The CAPM and the portfolio theory it is built on can be rewritten in various ways to highlight different insights. We shall analyze the differences between systematic and unsystematic risk a bit further, reformulate the CAPM for use as a discount rate or in certainty equivalent calculations, and look at some performances measures.
Systematic and unsystematic risk
The CML is a pricing relation for efficient portfolios:
E(rp)= rf +E(rm)− rf σm σp
where (E(rm)−rf)/σmis the price per unit of risk and σpis the volume of risk. The CML prices σpwhich includes all risks (systematic and unsystematic). It is therefore only valid when risk exclusively consists of systematic risk, i.e. for efficient portfolios in which all risk comes from the fraction of the market portfolio they contain. For other, inefficient portfolios, the CML uses the ‘wrong’ risk measure: for these portfolios σp comprises unsystematic risk that is not priced in the market.
The security market line (SML) only prices systematic risk, and it is therefore valid for all investments, including inefficient portfolios and individual stocks:
E(rp)= rf + (E(rm)− rf)βp
where βpis the volume of risk and (E(rm)− rf)is the price per unit of risk. We can write βp in terms of correlation (instead of covariance) as:
βp = σpm
σm2 = σpσmρpm
σm2 = σpρpm σm so that the SML becomes:
E(rp)= rf + (E(rm)− rf)σpρpm
σm , E(rp)= rf + E(rm)− rf
σm
σpρpm
Comparing this with the CML above we see that both relations use the same market price of risk E(rm)− rf /σm,but different measures for the volume of risk: σpin the CML and σpρpm in the SML. The difference is the correlation term, ρpm, that is ignored in the CML. It can be ignored in the CML because efficient portfolios only differ in the fraction of the market portfolio M they contain. This means that all efficient portfolios are perfectly positively correlated, because fractions of the market portfolio are perfectly pos-itively correlated with each other and with the market portfolio. If a portfolio is perfectly positively correlated with the market portfolio ρpm = 1 so that σpρpm = σp and the CML is equivalent to the SML. For inefficient portfolios the volume of risk is adjusted in the SML by the correlation coefficient. Inefficient portfolios are less than perfectly posi-tively correlated with the market portfolio: ρpm<1. So the adjustment is downwards: risk is reduced because unsystematic risk is eliminated. Figure 3.11 illustrates this for some assets. The assets labelled B, G, F and C are inefficient: their standard deviations contain a part unsystematic risk. This means that their standard deviations overstate the risk that is relevant for pricing, that is why they plot to the right of the CML in the picture on the left-hand side of Figure 3.11. Only the market portfolio M is efficient, its standard devia-tion contains no unsystematic risk so it plots on the CML. When the standard deviadevia-tions are ‘corrected’ for unsystematic risk through multiplication by the correlation coefficient, then all the remaining risk is priced and all assets plot on the SML, as the right-hand side of Figure 3.11 shows.
σp E[rp]
rf
Capital market line
B B
M
M
1 C
C F
F G
G
βp E[rp]
rf
Security market line
Figure 3.11 Systematic and unsystematic risk
Finally, for efficient portfolios βp= σp/σm or σp= βpσm, so the risk (standard devi-ation) of efficient portfolios is proportional to the risk of the market portfolio and the proportionality is measured by β.
CAPM and discount rates
In the introduction we said that there are three different ways to account for risk in valuation procedures:
75 3.3 The Capital Asset Pricing Model
1. By adjusting the discount rate to a risk-adjusted discount rate.
2. By adjusting the cash flows to certainty equivalent cash flows.
3. By redefining the probabilities that are incorporated in the expectation operator.
The CAPM can be used in the first two ways. Writing the CAPM in terms of a discount rate is straightforward. The CAPM gives the expected (= required) return on portfolio p as:
E(rp)= rf + (E(rm)− rf)βp
Return is also the expectation of the uncertain end-of-period value of portfolio p, VpT, minus the value now, Vp0, as fraction of Vp0, similar to the discretely compounded returns we calculated before:
E(rp)= E(VpT)− Vp0 Vp0
in which the end-of-period value and, hence, the return are uncertain. The discount rate that links the expected end-of-period value, E(Vp,T),to the value now, Vp,0,is found by equating the two expressions:
E(Vp,T)− Vp,0
Vp,0 = rf + (E(rm)− rf)βp and then solving for Vp,0which gives:
Vp,0= E(Vp,T)
1 + rf + (E(rm)− rf)βp
where rf represents the time value of money and (E(rm)− rf)βp is the adjustment for risk. Together they form the risk-adjusted discount rate.
