CAPM, beta and the security market line: required return and alpha for FRM candidates
The Capital Asset Pricing Model (CAPM) links the return investors should require from an asset to the part of its risk that cannot be diversified away. It appears in FRM® Part I (Foundations of Risk Management and Quantitative Analysis) and underlies performance measurement later on. See the FRM page for the other guides, and how the FRM exam is structured for where it sits.
The idea
Total risk splits into systematic risk, which moves with the whole market, and idiosyncratic risk, which is specific to the asset and can be diversified away. CAPM says the market pays a premium only for systematic risk. Beta measures an asset's exposure to it:
beta = Cov(R_i, R_m) / Var(R_m) = rho x (sigma_i / sigma_m)
where rho is the correlation of the asset with the market. A beta of 1 moves with the market, above 1 amplifies it, below 1 dampens it.
The security market line
The CAPM equation, drawn as a line against beta, is the security market line (SML):
E(R_i) = R_f + beta_i x (E(R_m) - R_f)
R_f is the risk-free rate and the bracket is the market risk premium. The SML plots expected return against beta (systematic risk). Do not confuse it with the capital market line, which plots expected return against total risk (standard deviation) and applies only to efficient portfolios.
An asset that sits on the line is fairly priced. Above the line it offers more than its required return, so it is undervalued; below it, overvalued.
Alpha
Jensen's alpha is the return achieved above what CAPM requires:
alpha = R_p - [R_f + beta_p x (R_m - R_f)]
A positive alpha is a position above the SML after adjusting for beta.
Sharpe versus Treynor
Both reward excess return per unit of risk, but the unit differs. Sharpe = (R_p - R_f) / sigma_p uses total risk, so it suits a whole portfolio. Treynor = (R_p - R_f) / beta_p uses systematic risk, so it suits a portfolio that is one part of a larger diversified one. They can rank portfolios differently.
Worked example
An invented fund, Vantrey Growth Fund, returned 11% over the year with a standard deviation of 18% and a beta of 1.2. Assume the risk-free rate is 3%, the market risk premium is 5.5% (so the market return is 8.5%) and the market standard deviation is 15%.
- Required return = 3% + 1.2 x 5.5% = 9.6%.
- Jensen's alpha = 11% - 9.6% = 1.4%.
- Sharpe = (11 - 3) / 18 = 0.44. The market's Sharpe is 5.5 / 15 = 0.37.
- Treynor = (11 - 3) / 1.2 = 6.67%. The market's Treynor is 5.5% (its beta is 1).
On both measures the fund beat the market. As a check on beta, an asset with correlation 0.65, volatility 28% and a market volatility of 16% has beta = 0.65 x 28 / 16 = 1.14.
Assumptions and their usual criticisms
CAPM assumes investors:
- care only about mean and variance of returns, and are rational and risk averse,
- share the same expectations and a single holding period,
- can borrow and lend unlimited amounts at the risk-free rate,
- face no taxes or transaction costs, with all assets tradable.
Critics point out that returns are not normal (see fat tails, skewness and kurtosis), borrowing rates exceed lending rates, and the true market portfolio is unobservable, which Roll's critique stresses. Empirical work finds that size, value and other factors explain returns beyond beta, which motivates multi-factor models. CAPM remains the standard teaching benchmark, but how well it describes real returns is debated.
Usual mistakes
- Using the market return in place of the market risk premium in the formula.
- Mixing up SML and CML, and using total risk where beta is needed.
- Applying Sharpe to a single stock held within a diversified portfolio, where Treynor is the better fit.
- Treating a positive alpha as skill without asking whether the benchmark or beta estimate is wrong.
- Using historical beta as if it were fixed. Estimates vary with the window and index.
How the exam typically tests it
Typically you compute a required return, a beta from a covariance and variance, an alpha, or choose between Sharpe and Treynor for a given investor. You may also be asked which assumption is violated in a scenario. Risk models built on a common factor are related to Value at Risk. Charta's practice questions for this topic let you rehearse the arithmetic.
Common questions
What does a beta of 1.5 mean?
The asset's returns tend to move 1.5 times as much as the market's, in the same direction. A market move of 2% is associated with about 3%, on average.
What is the difference between the SML and the CML?
The SML relates expected return to beta for any asset or portfolio. The CML relates expected return to standard deviation and only efficient portfolios, combining the risk-free asset and the market portfolio, lie on it.
When should I use Sharpe rather than Treynor?
Use Sharpe for a portfolio evaluated on its own, since it uses total risk. Use Treynor when the portfolio is one holding in a well-diversified whole, where only systematic risk matters.
Does a positive alpha mean the manager has skill?
Not necessarily. It may reflect luck, a mismeasured beta or a benchmark that omits other risk factors.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
The exam and the syllabus can change. Check GARP's current details before you book.