# What is the distribution of the trend-following strategy PnL?

Suppose you start with zero dollars; and a stock is at \$100 and goes up and down \$1 equally likely, i.e., both with probability 50%.

A trend-following strategy, during a period of 31 days, works as follows:

1. If the stock goes up you buy one share and sell it at the end of the 30-day period.
2. If the stock goes down you sell (short if necessary) one share and buy it back at the end of the 30-day period.

What is the distribution of your PnL at the end of the 31-day period?

• Not sure I understand the question. There is only outcome, so how can you have a distribution of returns? Dec 27, 2021 at 22:33
• @user42108 - the distribution over all possible realizations of up/down each day. There are $2^30$ possible trajectories, and order matters. Dec 27, 2021 at 23:58
• @rubikscube09 - thanks for the clarification. Perhaps the question should be rephrased so as to be more clear. Dec 28, 2021 at 3:50
• @michael - is this a homework question? There have been several papers on the distribution of trend following strategies and this is a very stylized example. Dec 28, 2021 at 3:51
• (1) A Monte Carlo simulation could help (if nothing else it would help to pin down the exact rules of the game, which are not clear to me. Do you enter the market only once at the beginning or once every day for 30 days you make a new trade?) (2) This does not seem to me to be how real life trend following works. Dec 28, 2021 at 11:33

• event +1: given the price goes up (at \$101) you buy and wait 30 days, the distribution of the PnL is a discretized Gaussian $${\cal N}(0, 30)$$ because the price at $$T=31$$ is centered at \$101 and with a std of $$\sqrt{30}$$.
• event -1: given the price goes up (at \$99) you sell and wait 30 days, the distribution of the PnL is a discretized Gaussian $${\cal N}(0, 30)$$ because the price at $$T=31$$ is centered at \$99 and with a std of $$\sqrt{30}$$.
Since these events occur with the same probs of $$1/2$$, the PnL is a mixture of these two independent Gaussians, it is itself a Gaussian $${\cal N}(0, 30)$$.
The scheme you describe implies that the number of shares in position at any time t is equal to the current stock price minus 100, since you buy another share every time it upticks and sell one every time it downticks. Therefore the cumulative p/l is of the form $$\Sigma{(S_t-100)dS_t}$$ and if you sum this you should get that the final p/l is of the form $$S_T^2 - E[S_T^2]$$. If $$S_T$$ is normally distributed, which your scheme would approximately satisfy , then the p/l is a chi squared distribution with an adjusted mean such that the expected p/l is zero.