Constructing Zero-Beta Portfolios Without a Risk-Free Asset
Summary
The document explains ways to identify a portfolio with zero beta when no risk-free asset exists. One practical approach begins with a portfolio of assets, estimates its beta by regressing its returns on market returns over a chosen horizon, then offsets the estimated market exposure by taking an opposing market position. Because the estimate can change, the hedge needs periodic recalculation. A second answer describes a graphical construction using the efficient frontier: a tangent from the market portfolio defines an intercept, and a horizontal projection identifies the corresponding zero-covariance portfolio.
The discussion distinguishes zero beta from automatically earning the risk-free rate. Under the CAPM, a zero-beta asset has the risk-free expected return when that asset and rate are part of the model. Without a risk-free asset, the zero-beta portfolio plays a related but not identical role, and actual returns may differ if CAPM assumptions do not hold. The regression hedge is an estimated, horizon-dependent construction rather than a permanent portfolio recipe.
Key ideas
- Estimate a portfolio’s beta by regressing its returns on market returns over a chosen period.
- Offset estimated market exposure to form a zero-beta position, and update the hedge as the estimate changes.
- A graphical efficient-frontier construction can identify a portfolio with zero covariance to the market portfolio.
- Zero beta implies the risk-free expected return under the CAPM, but a zero-beta portfolio is not itself a risk-free asset.
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Full text
# How can I identify a zero beta portfolio?
# 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.
## Answer by Mat2017 (score 3)
https://quant.stackexchange.com/a/31756
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 $b$.
## Answer by Alex C (score 3)
https://quant.stackexchange.com/a/31762
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.
## Answer by SRKX (score 2)
https://quant.stackexchange.com/a/31767
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.