Beta and the CAPM

◈ 11 cards

Beta is the slope of the stock on the market — SLOPE(stock, market) or COVARIANCE.S ÷ VAR.S; the CAPM prices only that risk, RF + β(RM − RF); and its r is the discount rate the Gordon model needs.

Beta — the slope against the market

Only systematic risk earns a premium (Lesson 13.3). Beta measures how much of it a share carries: the sensitivity of the share's return to the market's. Five years of Lakehead Robotics returns beside the S&P/TSX Composite:

Year	Lakehead	Market
1	10 %	6 %
2	−4 %	−2 %
3	7 %	4 %
4	15 %	9 %
5	2 %	1 %

Beta is the slope of the line through those five points, Lakehead on the vertical axis:

=SLOPE(stock_returns,market_returns) = 1.7077

The argument order matters: SLOPE(known_y, known_x) — the stock first, the market second. Reversed, it returns 0.5849, which is not a beta of anything. The same number by its definition:

$β = \frac{Cov ( r _{s} , r _{m} )}{Var ( r _{m} )} = \frac{0.003125}{0.001830}$

=COVARIANCE.S(stock,market) = 0.003125; =VAR.S(market) = 0.001830; the ratio is 1.7077. Both functions use n − 1, and the n − 1 cancels in the ratio, so COVARIANCE.P/VAR.P gives the same beta.

Reading it. When the market moves 1 %, Lakehead tends to move about 1.7 % the same way — an aggressive share. β = 1 is the market itself; β < 1 (a utility, a grocer) is defensive; β = 0 is a T-bill, which does not move with the market at all. Beta says nothing about firm-specific risk — that was diversified away, and the market does not pay for it.

The CAPM — pricing that risk

The capital asset pricing model says the return investors require of a share is the risk-free rate plus the share's beta times the market's premium over risk-free:

$E (r) = R_{F} + β (R_{M} - R_{F})$

$(R_{M} - R_{F})$ is the market risk premium — the reward for holding the average share, β = 1. The share's own premium is that, scaled by its beta. With the risk-free rate at 3.5 % (this course uses the Government of Canada T-bill yield as $R_{F}$ ), the expected market return at 9.5 % and a beta of 1.3:

=0.035+1.3*(0.095-0.035) = 0.113 — 11.3 %.

For β = 0.8: =0.035+0.8*0.06 = 8.3 %. The line from (0, 3.5 %) through (1, 9.5 %) is the security market line; every share's required return sits on it at its beta.

The chain — CAPM into the DDM

Module 11's Gordon model needs a required return $r$ , and this is where it comes from. Tamarack Foods: β = 1.3, next year's dividend $1.50, growth 5 %. Step one, the CAPM: r = 11.3 %. Step two, the Gordon model:

=1.50/(0.113-0.05) = $23.81

The paper sets this as one question with two marks: the r and the price. Use the wrong r — say the market return 9.5 % — and the price is 33.33, wrong twice. The callout names the trap.

Recompute

A share with β = 1.1, $D_{1}$ = 2.00, g = 4 %: r = =0.035+1.1*0.06 = 10.1 %; P₀ = =2/(0.101-0.04) = $32.79. Cold: β = SLOPE(stock, market); E(r) = RF + β(RM − RF); that E(r) is the DDM's r.

The security market line as a table of points: it starts at the risk-free rate at β = 0, passes through the market return at β = 1, and every share sits on it at its own beta. Tamarack at 1.3 requires 11.3 %.