Testing Strategy Alpha with a Benchmark Regression
Summary
The document describes testing whether a trading algorithm’s return above a risk-adjusted benchmark could plausibly be due to chance. It proposes regressing portfolio excess returns on benchmark excess returns, where the risk-free return is subtracted from both series. The estimated intercept represents alpha, and dividing it by its standard error gives a t-statistic for assessing whether that alpha differs significantly from zero. The example cites a Treasury bill assumption and a CAPM-style setup.
The answer clarifies that the regression residual is the unexplained return component, with a mean of zero under the fitted regression and nonzero dispersion. The factor loading is estimated as part of the regression rather than specified in advance. This is a basic framework, not a complete validation procedure: the note does not discuss return dependence, multiple testing, non-normality, or estimation choices that can affect standard errors and significance. A small p-value alone does not establish that a strategy will continue to perform.
Key ideas
- Regress portfolio excess returns on benchmark excess returns to estimate alpha and beta.
- The intercept measures performance not explained by the benchmark in this regression.
- The alpha t-statistic uses its estimated standard error to assess evidence against zero alpha.
- Regression residuals represent return variation left unexplained by the benchmark.
- Statistical significance alone does not establish persistent future strategy performance.
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# Answer by phdstudent (score 3, accepted)
# How to perform t-student test to validate whether the final cumulative wealth of a trading algorithm is achieved by luck?
As suggested by Huang et al. (2018), in order to test whether simple luck can generate the return of a trading algorithm, it is possible to conduct a statistical test to measure the probability of this situation.
> First, we separate the portfolio daily returns into two components: one benchmark-related and the other non-benchmark-related by regressing the portfolio excess returns against the benchmark excess returns. Formally, ${s_t} - {s_t}\left( F \right) = \alpha + \beta \left( {{s_t}\left( B \right) - {s_t}\left( F \right) + \epsilon\left( t \right)} \right)$ where ${s_t}$ stands for the portfolio daily returns, ${s_t}\left( B \right)$ denotes the daily returns of the benchmark (market index) and ${s_t}\left( F \right)$ is the daily returns of the risk-free assets (here, we simply choose Treasury bill and set it to 1.000156, or equivalently, annual interest of 4%). This regression estimates the portfolio’s alpha($\alpha$), which indicates the performance of the investment after accounting for the involved risk. Then, we conduct a statistical t-test to evaluate whether alpha is significantly different from zero, by using the t statistic $\frac{\alpha }{{SE\left( \alpha \right)}}$, where $SE\left( \alpha \right)$ is the standard error for the estimated alpha. Thus, by assuming the $\alpha$ is normally distributed, we can obtain the probability that the returns of the proposed strategy are generated by simple luck. Generally speaking, the smaller the probability, the higher confidence the trading strategy.
However, it is not specified what is the value of $\epsilon\left( t \right)$ or how they determine the value of $\beta$ in the equation above. Any suggestions?
## Answer by phdstudent (score 3, accepted)
https://quant.stackexchange.com/a/79224
Most likely the equation should be:
${s_t} - {s_t}\left( F \right) = \alpha + \beta ( {{s_t}( B ) - {s_t}( F ) ) + \epsilon ( t )} $
So $\epsilon ( t )$ are just the residuals of the regression. On average they will be zero. They will have positive standard deviation though.
This is just a simple CAPM type regression, with some weird notation.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.