Alpha as Benchmark-Adjusted Performance and Model Intercept

# What's the exact definition of alpha?


# What's the exact definition of alpha?












I am quite new to finance and I often hear people say 'I have 2 bps of alpha' or 'I have an alpha of two bps'

I don't quite understand what does this mean

For me alpha is about predicting power. At run time have a set of measurements/features X then I have a (linear) model f()

f(X) ->y gives me my estimated forward return y , which is in bps

When people say 2 bps of alpha. Does it mean y has normal distribution whose mean is 2 bps?

I am a newbie here , can anyone provide any insight? Thanks

## Answer by Alex C (score 3)

https://quant.stackexchange.com/a/32960

In Quant Finance we start with the assumption that (until shown otherwise) no one can outperform a simple, passive benchmark. Such a benchmark might be for example the S&P 500 index leveraged up or down by borrowing/lending.

To calculate your alpha we would obtain your monthly returns [actually excess returns $r-r_f$] for the past N months and regress them against the benchmark return, producing two estimated coefficients, Alpha (the constant) and Beta (the linear coefficient). Beta measures the degree of leverage implicit in your strategy. A positive Alpha of say 2bps per month implies that in the past you outperformed the passive strategy by 2bps on average. Alpha is also called "Jensen's Alpha" and you will find ample discussion of it under this name.

In practice the term Alpha is used widely and sometimes indiscriminately to refer to the ability to outperform "the market" and everyone claims to have it even when they don't have the mathematical background to explain what it means. (Try not to be such a person).

## Answer by madilyn (score 3)

https://quant.stackexchange.com/a/32967

As @Alex C had pointed out, the CAPM and subsequently Jensen were probably the original motivations of the term $\alpha$.

Bear in mind that $\alpha$ and $\beta$ are conventional notation for coefficients in a linear regression model, and quite easily as that, we can understand the intuition by thinking of this as an explanatory linear model of portfolio returns against market return. If you rewrite the conventional expression of $\alpha$,

$\underset{y}{\underbrace{r_p}} = \underset{c}{\underbrace{\left(\alpha + r_f\right)}} + \underset{m}{\underbrace{\beta}}\cdot\underset{x}{\underbrace{\left(r_m-r_f\right)}}$

where $r_p$, $r_f$, $r_m$ are your portfolio, risk-free and market rates of return. As you can see underneath my equation, this is really just the conventional linear model that we are familiar with.

The idea here is that you are trying to explain how much of your portfolio performance comes from simple market exposure, the risk-free rate and 'outperformance' of the market. Your outperformance should be a constant function of the market return so it should go into the intercept term. Here, we abuse the notation slightly and say that $\alpha$ is the amount by which the intercept that exceeds the risk-free rate.

This brings us to your question:

> Does it mean y has normal distribution whose mean is 2 bps?

No, this just means that we assume (by prescribing a linear model) that the residual $\epsilon$ is normally distributed:

$\hat{r}_p = \left(\alpha + r_f\right) + \beta\left(r_m-r_f\right) + \mathbb{O}\left(\epsilon\right)$

and $\alpha = 0.02\%$.

This is all quite simple, this leaves us remaining problem - why do people seem to refer to different things when they use the term nowadays?

The reason there's confusing definitions of $\alpha$ is that subsequently, people tried explaining $r_p$ with multiple regressors that we consider public information and should be priced into assets, e.g. price-to-book ratio (value factor), market capitalization (size factor), momentum etc. One would argue just as it requires no skill to operate a portfolio that has high return simply because of 1:1 exposure to a high market return, it requires no skill if your portfolio returns can be explained by these commonly accessible data points, so now $\alpha$ should be redefined as what additional 'secret sauce' you have modulo all these terms.

To take this to the extreme, one might argue that famed value investors like Warren Buffett have no skill at all as their alpha term would be close to 0 if they are concentrated on buying value stocks.