Systematic vs Unsystematic Risk

This reading adds a risk-free asset to a risky portfolio to derive the capital allocation line and the capital market line, separates systematic from unsystematic risk, and covers return generating models, the market model and the calculation of beta. It then derives the CAPM and the security market line, shows how to calculate required returns and identify mispriced securities, and how to compute and interpret the Sharpe ratio, M-squared, the Treynor measure and Jensen's alpha.

LOS 88.a — Adding a risk-free asset to a risky portfolio

A risk-free asset has a standard deviation of returns of zero, and its returns have zero correlation with the returns of any risky asset. When an investor puts a weight $w$ in a risky portfolio P and $1-w$ in the risk-free asset, the two-asset portfolio formulas reduce to two simple expressions:

• The expected return is a weighted average of the risk-free rate and the risky portfolio's expected return.
• The standard deviation is the risky weight times the risky portfolio's standard deviation, because the risk-free asset contributes no variance and no covariance.
• Both $E(R_C)$ and ${\sigma }_C$ are linear in $w$, so every combination plots on a straight line that starts at $R_f$ on the vertical axis and passes through P.

A single risky portfolio plus the risk-free asset therefore lets investors with very different degrees of risk aversion reach the level of risk they want. The resulting combinations can dominate risky portfolios that lie below the line.

LOS 88.b — The capital allocation line (CAL) and the capital market line (CML)

The capital allocation line (CAL) is the line of risk–return combinations available from the risk-free asset and one particular risky portfolio. Each risky portfolio produces its own CAL. Of all the CALs, the investor should use the one whose set of portfolios she prefers most. That is the steepest line from $R_f$, the one tangent to the efficient frontier, called the optimal CAL. The tangency point is the optimal risky portfolio: of all risky portfolios, it has the highest excess return per unit of standard deviation, $[E(R_P)-R_f]/{\sigma }_P$, which is the Sharpe ratio of LOS 88.i. Because every CAL starts at $R_f$, the optimal CAL offers a higher expected return than any other CAL at every level of risk above zero.

In the figure, $R_f=3\%$. B has a higher expected return than the global minimum-variance portfolio A and a steeper CAL: $(9.0-3)/20=0.30$ against $(4.6-3)/8=0.20$. T's CAL is steeper still, $(9.4-3)/16=0.40$, so T is the optimal risky portfolio. B lies inside the frontier, and T offers both less risk and more expected return than B. The investor then picks the point on the optimal CAL that maximizes her expected utility, where her highest indifference curve touches the line.

If investors had different forecasts, each would find a different tangency portfolio and a different CAL. Capital market theory assumes homogeneous expectations: everyone has the same estimates of expected returns, standard deviations and correlations. Everyone then faces the same efficient frontier and the same tangency portfolio, and that portfolio must be the market portfolio. The market portfolio is the portfolio of all risky assets in existence, held in market-value weights; it contains no risk-free asset. The common optimal CAL is called the capital market line (CML).

• The intercept is $R_f$. The slope, $\frac{E(R_M)-R_f}{{\sigma }_M}$, is the excess return per unit of total risk (standard deviation). It equals the Sharpe ratio of the market portfolio.
• $E(R_M)-R_f$ is the market risk premium. Writing the CML as $E(R_P)=R_f+[E(R_M)-R_f]({\sigma }_P/{\sigma }_M)$ shows that each unit of market risk ${\sigma }_M$ an investor accepts adds one market risk premium to her expected return.
• Risk on the horizontal axis is standard deviation (total risk). Beta is the risk measure of the security market line (LOS 88.f).
• Points between $R_f$ and M are lending portfolios, with part of the money invested at $R_f$. Points beyond M, further up the line, are borrowing portfolios: the investor borrows at $R_f$ and invests more than 100% of her equity in M (buying on margin). All portfolios on the CML hold the same risky portfolio (M), so their returns are perfectly positively correlated with each other and they carry no unsystematic risk.

