# Portfolio vs individual security Sharpe and Sortino ratios

For an individual security calculating it's Sharpe and Sortino ratios is straightforward.

What I'm curious about is the following:

Let's say I have a portfolio of several securities, which is a distribution of my total capital: for example Asset A has 25%, Asset B has 50%, and Asset C has 25%. At every timestep t, let's assume that I can adjust these percentages to maximize my profits, and that the total distribution always has to add up to 100%.

So at each timestep t my portfolio has a return of r_t, which is the dot product of the distribution vector (a) for each asset at time t with the vector of the change in price for each asset since time t-1.

If I want to calculate the Sharpe and Sortino for the portfolio, would I:

• Calculate the Sharpe and Sortino ratios for each individual security at time t and again take a dot product between my distribution vector a and the vector of each sharpe/sortino ratio for each security
• Directly calculate the Sharpe and Sortino ratios of the portfolio using the returns of the portfolio (r_t) across all timesteps t.

Another good question would be: are both of these approaches fundamentally the same?

Thanks in advance for your help!

## 1 Answer

If you want to calculate the Sharpe and Sortino ratios for the portfolio, you should

• directly calculate them using the returns of the portfolio

Even if the individual sharpe ratios for each of the $$N$$ assets being dot-multiplied by the portfolio weights is equivalent to the above approach, you would be calculating $$N$$ number of Sharpe/Sortino ratios when you could have just calculated the one that you want: the portfolio's ratio

• Could you clarify if ultimately the numerical result of both approaches is the same? I do understand that computationally your answer would be the most efficient. – Alex Pilafian Aug 23 '20 at 18:54
• you should try to program it to confirm whether they are equivalent approaches, but I would expect so, given that the portfolio performance ratio is computed from already aggregated (matrix) returns data, that is, the individual asset returns pre-weighted before the portfolio ratio is computed, whereas the one-by-one asset performance ratio approach, while not pre-weighted, is going at the same procedure except the weight vector is dot-applied, vector algebraically, to the sum of each individual asset's ratio. the redundancy for doing this will be why no one will even bother trying or respond – develarist Aug 23 '20 at 19:02
• This paper "Incremental Sharpe and other performance ratios" Benhamou and Guez (2018) explicitly states that the two methods produce the same result – Alex Pilafian Aug 24 '20 at 13:26