Posted On: 2015-04-06

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Optimal leverage is exactly what it sounds like - the optimal fraction of your capital to invest, based on the performance of your assets of choice. This fraction can easily be found mathematically, but has naturally a few shortfalls. One thing to be aware of is that it optimizes the growth rate, which can lead to long drawdowns and high volatility which the investor might not be comfortable with. In this article, we will delve into the mathematics behind optimal leverage and discuss some of its benefits and shortcomings.

Let's start with the most basic case, for which you have an asset (or portfolio) that we assume follows a geometric Brownian motion with leverage

Eq. 1.1

...where you can chose a leverage L, and L can be a fraction smaller than 1
meaning the account is de-leveraged. The instantanoues growth rate, based on investing L times the account equity, is found as

Eq. 1.2

To find the maximum of the instantaneous growth rate with respect to L is then trivial

Eq. 1.3

...and the optimal allocation is then L times the account size. It shows that optimal leverage is proportional to return and inversely proportional to variance.

Now, this example isn't very realistic. If you don't invest all of your equity
you would earn interest on it. Likewise, if you leverage your account you would
need to pay the borrowing interest. We'll therefore use a process to describe a leveraged asset, such as the one used in Giese (2010)

Eq. 2.1

...where r is the interest rate for borrowing. The formula simply states that the fraction of leverage exceeding the actual equity incurs the borrowing interest costs. The formula was originally derived to model leveraged assets, but the same assumptions can be applied to a leveraged trading account. This SDE has solution

Eq. 2.2

...and it can easily be verified with Ito's lemma that it is indeed the solution. Taking the expectancy of ln(S_{t}/S_{0}) we get the growth rate

Eq. 2.3

... and the optimal leverage, found the same way as in the simple case, end up being

Eq. 2.4

This formula is known as the Kelly formula, which is the Kelly criterion
but adjusted for continuous time rather than binary bets (although derived
differently). It's important to note that the drift is NOT the same as the
expected return of the asset, since a geometric brownian motion with drift
has the expected return

Eq. 2.5

The optimal leverage formula can be written to be based on the expected log
return rather than the drift, and becomes

Eq. 2.6

where u is the expected log return. The Kelly criterion, known as the "Optimal Growth Strategy", maximizes the expected growth rate of wealth asymptotically. What Kelly has proven mathematically, however, is that if your returns time series is stationary, in the sense of stationary returns, volatility, and therefore Sharpe ratio, then statistically the leverage given by Kelly's formula will generate the maximum growth rate.

Maximizing long term wealth is the same as maximizing the compounded growth rate. In the long term it can be shown to maximize your capital growth (assuming the price process moments are estimated correctly). It also avoids bankruptcy as the position is adjusted downwards when losses occurs. This comes from the objective of optimizing long term wealth - if there's a chance that the price reaches 0 at any point in time then the infinite time wealth is zero as well. The non-bankruptcy statement can be criticised as it requires continuous hedging intraday (if a large price move occurs we need to adjust our exposure as it happens, rather than in the end of the day).

The arguments against optimal leverage tend be related directly to the benets
of it. First of all, you'll be selling into a loss. As a loss occurs and your account gets smaller, the total exposure must be adjusted accordingly, meaning that you effectively locking in a loss. With an account size of \$100 000 which trades with 2x leverage means an exposure of \$200 000. If the price goes down 5%, bringing your account down to \$90 000, you need to adjust the exposure to \$180 000 instead. This is the effect of the non-bankruptcy property which will protect your account from reaching \$0.

Secondly, optimal leverage does not optimize sharpe ratio. Your equity
growth will not follow a smooth steady linear increase, but rather a jumpy
path that could have long periods of drawdowns. There are leverage formulas
with the objective of optimizing the sharpe ratio, but the aim of optimal leverage is higher leverage when a risky asset outperforms the risk free which in turn optimizes the long term growth rate.

Finally, the price series assumption used is not valid. It is well known that
asset prices does not have a log-normal distribution, even if this is the norm in finance to assume. The probability of very large price movements is therefore
underestimated, hence the calculations for the optimal leverage will assume a
theoretical risk smaller than the real market risk.

We all know that even if something works in theory, it might not work in practice. Even if it makes sense on paper we would not recommend anyone to invest blindly based on the optimal leverage formula. Before incorporating it in your trading, its wise to consider:

*Broker limitations* - A high optimal leverage output might not be possible,
as the broker restricts your leverage to 2-3 times you capital (or even less).
In addition to this different margin requirements can affect your maximum
leverage and when you have to take losses if the market moves against you.

*Risk of incorrect parameter estimation* - As estimated (or forecasted) return and variance is used in the formula there is a risk that these are
either incorrect or over-optimistic leading to a higher leverage than optimal. Common practice is to use more robust measures of return/risk, or
to use fractional optimal leverage to reduce the variance of your portfolio.
A popular approach is to use the half-Kelly ratio, which is half of the
optimal leverage suggested.

To illustrate the effect of optimal leverage in trading, we consider trading the S&P500 from 1995 with different levels of leverage compared with optimal leverage. The borrowing rate is set to 5%, and the leverage is bounded between 4 and 0.1. This avoids short selling in bad times, but allow us to hold an almost flat position if the optimal leverage suggest that. It's clear that optimal leverage would have outperformed any combination of fixed leverage.

*Quanters Group offers recommendations for optimal leverage in our available portfolio strategies online. The formula utilized in our strategies differs
from the generic one presented in this article, and is calculated with robust estimators, variance reduction techniques, and is adjusted for a realistic trading environment.*

Giese, G., 2010, On the risk-return profile of leveraged and inverse ETF's, *Journal of Asset Management*, Issue 11, pp.219-228

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