Zero Beta and Minimum Variance Portfolios
Team Latte
January 7, 2007

Long only fund managers sometimes work with zero beta portfolios and non-standard forms of Capital Asset Pricing Model. Recently, we had a very lively discussion on this subject with a couple of quantitative fund managers in a breakfast workshop session.

What is the objective of a fund manager? It is of course, to beat the index and generate above average returns. But at the same time the objective is also to minimize the variance of his portfolio. These objectives are very easily addressed in efficient frontier modelling and Sharpe's algorithm that result from standard form of CAPM. However, there is one major problem with CAPM. One of its main assumptions (like a couple of others) is untenable. CAPM says that investors can lend and borrow unlimited sums of money at risk free rate of interest. Of course, investors can lend unlimited sums of money which is equivalent of buying risk free government bonds, but they certainly cannot borrow at a risk free rate. This produces a kink in the model because the security market line takes risk free rate as the intercept of the market line (as shown below):

In the above, is the risk free rate. So if investors cannot borrow at a risk free rate and if the lending assumption is also withheld for the moment then risk free securities cannot form part of a security market line.

In one of the non-standard forms of CAPM a zero beta portfolio is used to replace the risk free security. In fact, under the no riskless lending and borrowing assumption, it can be shown, both intuitively and mathematically, that a zero beta portfolio lies on a straight line joining a risky portfolios with the market portfolio in an expected return-beta space. This is the so called zero beta version of the CAPM. This general equilibrium relationship is alternatively referred to as a two-factor model.

A zero beta security market line is represented as:

One of the defining features of a zero beta security market line is that the slope of the line at all points is given by the difference of the expected return on the market portfolio and the zero beta portfolio and the security market line always has a positive slope.

Variance Minimization Problem

The fund manager's goal is to choose a minimum variance portfolio for a certain expected return. The manager can invest a portion of his funds in zero beta portfolio and the balance in market portfolio in a way that the variance is minimized. If is such a portfolio and if and are variances of a zero beta portfolio and the market portfolio respectively then the variance of portfolio would be given by

In the above is the proportion of investment fund invested in the zero beta portfolio. Thus the fund manager's objective to invest his funds in the minimum variance portfolio which is a linear combination of a market portfolio and the zero beta portfolio reduces to finding the weight . The necessary condition for the minimum variance portfolio is that the first derivative of the portfolio's variance with respect to the weight should be equal to zero. This gives us the value of the weight as

Since both the variance of the market portfolio and the zero beta portfolio are positive numbers the minimum variance portfolio must involve positive weights on both the zero beta portfolio and the market portfolio. Further, since the expected return on a zero beta portfolio is always lower than the expected return on a market portfolio, the portfolios of and with positive weights must have higher expected return than . Therefore, the minimum variance portfolio has higher return and smaller variance than , the zero beta portfolio, cannot be on the efficient portion of the minimum variance frontier.

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