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Computing Asset Beta When the Market Portfolio Includes That Asset

Article Quant Q&A · Author: Lachezar Tilev
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Summary

The document works through an exam-style question about finding an asset’s covariance with a market portfolio when that portfolio is defined as a weighted combination of two assets. By applying covariance’s linearity, the answer expands the covariance into weighted terms. It then gives cases where asset A has a 75% or 50% portfolio weight and states the resulting weighted variance expressions.

This illustrates that an asset’s contribution to covariance with a portfolio depends on its weight and its covariance with every portfolio constituent. The shown results retain only the weighted variance of asset A; that simplification requires the covariance between A and B to be zero, or another stated assumption that removes the cross term. The document does not provide that assumption or the full exam setup, so the expressions should not be treated as generally valid. Beta also requires dividing the asset–market covariance by market variance, which is not worked out here.

Key ideas

  • Covariance with a weighted portfolio can be expanded as the weighted sum of covariances with its constituents.
  • When an asset is part of the portfolio, its own variance contributes to asset–portfolio covariance.
  • Covariances with the other portfolio constituents also contribute unless they are zero or otherwise specified away.
  • Beta is asset–market covariance divided by market variance, a further step not calculated in the example.

Tags

StatisticsPortfolio construction
Full text
# How to find beta from the information given?


# How to find beta from the information given?












This is an exam question. I know that to find beta I need the covariance between the portfolio and asset A but don't know how to find it.

## Answer by Andrew (score 1)

https://quant.stackexchange.com/a/39389

Investor I: market portfolio is consisting of 75% of asset A and 25% of asset B, i.p $r_m=3/4r_a+1/4r_b$ $\Rightarrow $ cov($r_a,r_m$)$=$cov($r_a,3/4r_a+1/4r_b$)=$3/4V(r_a)$.

Investor J: market portfolio is consisting of 50% of asset A and 50% of asset B, i.p $r_m=1/2r_a+1/2r_b$ $\Rightarrow $ cov($r_a,r_m$)$=$cov($r_a,1/2r_a+1/2r_b$)=$1/2V(r_a)$.

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.