Leverage and expected value

Published on December 1, 2015
This article is very much a work in progress and may contain errors. If you find any mistakes, I would be grateful if you could let me know at @imos.

Leverage and expected value

Intuitively, applying leverage raises your rate of return while also increasing the probability of losing assets. We calculate how much things actually fluctuate and consider how to decide on an appropriate amount of leverage.

The fluctuation model

Here we take the fluctuation model to be one that, using ε (a sufficiently small value), waits until the change becomes either 1+ε or \frac{1}{1+ε}. The probability of becoming 1+ε and the probability of becoming \frac{1}{1+ε} are the same (obviously so, since for any foreign-exchange position there is also someone taking the opposite side). This model has a log expected value of 1 and a linear expected value of 1+\frac{ε^2}{2}. Foreign-exchange fluctuations are generally said to follow a log-normal distribution, but by controlling the time axis this model is thought to achieve a simpler and more accurate model.

The leverage model

When applying leverage at a factor other than 1, you need to move assets in response to the latest fluctuation. When adopting a leverage of L times,
  • if a fluctuation of 1+ε occurs, assets become 1+Lε times (A), and
  • if a fluctuation of \frac{1}{1+ε} occurs, assets become 1-\frac{Lε}{1+ε} times (B).
Through this single operation, assets have
  • a log expected value of 1+(1-L)L\frac{ε^2}{2}\ (~ \sqrt{AB}) times, and
  • a linear expected value of 1+L\frac{ε^2}{2}\ (~\frac{A+B}{2}) times.

The leverage model with drift

Suppose that, in the absence of drift, the investment target is expected to become 1+Δ times during the time it takes to change by ε. Then,
  • if an upward fluctuation occurs, assets become 1+Lε+LΔ times (A), and
  • if a downward fluctuation occurs, assets become 1-\frac{Lε}{1+ε}+LΔ times (B).
Through this single operation, assets have
  • a log expected value of 1+LΔ+(1-L)L\frac{ε^2}{2}\ (~\sqrt{AB}) times, and
  • a linear expected value of 1+LΔ+L\frac{ε^2}{2}\ (~\frac{A+B}{2}) times.
The leverage L that maximizes the log expected value is 0.5+\frac{Δ}{ε^2}, while the linear expected value keeps rising the more leverage you apply.

Appropriate leverage for foreign exchange and stocks

Appropriate leverage for the Nikkei index

The daily fluctuation of the Nikkei index varies by period, but as of 2015 it is roughly 1.0% to 2.0%, so here we use 1.5%. The rate of return of the index varies greatly by period, but it is generally said to be around 6% to 8%, so here we use 7%. There are about 250 business days per year, so we can consider the daily rate of return to be about 0.027%. The Nikkei index is also said to follow geometric Brownian motion, so we assume this model can be applied.
Taking the parameters to be Δ=0.00016,\ ε=0.015, the expected annual return with a leverage of 1 is 7%. The log expected value at this point, converted to an annual return, is about 4.1%. The leverage that maximizes the log expected value is 1.2 times, giving a log expected annual return of about 4.2%, a slight increase compared to 1 times. Meanwhile, the linear expected value is 8.5%, which is also a slight increase. Running an actual program simulation yielded similar results.

Leverage under active management of the Nikkei index

We consider how things change under active management using the algorithm from A long-term investment strategy using the Nikkei average. Estimating generously, a 7% return can be achieved over about 200 business days, so we can consider the daily rate of return to be about 0.033%. Taking the parameters to be Δ=0.00023,\ ε=0.015, the linear expected annual return at a leverage of 1 is 7%, and the log expected annual return is 4.7%. The leverage that maximizes the log expected value is 1.5 times, giving a linear expected annual return of 10.8% and a log expected annual return of 5.4%. Compared with passive management, the log expected annual return improves, and we can see that it becomes easier to apply more leverage.

Leverage when diversifying investments

The log expected value of the result of investing in multiple independent investment targets does not coincide with the log-average of the individual log expected values. This is clear from the fact that if there are infinitely many independent investment targets and you diversify across them, the variance approaches 0 without limit and the log expected value nearly coincides with the linear expected value.
To verify this, we compute using N independently fluctuating investment targets. In this model,
log expected value =\exp(E_{s \in \{0,1\}^N}\log(1+{\rm Σ}_i(L\frac{ε^2}{2}+(-1)^{s_i}(Lε-L\frac{ε^2}{2}))))
=\exp( NL\frac{ε^2}{2} -E_{s \in \{0,1\}^N} \frac{1}{2}({\rm Σ}_i((-1)^{s_i}(Lε)))^2)
=\exp(NL(1-L)\frac{ε^2}{2})
The leverage L that maximizes the log expected value is 0.5 for each investment target, and we can see that when there are two or more investment targets, the log expected value can be maximized by investing more than your margin.

Fluctuation of the home currency

In reality, the home currency also fluctuates just like other currencies, so things do not fluctuate exactly as the earlier theory predicts. Taking this into account,
log expected value = \exp(E_{s \in \{0,1\}^N}E_{S \in \{0,1\}}\log(1+NL\frac{ε^2}{4}+N(-1)^S(L\frac{ε}{\sqrt{2}}-L\frac{ε^2}{4})+{\rm Σ}_i(L\frac{ε^2}{4}+(-1)^{s_i}(L\frac{ε}{\sqrt{2}}-L\frac{ε^2}{4}))))
= \exp(NL(1-\frac{N+1}{2}L)\frac{ε^2}{2})
The leverage L that maximizes the log expected value is \frac{1}{N+1} for each investment target, and we can see that the log expected value is maximized when the margin is spread evenly, including across the home currency.

References