Optimal Market Exposure

Having established in the previous post (Logarithmic Utility of Wealth) that the utility of a rational, small, long-term trader’s account is logarithmic in the account’s size, we are in a position to mathematically optimize how much risk (exposure) the trader should take on a given opportunity. We do this simply by maximizing expected utility with respect to investment size.

Background

Let us consider the typical trade, which has two possible outcomes: After opening the position, the market can move in the trader’s favor and hit the TP (take profit) level, or the market can move against the trader and hit the SL (stop loss) level. In either event, the position is closed with some predetermined profit or loss.

For a trade with given SL and TP levels, let p be the probability of a win if the efficient-market hypothesis is true (i.e. if the market had zero predictability, called a random walk, or Brownian motion). Then 1-p is the probability of a loss under the same condition.

Of course, trading on a market with zero predictability is always a pure gamble. More mathematically, it can be shown that any trade on a random walk market, yields zero expected profit**. That is to say:

\large{pT-(1-p)S = 0}

where T is the TP and S is the SL, both positive and measured from the level where the position was opened. Solving for p, we get:

\large{p=\frac{S}{S+T}}

Loosely speaking, p represents the ”outside world’s belief” of the probability of a win, for the given trade. Let b be our belief of the probability of a win. If we believe in the efficient-market hypothesis, we agree with the “outside world”, and b=p. If we make a trade with the belief that the market’s future is biased in our trade’s favor, then b>p.

Optimization

Before getting to the optimization, we summarize our definitions:

  • T = TP (measured in same unit as m, from opening position)
  • S = SL (measured in same unit as m, from opening position)
  • p = S/(S+T)

And we need a few more definitions:

  • x is the account size (equity)
  • y is the account utility
  • f is the fraction (or multiple) of account size invested in the trade (this is bounded by leverage available)
  • m is the market position (initially, the trade’s opening price)
  • s is the spread and commission (in same unit as m)
  • ∆… denotes change resulting from the trade
  • E{…} denotes expectation (probability-weighted average)

We start with our logarithmic utility function of wealth (from previous post)

\large{y=\ln x}

The account utility after the trade is

 \large{y+\Delta y = \ln(x+\Delta x) = \ln x + \ln(1+\frac{\Delta x}{x})}

So the change in utility is

 \large{\Delta y = \ln(1+\frac{\Delta x}{x})}

The fractional (%) change in account size is

 \large{\frac{\Delta x}{x}=f \frac{\Delta m-s}{m}}

So substituting above, we have

 \large{\Delta y=\ln(1+f \frac{\Delta m-s}{m})}

 \large{E\{\Delta y\}=b \ln(1+f\frac{T-s}{m})+(1-b) \ln(1+f\frac{-S-s}{m})}

In order to determine the optimal exposure, we need to maximize this (E\{\Delta y\} ) with respect to f. If E\{\Delta y\} <= 0 then the trade should not be opened at all.

We find the maximum by locating the stationary point, where

 \large{\frac{dE\{\Delta y\}}{df}=\frac{b(T-s)}{m+f_{best}(T-s)}-\frac{(1-b)(S+s)}{m+f_{best}(S+s)}}=0

yielding

 \large{f_{best}=m\frac{b(T-s)-(1-b)(S+s)}{(T-s)(S+s)}=m\frac{E\{\Delta m\}-s\text{sgn} T}{TS+s(T-S)-s^2}}

which answers our original question: how much exposure to take. This result is only valid if |E\{\Delta m\}| > s . Otherwise E\{\Delta y\}\leq 0 and the trade should not be opened at all.

If spread and commission are small, then

 \large{f_{best}\approx m\frac{E\{\Delta m\}}{TS}}

Application

In order to use this result, you need to first know how confident you are in the trade. There are two equivalent alternative ways of expressing your confidence:

  • What your belief is on the probability of success (b)
  • How many pips per trade you expect to make, in the long run, on trades such as this. (E{Δm})

The table below shows how to use this result in the EUR/USD market assuming

  • that it is currently trading around 1.5
  • that our account size is $10000 (0.1 lots)
  • that we want to place TP of 0.0015 (15 pips)
  • that we want to place SL of 0.0015 (15 pips)
  • that the spread and commission amount to 1 pip on every trade
p (world’'s confidence)b (your confidence)E{Δm} (“pips per trade”)f (optimal leverage)optimal investment
50%50.00%0-67N/A
50%53.30%100.00 lots
50%55.00%1.533.53.35 lots
50%56.70%2676.70 lots
50%58.30%2.5100.410.04 lots
50%60.00%3133.913.39 lots
50%63.30%4200.920.09 lots
50%100.00%15937.593.75 lots

Remember that this entire table needs to be adjusted if the assumptions above it change. Also, you need to be careful to not overestimate confidence (or pips per trade) as this will quickly result in very high exposures. For example, if a particular scenario has given 60% wins in a finite history, it does not mean you can expect anywhere near 60% wins in future, but that is another whole statistical story, for another day.

In this post, we saw how trading requires superior information to the “outside world’s belief”, in order to be profitable. In the next post (Profitability and Information), we will demonstrate that profitability (maximum achievable expected utility gain) is in fact exactly equivalent to information (in the Shannon entropy sense).

** if spread and commission are zero

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