Finance-Theory · Unit 18 · Video 4 · Interactive Practice

Portfolio Beta Is a Weighted Average: Microsoft vs Gillette

IKey Formulas

Formula	Name	What it says
$\beta_p = \omega_1 \beta_1 + \cdots + \omega_n \beta_n$	Portfolio beta	Covariance is linear in each argument, so dividing $\text{Cov}[R_p, R_m] = \sum_i \omega_i \text{Cov}[R_i, R_m]$ by $\text{Var}[R_m]$ gives a weighted average of betas
$\sigma_p^2 = \sum_i \omega_i^2 \sigma_i^2 + \sum_{i \ne j} \omega_i \omega_j \text{Cov}[R_i, R_j]$	Portfolio variance	Not linear: the cross products mean it cannot be read off the components one at a time
$\text{E}[R_p] = R_f + \beta_p \left( \text{E}[R_m] - R_f \right)$	SML for a portfolio or project	The weights sum to one, so $R_f$ appears once; the risk adjustment is $\beta_p$ times the market risk premium
$\Delta \sigma_p^2 \approx 2\omega_i \sigma_m^2 (\beta_i - 1)$	Small weight $\omega_i$ moved from the market fund into stock $i$	Variance rises when $\beta_i > 1$ and falls when $\beta_i < 1$

Key Insight: The required rate of return, the opportunity cost of capital and the risk-adjusted discount rate are one quantity. For a project, take the beta of comparable stocks, put it into the security market line, and the result is the project's discount rate.

IIBeta Averages, So the Portfolio Lands on the SML

With $R_f = 4\%$ and a $7.5\%$ premium, a mix of three stocks is priced at its weighted-average beta.

IIIA Sliver of One Stock in the Market Fund

Moving a small weight from fund $M$ into one stock raises your variance, to first order, exactly when $\beta_i > 1$ .

Stock Weight ω_i 5%

💡 The approximation holds for a small weight $\omega_i$ : it drops the terms in $\omega_i^2$ , which shrink much faster than $\omega_i$ . Those terms add up to $\omega_i^2\left(\sigma_m^2 - 2\,\text{Cov}[R_i, R_m] + \sigma_i^2\right) = \omega_i^2\,\text{Var}[R_i - R_m]$ , which is never negative, so for a larger weight even a $\beta_i < 1$ stock can end up adding variance. Whether the trade is worth making also depends on the expected return the SML assigns to that extra risk.

IVPricing a Project from Its Comparables

A project's discount rate is the SML return at its comparable stocks' beta: $R_f$ plus a risk adjustment.

Comparable stocks Your guess 10%

VQuiz Questions

Problem 1 · Portfolio Beta

Given: A portfolio holds $20\%$ in a stock with $\beta = 1.3$ , $30\%$ in a stock with $\beta = 0.6$ and $50\%$ in a stock with $\beta = 0.9$ . Find the portfolio's beta.

✅ Correct!

0.2(1.3) + 0.3(0.6) + 0.5(0.9) = 0.26 + 0.18 + 0.45 = 0.89

. Each beta is weighted by its own stock's share.

❌ Not quite. That pairs the weights with the wrong betas (

0.2 \times 0.6 + 0.3 \times 0.9 + 0.5 \times 1.3

). The

50\%

weight belongs to the

\beta = 0.9

stock.

❌ Not quite.

(1.3 + 0.6 + 0.9)/3 \approx 0.93

is the equal-weighted average. Portfolio beta weights each beta by the fraction of the portfolio in that stock.

❌ Not quite.

0.5 \times 0.9 = 0.45

is only the largest holding's term. Add the terms for the other two stocks.

❌ Not quite. Use

\beta_p = \omega_1 \beta_1 + \omega_2 \beta_2 + \omega_3 \beta_3

Show solution

Covariance with the market is linear in the weights, so beta is too:

\beta_p = 0.2(1.3) + 0.3(0.6) + 0.5(0.9) = 0.26 + 0.18 + 0.45 = 0.89

The weights are non-negative and sum to one, so $\beta_p$ always lies between the smallest and the largest beta in the portfolio.

Problem 2 · Required Return of a Portfolio

Given: $R_f = 4\%$ per year and a market risk premium $\text{E}[R_m] - R_f = 7.5\%$ per year. A portfolio holds $30\%$ in a stock with $\beta = 1.3$ , $50\%$ in a stock with $\beta = 0.6$ and $20\%$ in a stock with $\beta = 0.9$ . Find the expected return the security market line requires of the portfolio.

✅ Correct!

\beta_p = 0.87

, so

\text{E}[R_p] = 4\% + 0.87 \times 7.5\% = 4\% + 6.525\% = 10.525\%

per year.

❌ Close, but…

6.525\%

is only the risk adjustment

\beta_p \times 7.5\%

. Add the risk-free rate: the weights sum to one, so

R_f

appears exactly once.

❌ Not quite. That uses the equal-weighted beta

(1.3 + 0.6 + 0.9)/3 \approx 0.933

. Weight each beta by its holding:

\beta_p = 0.87

❌ Not quite. That is

4\% + 0.87 \times 11.5\%

: it multiplies

\beta_p

by the whole expected market return,

11.5\%

. The SML multiplies beta by the premium

\text{E}[R_m] - R_f = 7.5\%

❌ Not quite. Find

\beta_p

first, then use

\text{E}[R_p] = R_f + \beta_p(\text{E}[R_m] - R_f)

Show solution

Step 1: portfolio beta.

