UTS16657 Chap.10 Pricing Investment with CAPM and APT
Pricing Investment with CAPM and APT
Combining a security with others reduces the risk of holding it alone, but it does not eliminate all risk, and the split between what goes and what stays is the foundation of asset pricing. Unsystematic or diversifiable risk relates to events affecting individual securities alone and can be diversified away.
Systematic or non-diversifiable risk relates to broad macroeconomic events affecting the prices of all securities and cannot. If a risk can be escaped by holding more assets, the market will not pay you to carry it; if it cannot, you must be compensated. That is the entire logic of the capital asset pricing model.
Quantifying the surviving risk starts with a line.
Plot an asset's returns against the market's, fit a line of best fit, and its slope is beta. A beta of one says the share carries exactly the market's risk, above one says more and below one says less, so a portfolio whose beta is 1.50 is expected to return 15% where the market returns 10%. That is not regarded as skilled construction, because the manager simply took more market risk.
The vertical distance between each plotted pair and the fitted line is residual variance, which is the diversifiable part.
High financial leverage, thin operating margins, daily repricing rather than secured income, discretionary products and sovereign risk are the named reasons some assets carry high betas.
Introducing a risk-free asset produces the capital market line, running from the risk-free rate through the market portfolio, on which an investor's position depends on risk appetite.
Substituting the equilibrium relationship that a portfolio's beta equals its standard deviation over the market's converts that statement about total risk into the capital asset pricing model.
On the security market line every beta has a required return, an asset plotting above it is underpriced and one plotting below is overpriced.
Arbitrage pricing theory generalises the single market factor into several, using single-factor or multi-factor models estimated by ordinary least squares, with factors drawn from external variables, firm characteristics or portfolios of assets.
What this chapter covers
- 01
Unsystematic against systematic risk, and why only one is priced
- 02
The characteristic line and reading a beta
- 03
Residual variance, and what makes a beta inherently high or low
- 04
The capital market line and the substitution that produces CAPM
- 05
The security market line as a pricing rule
- 06
The six assumptions, and the criticisms built on them
- 07
Single-factor and multi-factor models, and the three families of factor
Two required returns and a mispricing verdict
- +1Market risk premium: 8.40 − 3.85 = 4.55 percentage points, the extra return for carrying one unit of market risk.
- +1Required return on V: 3.85 + 4.55(0.68) = 3.85 + 3.094 = 6.944%. Required return on W: 3.85 + 4.55(1.24) = 3.85 + 5.642 = 9.492%.
- +1Compare each with what is on offer. V offers 7.60% against a requirement of 6.944%, a surplus of 0.656 points, so it plots above the security market line and is underpriced. W offers 8.90% against 9.492%, a shortfall of 0.592 points, so it plots below and is overpriced.
- +1Say what the verdict means. Underpriced means the price is too low for the return on offer, so buying pushes the price up until the offered return falls back to the line. The model identifies the gap and makes no promise about when it closes.
Key terms
- Unsystematic risk
- The component of total risk arising from events affecting individual securities alone. It does not move every security, so holding enough unrelated names removes it and the market pays no premium for carrying it.
- Characteristic line
- The line of best fit through an asset's returns plotted against the market's returns. Its intercept is a constant and its slope is the asset's beta.
- Residual variance
- The portion of an individual investment's return variation not explained by the market, shown on a scatter as the vertical distance between each plotted pair and the fitted line.
- Capital market line
- The set of efficient combinations of the risk-free asset and the market portfolio, drawn against standard deviation, on which an investor's position depends on risk appetite.
- Security market line
- The linear relationship between beta and required return. In equilibrium every asset plots on it; above it an asset is underpriced and below it overpriced.
- Factor loading
- The coefficient on a factor in an arbitrage pricing model, measuring the sensitivity of an asset's return to changes in that factor.
- Single-factor model
- A model relating an asset's return to one explanatory variable plus an error term, estimated as a linear regression by ordinary least squares or in a spreadsheet.
- Principal component analysis
- A technique transforming a number of possibly correlated candidate variables into a smaller number of uncorrelated ones, used to choose the factors for a multi-factor model.
Pricing Investment with CAPM and APT FAQ
Why does the model price only systematic risk?
Because the other component can be removed for free. Under the efficient market hypothesis every investor can diversify cheaply, so nobody needs compensation for bearing a risk they could have eliminated by holding more securities. Only the non-diversifiable part commands a premium, which is why the required return is expressed as the risk-free rate plus beta times the market risk premium.
For an investor holding a single undiversified stock the implication is uncomfortable: they bear the full standard deviation while the market prices the stock as though only its beta mattered.
What are the main criticisms of the model?
They attack the assumptions rather than the algebra. The simple form is robust and the accusation is that the assumptions are oversimplified. Not all investors share identical expectations about risk and return, with contrarian investors the standing counter-example, although in an informationally efficient market it is the fully informed investors whose views end up in prices, not the consensus of everyone.
And tax treatment of income and of capital gains differs from one jurisdiction to the next, so a portfolio efficient after tax for one investor will not be for another.
What does arbitrage pricing theory add?
It replaces the single market factor with several, which reduces the number of parameters a mean-variance approach would otherwise need as the asset count rises. Four or five factors explain most of the covariance between stocks, and the payoff is the ability to forecast a portion of the residual risk that the capital asset pricing model treats as unexplained.
Some residual always remains, so diversification is still necessary. There is no fixed list of factors: the aim is to name whatever underlying risk actually drives the asset you care about.
How do I choose factors without simply mining the data?
State the expected sign before you run the regression. The subject groups candidate factors into external variables such as output growth, unemployment, interest rates, mortgage approvals and inflation; firm characteristics such as the price-earnings ratio, dividend yield, earnings forecasts, a size proxy and industry indicators; and portfolios of assets such as small against large stocks or sector portfolios.
An analyst worried about a geared trust would include the interest rate and expect a negative relationship, for the reasons the leverage chapter set out.
Exam move
This is the densest examinable chapter and it rewards drawing over reading. Sketch the security market line from memory with the risk-free intercept, the market point at a beta of one, and an underpriced and an overpriced point, and label what each distance means.
Then work required returns until the substitution is automatic, because the calculation itself is only worth a mark and the marks sit in reading the verdict and in reciting the assumptions when asked to criticise the model.
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