Reverse Optimization: A Glimpse into an Investor’s Expected Returns?

Investment professionals who know what “reverse optimization” means typically know of two methods. The more widely used method is known as the Black-Litterman method. That approach is well covered elsewhere, especially on StyleAdvisor.

The less widely-used method, described by Bill Sharpe in 1974 in “Imputing Expected Security Returns from Portfolio Composition,” (linked at bottom of this post) merely solves CAPM inductively: given a set of asset weights, volatilities, correlations, and a return expectation for the entire portfolio, you can infer the expected returns on the assets that make this set of weights optimal.

Theoretically this technique gives you a glimpse into an investor’s return expectations. In reality, that’s only true if the investor used a single Mean-Variance Optimization (MVO) to arrive at these portfolio weights, which may be unlikely.

Regardless of whether it allows you to read an investor’s mind, the Sharpe technique is very useful for understanding how a typical MV optimizer works.

Try this Reverse Optimization Spreadsheet
(Update: The method described in this workbook finds the return expectations that make this set of portfolio weights the highest Sharpe ratio portfolio. TMA, 6/15/2011)
I’ve embedded an interactive spreadsheet below that invites you to play with different assumptions about your asset classes. Change your assumptions about their volatilities, correlations, weights, or return premium for the portfolio as a whole, and you change the respective implied return premia for the assets.

In the spreadsheet below, the Blue font cells are editable and the rest should be locked. The cells highlighted in yellow reveal the return expectations consistent with the set of weights, volatilities, and correlations. Go ahead and plug them into your own optimizer to verify.

MV optimizers typically give you a frontier of different portfolio allocations that all share the same quality: the ratio of return (or excess return) to contribution to portfolio risk is equal for all assets. The spreadsheet embedded here simply takes the weights, volatilities, and correlations, and measures the total portfolio variance. It then assigns each asset’s expected return above cash to be proportionate to the ratio of its covariance with the portfolio to the portfolio variance.

As you’re entering the same type of return (total or excess) into your mean-variance optimizer, it offers a convenient glimpse into how mean-variance optimizers work.

Play around with the return for the total portfolio and you’ll notice the returns for the assets simply change proportionately. But if you change weights, correlations, or volatilities, the returns necessary to make this an optimal portfolio can change dramatically.

PS: This is my first time embedding an editable spreadsheet onto a blog page using Zoho, so please alert me if something doesn’t seem to be working as you think it should.

Update 9/2011: I’ve been pleased to discover that Zoho’s array functions appear to work nicely now, so I’ve updated this workbook to use them. Using matrix functions should make the workbook work more efficiently now. Tom