Two-fund separation. Combining the CML with an investor's indifference curves splits portfolio choice into two separate decisions. Which risky assets to hold is the same decision for everyone: hold the market portfolio. How much risk to take, meaning how much to lend or borrow at $R_f$, depends on the investor's own risk aversion. This is the two-fund separation theorem (the separation theorem): every investor's optimal portfolio combines just two funds, the risk-free asset and the optimal risky portfolio. Security selection thereby reduces to building one well-diversified portfolio.

Passive vs. active. Investors who believe market prices are informationally efficient follow a passive investment strategy. They hold an index fund as a proxy for M and split their money between it and the risk-free asset. Investors who think they can identify mispriced securities use active portfolio management, overweighting securities they consider undervalued and underweighting those they consider overvalued.

LOS 88.c — Systematic and unsystematic risk

Total risk is measured by the standard deviation (or variance) of returns. Exam convention: total risk = systematic risk + unsystematic risk is stated as a simple sum, with total risk measured by standard deviation. Current practice: the sum is exact for variances, as in the market model ${\sigma }_i^2={\beta }_i^2{\sigma }_m^2+{\sigma }_e^2$; the standard deviations of the two parts do not add up to the total standard deviation.

Systematic risk applies to single securities as well as to portfolios. Makers of luxury goods, such as sports cars and premium motorcycles, respond strongly to the business cycle and have high systematic risk. Utilities respond little to market-wide factors and have low systematic risk. As the number of randomly chosen stocks in a portfolio grows, total risk declines and approaches market risk. According to one study, about 12 to 18 stocks capture roughly 90% of the maximum diversification benefit; another put the number at about 30 stocks. Either figure is far smaller than the number of securities in the market.

As unsystematic risk is diversified away, the portfolio behaves more and more like the market, so its correlation with the market moves toward +1. The market portfolio is perfectly diversified, and all of its risk is systematic.

Bonds. The same reasoning applies to a bond portfolio. A surprise that hits one issuer, such as a downgrade, can be diversified away by holding bonds of many issuers, while economy-wide changes, such as a recession or a rise in interest rates or inflation, affect all bonds and cannot be diversified away.

Why unsystematic risk earns no premium. Capital market theory assumes diversification is costless; a low-cost index fund comes close. Investors will not be paid for bearing a risk they could remove for free, so in equilibrium the expected (required) return depends only on systematic risk. A single-drug biotech stock can have very high total risk yet low systematic risk, and therefore a low equilibrium return. A steady machine-tool maker with modest total risk but high sensitivity to the business cycle requires a higher return. A portfolio of 50 or 100 such biotech stocks would carry little of that firm-specific uncertainty, because the successes and failures offset one another. A security's systematic risk is measured by what it contributes to the risk of a well-diversified portfolio.

LOS 88.d — Return generating models and the market model

A return generating model estimates an asset's expected return from its exposure to one or more factors. Factors may be macroeconomic (GDP growth, inflation, consumer confidence), fundamental (earnings, earnings growth, firm size, research spending) or statistical. In practice, most models rely mainly on macroeconomic and fundamental factors. Statistical factors frequently lack any grounding in finance theory. They may capture a pattern specific to one sample period that was found by data mining (testing one data set over and over).

The multifactor model in excess-return form is:

Each $\beta$ is a factor sensitivity (factor loading). The first factor is often the market's excess return.

• The Fama and French model uses three factors: the market's excess return, firm size, and the book-to-market (book value to market value) ratio.
• Carhart adds a fourth factor, price momentum, based on prior-period returns. The four factors explain a good share of the differences in US equity returns over the period in which the model was estimated.

A single-factor model uses just one factor, whatever that factor is (for example, only the book-to-market ratio, or only the earnings growth rate). The single-factor model whose factor is the market's excess return is the single-index model: $E(R_i)-R_f={\beta }_i[E(R_m)-R_f]$.