\beta_p = 0.3(1.3) + 0.5(0.6) + 0.2(0.9) = 0.39 + 0.30 + 0.18 = 0.87

Step 2: security market line.

\text{E}[R_p] = 4\% + 0.87 \times 7.5\% = 4\% + 6.525\% = 10.525\%

Check: the stocks' own SML returns are $13.75\%$ , $8.5\%$ and $10.75\%$ , and $0.3(13.75\%) + 0.5(8.5\%) + 0.2(10.75\%) = 4.125\% + 4.25\% + 2.15\% = 10.525\%$ , the same number.

Problem 3 · Adding to the Market Fund

Given: You hold only the market fund $M$ , with variance $\sigma_m^2$ . You move $5\%$ of your money out of $M$ and into a stock with $\beta = 0.6$ . Find the approximate change in your portfolio's variance.

✅ Correct!

2(0.05)\sigma_m^2(0.6 - 1) = -0.04\,\sigma_m^2

: the stock's covariance with

M

0.6\,\sigma_m^2

, is below the variance you already carry.

❌ Check the sign.

\beta_i - 1 = 0.6 - 1 = -0.4

is negative, so variance falls. A stock with

\beta < 1

adds less covariance than the

\sigma_m^2

it replaces.

❌ Not quite.

2(0.05)(0.6) = 0.06

uses

\beta_i

instead of

\beta_i - 1

. The weight comes out of

M

, so what matters is how the stock's covariance compares with

\sigma_m^2

❌ Close, but… the sign is right, but the factor

2

is missing: the cross term is

2\omega_i(1 - \omega_i)\text{Cov}[R_i, R_m]

❌ Not quite. Use

\Delta \sigma_p^2 \approx 2\omega_i \sigma_m^2 (\beta_i - 1)

Show solution

With weight $\omega_i$ in the stock and $1 - \omega_i$ in $M$ :

\sigma_p^2 = (1 - \omega_i)^2 \sigma_m^2 + 2\omega_i(1 - \omega_i)\text{Cov}[R_i, R_m] + \omega_i^2 \sigma_i^2

Dropping the terms in $\omega_i^2$ , the change from $\sigma_m^2$ is

\Delta \sigma_p^2 \approx 2\omega_i\left(\text{Cov}[R_i, R_m] - \sigma_m^2\right) = 2\omega_i \sigma_m^2 (\beta_i - 1)

= 2(0.05)\sigma_m^2(0.6 - 1) = -0.04\,\sigma_m^2

Variance falls by about $4\%$ of the market variance.

Problem 4 · The Firm's Rate or the Project's?

Given: A firm has three divisions making up $50\%$ , $30\%$ and $20\%$ of its value. Comparable stocks for the three businesses have betas $0.9$ , $1.3$ and $0.6$ . $R_f = 4\%$ and the market risk premium is $7.5\%$ , both per year. Find the firm's beta, then the discount rate for a new project in the $\beta = 0.6$ business.

What is the firm's beta?

What is the project's discount rate?

✅ Correct! The firm's beta is

0.96

(cost of capital

11.2\%

), but the project carries the risk of its own business:

4\% + 0.6 \times 7.5\% = 8.5\%

❌ Not quite.

(0.9 + 1.3 + 0.6)/3 \approx 0.93

treats the three divisions as equal. Weight each beta by its division's share of firm value.

❌ Not quite.

0.5(1.3) + 0.3(0.6) + 0.2(0.9) = 1.01

pairs the shares with the wrong betas. The

50\%

division has comparables with

\beta = 0.9

, the

30\%

one

\beta = 1.3

, the

20\%

one

\beta = 0.6

❌ Not quite.

1.3

is the riskiest division's beta. The firm is a portfolio of its divisions, so its beta is their value-weighted average.

❌ Check the firm's beta. Weight each division's beta by its share of value:

0.5(0.9) + 0.3(1.3) + 0.2(0.6)

❌ Not quite.

11.2\%

is the firm's own cost of capital, priced at

\beta = 0.96

. A project is priced by the beta of its own comparables, not by the average risk of the firm that undertakes it.

❌ Close, but…

0.6 \times 7.5\% = 4.5\%

is only the risk adjustment. Add

R_f = 4\%

❌ Not quite.

10.75\%

prices the largest division (

\beta = 0.9

). The project belongs to the

\beta = 0.6

business.

❌ Not quite. Put the comparables' beta into

R_f + \beta(\text{E}[R_m] - R_f)

Show solution

Firm beta (a portfolio of its divisions):

\beta_{\text{firm}} = 0.5(0.9) + 0.3(1.3) + 0.2(0.6) = 0.45 + 0.39 + 0.12 = 0.96

The firm as a whole requires $4\% + 0.96 \times 7.5\% = 4\% + 7.2\% = 11.2\%$ .

Project rate. The project's risk is that of the $\beta = 0.6$ business, so its comparables set the rate:

\text{E}[R] = 4\% + 0.6 \times 7.5\% = 4\% + 4.5\% = 8.5\%

Discounting it at $11.2\%$ would undervalue it, because its cash flows are less exposed to the market than the firm's average business.

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