Finance-Theory · Unit 23 · Video 6 · Interactive Practice

Course Summary II: Risk, the CAPM and When to Trust Prices

IKey Formulas

FormulaNameWhat it gives
E[Rp]=Rf+σpσm(E[Rm]−Rf)\mathsf{E}[R_p] = R_f + \frac{\sigma_p}{\sigma_m}\left(\mathsf{E}[R_m] - R_f\right)Capital market lineExpected return of an efficient portfolio, priced by its volatility relative to the market's
E[Ri]=Rf+βi(E[Rm]−Rf)\mathsf{E}[R_i] = R_f + \beta_i\left(\mathsf{E}[R_m] - R_f\right), with βi=Cov[Ri,Rm]Var[Rm]\beta_i = \frac{\text{Cov}[R_i, R_m]}{\text{Var}[R_m]}Security market lineThe risk-adjusted discount rate for any security, portfolio or cash flow
αi=E[Ri]−{Rf+βi(E[Rm]−Rf)}\alpha_i = \mathsf{E}[R_i] - \{R_f + \beta_i(\mathsf{E}[R_m] - R_f)\}AlphaReturn delivered minus return the line requires: the measure of a manager's performance
NPV=CF0+CF11+r1+⋯+CFT(1+rT)TNPV = CF_0 + \frac{CF_1}{1+r_1} + \cdots + \frac{CF_T}{(1+r_T)^T}NPV ruleEach rate rt=Rf+βproject(E[Rm]−Rf)r_t = R_f + \beta_{\text{project}}(\mathsf{E}[R_m] - R_f), from the project's beta, not the company's

Key Insight: The market supplies every input of the discount rate (RfR_f, the market risk premium and β\beta), so the rate is only as trustworthy as the market that set it: reliable in calm periods, not when the crowd becomes an angry mob, as in the autumn of 2008.

IIDiversification Needs ρ<1\rho \lt 1

A long-only mix of two stocks is less volatile than the weighted average of their volatilities whenever ρ<1\rho \lt 1.

IIIAlpha Is Measured From the Line

Funds with the same total return can have very different alphas, because the SML requires more of higher-beta funds. (The three funds are illustrative, our numbers.)

IVDiscount at the Project's Beta

A project's beta fixes its discount rate on the SML, and that rate decides whether its NPV is positive.

💡 The same line applied to the firm's own beta gives the firm's cost of capital, which is the right rate only for a project of the firm's average risk.

VQuiz Questions

Problem 1 · The Market's Discount Rate

Given: Rf=2.5%R_f = 2.5\%, E[Rm]=8.5%\mathsf{E}[R_m] = 8.5\% and a stock with β=1.25\beta = 1.25. Find the expected return the security market line assigns to the stock, which is the risk-adjusted discount rate for its cash flows.

✅ Correct! 2.5%+1.25×6%=10.0%2.5\% + 1.25 \times 6\% = 10.0\%: the riskless rate plus beta times the market risk premium.
❌ Not quite. That is 1.25×8.5%1.25 \times 8.5\%: beta multiplies the market risk premium E[Rm]−Rf\mathsf{E}[R_m] - R_f, and RfR_f is then added back.
❌ Not quite. That is 2.5%+1.25×8.5%2.5\% + 1.25 \times 8.5\%: beta multiplies the premium E[Rm]−Rf=6%\mathsf{E}[R_m] - R_f = 6\%, not the market's total return.
❌ Not quite. 8.5%8.5\% is the rate for β=1\beta = 1; a stock with more systematic risk than the market must offer more.
Show solution

The market risk premium is E[Rm]−Rf=8.5%−2.5%=6%\mathsf{E}[R_m] - R_f = 8.5\% - 2.5\% = 6\%. On the security market line,

E[R]=Rf+β(E[Rm]−Rf)=2.5%+1.25×6%=2.5%+7.5%=10.0%\mathsf{E}[R] = R_f + \beta\left(\mathsf{E}[R_m] - R_f\right) = 2.5\% + 1.25 \times 6\% = 2.5\% + 7.5\% = 10.0\%

All three inputs come from the market: the riskless rate, the premium and the stock's beta.

Problem 2 · Reading a Fund's Beta From Its Alpha

Given: Rf=2.5%R_f = 2.5\% and E[Rm]=8.5%\mathsf{E}[R_m] = 8.5\%. A fund earned 7.0%7.0\% last year, and its manager reports an alpha of +1.8+1.8 percentage points, measured from the security market line. Find the fund's beta.