The market model is a simplified single-index model estimated by regressing the asset's return on the market's rate of return:

• ${\alpha }_i$ and ${\beta }_i$ are estimated from historical returns. The expected return is ${\alpha }_i+{\beta }_{i}E(R_m)$.
• It is used to estimate ${\beta }_i$ and to measure abnormal return, $e_i=R_i-({\alpha }_i+{\beta }_i{R}_m)$: the actual return minus the return expected given the actual market return.
• For consistency with the single-index model, ${\alpha }_i$ can be set to $R_f(1-{\beta }_i)$.

LOS 88.e — Calculating and interpreting beta

Beta is a standardized measure of the covariance of an asset's return with the market's return. It measures systematic risk only:

• The market's beta is 1, because the covariance of the market's return with itself is its variance: $Cov(R_m,R_m)/{\sigma }_m^2=1$. The (value-weighted) average beta of all stocks is also 1. A beta of 0 means the asset's return is uncorrelated with the market. Negative betas are possible.
• In practice beta is estimated as the slope of a regression of the asset's excess returns on the market's excess returns. The least squares line, which minimizes the sum of squared vertical distances of the points from the line, is the security characteristic line. Its slope is $Co{v}_{i,m}/{\sigma }_m^2$. A slope steeper than 45° means $\beta >1$: the asset's return reacts more strongly to systematic factors than the market's does.

Backing beta out of the CAPM. The CAPM (LOS 88.f) states $E(R_i)=R_f+{\beta }_i[E(R_m)-R_f]$. Rearranging it gives ${\beta }_i=[E(R_i)-R_f]/[E(R_m)-R_f]$. If a stock is expected to earn 12% with $R_f=2\%$ and $E(R_m)=7\%$, $\beta =10/5=2.0$.

LOS 88.f — The CAPM, its assumptions, and the security market line (SML)

Only systematic risk is priced, so the relevant risk of any asset is its covariance with the market. Plotting expected return against $Cov(R_i,R_m)$, or against beta after standardizing by the market variance, gives the security market line (SML). The equation of the SML is the capital asset pricing model (CAPM):

The intercept is $R_f$ (beta 0), the market plots at $\beta =1$, and the slope is the market risk premium $E(R_m)-R_f$. With a positive market risk premium, expected return rises linearly with systematic risk.

Assumptions of the CAPM

Comparing the CML and the SML

The two lines are drawn against different risk measures, so there is no point at which they "converge". In equilibrium an inefficient portfolio or a single stock plots below the CML but on the SML. Every asset with a beta of 1 plots at the same point on the SML as the market, whatever its total risk.

LOS 88.g — Expected (required) return with the CAPM

Substitute the inputs into $E(R_i)=R_f+{\beta }_i[E(R_m)-R_f]$. Check whether the question gives the expected market return $E(R_m)$ or the market risk premium $E(R_m)-R_f$.

Useful rearrangements and relationships:

• Market risk premium: $E(R_m)-R_f=[E(R_i)-R_f]/{\beta }_i$. The same equation solved for beta is shown under LOS 88.e.
• Nominal risk-free rate $\approx$ real risk-free rate + expected inflation.
• With a positive market risk premium ($E(R_m)>R_f$), a negative beta gives a required return below $R_f$. If $E(R_m)<R_f$, a negative beta gives a required return above $R_f$. For example, with $\beta =-0.5$, $R_f=4\%$ and $E(R_m)=2\%$, the required return is $4-0.5(-2)=5\%$.
• Holding everything else constant and assuming a positive market risk premium, a higher beta raises the required return, so the price investors will pay today falls. With a negative market risk premium the beta ranking reverses.

LOS 88.h — Applying the CAPM and the SML: finding mispriced securities

In equilibrium, a security's expected return equals its required return from the SML. An analyst compares her own forecast with the CAPM required return:

The forecast minus the required return is the security's expected alpha, the size of the mispricing. When forced to sell one of two undervalued holdings, sell the one with the smaller alpha.

