FRM 5: Modern Portfolio Theory and the Capital Asset Pricing Model
Future market moves cannot be observed in advance, so measuring market risk means working through a model, and a model earns its keep by forecasting well.
Harry Markowitz set out the first of them in his 1952 doctoral work, published as “Portfolio Selection” in the Journal of Finance, and the Nobel Prize in Economics followed. Modern Portfolio Theory, or MPT, asserts how investors should construct portfolios, resting on assumptions about investor behaviour and capital markets.
The rational investor and the mean-variance rule
The investor in the theory is rational in a narrow sense, risk averse and out to maximise utility, with the Von Neumann-Morgenstern utility theorem supplying the formal backing. Such an investor ranks allocations on the mean and the variance of the return distribution, and between two holdings with the same expected return the lower variance wins.
The assumptions Markowitz needs
Two blocks of assumption carry the theory. Capital markets are perfect: no taxes or transaction costs, costless access to all available information for every trader, and perfect competition among market participants. Returns are normally distributed, and normality collapses the problem into two numbers, the mean for performance and the variance for risk.
Diversification and covariability
Variance falls when the assets held do not move in lockstep, which is to say when they are not perfectly correlated, and diversification cancels the specific risk exposures attached to individual holdings. No asset is judged alone: what a security contributes to the mean and variance of the whole decides how much of it belongs, so what matters is the covariability of its return with the returns of everything already held.
Plot every portfolio that can be built from the available risky assets, standard deviation on the horizontal axis and expected return on the vertical. Along the left-hand boundary of the resulting cloud sits, for each level of risk, the portfolio with the highest expected return.
Only the upper half of that boundary is of interest. Its turning point is the global minimum variance portfolio, the least risky combination the asset universe allows, and below that point every portfolio is dominated by the one directly above it. The upper branch is the efficient frontier.
Where the frontier sits follows from the correlations among the assets. Assets that move together flatten it towards the straight line joining them, while weak or negative correlations bow it to the left. Which point on the curve an investor picks depends on risk tolerance, since a higher expected return can be bought only with a larger standard deviation.
MPT reorganised the theory of portfolio selection, and the objections fall into two groups: unwarranted assumptions, and fragile implementation.
The assumptions do not survive the data
Normality is the weakest link. Empirical work across asset classes and countries mostly fails to support it, finding fat tails, meaning more observations in the tails than a normal distribution allows, and asymmetry besides. An investor looking only at the mean and the variance also ignores skewness, which leaves the estimated mean and variance themselves incorrect. Campbell Harvey, and Bekaert together with Harvey, found that skewness matters in asset pricing.
Estimation error swamps the optimiser
The second difficulty arrives at implementation. Running the model requires the mean and variance of returns for every asset plus the correlations between them, all estimated from historical data over some window. The theory does not identify which window, and the allocation can differ greatly depending on which stretch of history is used.
Even small deviations in the input values move the resulting portfolios a long way, so mean-variance efficient portfolios are highly sensitive to their inputs. Errors in estimating expected returns cause the most concern, and the research puts them at a minimum of 10 times the weight of errors in the variances and covariances.
Methods exist for handling that uncertainty, the best known being robust portfolio optimization, which builds estimation error directly into the optimisation process.
Correlations move, and they rose after the global financial crisis
Correlation between assets is the input mean-variance analysis leans on hardest, and it is not a fixed property. Craig Israelsen tracked rolling correlations between large capitalisation United States stocks and the other major asset classes around the global financial crisis, finding them higher afterwards than before, still elevated once conditions calmed, and in some pairings later reversed. The reason usually offered is basket trading: index-tracking mutual funds and exchange-traded funds move large baskets built on benchmark indices at once, independently of any analyst view on the names inside them.
One response is to identify risk regimes and optimise a separate allocation for each. A worried market shows up as higher equity volatility and wider credit spreads, and such spells give way to quieter ones. A manager expecting a high-risk regime shifts towards money market funds and similar low-risk holdings, while one expecting calm can favour equities, emerging markets, commodities and high-yield bonds.
