# Hedge backtesting: ex-ante Beta vs observed Beta (is this even possible?)

A global equity portfolio has for objective to outperform a benchmark (MSCI World). I hedge the sensitivity of the portfolio to MSCI World (the beta) so that only the alpha remains unhedged.

The ex-ante beta is calculated by looking at the covariance and volatility of the portfolio and MSCI World for the past 2 years. So if the Beta is say 0.98, I short 0.98*portfolio exposure (beta hedging).

Now I want to backtest how accurate the model has been in "predicting" the right beta by comparing the observed beta to the ex-ante (predicted by the model) beta.

How do I decompose the portfolio return in Beta and Alpha, so that I can check if the model was good when predicting beta?

(I must precise the portfolio is re balanced monthly, this means I don't have a lot of ex-post data to play with before another beta / hedge ratio is calculated and implemented)

An answer followed by a criticism.

Answer: So you've calculated the beta of the portfolio as of a point in time let's call it t=0, using the last two years of data. Let's call this beta_-24m_0m. One month from now, use just the previous month's data to calculate the beta of the portfolio over what is then the past month, let's call this beta_0m_1m. This assumes that you didn't rehedge over that time period. The difference in the betas is what you could call the ex-post - ex-ante beta difference. Then you can look at what pnl would have been if your hedge at time 0 had been the ex-post figure as opposed to the ex-ante figure.

Criticism: Using the last two years of data to calculate your beta is basic, in a bad way. I'm sorry to say, but it's true, but now you know. There are so many ways that are much better. Take a look at the James-Stein estimator to understand why betas should be biased towards 1. And take a look at this paper, you should really be doing this or something else good if you're trying to minimize the error: http://www.ledoit.net/honey.pdf

• Thanks for your input R down by the bay. I will look into this. However the Beta is calculated by a third party model (MSCI) so it may be more sophisticated than I think. The issue here is not the calculate the ex-ante Beta but the realized Beta over 1 month of data only. – tweedi Apr 22 at 21:32
• why can't you calculate the realized beta? there's something here that I'm not getting... don't you only need literally two days of returns for the market and your portfolio to be able to estimate the two unknowns of your portfolio, the alpha and the beta? – RWP - Down by the Bay Apr 23 at 23:44
• What is your method to do this? Do you first calculate alpha by doing the difference between the portfolio and the market? and then you have an equation of type Portfolio return = Alpha + [Beta*Market Return] (+an error term)? – tweedi Apr 27 at 20:22
• it's a one step process. you don't calculate anything first. it's just a linear regression where portfolio return = alpha + beta * mkt return + error term. the unknowns here are the alpha and beta so as long as you have two days of returns then you can estimate alpha and beta. now, your estimates will be "better" if you have more days of data, however, you can do this with as few as two days. – RWP - Down by the Bay Apr 29 at 13:53

My issue was how to calculate Beta for the past month with only a month of daily data. I thought this was only done via linear regression and a month was insufficient.

I believe I found the answer:

First I calculate the active return (alpha) of the portfolio over the period:

(Portfolio return - Market return). This is my over or under performance compared to the market.

Then I find how many contracts I should have shorted at the beginning of the period to land on the same active return. From the # contracts I find the Beta = Portfolio value to hedge / (# contract * contract size * future price).

• beta can be found in a shorter time frame using regression, just might be less accurate. Also you could use ur formula above. – JazKaz May 22 at 22:40