters, because more speed can be bought.
The Sharpe ratio is return for the risk taken: the return above the risk-free rate, divided by volatility. Leverage is the faster car. Borrow to run a strategy twice over and both its excess return and its volatility double, so the Sharpe stays the same, except that borrowing costs a financing spread above the risk-free rate, which wears the Sharpe down.
Sharpe = (return − risk-free) ÷ volatility. Levered L times: volatility = L × volatility; return = risk-free + L × excess return − (L − 1) × financing spread. Leverage for a target volatility = target ÷ volatility.
Why it works
- The conventions here: annual figures. Levering L times scales volatility by L. Return = risk-free + L × excess return − (L − 1) × the financing spread, the extra the fund pays above the risk-free rate on what it borrows.
- With free borrowing, leverage leaves the Sharpe unchanged: excess return and volatility scale by the same L.
- Each borrowed turn pays the spread, so the levered Sharpe falls as leverage rises.
- That is why allocators compare Sharpe ratios first: a high Sharpe at low volatility can be levered into a higher return than a low Sharpe at high volatility, for the same risk.
- Rough benchmarks: the stock market has managed about 0.4 to 0.5 over the long run; around 1.0 over a full cycle is good for a hedge fund.
- Leverage assumes the history continues: that financing stays available, volatility stays low and nothing breaks at the larger size. Levered low-volatility strategies are the ones that have blown up when that failed.
| Sharpe: (8% − 3%) ÷ 5% | 1.00 |
| Levered excess return: 3 × 5% − 2 × 1% | 13% |
| Levered return: 3% + 13% | 16% |
| Levered volatility: 3 × 5% | 15% |
| Levered Sharpe: 13% ÷ 15% | 0.87 |
| With free borrowing: 3 × 5% ÷ 15% | 1.00 |
A fund returning 15% at 15% volatility has a Sharpe of 0.80, so the levered 8% strategy beats it at the same risk, even after financing.
The formulas
Excess return per unit of risk.
Risk scales with leverage.
The borrowed turns pay the spread.
Unchanged only if the spread is zero.
Run it backwards.
Worked example
Scale the excess return and take off the financing on the borrowed turns; scale the volatility; divide one by the other.
See it move
Same strategy. Change its return and volatility, the risk-free rate, the financing spread, the leverage and the volatility an allocator wants.
- Raise the leverage. Volatility rises and the levered Sharpe falls; with free borrowing it would not move.
- Raise the financing spread. The levered Sharpe falls; the unlevered Sharpe does not move.
- Raise the unlevered volatility. The Sharpe falls, and less leverage is needed to reach the target.
- Raise the target volatility. More leverage is needed, and the Sharpe delivered at the target falls.
Run it backwards
Same strategy, reversed: an allocator wants it run at a set volatility. What leverage does that need, and what return and Sharpe does it deliver?
Volatility scales with leverage, so leverage = target volatility ÷ the strategy's volatility. Then apply the levered return formula at that leverage.
The Sharpe delivered is a little below the unlevered one because of the financing spread. The bigger question is whether the strategy can really be financed and run at that size.
Traps
Say it in the interview
“Fund A returns 8% at 5% volatility; fund B returns 15% at 15%. With 3% risk-free, which would you rather own?”
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.
- Sharpe = (return − risk-free) ÷ volatility.
- Levered volatility = L × volatility.
- Levered return = risk-free + L × excess − (L − 1) × spread.
- Free borrowing leaves the Sharpe unchanged; real borrowing lowers it.