# Measuring alpha (Academia vs the Industry)

During academia, I learned to evaluate the performance of a portfolio by calculating alpha as the following:

$\alpha_{i} = (R_{it}-R_{ft})-[\beta_i(R_{BMK_t}-R_{ft})]$

where $\alpha_i$ and $\beta_i$ are obtained through regressing past excess returns with the benchmarks excess returns.

Using such approach, I would measure alpha after capturing market risk premium.

In the industry, however, I noticed that portfolio managers calculate their alpha when distributing their factsheets as the following:

$\alpha_i = R_{it} - R_{BMK_t}$

Simply by subtracting the benchmarks (average or cumulative) returns from the portfolios (average or cumulative) returns.

One could conclude that using the such approach is somewhat misleading since the portfolio return is not corrected for market risk premium, $\beta$. Therefore, a portfolio manager could simply hold a portfolio with $\beta > 1$, and generate alpha when the benchmark goes up.

I noticed that many fund managers claim that they are outperforming their benchmarks with significant alphas, however, when I correct for market risk premium, the alpha becomes either negative or very insignificant.

My questions are:

1) Why the industry use such methodology to present their alpha and does it mislead retail and institutional investors thinking that the manager is outperforming his benchmark through skill?

2) Do you think measuring alpha as $\alpha_i = R_{it} - R_{BMK_t}$ is sufficient to show skilled managers.

This is an open discussion so feel free to comment and share your thoughts.

Kind Regards,

Both questions are not as straightforward as @Hui (and most academics and practitioners) would immediately think. I would try to put in my two cents to answering your question 1.

Short answer: It might have to do with the bias-variance tradeoff, as measuring the alpha precisely is a tricky task in small samples (and young funds do have short histories). Simplicity might be your friend here.

Long answer: let's write down what we are measuring. It's the alpha here:

$$R_t = \alpha + \beta R_{mt} + \varepsilon_t$$

I hope you are familiar with the difference between the 'true', or unobserved, or god-given alpha in the formula above, and its estimate $\hat{\alpha}$ that you calculate possibly as OLS — the 'academic' — estimate $\hat{\alpha}_a$ or as... let's call it industry estimate $\hat{\alpha}_i$:

$$\hat{\alpha}_i = \frac{1}{T} \sum_t (R_t - R_{mt}).$$

The OLS estimate on a sample of size $T$ (all the usual assumptions apply) has the following properties (without proof):

$$E[\hat{\alpha}_a] = \alpha, \\ Var[\hat{\alpha}_a] = \frac{1}{T}(\sigma^2 + SR_m^2\sigma^2),$$

that is, it is unbiased and has variance proportional to the variance of residuals $\sigma^2$ and the benchmark Sharpe ratio $SR_m$. The industry estimator has the following properties (work out the proof, it's moderately fun):

$$E[\hat{\alpha}_i] = \alpha + (\beta - 1)E[R_{mt}], \\ Var[\hat{\alpha}_i] = \frac{1}{T} \left( \sigma^2 + (\beta-1)^2 Var[R_{mt}] \right),$$

that is, it is biased and has variance proportional to the variance of residuals and the variance of the benchmark scaled by how much the beta differs from 1.

Now, it is usually the case that the beta of a fund manager is close to 1: everyone is tracking their favorite MSCI or whatever close enough. Comparing the variances, you see that the variance of the 'indstry' estimate can be so much-much-much lower that the small bias can be tolerated.

1. Normally, alpha is the excess return beyond the benchmark, meaning if the benchmark return is 0, your portfolio will still have a return of alpha - the easiest way to understand. However, your second alpha, αi=Rit−RBMKt, is incorrectly addressing what alpha is, at least from most people's understanding. Could you provide any papers that use (αi=Rit−RBMKt) ?

2. This is definitely insufficient or I should say it's wrong. One of most widely used performance measurement for portfolio managers is sharpe ratio, which is a measure that indicates the average return minus the risk-free return divided by the standard deviation of return on an investment.So if a portfolio manager's strategy is based on extremely high volatility, then no matter how high the return is, he/she will never be qualified as a good portfolio manager.

As you are saying it's an open discussion. I like to add two more cents that sharpe ratio could be lying as well. A short premium strategy(especially deep out of the money) could derive a very high sharpe ratio if there no major market events or volatility shocks. However, the true risk is horrible. So understanding their strategies is more important than solely looking at the performance number.

Moreover, if we talk about benchmarked equity portfolios. Then your portfolio will have a beta of roughly $1$ and over the last years the influence of risk-less interest is neglible.