# How much correlation between alpha and forward returns can be considered good

There are many way to quantify the quality of an $\alpha$ (prediction of the future return of a security). I heard/do many things, most of them are equivalent to correlation.

Typically $corr(\alpha,fr)$ (where $fr$ is the vector of returns that got realized) is a simple way to express adequation. Of course if you predict the next tick or the next close the numbers are totally different, similarly calculating a cross sectional correlation on 1000 instruments or 10 might change everything.

For example, for a one-day horizon (say you predict the close of day D+1 just before the close of day D), on a universe of >200 instruments... what correlation qualifies as "good alpha" ?

# Introduction

The answer below measures performance against a benchmark. Alternatively, one can ignore benchmarks and use the Sharp Ratio. I'm not aware of any generally accepted and published ranking for a 'good' SR.

The Level II CFA curriculum contains a number of chapters from the book Active Portfolio Management: A Quantitative Approach for Providing Superior Returns and Controlling Risk by Richard C. Grinold and Ronald N. Kahn that suggest an approach to the problem of measuring portfolio manager performance. It is a formalized version of what you propose in your question. The focus of the book is portfolio management and performance is compared against a benchmark. I've summarized the discussion in the book below.

# Definining $\alpha$

The first step is to calculate the residual or risk adjusted return, in the CAPM for a portfolio $p$ you would have

$$R_p(t) = \alpha_p + \beta_p \cdot r_\beta(t) + \varepsilon(t)$$

with $\varepsilon(t)$ a zero mean random variable. Of course, you can also use a multi-factor model but lets keep it simple. The residual returns are defined as

$$\theta_p(t) = \alpha_p(t) + \varepsilon(t).$$

Clearly, $\mathrm{E}[\theta_p(t)] = \mathrm{E}[R_p(t) - R_B(t)] = \alpha_p(t)$ where $R_B(t)$ is the expected benchmark return. They convincingly argue that $\alpha_p(t)$ is insufficient to judge performance and risk also should be taken into account.

# Defining risk

As said, performance is compared against a benchmark and so the risk is also measured against the benchmark. Risk is denoted by $\omega_p$ and is simply the standard deviation of the residual return

$$\omega_p = \sqrt{\mathrm{Var}[R_p - R_B]}$$.

# The information ratio

The information ratio of a portfolio is $$\mathrm{IR}_p = \frac{\alpha_p}{\omega_p}$$ and the information ratio of a portfolio manager is $$\mathrm{IR} = \max\{IR_p|p\}.$$ The information ratio has one very convenient property:

The information ratio is independent of the manager's level of aggressiveness (emphasis adjusted by me).

# What's good?

According to Grinold and Kahn an 'Exceptional' manager has IR = 1.00, a 'Very good' one has 0.75 and a 'Good' manager has IR = 0.50 (CFA Institute Level II 2014 Volume 6 Derivatives and Portfolio Management, Reading 58, Table 3).

# The Fundamental Law

An alternative way of defining and obtaining(!) the IR is the following: $$\mathrm{IR} = \mathrm{IC} \cdot \sqrt{\mathrm{BR}}$$ where $\mathrm{IC}$ is the Information Coefficient and $\mathrm{BR}$ is breadth. This tells you how to obtain a good information ratio. In very few words: $\mathrm{IC}$ is the correlation of your bets with the ex post realization and $\mathrm{BR}$ the number of independent bets you can make. For example, the breadth of a stock picker depends on the number of stocks he follows and the number of revisions he makes.

By the way, Grinold and Kahn also elaborate on the implementation of your strategy but that's a whole different chapter.

• This is mostly from the top of my head with a bit of help of the CFA curriculum. The book itself is a tome, I probably have forgotten some stuff. If I remember again I will edit. – Bob Jansen Aug 30 '14 at 20:14
• Ok so according to the last rule say you get a 10% correlation so $IC=.1$ and you get this on $BR=100$ then you get an $IR=1$... I guess $BR$ is a number of stocks, otherwise trading a billion times would explode the IR. I feel this rule is a bit weak as it does not take into account the horizon. I can easily a IC of 20% on the next 30 seconds for 500 stocks... of course if you look at this you need to think about how much does the spread cost you... – statquant Aug 30 '14 at 20:25
• I've added that the bets should also be independent and remember that it's the ex post realization (Grinold and Kahn say actual outcome). This realization does include trading costs. – Bob Jansen Aug 30 '14 at 20:45
• That the correlation include the cost is natural I still think there must be a time-scale component... But anyway that's a good answer – statquant Aug 30 '14 at 20:51
• This metric is also not really made for backtest I think as, as you point out the correlation should be taken from the realised returns not the alpha but still... – statquant Aug 30 '14 at 20:55