Backtesting Ex-Ante Beta Estimates for Equity Portfolio Hedges
Summary
The document asks how to assess a monthly equity portfolio’s estimated market beta and hedge ratio against the beta observed after the hedge period. The portfolio estimates beta from a two-year covariance and volatility window, then hedges its benchmark exposure. One response suggests comparing that estimate with a beta measured over the subsequent month, assuming the hedge was left unchanged, and comparing the resulting profit and loss with what a hedge based on the later estimate would have produced.
A second response reframes the monthly comparison through active return and infers the hedge exposure consistent with that return. The discussion also criticizes a simple historical beta estimate and points to shrinkage as a possible improvement. Its evidence is conceptual rather than a worked backtest, and it does not settle how to estimate beta reliably from a short monthly sample, account for changing portfolio holdings, or separate alpha from benchmark exposure in a full return decomposition.
Key ideas
- The proposed backtest compares a beta estimated before a month with beta inferred over the hedge period.
- A hedge left unchanged during the period makes the ex-post comparison interpretable.
- Profit and loss can be compared with the outcome under a hedge based on the later beta estimate.
- The discussion raises shrinkage toward one as a possible improvement over a basic historical estimate.
- Monthly observations and portfolio rebalancing limit the evidence available for evaluating hedge accuracy.
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Full text
# Hedge backtesting: ex-ante Beta vs observed Beta (is this even possible?) # 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) ## Answer by RWP - Down by the Bay (score 1) https://quant.stackexchange.com/a/53350 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 ## Answer by tweedi (score 1) https://quant.stackexchange.com/a/53539 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).
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