Skip to content
All library documents

Portfolio Beta for a Market and Risk-Free Asset Mix

Article Quant Q&A · Author: armara
Read the original

Summary

The document explains how to calculate beta for a portfolio invested in both the market portfolio and a risk-free asset under the Capital Asset Pricing Model (CAPM). CAPM describes expected asset return as the risk-free rate plus beta times the market risk premium. The market portfolio has beta one, while the risk-free asset has beta zero.

Portfolio beta is the weighted average of the component betas, so the market allocation contributes in proportion to its weight and the risk-free allocation contributes no market beta. The response also cautions that CAPM is a theoretical pricing model whose empirical support is disputed. The example illustrates beta arithmetic under the model’s assumptions; it does not demonstrate that CAPM reliably explains observed returns.

Key ideas

  • CAPM relates an asset’s expected return to the risk-free rate and its beta-scaled market risk premium.
  • The market portfolio has beta one, and the risk-free asset has beta zero.
  • A portfolio’s beta is the weighted sum of its component betas.
  • CAPM is a theoretical framework, and its empirical validity is contested.

Tags

Portfolio constructionRisk managementStatistics
Full text
# Calculating beta when holding market portfolio


# Calculating beta when holding market portfolio












Suppose that CAPM holds and that you hold a portfolio of the market portfolio and the risk-free asset with weights equal to 0.74 and 0.26 respectively. What is the beta of your portfolio?

My questions are;

- What does it mean that "CAPM holds"? I've googled this but can't seem to find any answer.

- When I tried calculating beta, this is what I did;

$$ r_i=r_f+\beta(r_m-r_f)$$ $$ r_i-r_f=\beta(r_m-r_f) $$ but since the portfolio is the market portfolio, I thought that I could use $r_i=r_m$, so I get

$$\beta=\frac{r_i-r_f}{r_m-r_f}=\frac{r_m-r_f}{r_m-r_f}=1 $$.

When looking at the answer, they say that the beta of my portfolio is 0.74, but I don't understand why. Any help would be greatly appreciated.

## Answer by skoestlmeier (score 2, accepted)

https://quant.stackexchange.com/a/43533

The phrase "The CAPM holds" refers to the assumption, that any asset return $r_i$ fulfills the pricing relation $r_i=r_f+\beta_i(r_m-r_f)$, where $r_m$ denotes the market return, $r_f$ the risk-free rate and $\beta_i$ the beta-factor of the asset. The CAPM is an economic theory, but be aware that plenty of empirical research does not support the CAPM, it is long ago "shot dead" by academics.

Your calculation is right, that the beta-factor of the market portfolio equals 1. But further, your portfolio is a linear combination of 74% market portfolio and the remaining investment in the riskless asset. As the latter one has a beta of zero, your portfolio beta is $0.74\cdot 1 + 0.26 \cdot 0 = 0.74$.

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.