During the 1960s William Sharpe, John Lintner and Jan Mossin furthered MPT by incorporating overall capital market equilibrium. The model they derived is the capital asset pricing model, or CAPM, relating the risk and the expected return of a risky asset. Sharpe took the Nobel Prize in 1990; Lintner and Mossin had died and could not share it.
Splitting total risk in two
The starting move splits total risk, as measured by the standard deviation of returns, into two components. The first is systematic risk, which under the CAPM is market risk and is proxied by beta. The second belongs to the asset alone: for a stock, a strike, an adverse turn in regulation, litigation alleging wrongdoing. It goes under several names, idiosyncratic risk and unique risk among them, and because a portfolio eliminates it, diversifiable risk as well. Systematic risk survives even a well-diversified holding, so it is equally known as non-diversifiable risk, and it alone earns compensation.
How beta measures systematic risk
Beta is the covariance of asset returns with market returns divided by the variance of market returns, which is also the correlation between the two series times the ratio of their standard deviations:
The numerator is what the asset adds to the risk of the market portfolio and the denominator is the total risk of that portfolio, so beta reports a share. That correlation also splits the variance of the asset: at a correlation of 0.6 with the market, 36% is systematic and 64% specific.
The assumptions
Equilibrium requires a demanding list, much of it inherited from Markowitz. Information reaches every market participant free of charge and is absorbed the instant it arrives. All participants hold the same expectations and decide on the mean and the variance of returns. Frictions are absent: no transaction costs, no taxes, nothing standing between a decision and a trade. Investments are perfectly divisible, so an allocation of any partial amount can be made. Everyone can lend and borrow at one common risk-free rate of interest, usually the rate on a government obligation. And no single investor is big enough for an allocation decision to move market prices. Researchers have since relaxed all of them.
The equation
Under those conditions the expected rate of return on asset i over a given holding period is:
Here E(Ri) is the expected return on asset i across the holding period and r is the rate the risk-free asset pays. Market risk premium is E(RM) minus r, beta is the quantity of market risk, and their product is the premium above the risk-free rate that investors require.
Both inputs need proxies. E(RM) stands for a portfolio holding every risky asset, so a broad index of traded shares, the S&P 500 for instance, stands in, though broadness is subjective and the size of the premium is debated. The risk-free rate is commonly estimated by the three-month U.S. Treasury rate.
Applying the model is arithmetic once the risk-free rate, the expected market return and beta are in hand. Beta is the only input that changes from one security to the next, so two stocks sharing a beta share an expected return in equilibrium. A stock whose beta exceeds 1 is aggressive, carrying more systematic risk than the market, and one below 1 is defensive. Utility companies in the United States are often extremely defensive, with betas of around 0.5.
Take a risk-free rate of interest of r = 5%, with the market portfolio characterised by E(RM) = 13%. Stock A has a beta of 0.5, stock B a beta of 1.0 and stock C a beta of 2.0.
Beta of a portfolio
Portfolio beta is the weighted average of the betas of the holdings, using the portfolio weights. That linearity is also why systematic risk cannot be diversified away: averaging exposures to the market leaves an exposure to the market. Weight the covariances the same way across every asset in the market portfolio and they sum to the variance of the market, so the weighted betas sum to one: the market portfolio has a beta of one by construction.
A portfolio holds three stocks whose betas are 3M at 1.14, IRobot Corporation at 1.49 and Applied Materials Inc. at 1.64, with 40% of its value in 3M, 35% in IRobot and 25% in Applied Materials. The risk-free rate is 5% and E(RM) is 13%.
Return to the efficient frontier and introduce the risk-free rate r. That asset carries no standard deviation, so it lies on the vertical axis, and a line drawn from it runs tangent to the frontier at a single point M, the tangency portfolio. That line is the capital market line, or CML.
Every portfolio on the line except M offers a higher expected return at its level of standard deviation, so the line dominates the curve. The tangency portfolio holds every asset in proportion to its market value, which makes it the market portfolio of the CAPM.
