# Is portfolio beta truly just the weighted average of stock betas? If so, is there a way to distinguish betas based on composition?

For instance, assume you have three portfolios.

Portfolio A is 100% invested in a market index with β=1.

Portfolio B is a 50/50 mix of two stocks (Stock A β=1.5 and Stock B β=0.5). The weighted average of these two gives us a portfolio β=1.

Portfolio C is a 50/50 mix of one stock and cash (Stock C β=2.0 and cash β=0). The weighted average of these two gives us a portfolio β=1.

Assuming a starting value of \$100, if all of these portfolios have β=1, why is it that if we experienced sequential returns of -10% followed by +11.11% Portfolio A would finish with a value of \$100 whereas Portfolio B & C end up with values of \$99.72 and \$98.89 respectively?

Is there a way to indicate the varying compositions of these portfolios with the same portfolio β=1? Or am I calculating β wrong to begin with?

• Beta measures the market volatility of the stock. It says nothing about market return. High beta stocks have the potential for higher returns, but nothing is guaranteed. Stocks can have a very high beta and lots of negative returns. – Peter K. Apr 22 '17 at 20:25
• Thanks for the response. I should have clarified that I'm making a simplifying assumption here that R^2=100. That would change things, correct? Since a stock with β=1.5 and R^2=100 would experience returns 150% more extreme than the market. That's obviously not realistic in the real world, but I'm more interested in understanding the divergence between the portfolios in the overly simplified example above. – John Jones Apr 23 '17 at 18:37
• β is an instantaneous correlation. As the portfolios change over time, β changes too unless you rebalance the portfolios to keep it constant. E.g. in Portfolio C, after the -10% returns, the stock will be at 80% of its value so your portfolio will be 44%/56%, with β = 0.88. And in reality Stock C's β will probably have changed too. – Tavian Barnes Apr 24 '17 at 21:40

The answer to your initial question is yes. The portfolio beta is the weighted average of the individual betas. This is not true for many portfolio statistics (such as volatility) but it is true for beta. You can prove this for yourself by writing out the two regression equations and then adding them together.

The answer to your second question (in the title) is that there are many ways to distinguish two portfolios that have the same beta. But beta, by definition, is not one of those ways. Beta is just a single statistic. If you have two portfolios that have the same beta, then of course you can't use beta to tell them apart. You will have to use something else.

Now, why your example doesn't work mathemetically: Beta is a single-period concept (it is a marginal effect) and the CAPM equation holds over single periods only. This is why your example works perfectly in the first period but breaks down in the second.

This is not particular to beta. It's a feature of compounding returns. It is not possible for one stock to always earn twice what another does at more than one horizon, which is what your example assumes. A simpler example illustrates this: Suppose Y earns twice what X does in every period and that X earns R1 and then R2. Then over two periods, an investment in X earns a return of

(1+R1)(1+R2) - 1 = R1 + R2 + R1*R2

and an investment in Y earns a return of

(1+2*R1)(1+2*R2) -1 = 2R1 + 2R2 + 4*R1*R2

Notice that the return on a Y investment over two periods is not twice the return on an X investment. This nonlinearity of return aggregation is what makes your 2-period example fail to work.

Intuitively this may seem odd. I find it best to think of beta as a measure of the levered exposure to market risk. This means betas are not constant, just as leverage is not constant. If a levered stock loses money, its leverage increases mechanically. Stock A in portfolio B will no longer have a beta of 1.5 after the first return. We say ex ante that we expect a return of 1.5 times the market in the second period only because we do not know what will happen in the first period. The weights within the portfolios you have constructed will also change after the first period, by the way.

Bottom line: Using a CAPM-style equation to predict returns relative to the benchmark works only in expectation and over a single period.