Why would I hold a zero beta stock in my portfolio?
Some Background
In class we wondered why you would hold a zero beta stock, that is a stock that has the same expected return as the risk free rate yet has its return is uncertain, but the risk free return is certain. The only way you would in fact hold such a stock is if it gave you something the risk free return did not give you. I said this was because this stock provided diversification, and this example illustrates this point. Before I work the example, there is something you should think about. If the CAPM is true, the only stocks that are held are those in the market portfolio, so by holding a zero beta stock I am also saying it is part of the market portfolio.

The Example
Assume the CAPM holds with a riskless asset and the market porfolio contains a zero beta stock. Since everybody holds the market, this means that everybody willing chooses to hold a zero beta stock. What I want to know is why people just don’t drop this stock out of their portfolios and replace it with the risk free security.

What would happen to the expected return of my portfolio if I took the 5% of my wealth out of the zero beta stock and put it in the risk free asset instead?
Nothing, since the from the CAPM equation we know that the expected return of the zero beta stock must be the same as the expected return of the riskless asset.

What would happen to the standard deviation of my portfolio if I replace the zero beta asset with the riskless asset? We first need to know the correlation between portfolio Z and portfolio A
To figure this out we use the fact that Z is a zero beta portfolio, that is, Cov(M,Z)=0 so Cov(0.80 A + 0.20 Z, Z)=0 so 0.8 Cov(A,Z)=-0.20 Cov(Z,Z)=-0.2 Var(Z)=-0.2 0.152 Now, the Cov(A,Z) = r 0.26 0.15. Puting this together and solving for the correlation r gives:
In[14]:=

- 0.2 0.152 r = €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ € 0.8 0.26 0.15 - 0.144231

Out[14]=

2

zerobeta.nb

Now we need to compute the standard deviation of the market portfolio

In[15]:=

SDofMarket =
0.205825

0.82 .262 + 0.22 0.152 + 2 .15

.26 0.8 0.2 r

Out[15]=

So if you originally hold the market and decide to replace portfolio Z with the riskless asset then the new standard deviation of your portfolio will be
In[16]:=

0.26 0.8
0.208

Out[16]=

This is larger than just holding the market alone, so you will not want to substitute the riskless asset for the zero beta portfolio!