# Performance measurement

When I regress the excess performance of a portfolio on the MKT Factor using daily data. I get a Beta of 0.95 and an alpha of 0.00011 that I annualize *252 = 2.77%

I know that the annualized return of the MKT Factor is 8.5% for the period and the annualized performance of the excess return of the portfolio is 11%. When I add up 2.77% + 0.95*8.5% = 10.85% , I don't get the 11% annnualized performance of the portfolio. Why is that? Is my alpha correctly annualized?

Edit : The return of the MKT is annualized using : (1+Return)^(252/Number of days)-1

When Changing for 365 days instead of 252 days I go over the 11% return. Why is that? Alpha is annualized using 365 days and MKT return. Beta stays constant.

I'm guessing you are regressing excess returns $$R_i$$ on asset $$i$$ (so returns $$r_i$$ minus the risk-free rate $$r_f$$). Then, for market excess returns $$R_M=r_M-r_f$$, we have: \begin{align} R_i &= r_i - r_f = \alpha_i + \beta_i R_M + \epsilon_i \quad \text{or} \\ r_i &= r_f + \alpha_i + \beta_i R_M + \epsilon_i, \\ \implies \bar{R}_i &= \hat\alpha_i + \hat\beta_i \bar{R}_M. \end{align} So $$R_M$$ = 8.5%, $$\hat\beta$$ = 0.95, and $$\hat\alpha_i$$= 2.77%.

The one possible bit of wiggle room is in the values you have given. These are surely rounded off. If we consider the values that are possible for the numbers you gave, we can get an idea of how much round-off error might change the results.

\begin{align} \text{Lower: } \bar{R}_i &= 0.000105\cdot252 + 0.945\cdot 8.45\% = 10.63\% \quad \text{and} \\ \text{Upper: } \bar{R}_i &= 0.0001149\cdot252 + 0.9549\cdot 8.549\% = 11.06\%. \end{align}

So, round-off error is a likely culprit.

• By doing so, the return of the portfolio increases, so nothing changes – Circus_beta Aug 13 '20 at 23:10
• The 11% corresponds to Ri , it already has the rf substracted from it. If the risk free is removed the return of the portfolio increases – Circus_beta Aug 13 '20 at 23:20
• Ah! Yes, my comment was wrong; I will delete it so as not to mislead. Hmmm... I think this cold be due to round-off error. Will update my answer to show how. – kurtosis Aug 13 '20 at 23:24
• @Circus_beta Thanks for catching my mistake! Hopefully this explains things. I bet if you dig into a few more digits of your estimate, things will come out closer. – kurtosis Aug 13 '20 at 23:33
• I will look into the round off error. I doubt it's the error since the decimal value in excel goes up to 9. But thanks ! – Circus_beta Aug 13 '20 at 23:37

Alpha is a risk measurement & is not equal to excess return because of the beta.