# Computing Sharpe ratio correctly when adding more funds

I've a very basic question:

Assume, as time $$t_0$$, I started a algorithmic trading strategy with initial capital $$X_0$$. My strategy on subsequent $$N$$ time frames generated returns $$r(t_1), r(t_2),..r(t_N)$$. Now at time step $$t_{N+1}$$, I added an additional $$X_{N+1}$$ capital to my algorithm for trading and also increased the position size (irrelevant I assume).

What do I need to do to correctly compute the annualized Sharpe ratio?

• Sharpe ratio is not based on capital. It's based on returns. Putting more money into a strategy won't change its level of return and therefore won't change its Sharpe ratio – develarist Aug 24 '20 at 7:09
• Here is where I am stuck... How do you compute the excess return of a timestep without knowing the initial capital? For example if my portfolio value is p(1), p(2), ...p(n), then the return will be: (p(2)-p(1))/p(1), (p(3)-p(2))/p(2), .., (p(n) - p(n-1))/p(n-1), correct? In this case, how do I get the original portfolio value without the initial capital? – joshi Aug 24 '20 at 7:26
• excess return is the return generated by your strategy minus some benchmark reference rate like the return from a risk-free asset. so far, you have not mentioned anything about having a benchmark so don't see what you want your returns to be in excess of. Besides this, your return formula is right, but capital is computed as $p\times q$, not $p$ alone, so you're again confusing returns and capital. when you said "added additional capital" and "increased position size", are these two separate activities in that sense that a constituent asset (of many) was given a higher weight at $t_{N+1}$? – develarist Aug 24 '20 at 12:47