Certainty equivalent formulation
The second way to account for risk is to adjust the uncertain end-of-period value of our portfolio into a certainty equivalent value, which can (and should) be discounted at the risk-free interest rate. This requires some calculations, and we start with the expression that equates the two formulas for expected portfolio return:
E(Vp,T)− Vp,0
Vp,0 = rf + (E(rm)− rf)βp
We know that βp = covar(rp, rm)/σm2 and rpcan be written as (Vp,T − Vp,0)/Vp,0= (Vp,T/Vp,0)− 1. Substituting both in the right-hand side expression of portfolio return above we get:
E(Vp,T)− Vp,0
Vp,0 = rf + (E(rm)− rf) covV
p,T
Vp,0 − 1, rm
σm2
We now redefine the market price of risk (λ) as the price per unit of variance, instead of the price per unit of standard deviation that we used before:
λ= E(rm)− rf
σm2
Since σmis a constant in equilibrium this redefinition is trivial. Substituting this we get:
E(Vp,T)− Vp,0
Vp,0 = rf + λcov Vp,T
Vp,0 − 1, rm
The multiplicative constant 1/Vp,0can be written before the cov operator and the additive constant −1 can be omitted from it:
E(Vp,T)− Vp,0
Vp,0 = rf + λ 1
Vp,0cov Vp,T, rm Solving for Vp,0gives:
E(Vp,T)− Vp,0= rfVp,0+ λcov Vp,T, rm Vp,0= E(Vp,T)− λcov Vp,T, rm
1 + rf
This is the certainty equivalent formulation of the CAPM. We see that the uncertain end-of-period value is adjusted by the market price of risk, λ, times the volume of risk, which is the covariance of the uncertain end-of-period value with the return on the market portfolio. The resulting certainty equivalent value is discounted at the risk-free rate to find the present value.
Performance measures
The risk-return relationships in the CML and the SML can be reformulated to measure ex post, risk-adjusted performance. Such measures relate historical, realized returns to observed, historical risks. They are very useful to evaluate the performance of port-folios and portfolio managers because they integrate return and risk in one measure.
Sharpe (1966) suggested to use the slope of the CML as performance measure. The CML in (3.8):
E(rp)= rf +(E(rm)− rf) σm
σp can be rewritten as:
E(rp)− rf
σp = (E(rm)− rf) σm
The ex post, empirical form of the left-hand side of this equation is known as the reward-to-variability ratioor the Sharpe ratio:
Sharpe ratio: SRp = rp− rf
σp (3.10)
in which SRpstands for the Sharpe ratio of portfolio p, and rpis the portfolio’s historical average return over the observation period: rp =
trpt/T. The average risk-free interest
77 3.3 The Capital Asset Pricing Model
rate, rf, is calculated in a similar way, andσp is the standard deviation of portfolio returns: σp =
t(rpt− rp)2/T. The summations are over the T sub-periods (e.g.
weekly returns over the past two years). A set of portfolios (or funds) can be ranked according to the Sharpe ratios. Where appropriate, the ranking can be used to identify portfolios that e.g. failed to diversify properly (too highσp) or charged too high fees (rp
too small). The Sharpe ratio can be adapted so that the risk premium is not measured relative to the risk-free rate, as in (3.10), but relative to some benchmark portfolio. This performance measure is known as the information ratio.
The Treynor ratio takes its risk measure from the SML and scales the risk premium by β:
Treynor ratio: T Rp = rp− rf
βp (3.11)
where βpis estimated from the historical returns over the observation period (see equation (3.13) later on). The most obvious figure to compare the Treynor ratio with is the risk premium on the market portfolio rm− rf, which is the Treynor ratio for a portfolio with a β of 1. If the CAPM obtains, all assets and portfolios lie on the SML and have the same Treynor ratio.
A second performance measure that is directly based on the CAPM is Jensen’s alpha.
It measures the return of a portfolio in excess of what the CAPM specifies:
Jensen’s alpha: αp = rp− (rf + Bp(rm− rf)) (3.12) The alpha can be obtained by regressing the portfolio’s risk premium on the risk premium of the market portfolio:
rpt− rf t =αp+ Bp(rmt− rf t)+εpt
Taking averages over the observation period replaces the time-indexed variables with their averages and makes the error term disappear (its average is zero by the definition of a regression line): rp − rf =αp+ Bp(rm− rf). Rearranging terms gives Jensen’s alpha as in (3.12). Regression analyses also provide the statistical significance of the estimated parametersαpand Bp.
Comparing these performance measures we see that the Sharpe ratio uses total risk (σ ), while the Treynor index and Jensen’s alpha take only systematic risk (β) into account.
This makes the Sharpe ratio better suited to evaluate an investor’s total portfolio. When a portfolio is split into sub-portfolios by e.g. country, industry or portfolio manager, eval-uating these sub-portfolios with the Sharpe ratio would ignore the correlations between them and overstate risk. The Treynor ratio is more appropriate for such sub-portfolios, particularly if they are well diversified. Jensen’s alpha uses the CAPM as a benchmark.
This makes alpha very easy to interpret: it is the return in excess of (or below) what the CAPM specifies. A major disadvantage is that alphas are difficult to compare between portfolios with different risk: 2 per cent excess return may be much when the expected return is 6 per cent, but not when it is 24 per cent. By contrast, the Sharpe and Treynor ratios can be used to compare, i.e. rank, portfolios with different levels of risk, but it is difficult to interpret their values as high or low. Finally, using systematic risk makes the risk measures in the Treynor index and Jensen’s alpha dependent on the choice of
market index or market portfolio. This makes them vulnerable to Roll’s critique, as we shall see later on.