# How to compute a portfolio PnL and Sharpe?

I understand this is quite the common question but I haven't been able to understand this concept through the previous posts.

My situation is that each day, I'm interested in buying/selling one financial instrument (always the same). Every day, I have a signal that tells me if I should buy or sell or do nothing. In order to test this signal, I'm trying to backtest the strategy on a 5 years period and I have two questions:

1. How do I compute the PnL? I believe I should specify an initial fortune $$V_0$$ that I'm willing to invest on day 0. Then, if say my first signal tells me to buy, should I substract the today's price of the instrument $$S_0$$ to $$V_0$$ and my PnL today would be $$V_0 - S_0$$? What about the other days and the total PnL?

What if my first signal tells me to sell? How can I do that when I don't have any positions?

1. My signals are the expected value $$\mu_t$$ and the risk $$\sigma_t$$ of the instrument's return on each day $$t$$. I can from then compute a daily Sharpe ratio $$\text{Sharpe}_t = \mu_t/\sigma_t$$ . If I want to have a Sharpe ratio of at least $$1$$ at the end of the 5 years, then I believe I should buy/sell on day $$t$$ if $$|\text{Sharpe}_t| > 1/\sqrt{5\times250}$$, is that right?

• Regarding Q1, at time 0 the price hasn't changed yet, so your holdings would be what you've spent buying stock ($kS_0$, with $k$ the number of shares) plus what you have left over: $V_0 - kS_0$. In any case, your day-0 wealth is your initial capital $V_0$. In terms of PnL, you select a "benchmark" (typically your initial wealth $V_0$) and use $V_t-V_0$ as your PnL (so how much you've made so far). You can also use percentages for this, e.g. $(V_t-V_0)/V_0$ or log-returns, etc. – Alex Lostado Apr 8 at 17:11
On any day you should keep track of two things: The cash on hand $$V_t$$ and the amount of the asset you own $$A_t$$ ($$A$$ can be negative if you allow short positions. Or if you want you can restrict $$A$$ to a range such as $$0\le A_t \le U$$ for example, or $$-1 \le A_t \le 1$$ for a simple startegy that is either neutral or long/short 1 unit). You don't allow the cash $$V_t$$ to go negative (print an error message if this happens).
When you buy the asset, $$V$$ decreases by the cost of the asset and $$A$$ increases by 1. When you sell, the opposite: the cash $$V$$ increases by the price of the asset and $$A$$ decreases by 1. It does not matter which event come first.
On any day your wealth is $$W_t=V_t+P_t*A_t$$, where $$P$$ is the price of the asset. The P&L is $$\pi_t=W_t-W_{t-1}$$ and the return is $$r_t=\frac{W_t-W_{t-1}}{W_{t-1}}$$