## Optimal leverage and compound growth

Chapter 5, Sushil, talks about optimal leverage and compound growth. The main takeaway is that investors are better off maximising **expected logarithmic return** (*not expected return*), and this often means lower leverage than you think.

In my view expected return (the mean) is often a poor objective, because the distribution of terminal wealth from any long period of compounding is likely to be very positively skew – a small number of very high outcomes contribute a lot to the mean. **Median** terminal wealth – found by *summing the logarithmic returns, and then taking the anti-logarithm* – seems a better measure of ‘typical’ results than the mean.

For 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 (=1.2^{3}). If you used the mean as your objective, the investment looks good. ** But 7 of 8 possible outcomes are below this. ** The median terminal wealth of exp –{0.11x3} = 0.72 seems a better measure of 'typical' results.

If we extend the scale of the graph in the book, we can show the expected compound growth (the blue line) as well as median compound growth (the red line). Adding leverage always increases the expected (the blue line slopes upwards), but increases the median only up to a certain point* L** (the red line has a maximum). And with too much leverage, the red line eventually goes negative - which means that in the long term, you almost surely go broke.

We can also draw a graph for the example Sushil talks about: an investment which rises 25% or falls 20% in each period. With no leverage (*L *= 1), the expected compound growth is +2.5% per period (blue line intercept on the y-axis), but median compound growth is zero (red line). The optimal leverage in this case is *L** = 0.5, that is, invest only half your bankroll, keeping the other half in cash (assumed nil return here, for simplicity).

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

- mean return is what casual intuition leads to, but this is not the same as median return, which is usually lower
- median return is the best measure of 'typical' results over long periods
- if we want to maximise median return, we need to maximise expected logarithmic 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 expected log return of a leveraged investor is then

** ***LE * – *L*^{2 }*V* / 2 .

*LE*–

*L*

^{2 }

*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* = 0.2^{2} = 0.04). The theoretically optimal leverage is 0.06/0.04 = 1.5x. In practice I would halve the leverage above 100% for safety, ie 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. Quite a few hedge funds operate with leverage > 3x !

This quick estimate *(LE * – *L*^{2 }*V* / 2) for the expected log return is often enough to highlight the lack of safety in common financial structures, including some cases of hedge funds, spread betting and CFD leverage, and split capital investment trusts.

Nice explanations … but isn’t this essentially about the Kelly formulae?