* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 16-CAPMI - BYU Marriott School
Survey
Document related concepts
Securitization wikipedia , lookup
Rate of return wikipedia , lookup
Pensions crisis wikipedia , lookup
Short (finance) wikipedia , lookup
Systemic risk wikipedia , lookup
Credit rationing wikipedia , lookup
Mark-to-market accounting wikipedia , lookup
International asset recovery wikipedia , lookup
Greeks (finance) wikipedia , lookup
Business valuation wikipedia , lookup
Stock trader wikipedia , lookup
Stock valuation wikipedia , lookup
Private equity secondary market wikipedia , lookup
Investment fund wikipedia , lookup
Modified Dietz method wikipedia , lookup
Beta (finance) wikipedia , lookup
Financial economics wikipedia , lookup
Harry Markowitz wikipedia , lookup
Transcript
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 P k e P 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 Arbitrage Pricing Model Requires many assumptions Model is very specific Requires light assumptions Model is more general Accounting Ratios Can be difficult to interpret Relies more on gut instinct Expected return E[Get ] Expected return = 1 Pay 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 If E[r]<rf+b(E[rM]-rf) then everyone should tilt away Increases price, lowers the expected return Lowers price, lowers the expected return In equilibrium: E[r]=rf+b(E[rM]-rf) for any asset 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 1 Aggregation 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” (the R-investor) CAPM: Assumption 2 Preferences Assumption The representative investor only cares about maximizing the Sharpe ratio of his portfolio Result (2.1): In equilibrium, the R-investor holds the tangency portfolio, or the portfolio with maximum Sharpe ratio. CAPM: Assumption 2 Preferences Assumption Result (2.2): For any asset i in equilibrium: E[ri ] rf bi ( E[r ] rf ) * p bi is the slope coefficient in the regression ri ai bi rp* ei ri return for stock i observed at time t rp* return for tangen cy portfolio observed at time t CAPM: Assumption 2 Preferences Assumption What if the R-investor finds an asset i such that * E[ri ] rf bi ( E[rp ] rf ) ? He knows he can increase his Sharpe ratio by tilting towards this asset. But he can’t tilt He already owns all shares of all assets! Supply of shares is fixed. CAPM: Assumption 2 Preferences Assumption CAPM: Assumption 2 Preferences Assumption What if the R-investor finds an asset i such that * E[ri ] rf bi ( E[rp ] rf ) ? He knows he can increase his Sharpe ratio by tilting away from this asset. But he can’t tilt He owns all shares of all assets. No one around to borrow from! CAPM: Assumption 2 Preferences Assumption CAPM: Assumption 3 Supply=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: Representative Investor’s total equity in the tangency portfolio: pricei*sharesi = market-cap of asset i 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). ri ai b i rm ei ri return for stock i observed at time t rm* return for value - weighted portfolio observed at time t CAPM In equilibrium, expected returns, or discount rates are given by: E[ri ] rf bi ( E[rM ] rf ) * 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. Pumpkin Crusher You need $2 to feed your family tonight Assets: Two green pumpkins: One white pumpkin Each when opened, has $0.50 inside When opened, has $1 inside 75 cents Pumpkin crusher chooses pumpkins to crush at random. If a pumpkin is crushed, you get nothing PC crushes white with 50% probability PC crushes gold with 50% probability PC never crushes more than 1 color Pumpkin Crusher If PC crushes white, how much do you have tonight? $1.75 (not enough to feed family) If PC crushes gold, how much do you have tonight? $2.75 (more than enough to feed family) Pumpkin Crusher Assume you can buy gold, white, or green pumpkins for 45 cents. What to do? Suppose you buy white If gold is crushed: $3.30 If white is crushed: $1.30 Suppose you buy Green If gold is crushed: 2.80 If white is crushed: 1.80 Pumpkin Crusher Suppose you buy gold: If white is crushed: 2.30 If gold is crushed: 2.30 At 45 cents expected returns are White: 50/45-1=0.111 Gold: 50/45-1=0.111 Green: 50/45-1=0.111 Pumpkin Crusher You probably would be willing to pay more for gold than either white or green I Gold acts as a hedging instrument – “insurance” against bad times For such assets, we are willing to accept low or even negative expected returns. Pumpkin Crusher You probably would pay more for green than white. Green makes you better off in bad times than white does, although not as much as gold. So: P(gold)>P(green)>P(white) This implies for expected returns: Gold<green<white But then Gold has a negative Sharpe ratio, and we want to buy it! Stock Return Stock Return Intuition x x y y Market Portfolio Return Asset with a positive Beta Market Portfolio Return Asset with a negative Beta “Hedging Instrument” CAPM Intuition 2 ri ai b i rm ei Var (ri ) b i2Var (rm ) Var (ei ) (by rule #6) 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. CAPM Intuition 2 Total Risk of a Stock: Var (ri ) b i2Var (rm ) Var (ei ) Systematic Risk of a Stock: b i2Var (rm ) Idiosyncratic Risk of a Stock: Var (ei ) Idiosyncratic risk gets diversified away Beta indicates the amount of risk the stock is contributing to the portfolio. Example Standard Deviation A .9 B .2 Market .18 Systematic Risk Idiosyncratic Risk A: (.05^2)*.18^2=0.00008 B: (1^2)*.18^2=.0324 A: B: 0.9^2-.00008=0.81 0.2^2-.0324=0.0076 Percent of risk that is systematic A: .00008/(.9^2)=.000010 B: .0324/.2^2=0.81 beta .05 1 Example “A” has more total risk, but virtually all of this is diversified away in the market portfolio. The representative should not require a premium to bear this risk Therefore, the representative investor should demand a higher expected return to hold “B” even though its total risk is lower than that of “A”. 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 skilled, 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 should not tilt towards the fund manager’s portfolio – rather we should invest in a passive value-weighted index. Alpha If a fund manager has been adding value in the past then rP rf b P (rM rf ) Alpha is defined as a rP rf b P (rM rf ) If a>0 then by tilting our portfolio toward the fund manager’s portfolio, we can increase our Sharpe ratio, if the past somehow reflects the future. 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: Assume .18-.05-1.8(.13-.05) = -1.4% 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