# How can I identify a zero beta portfolio?

Suppose that there is no risk free asset whatsoever. In addition I have a market portfolio which consists of 4 stocks. How can I identify a zero beta portfolio? Shouldn't it actually have expected return which is equivalent to the risk free asset? I am a bit confused.

There is a graphical method of solving this problem. Any portfolio say P, Q, M has a counterpart on the frontier that has zero covariance with it (with one exception).

From the given portfolio say M draw a tangent to the frontier that intersects the y axis at a point $R_M$. This point plays a role similar to the risk free rate. From this point draw a horizontal line until it intersect the frontier at $Z(M)$. This portfolio has zero covariance with M and therefore when M is the market portfolio $Z(M)$ is the zero-beta portfolio. It is on the inefficient side of the frontier.

See diagram here

http://images.slideplayer.com/6/1619405/slides/slide_42.jpg

Note: the concept of the zero-beta portfolio was invented by Fischer Black to deal with the situation where there is no risk free asset. In this situation the zero beta portfolio plays a role similar (but not exactly the same) as the risk free asset if the risk free asset existed.

Suppose you construct a portfolio $P$ using the 4 stocks with some weights $w_1$, $w_2$, $w_3$ and $w_4$. Then :

• Compute your portfolio's PnL: $P(t)$,
• Regress $P(t)$ on $M(t)$ (where $M(t)$ is the market return) on a specified time-horizon that you must choose. You obtain a coefficient $b$.
• Then consider the new portfolio $P'$ defined as $P'(t)=P(t)-bM(t)$.

$P'$ is then a zero-beta portfolio. However you must frequently compute the regression to adjust the coefficient $b$.

How to build a 0-$\beta$ portfolio was addressed by the two other answers.

Regarding the second part of your question:

Shouldn't it actually have expected return which is equivalent to the risk free asset?

Under the CAPM:

$$\mathbb{E}(r_a - r_f) = \beta(r_m-r_f) \Longleftrightarrow \mathbb{E}(r_a) = r_f+\beta(r_m-r_f)$$

In this case, indeed $\beta=0 \Longrightarrow \mathbb{E}(r_a) = r_f$.

Nevertheless, a lot of people either do not believe in the CAPM or at least want to exploit small arbitrage opportunities around it. These people would expect to have another return on top of the risk-free rate. A typical example of people doing this for living are market-neutral hedge-funds.