Optimal Leverage Calculator

Definitions

• Interest Rate - the real annual interest rate
• epsilon - the elasticity of utility relative to wealth; set to 0 to assume utility is proportional log welath; for more information see here
• Wealth Available - the percent of the net present value of all your wealth (including future labor income) that is available to you
• Adjustment - This shifts all the S&P 500 annualized returns used in the model up or down this many percentage points. This can be useful if you think the outlook for stocks is better or worse than the historical average. For context, the average real total annual return since 1871 was 7.05%.
• Model Period - All the text on this page refers to a model that assumes historic quarterly returns are independent and models price movements within quarters using geometric Brownian motion. However, you can change this to monthly or yearly data. Note: all inputs and outputs of the model are always annualized.
• Optimal Leverage - the optimal amount of leverage from an expected utility perspective
• Equivalent Alpha - the amount of unleveraged alpha that is equivalent to using the optimal leverage above rather than no leverage from an expected utility perspective
• Unwind Risk - Annual risk of a forced unwind of your position.

Details

This computes the amount of leverage that would have historically maximized your expected utility based on total real returns on the S&P 500 from 1871 to the present Shiller. For the purpose of computing unwind risk, the model assumes returns within each quarter follow geometric Brownian motion pattern Geometric Brownian motion Brownian bridge, and that you unwind your position if your net worth hits zero. You are assumed to rebalance each quarter.

While the assumption that returns follow geometric Brownian motion is common Black–Scholes model, real life returns are skewed left, which means the simulation will be slightly towards more leverage. The fact that the model considers it impossible to have a quarter worse than the worst quarter experienced so far also biases the results somewhat towards more leverage.

On the bright side, through the magic of math (see citations and source code), this simulator does need any randomness: it considers all possible Brownian motion paths at once Zeitouni to compute the unwind risk.

The model does not consider leverage ratios below 1.0.

Take Aways

The optimal amount of leverage generally follows an S-shaped curve relative to the interest rate. To generalize:

• The "adjustment" parameter shifts the S cure left and right
• The epsilon and the available wealth change the "spread" of the curve. When these parameters are small, then you are effectively a risk-neutral investor who will basically take leverage if and only if the interest rate for it is lower than the expected stock returns.

Also, you should never leverage more than 3.5x.