Annualizing Regression Alpha and Reconciling Portfolio Returns
Summary
The document examines why annualized portfolio excess return may not equal annualized regression alpha plus beta times the annualized market factor return. It describes a daily regression of portfolio excess returns on market excess returns, then compares the reported alpha, beta, market return, and portfolio return. The key relationship is that the mean portfolio excess return equals alpha plus beta times the mean market excess return, subject to consistent return definitions and measurement periods.
The response suggests that rounding the reported inputs can account for the observed gap: using plausible unrounded values within the stated precision produces estimates spanning the reported portfolio return. The discussion also distinguishes alpha, a regression intercept or risk-adjusted performance measure, from excess return itself. It does not establish that changing the annualization basis from 252 to 365 days is appropriate; compounding conventions and consistent annualization of all quantities matter. The numerical reconciliation is illustrative, and the document does not provide the underlying daily observations needed to verify the regression or pinpoint the discrepancy.
Key ideas
- Regression alpha is an intercept in a model of portfolio excess returns against market excess returns.
- The portfolio’s mean excess return is related to alpha plus beta multiplied by the market’s mean excess return.
- Rounded alpha, beta, or market return inputs can create a noticeable mismatch in a reconstructed portfolio return.
- Annualization comparisons require consistent return definitions, periods, and conventions.
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Full text
# Performance measurement
# 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.
## Answer by Circus_beta (score 1, accepted)
https://quant.stackexchange.com/a/57338
Alpha is a risk measurement & is not equal to excess return because of the beta.
See link : https://www.google.com/amp/s/freefincal.com/alpha-not-excess-return/amp/
## Answer by kurtosis (score 1)
https://quant.stackexchange.com/a/57325
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.