Case 034Position sizing and bankrollCore

A strategy has expected excess return of 8% and volatility of 16% a year. What is the full-Kelly leverage, what growth do full and half Kelly give, and what gross exposure would you run on Rs 200 crore?

The situation

Neelkosh Capital runs a diversified futures strategy whose backtest and live record suggest an expected return of 8% a year above the funding rate, with 16% annual volatility at one times leverage. It has Rs 200 crore of capital and can scale exposure freely using futures.

The portfolio manager wants to run at the growth-optimal leverage. The chief risk officer wants a number she can defend to investors.

Your task

Compute the full-Kelly leverage and the long-run growth rate at full and half Kelly, then argue for a gross exposure on Rs 200 crore.

Quick check

Half Kelly halves the leverage. Roughly what share of full Kelly's growth rate does it keep?

Worked solution

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

30-second answerThe answer to give first

Full Kelly is 8% divided by 16% squared, 3.12 times, for long-run growth of 12.5% a year; half Kelly at 1.56 times keeps 9.38%, three quarters of it, at half the volatility. Because the 8% is an estimate, I would size at half Kelly on a return shaded to 6.4%, about 1.25 times, or roughly Rs 250 crore of gross exposure on Rs 200 crore of capital.

Step 1Where does the Kelly leverage come from?

Compounding punishes volatility. If you gain 50% then lose 50%, you are down 25%, even though the average move was zero. The long-run growth rate of a levered strategy is the leverage times the expected return, minus half the leverage squared times the variance; the first term grows in a straight line, the second as a square, so there is a peak. Setting the slope to zero gives the Kelly leverageThe leverage that maximises the long-run growth rate of capital: expected excess return divided by variance, for returns that are roughly normal. of 0.08 divided by 0.0256, which is 3.125 times.

The relationship

g(L) = L\mu - \tfrac{1}{2}L^2\sigma^2, \qquad L^* = \frac{\mu}{\sigma^2} = \frac{0.08}{0.16^2} = 3.125, \qquad g(L^*) = \frac{\mu^2}{2\sigma^2} = 12.5\%

L	leverage, gross exposure over capital
\mu	expected excess return, 8%
\sigma	volatility at one times leverage, 16%
g	long-run growth rate of capital

What it says in wordsGrowth rises with leverage until the volatility drag, which grows with the square of leverage, overtakes it at 3.125 times.

Step 2What does half Kelly cost, and what does it buy?

Plug in half the leverage. At 1.5625 times, growth is 1.5625 x 8% minus half of 1.5625 squared x 2.56%, which is 9.375%. Half Kelly keeps three quarters of the growth while halving the volatility, from 50% a year to 25%. At full Kelly a 50% volatility means routine drawdowns of a third of the capital or more, which no outside investor will sit through. That trade, a quarter of the growth for half the pain, is why desks almost never run at the theoretical optimum.

Neelkosh's growth rate peaks at 12.5% at 3.12 times leverage, half Kelly at 1.56 times still grows at 9.38%, and if the true edge is only 4%, running 3.12 times produces zero growth.

Step 3What gross exposure would you actually run on Rs 200 crore?

The 8% is an estimate, and the formula is merciless about errors in it. If the true edge is 4%, the true Kelly leverage is 1.56 times, and running 3.12 times is twice Kelly, where growth is exactly zero: all the volatility, none of the compounding. Overbetting hurts much more than underbetting, so the sizing should assume the estimate is too high. Shade the expected return by a fifth to 6.4%, which gives a Kelly leverage of 2.5 times, and run half of that: 1.25 times, or about Rs 250 crore of gross exposure, with annual volatility near 20%. A household that budgets on last year's bonus rather than its salary is making the same mistake the full-Kelly portfolio manager makes.

Leverage	Exposure, Rs crore	Volatility	Growth if edge is 8%	Growth if edge is 4%
1.00x	200	16%	6.72%	2.72%
1.25x	250	20%	8.00%	3.00%
1.56x	312	25%	9.38%	3.12%
3.12x	625	50%	12.50%	0.00%

At the chosen 1.25 times, growth stays positive whether the edge is 8% or 4%; at full Kelly, a halved edge leaves growth at zero with 50% volatility.

State the limits of the formula. It assumes normal returns, continuous rebalancing and a known, constant edge. Fat tails and sudden correlation jumps make the true optimum lower still, and the approximation breaks down for strategies with large single-period losses, where the discrete Kelly formula on the actual payoff distribution is the honest check.

Where candidates lose it

The usual loss is dividing the return by the volatility, not the variance, and getting 0.5 times, which is the Sharpe ratio, not a leverage. Kelly divides by sigma squared.

The second is presenting full Kelly as the answer to run. The question asks what you would run, and the interviewer wants to hear that estimation error and investor tolerance both push you well below the theoretical optimum, with a number attached.

What the interviewer asks next

At what leverage does the growth rate turn negative?
How would you set leverage for two uncorrelated strategies run together?
Why does a fractional Kelly bettor have a much smaller chance of a 50% drawdown?
How does a drawdown stop change the optimal leverage?

Company names and figures are illustrative.