Capital Asset Pricing Model

Capital Asset Pricing Model BKM: Chapter 7

Optimal Portfolios • Given current expected returns, standard deviations, and correlations: • Choose risky portfolio to maximize Sharpe ratio. • Tailor the risk by investing (long or short) in the risk-free asset. • This is the right approach whether or not markets are “efficient”.

Beating the Market • Beating the market means more than earning a higher average return • You could do this by simply investing in small-cap stocks. • Beating the market means developing a portfolio with a higher Sharpe ratio.

Beating the market • To beat the market: you should continually be searching for stocks such that • E[r]>rf+bE[rM]-rf) tilt towards these • E[r]<rf+b(E[rM]-rf) tilt away from these • By tilting the market portfolio as above, you increase the Sharpe ratio of the portfolio.

Efficient Markets • If markets are efficient • We can’t predict “e” – our best guess is zero • k is then the expected return • What determines k across stocks?

Models of k • Capital Asset Pricing Model • Requires many assumptions • Model is very specific • Arbitrage Pricing Model • Requires light assumptions • Model is more general • Accounting Ratios • Can be difficult to interpret • Relies more on gut instinct

Expected return • Expected return = • Anything that raises the price, lowers the expected return • Anything the lowers the price, raises the expected return

Intuition of the CAPM • Basic Idea: everyone is trying to beat the market • If E[r]>rf+bE[rM]-rf) then everyone should buy these • Increases price, lowers the expected return • If E[r]<rf+b(E[rM]-rf) then everyone should tilt away • Lowers price, lowers the expected return • In equilibrium: E[r]=rf+b(E[rM]-rf)

Intuition of the CAPM • But what is the market portfolio? • In equilibrium, the “market portfolio” should be the portfolio with highest Sharpe ratio. • To answer this question, let’s look at a sketch of the proof of the CAPM.

CAPM: Assumption 1Aggregation Assumption • Assumption: all investors have the same forecasts of variances, covariances, and expected returns. • Very strong assumption • Although not true, there may be considerable consensus among investors on what these parameters are. • Result (1): • We can think of the aggregate investing community as one big investor called the “representative investor”

CAPM: Assumption 2Preferences Assumption • The representative investor only cares about maximizing the Sharpe ratio of her portfolio • Result (2.1): • The representative investor seeks to hold the tangency portfolio, or the portfolio with maximum Sharpe ratio.

The Tangency Portfolio • Result (2.2): For any asset i in equilibrium: • bi is the slope coefficient in the regression

CAPM: Assumption 3Supply=Demand Assumption • The representative investor must hold all shares of all risky assets (supply=demand) • Result (3) • Representative Investor’s total equity in asset i: • pricei*sharesi = market-cap of asset i • Representative Investor’s total equity in the tangency portfolio: • total market-cap of all risky assets. • Portfolio weight of asset i in tangency portfolio: • Value weight

CAPM • Given the assumptions, the tangency portfolio held by the representative investor is the value-weighted portfolio of all assets. • Notationally, we call the slope coefficient of any asset on the value-weighted portfolio “beta” (b).

CAPM • In equilibrium, expected returns, or discount rates are given by:

CAPM: Intuition 1 • Assets with negative betas tend to pay high realized returns when the market tanks • Assets with positive betas tend to pay low realized returns when the market tanks • Asset with negative betas are like hedging instruments (insurance contracts). They should therefore have high prices and low expected returns.

Intuition Stock Return Stock Return Market Portfolio Return Market Portfolio Return Asset with a negative Beta “Hedging Instrument” Asset with a positive Beta

CAPM Intuition 2 • But we know any fluctuations in ei are “diversified away” • The market does not require compensation to bear this type of volatility risk. • The only firm-specific characteristic that matters in determining expected returns is the firm’s beta.

Using the CAPM • How can we evaluate a fund manager? • Is manager’s Sharpe ratio above that of the value-weighted portfolio? • Many fund managers try to create value by specializing in a certain style: • Small-cap • Growth • Emerging market equity

Using the CAPM • Fund managers who are style focused create “separate pieces” that we can combine to create portfolios with high Sharpe ratios if they are consistently finding securities such that • E[r]>rf+bE[rM]-rf) and buying these • E[r]<rf+b(E[rM]-rf) and shorting these

Using the CAPM • If a fund manager is “adding value” then for the portfolio he manages: E[rp]>rf+bp(E[rM]-rf) • We can increase the Sharpe ratio of our portfolio by tilting toward the fund manager’s portfolio. • If the above condition is not satisfied, then we are just as well off in a passive index.

Alpha • If a fund manager has been adding value in the past then • Alpha is defined as • If a>0 then by tilting our portfolio toward the fund manager’s portfolio, we can increase our Sharpe ratio.

Example • Suppose a small cap manager has earned an average return of 18% over the last 10 years. • CAPM beta of portfolio = 1.8 • risk-free rate = 0.05 • Average return on value-weighted portfolio = 0.13 • What is fund manager’s alpha? • How could you create a portfolio using t-bills and the market portfolio with the same beta?

Example • Fund manager’s Alpha: • .18-.05-1.8(.13-.05) = -1.4% • Assume • w1 is portfolio weight in asset 1 • w2 is portfolio weight in asset 2 • … • wn is portfolio weight in asset 3 • Beta of portfolio is • w1b1+ w2b2+…+wnbn

Example • Beta of value-weighted portfolio=1.0 • Beta of risk-free bonds = 0.0 • Portfolio with same beta w(1.0)+(1-w)(0)=1.8 w=1.8 So invest 180% of equity in market portfolio by shorting the t-bill. • Expected Return of this portfolio: • .05+1.8*(.13-.05)=19.4% • 1.4% higher than fund

Capital Asset Pricing Model