The equation for the CML is:
The bracketed term is the slope, the fair equilibrium compensation per unit of volatility. One consequence is the two-fund separation theorem: every investor holds the risk-free asset alongside the market portfolio, with the split between them set by risk appetite alone.
From the CML to the security market line
A second relationship developed from the CML is the security market line, or SML, relating expected return and risk for individual assets rather than portfolios. Its risk measure is systematic risk proxied by beta, not standard deviation. Plotted against beta it is the CAPM equation itself, starting at r and passing through the market portfolio at a beta of 1.
Beta is estimated rather than observed, and in practice the estimate comes from a simple linear regression:
Subtracting the risk-free rate from both sides of the CAPM produces this form, which lowers the intercept but leaves the slope alone. The two are still often confused. The market model is empirical and built from realised returns, whereas the CAPM rests on expected and unobserved variables, and it splits returns into a systematic component and an uncorrelated residual.
A worked estimate: J.P. Morgan against SPY
Take J.P. Morgan stock on monthly returns running from June 2008 through May 2019, with the ETF SPY, designed to track the S&P 500, as the market proxy. The calculation ignores dividends, uses log returns for the equities and nets the one-year U.S. Treasury bill rate against those figures. The fitted line is y = 0.3639x – 0.0014 with R2 = 0.4571, so the raw, unadjusted beta is 0.36. A 1% change in market excess returns corresponds to a 0.36% change in excess returns for J.P. Morgan, which makes it a defensive stock.
A beta calculated solely on the basis of historical data is a raw beta; adjusting it, say to reflect mean reversion properties, gives an adjusted beta. Either way the result is an estimate of a true beta that stays unknown, so it carries statistical estimation error and can be placed only inside a confidence interval at a chosen confidence level. A different window or market proxy moves it.
The equity beta of a company reflects the debt it carries; removing that effect of leverage gives the unlevered beta:
Where beta is used, and where the CAPM falls short
Beta matters beyond portfolio management. Many corporations set a minimum rate of return before committing to a new venture, and that hurdle rate is often based on beta analyses.
The model has little empirical support despite its critical role in financial theory, and one finding against it is that factors other than the market drive security returns. The original CAPM was built for discrete time intervals, one-year or one-month horizons for instance; Merton extended it to continuous time by assuming trades are continuously executable and price changes smooth.
The three CAPM-based measures all judge performance against the return on a risk-free asset. Two further measures use something the client specified instead.
Tracking error and the information ratio
Tracking error, TE, measures how far portfolio returns drift from the benchmark it was built to mimic or beat. The gap in each period between the portfolio return and the benchmark return the client specified is the active return, and tracking error is its standard deviation.
An indexed portfolio, built to mimic the benchmark, has a tracking error near zero, while active managers run larger ones that grow the further the portfolio strays. Since tracking error says nothing about whether the deviation paid off, the information ratio divides average active return by it.
Sortino ratio
The Sortino ratio, SR, modifies the SPI for an investor focused on downside risk. The numerator replaces the risk-free rate with T, the target or required rate of return, also known as the minimum accepted rate of return, or MAR. The denominator becomes the downside deviation, the standard deviation of returns below the target:
Because performance is measured against a client-specified return rather than a risk-free asset, portfolio managers often prefer it to Sharpe and Treynor.
A fund reports annual returns over five years of 9%, -3%, 12%, 5% and 1%. Against its benchmark those years give active returns of 1.8%, -0.5%, 2.2%, 0.4% and -0.9%. The client sets a minimum accepted rate of return of 2%, and the risk-free rate is also 2%.
| Measure | Numerator | Risk in the denominator | Compared against |
|---|---|---|---|
| Sharpe performance index | Excess over the risk-free rate | Standard deviation | Slope of the capital market line |
| Treynor performance index | Excess over the risk-free rate | Beta | Market excess return |
| Jensen performance index | Regression intercept | Beta, through the regression | Zero alpha |
| Information ratio | Average active return | Tracking error | Zero active return |
| Sortino ratio | Excess over the target T | Downside deviation | Zero excess over T |
Source: the performance measures as defined in the chapter.