Optimal leverage and compound growth

In Chapter 5 of my book Free Capital, Sushil talks about optimal leverage and compound growth.  The main takeaways are that (i) it may be more sensible for an investor to maximise expected log return rather than expected return, and (ii) maximising log return will generally prescribe lower leverage than casual intution based on expected return might suggest.

In my view the expected terminal wealth at some arbitrary time horizon is often a poor investment objective, because the distribution of terminal wealth from any long period of compounding is very positively skew – a small number of very high outcomes contribute a lot to the mean.  The median (50th percentile) of terminal wealth tends to give a better indication of ‘typical’ results. 

Example 1

As a toy example, consider an investment which either doubles or falls 60%, each with probability 50% in each period, over a time horizon of 3 periods.  This gives eight equi-probable outcomes for terminal wealth levels: 0.064, 0.32 (three times), 1.6 (three times), 8.  Taking the mean of these, we find mean terminal wealth as 1.728 (=1.2³, so mean return = 20% per annum).  If you use the mean as your criterion, this investment looks great!   But 7 of 8 possible outcomes are below the mean.  The median terminal wealth – somewhere in between 0.32 and 1.6 – seems a better measure of ‘typical’ results.

Under very general assumptions, maximising median terminal wealth is asymptotically equivalent to maximising the log mean (not mean) of any sub-period (e.g. one-year) rates of return. The relevance of logarithms arises from the geometric nature of compounding: log returns (not returns) are additive over time.

In the toy example, the mean log return is -0.1116 per annum (that’s (log 2 + log 0.4)/2) = -0.1116), which gives exp(-0.1116 x 3) = 0.7155 over the time horizon of three years. This is equivalent to the median I just characterised as “somewhere in between 0.32 and 1.6”, where the “somewhere in between” is evaluated geometrically: sqrt(0.32 x 1.6) = 0.7155.

An investor who focuses on expected returns might think that this investment can be judiciously leveraged. Not too much, we want to be safe, we’re sensible investors, but how about say 50%? That will increase the expected return from 20% per annum to 30%. Sounds great! But in 4 of the 8 outcomes, this investor goes bust: the terminal proceeds after three years are less than the amount owed. Maximising the mean return is a dangerous criterion: it tends to push you towards taking too much risk.^{[1]“Too much” risk is of course subjective. It is always ultimately a matter of preferences. But I suspect that many investors who might implement something like the 50% leverage just … Continue reading}

Example 2

We can draw a graph for the example Sushil talks about: an investment which rises 25% or falls 20% in each period.  The blue line shows the expected return on the y-axis as a function of leverage on the x-axis. Note that tnis this increases without limit. If mean return is our criterion, then the more leverage, the better. The red line shows the mean log return. Note that this initially increases with leverage, but up to a leverage of L*: If mean log return is our criterion, there is an optimal leverage level.

With no leverage (L = 1), the mean return is [(1.25 +0.8)/2] – 1 = +2.5% per period (blue line intercept on the y-axis). But the mean log return is (log 1.25 + log 0.8)/2 = 0% per period. The optimal leverage in this case is L* = 0.5, that is, invest only half your funds, keeping the other half in cash (assumed nil return here, for simplicity); and then periodically rebalance the portfolio of asset and cash to the same 50:50 proportions. Any leverage above 1.0 – that is, any borrowing at all – gives a negative mean log return (i.e. median terminal wealth less than what we started with). This investment should not be leveraged. ^{[2]The investment might sensibly be leveraged as part of a portfolio, if it is negatively correlated with other assets in the portfolio. But that is beyond the scope of this single-asset toy example.}

 Of course, these toy examples – repeated trials of a single investment with only two equi-probable outcomes in each period – don’t correspond to any real-world problem.  The value of the toy examples is to highlight the following points:

• mean terminal wealth at a long horizon is heavily influenced by a very few extreme positive values
• median terminal wealth gives a better indication of ‘typical’ results
• to maximise median terminal wealth, we need to maximise the mean log return (not return) over sub-periods.

A useful approximation for log return

Looking up logarithms is inconvenient for mental arithmetic, so it helps to have a simpler approximation.  Suppose E  is the expected return of the investment, and V is the variance of return.  For short periods E will typically be small, so LE will be small, and log(1+LE) ≈ LE, and (1+ LE)² ≈1; and if we use these approximations for the first and second terms of the Taylor expansion of log (1+LE), the errors in the two approximations are of opposite sign.  A quick mental arithmetic approximation of the expectedlog return of a leveraged investor is then

Log return ≈  LE  –  L² V / 2 .

This expression is maximised when L = E / V. For example, suppose you invest in equities with an expected return (net of any charges) of 6%pa and standard deviation of return of 20%pa (ie V = 0.2² =  0.04). The theoretically optimal leverage is 0.06/0.04 = 1.5x.  In practice I would halve the leverage above 1.0 (= no leverage) for more safety, i.e. maximum 1.25x (partly because I don’t know if my estimates of E and V are correct).   Leverage of L > 3x gives negative expected log return, ie you would almost surely go broke in the long run. 

This quick estimate (LE  –  L² V / 2) for the expected log return is often enough to highlight the folly of many common financial structures, including highly leveraged hedge funds, structured products and spread betting.

Summary

So median terminal wealth gives a better indication than mean terminal wealth of ‘typical’ results, and maximising it (equivalently: maximising log return) helps to keep you out of leverage trouble. But there is, in the end, nothing wholly sacrosanct about the median (or the log return). It is just one measure of central tendency (or one measure of return). Depending on your subjective preferences, you might alternatively prefer to maximise the 10th, 25th, 75th or any other percentile of terminal wealth. Arguably, to make good decisions, you really need to think about the whole distribution of terminal outcomes, not just a single point or statistic of the distribution.

But there is also nothing sacrosanct about the mean. And if we are going to use a single statistic, for problems involving compound growth, the median of terminal wealth is generally a better guide than the mean (equivalently: log returns are a better guide than mean returns). ^{[3]For more on the many good properties (as well as a few less good ones) of log return as a maximand, see the paper Good and bad properties of the Kelly Criterion.}