# Comparison of Carhart alphas (four-factor model)

I would like to compare two strategies through the alpha Carhart obtained. The idea is to find out which one is more profitable (or the least bad). For the first strategy, I obtained a significant negative alpha. For the second one, the alpha is positive but not significant.

I would like to be sure to interpret this correctly: does this mean that the first strategy underperforms the second strategy? Given that both strategies (their returns) are regressed with the same model (4-F) at the same period.

What do you think?

Thank you very much!

# Setup

From your question, I assume you have two strategies $$A$$ and $$B$$ with excess returns $$R_{A,t}, R_{B,t}$$. You model these excess returns using the Carhart 4-factor models (FF3 factors + the fourth monthly momentum factor which looks at winners minus losers, WML):

\begin{align} R_{A,t} &= \alpha_A + \beta_{A,M} R_{M,t} + \beta_{A,HML} HML_t + \beta_{A,SMB} SMB_t + \beta_{A,WML} WML_t + \epsilon_{A,t}, \\ R_{B,t} &= \alpha_B + \beta_{B,M} R_{M,t} + \beta_{B,HML} HML_t + \beta_{B,SMB} SMB_t + \beta_{B,WML} WML_t + \epsilon_{B,t}. \end{align}

You find $$\hat\alpha_A<0$$ and is statistically significant while $$\hat\alpha_B>0$$ but not statistically significant.

# Questions

You ask if you can infer if:

• one strategy is more profitable than the other and, later,
• if A "underperforms" B.

# Profitability

The answer to the first question is that you cannot infer one strategy will be more profitable than the other. Depending on the betas and the factor returns, one strategy might outperform the other over any time period.

For example, suppose all betas are 1 except $$\hat\beta_{A,SMB}=1.5$$ and $$\hat\beta_{B,HML}=1.5$$. If the SMB factor has a high return and the HML factor has a return of 0, the additional exposure to SMB will allow strategy $$A$$ to overcome the negative $$\alpha_A$$. However, if the HML factor has a high return and the SMB factor has a return of 0, the additional exposure to HML will allow strategy $$B$$ to return more that strategy $$A$$.

# Outperformance

You also ask if one model "outperforms" the other. This can be interpreted as:

• profitability being higher (indeterminate, as discussed above),
• return per unit of risk being lower (also indeterminate without knowing factor betas), or
• having significantly higher alpha.

The latter definition is especially useful if we are measuring hedge fund performance -- since nobody should pay high fees for factor exposure which can be obtained cheaply.

### Pooled $$t$$-test

If we use the latter definition of outperformance, we can test the difference between $$\hat\alpha_A$$ and $$\hat\alpha_B$$ using a pooled $$t$$-test: check if $$\left|\frac{\hat\alpha_A-\hat\alpha_B}{\sqrt{s.e.(\hat\alpha_A)^2 + s.e.(\hat\alpha_B)^2}}\right| > t_{df,97.5\%}$$.

### Paired $$t$$-test

Alternately, we could construct a paired $$t$$-test by estimating the model $$D$$:

$$R_{A,t}-R_{B,t} = \alpha_D + \beta_{D,M} R_{M,t} + \beta_{D,HML} HML_t + \beta_{D,SMB} SMB_t + \beta_{D,WML} WML_t + \epsilon_{D,t}.$$

You then check the difference alpha for significance: $$\left|\frac{\hat\alpha_D}{s.e.(\hat\alpha_D)}\right|>t_{df,97.5\%}$$.

### Result

In this case, we know $$\hat\alpha_A<-$$ is significantly different from 0 and $$\hat\alpha_B>0$$. Therefore, we can conclude that for an alpha-based measure of outperformance, strategy $$B$$ outperforms strategy $$A$$.