✅ Correct! The line required 7.0%−1.8%=5.2%7.0\% - 1.8\% = 5.2\%, and 5.2%=2.5%+β×6%5.2\% = 2.5\% + \beta \times 6\% gives β=0.45\beta = 0.45.
❌ Not quite. That puts the whole 7.0%7.0\% on the line, as if alpha were zero. The line requires only the return minus alpha.
❌ Not quite. That adds the alpha to the return. Alpha is delivered minus required, so the required return is 7.0%−1.8%7.0\% - 1.8\%.
❌ Close, but… beta multiplies the premium E[Rm]−Rf=6%\mathsf{E}[R_m] - R_f = 6\%, not the market's total return of 8.5%8.5\%.
Show solution

Alpha is the return delivered minus the return the line requires, so the required return is

Rf+β(E[Rm]−Rf)=7.0%−1.8%=5.2%R_f + \beta\left(\mathsf{E}[R_m] - R_f\right) = 7.0\% - 1.8\% = 5.2\%

With the premium 8.5%−2.5%=6%8.5\% - 2.5\% = 6\%:

2.5%+β×6%=5.2%⟹β=5.2−2.56=2.76=0.452.5\% + \beta \times 6\% = 5.2\% \quad\Longrightarrow\quad \beta = \frac{5.2 - 2.5}{6} = \frac{2.7}{6} = 0.45

A low-beta fund with a modest total return can still add value: the line asks little of it.

Problem 3 · The Project's Beta, Not the Company's

Given: Rf=2.5%R_f = 2.5\%, E[Rm]=8.5%\mathsf{E}[R_m] = 8.5\%. A company whose own beta is 0.450.45 considers a project in a riskier line of business with βproject=1.25\beta_{\text{project}} = 1.25. It costs $800 today and pays $220 at the end of each of the next 5 years (after-tax cash flows). Find the project's NPV, discounting with annual compounding.

✅ Correct! At the project's rate of 10%10\% the five payments are worth about $834, so the NPV is about +$34.
❌ Close, but… that discounts at the company's rate, 2.5%+0.45×6%=5.2%2.5\% + 0.45 \times 6\% = 5.2\%. The rate must match the project's own risk.
❌ Not quite. That discounts at Rf=2.5%R_f = 2.5\%, treating risky cash flows as riskless.
❌ Not quite. That discounts at E[Rm]=8.5%\mathsf{E}[R_m] = 8.5\%, the rate for β=1\beta = 1; this project's beta is 1.251.25.
Show solution

Step 1: the project's discount rate.

r=Rf+βproject(E[Rm]−Rf)=2.5%+1.25×6%=10%r = R_f + \beta_{\text{project}}\left(\mathsf{E}[R_m] - R_f\right) = 2.5\% + 1.25 \times 6\% = 10\%

Step 2: present value of the five payments (an annuity, amounts in dollars):

220×1−1.10−50.10≈220×3.7908≈833.97220 \times \frac{1 - 1.10^{-5}}{0.10} \approx 220 \times 3.7908 \approx 833.97

Step 3: NPV.

NPV≈−800+833.97=33.97NPV \approx -800 + 833.97 = 33.97

The NPV is about +$34, so the project is worth taking. Discounting at the company's 5.2%5.2\% would overstate it at about +$147.

Problem 4 · When to Trust the Market's Numbers

Given: In October 2008 a firm wants the discount rate for a 10-year plant. Its analyst proposes estimating the market risk premium and the plant's beta from the past few weeks of prices. Which response matches the course's view of when market inputs can be trusted?

✅ Correct! Markets work well most of the time, but the emotional state of the market matters; autumn 2008 was a period when it did not set prices calmly.
❌ Not quite. The answer to "can you trust market data?" is "sometimes": bubbles, crashes and excess volatility are part of normal markets.
❌ Not quite. Markets work well most of the time, and payback is one of the alternatives to be wary of: know how to compute it, but decide on NPV.
❌ Not quite. The rate must match the project's own risk: βproject\beta_{\text{project}}, not βcompany\beta_{\text{company}}, in a crisis or not.
Show solution

The SML rate is built entirely from market inputs: RfR_f, the market risk premium and β\beta. When markets are efficient, prices fully reflect all available information and those inputs are compelling. But price discovery is neither costless nor effortless, and in some periods, such as the autumn of 2008, the crowd becomes an angry mob.

The adaptive markets hypothesis covers both: the rational theory of the course applies in calm periods, while in a crisis emotion overwhelms deliberation. A few weeks of crisis prices are the wrong basis for valuing a 10-year plant. The decision rule itself (NPV with the project's beta) does not change; what changes is how much weight the latest market readings deserve.

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