MATH2070 Chap.10 CAPM and Riskless Assets
CAPM and Riskless Assets
Introducing a riskless asset changes the portfolio opportunity set from a curved risky frontier to straight capital allocation lines. The tangency portfolio supplies the greatest excess return per unit of risk. CAPM then links an asset's expected excess return to beta, its exposure to market movements, rather than to total variance.
The security market line provides a pricing benchmark: assets above or below it appear underpriced or overpriced relative to their systematic risk under the model assumptions. The capital-market line concerns efficient portfolios and uses total standard deviation, while the security-market line applies more broadly and uses beta.
Beta divides covariance with the market by market variance, isolating systematic exposure from diversifiable volatility. Alpha measures a return difference relative to the CAPM benchmark and therefore depends on model inputs rather than guaranteeing profit. Borrowing or lending at the risk-free rate changes the chosen exposure to the market portfolio without changing the composition of that risky fund under the model.
What this chapter covers
- 01
Riskless borrowing and lending
- 02
Capital allocation lines
- 03
Tangency portfolios
- 04
Market portfolios
- 05
Beta and systematic risk
- 06
Capital Asset Pricing Model
- 07
Security market line
Worked example · free
Price risk with beta
- 1Find the market risk premium from the market and risk-free returns.
- 1Multiply the market premium by the asset beta.
- 1Add the risk-free rate to obtain the CAPM benchmark.
- 1Compare the observed expected return with the benchmark and interpret the gap.
Key terms
- Risk-free rate
- The return on an asset treated as having no uncertainty over the relevant horizon.
- Tangency portfolio
- The risky portfolio where a line from the risk-free rate is tangent to the efficient frontier.
- Beta
- An asset's covariance with the market divided by market variance.
- Security market line
- The CAPM relation between expected return and beta.
- Market premium
- The expected market return minus the risk-free rate. CAPM multiplies this excess return by an asset's beta to determine the compensation associated with its systematic market exposure.
- Alpha
- The difference between an expected or observed return and the CAPM benchmark implied by beta. It is conditional on the model and estimates, rather than a guaranteed riskless profit. Reporting it in percentage points keeps the subtraction distinct from a percentage change in return.
CAPM and Riskless Assets FAQ
Why does CAPM use beta rather than total variance?
Investors can diversify asset-specific risk in broad portfolios, so the model rewards only systematic exposure that co-moves with the market. Beta measures that exposure relative to the market portfolio.
What is special about the tangency portfolio?
Among risky portfolios, it gives the greatest slope when joined to the risk-free rate. That slope is excess expected return per unit of risk, so combining it with the riskless asset spans the best capital allocation line.
What does a point above the security market line mean?
Its expected return exceeds the CAPM return implied by beta. Within the model, that positive alpha indicates apparent underpricing, though the conclusion depends on the inputs and CAPM assumptions being appropriate.
Exam move
Keep the capital-market line and security-market line on separate cards. Label both vertical axes as expected return, but label the CML horizontal axis with total standard deviation and the SML axis with beta. State that the CML describes efficient combinations of the riskless asset and market portfolio, whereas the SML provides a CAPM benchmark for individual assets and portfolios.
Recreate both diagrams from memory before attempting calculations. For a riskless-plus-risky allocation, write the risky weight and the residual risk-free weight, then derive expected return and standard deviation. Interpret weights above one as leveraged borrowing only when borrowing is allowed. Compare slopes to identify the tangency portfolio rather than choosing the highest-return risky asset.
Relate the slope to excess return per unit of total risk and explain why the market portfolio is special under the model assumptions. For CAPM pricing, calculate beta from covariance divided by market variance, compute the market risk premium and multiply it by beta exactly once before adding the risk-free rate. Keep all rates in decimals or all in percentage points.
Compare a forecast return with the required return to obtain alpha, and describe positive or negative alpha as model-relative rather than as a guaranteed profit. Include cases with beta below zero, between zero and one, equal to one and above one.
Use contrast questions to strengthen selection: an inefficient portfolio with known standard deviation does not belong on the CML; a volatile individual stock is priced through beta on the SML; idiosyncratic risk affects realised outcomes but earns no CAPM premium. Keep an error ledger for line choice, horizontal-axis variable, beta denominator, market premium and alpha sign.
Finish each answer with one sentence stating whether the result concerns total portfolio risk or systematic market exposure. That sentence often reveals a formula applied to the wrong economic object.
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