Sharpe Ratio Calculation: Consistent Return Frequency and Risk-Free Rates

# Sharpe Ratio - my own calculation differs from Yahoo finance, Morningstar


# Sharpe Ratio - my own calculation differs from Yahoo finance, Morningstar












I am trying to compute the Sharpe ratio for my portfolio. To check that I am doing this correctly, I am first trying to compute it for SPY (the S&P 500 index).

```
S.R. = mean({SPY return j - risk-free return j})/std dev({SPY return j})
```

I am using annualized monthly returns that I compute from the SPY index itself. For the risk-free return, I am using the 10-year T-bill. Again, I am using monthly data, same as Yahoo! and Morningstar.

This is how I compute the annualized monthly return for month j:

```
temp = [ ( (value at month j) - (value at month j-1) ) ] / (value at month j-1)
return for month j = (1+temp)^12 - 1
```

For some reason, I get 0.61 for the Sharpe Ratio. Yahoo! and Morningstar report it to be about 1.3, again amortized monthly. What am I doing wrong? Here is some sample data:

```
        S&P price   T-bill price      S&P return   T-bill return
...
Jan-14  176.55      1165.31     
Feb-14  184.59      1209.48           0.706417809  0.562739592
Mar-14  186.12      1212.98           0.104125623  0.035283725
Apr-14  187.41      1198.61           0.086417122  -0.133255532
May-14  191.76      1218.43           0.316991962  0.217509175
Jun-14  195.72      1230.87           0.277986402  0.129637856
...
```

For some reason the computation doesn't work out for me. Can anyone see what I am doing wrong?

## Answer by Simon (score 7, accepted)

https://quant.stackexchange.com/a/14038

Your approach of computation is not very standard. Specifically, you do not need to compute the annualized monthly return. One can compute the annualized Sharpe ratio from return sampled at any frequency using the following Generalized formula:

$$ Sharpe = \frac{E|R_p - R_{rf}|}{\sqrt{var(R_p - R_{rf})}} * \sqrt{N}$$ where $R_{rf}$ is the benchmark/ risk-free return, $R_p$ is the portfolio return, $N$ is the number of sampling periods in a year. The portfolio return and risk-free rate can be of any interval (daily, weekly, monthly, etc), as long as they are consistent with each other.

In your case, the risk-free rate is effectively 0 nowadays, and N is 12 assuming you are using monthly returns, and N is 252 if you are using daily returns.

Hope it helps.

## Answer by RRL (score 2)

https://quant.stackexchange.com/a/14037

To compute Sharpe Ratio the risk-free rate has to be proxied by something like the 1-month T-Bill yield. The 10-year Treasury Bond return is not a "risk-free" return.