• Stock P: $\beta =0.8$, bought at 50, forecast price 53 plus a 1.50 dividend. Forecast $=(53-50+1.50)/50=9.0\%$; required $=3+0.8(6)=7.8\%$. P is undervalued by 1.2% and plots above the SML.
• Stock Q: $\beta =1.3$, forecast 10.0%; required $=3+1.3(6)=10.8\%$. Q is overvalued by 0.8% and plots below the SML.

The graph and the algebra always give the same conclusion, because the SML is the graph of the CAPM equation.

LOS 88.i — Sharpe ratio, Treynor measure, M-squared and Jensen's alpha

Performance evaluation analyzes the risk and return of an active manager's portfolio. Attribution analysis, part of performance evaluation, looks for the sources of the difference between the active portfolio's return and a passive benchmark's return. A higher return than the benchmark proves little on its own: a portfolio with more risk (especially more beta) than the benchmark should earn more over time, so returns must be adjusted for risk.

• The Sharpe ratio can be used ex ante (expected values) or ex post (realized means and sample standard deviation). Its value is meaningful only in comparison with another portfolio's Sharpe ratio.
• M² produces the same rankings as the Sharpe ratio but is stated in percentage terms. Because $M^2=S{R}_P\times {\sigma }_M+R_f$, M² exceeds $R_M$ (M² alpha > 0) exactly when P's Sharpe ratio exceeds the CML slope. The Treynor measure and Jensen's alpha are the beta-based analogues: Treynor is a slope, and Jensen's alpha is a percentage.
• The choice of risk measure depends on whether the portfolio carries unsystematic risk. A single manager's portfolio or a concentrated holding should be judged with total-risk measures (Sharpe or M²). A well-diversified fund, such as one spread over many managers, can be judged with beta-based measures (Treynor or Jensen's alpha).
• Portfolios above the CML have higher Sharpe ratios than any CML portfolio and positive M² alphas. Portfolios above the SML have higher Treynor measures than any asset on the SML and positive Jensen's alphas.

• Sharpe $=8/20=0.40$, above the CML slope of $6/16=0.375$, so P beat the market on a total-risk basis.
• M² $=3+0.40(16)=9.4\%$; M² alpha $=9.4-9.0=0.4\%$.
• Treynor $=8/1.1=7.27\%$, against a market Treynor measure of 6%.
• Jensen's alpha $=11-[3+1.1(6)]=11-9.6=1.4\%$.

	Return	σ	β	Sharpe ratio	M² alpha	Treynor measure	Jensen's alpha
Fund X	12%	24%	0.9	9/24 = 0.375	3 + 0.375(16) − 9 = 0.0%	9/0.9 = 10.0%	12 − [3 + 0.9(6)] = +3.6%
Fund Y	10%	14%	1.2	7/14 = 0.50	3 + 0.50(16) − 9 = +2.0%	7/1.2 = 5.8%	10 − [3 + 1.2(6)] = −0.2%
Market	9%	16%	1.0	6/16 = 0.375	0.0%	6/1.0 = 6.0%	0.0%

Fund Y wins on the total-risk measures (Sharpe 0.50 against 0.375; M² alpha +2.0% against 0.0%). Fund X wins on the beta-based measures (Treynor 10.0% against 5.8%; Jensen's alpha +3.6% against −0.2%). X has a low beta but a high standard deviation, so much of its risk is unsystematic. If X is the investor's whole portfolio, that risk matters and Y is better. If X is one small part of a well-diversified fund, its unsystematic risk is diversified away and X is better. Sharpe and Treynor values are read by comparing them with another portfolio or with the market; M² alpha and Jensen's alpha are read directly, where a value above zero means the fund beat the market after adjusting for risk.

On a graph of return against beta, P's Treynor measure is the slope of the line from $R_f$ through P, and its Jensen's alpha is the vertical gap between P and the SML at $\beta =1.1$.

The inputs to these measures, including the expected market return (and hence the market risk premium) and betas, are estimated with error. The expected market return need not equal its historical average. Results from the models are only as good as those estimates.