Why a Market Benchmark Is Needed for a Beta-Constrained Portfolio

Summary

The document considers choosing weights for two stocks to minimize portfolio variance while targeting a beta of one. It supplies expected returns, volatilities, a correlation, and betas measured against an index, and shows that the covariance information can be used to describe portfolio variance. The key issue is that the stated betas are not necessarily betas to the market required by the constraint.

The answer explains that the market must first be defined. A covariance matrix can support construction of an efficient frontier, but it cannot establish a meaningful market-beta constraint without specifying the benchmark, such as a capitalization-weighted index or a portfolio formed from the assets. The document does not provide a final weight ratio or numerical solution. Its lesson is about the limits of the question as posed: the portfolio optimization depends on a benchmark definition that the given inputs do not establish.

Key ideas

• A portfolio variance can be expressed using asset variances, covariance, and portfolio weights.
• A beta constraint is meaningful only relative to a defined benchmark.
• Betas measured against an unspecified index do not determine market beta.
• The covariance matrix can describe an efficient frontier but cannot supply the missing market definition.

Tags

# Build a portfolio with $\beta=1$ and minimize $\sigma^2$ using CAPM


# Build a portfolio with $\beta=1$ and minimize $\sigma^2$ using CAPM












Suppose there are two stocks A and B:

- expected returns are $E[R_A]=0.1$, $E[R_B]=0.15$;

- standard deviations are $\sigma_A=0.1$, $\sigma_A=0.2$;

- correlation is $corr(A,B)=0.6$;

- their betas to some index (not the market) are 0.45 and 0.9, respectively.

If we want to construct a portfolio using stock A and B such that portfolio beta to the market is 1 and sigma as small as possible, what would the ratio between weight of stock A and weight of stock B in this portfolio?

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I could go as far as getting the covariance between A and B ($\sigma_{AB}=0.012$), and let weight of stock A $= X$, express the portfolio's variance in terms of X = $0.026X^2+0.016X+0.04$. But I have no idea how to switch from index beta to market beta.

## Answer by Preston Lui (score 2, accepted)

https://quant.stackexchange.com/a/59938

The problem is, what do you define as the market?

You can easily construct the efficient frontier using the covariance matrix. You can easily show the efficient frontier contains the solution, maybe with or without leverage.

Without the definition of the market (e.g. the cap-weighted index, tangency portfolio of the 2 assets), the question is however not too meaningful.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.