LECTURE NOTES 3 ARIEL M VIALE I Markowitz-Tobin Mean-Variance Portfolio Analysis Assumption Mean-Variance preferences Markowitz 95 Quadratic utility function E [ w b w ] { = E [ w] b V ar w + E [ w] } by definition of variance Normally distributed asset returns Assume k risky assets with R = R R K vector of gross returns ι + r R = R R K vector of expected returns ι = vector of K ones w = w w K vector of portfolio weights a function of wealth invested in each asset { R p = w R w R + ι w R f where R f = + r f If there is no riskless asset then we require ι w = otherwise ι w = fraction of wealth invested in the riskless asset could be negative ie short position Define [ ] V = E R R R R = var cov &K cov K& var K as the variance-covariance matrix of the k risky assets Also { w R p = R w R + ι w R f Date: 09/0/09 - Rev: 09/0/3
LECTURE NOTES 3 And [ σp = E Rp R ] p [ ] = E w R w R [ { } ] [ = E w R R ] = E w R R R R w [ ] = w E R R R R w = w V w Note that V is symmetric and psd positive semi-definite Claim If it is not definite then there is no risk-free portfolio and no perfect hedge in the economy Proof By example Let K = σ V = ρσ σ ρσ σ σ where ρ = σ σ σ is the correlation coefficient Note that there exists a zero-risk portfolio iff ρ = ± Why? V ar R p = wσ + wσ + w w ρσ σ If ρ = then V ar R p = w σ + w σ = 0 if w = wσ σ If ρ = then V ar R p = w σ w σ = 0 if w = wσ σ The Portfolio Choice Problem Find the portfolio with minimum variance for a given R p 3 Min w w V w st w R + ι w R f = R p Assuming an interior solution exists the Lagrangian function can be written as L w V w λ w R + ι w R f R p with first order necessary conditions FONC: wrt w L w derivative of matrix w = λv R Rf ι where V has full rank w V λ R R f ι = 0 V w λ R Rf ι = 0 wrt λ R Rf ι w = Rp R f λ R Rf ι V R Rf ι = R p R f after substituting the optimal w
LECTURE NOTES 3 3 λ = R p R f R R f ι V R R f ι where R Rf ι is the vector of risk premiums R 4 w p R f = R Rf ι V R Rf ι V R Rf ι k scalar R Rf ι V R Rf ι V w R = p R R R f ι η w T V w = w T R Rf ι f scalar R k R f = η cov of asset k with efficient portfolio k σ p = R p R f η If R p = R f σ p = 0 if R p > R f σ p directly proportional to the portfolio s risk premium The CML The risk-return tradeoff is linear and positive Definition Sharpe s ratio We define the Sharpe Ratio as R p R f σ p risk premium stdev = That is the equilibrium excess return on the efficient portfolio per unit of portfolio risk ie the slope of the CML Claim The efficient tangency portfolio is the one that maximizes Sharpe s ratio Proof Follows from the previous analysis II CAPM The Markowitz-Tobin mean-variance analysis is normative as it prescribes the best way to allocate wealth among various assets Alternatively we can interpret it as a positive or descriptive theory of what investors actually do Under this interpretation we can extend the portfolio problem to that of asset pricing under equilibrium Note that the mean-variance analysis does not assume a RA with SI or TI utility function and initial aggregate endowment/wealth It does assume that investors have the same expectations regarding the probability distributions of asset returns all assets are tradable there are no indivisibilites in asset holdings and there are no limits to borrowing and lending at the risk-free rate of return wealth of investor i Let w i =portfolio of investor i and ς i = total wealth investor s i fraction of total wealth Then the market portfolio is w m = i ς iw i i I This market portfolio is a claim to the total future endowment of the economy regardless of the state of the world By holding a share of the market portfolio the investor assures a fraction ς i of total future endowment almost sure Thus the equilibrium relationship 4 can be interpreted as a relation between expected excess returns on any asset and the expected excess return on the broad market portfolio with gross return R m proxying for systematic risk Assume riskless asset Define X k = R k R f as the excess return on asset k so X k = R k R f is the risk premium on asset k and X m = R m R f is the market excess return The behavioral problem of the ith investor is now 5 = Min w w V w
LECTURE NOTES 3 4 st w X = Xm Assuming an interior solution exists the Lagrangian is L w V w λ w X Xm where the FONC are wrt w L w V w λ X = 0 and wrt λ w X Xm = 0 w X = Xm and λv X = w Substituting the latter in the former gives λ X V X = Xm λ = X m X V X X m X V X V X Rearranging terms and multiplying by w T on both w = sides of the equal sign as before leads to 6 k Rk R f = η cov Rk R m where Rm = w R + ι w R f And the standard deviation of the market mean variance efficient portfolio is λ X V X The Sharpe ratio is X V X by definition ie the slope of the CML that now passes through the origin and ι w = Two-fund allocation ie investors are long on the market portfolio consisting only of risky assets and short on the risk free asset The Sharpe ratio can be interpreted now as the market price of systematic or non-diversifiable risk Recall that η = X m var R m the latter in 6 gives the CAPM or SML cov Rk R m 7 Rk R f = Rm σm }{{ R f k } β k Thus substituting Equation 7 shows that the required excess expected return for a risky asset is a linear function of systematic risk with the price of systematic risk as slope Also Empirical Test of the CAPM Consider an orthogonal projection of Rk on R p where p is some portfolio proxying for the market portfolio The empirical SML Then write 8 Rk = a k + b k Rp + ε k k where E [ˆε k ] = 0 and E [ˆε k R ] p = 0 Note R k = a k + b k Rp [ E Rk a k b k Rp R ] p = 0 cov Rk R p b k var Rp = 0 b k = cov R k R p var R p and substituting a k = R k cov R k R p var R p R p Condition If R p is minimum variance mv-efficient then a k = 0 k From 6 the standard deviation of the market portfolio return is V ar Rm = X mη From the FONC wrt λ we know that X m = λ X V X and V ar Rm = λ X V X η Notice that λ = η and V ar Rm = λ X V X
LECTURE NOTES 3 5 Remark The test using the SML is a test of efficient mean-variance preferences not the CAPM 3 Roll s Critique i If the market is assumed to be ex ante mean variance efficient then the CAPM holds exactly ii As the return of the market portfolio is unobservable the best we can do is an ex post test of mean variance efficiency on some index portfolio we choose to proxy for the market portfolio without actually testing the CAPM III Black s Zero-Beta CAPM Rp α k Pick any mv-efficient portfolio w Then α st Rk α = cov R k R p σp where α is the expected return of any portfolio or asset with zero covariance with the mv-efficient portfolio Recall that the projection of R k on R p is R k = a k + cov R k R p R var R p p + ε k Notice that R k = α b k α+b k Rp if b k = cov R k R p var R Thus the restriction to be tested in the p zero-beta CAPM is now a k = b k α = 0 a = a = = a n b b IV Arbitrage Pricing Theorem } {{ b n } n restrictions We assume N risk factors and K assets in the economy such that K > N Definition Risk factor f n is the random realization of the nth risk factor Definition 3 Factor sensitivity b kn is the sensitivity of the kth asset to the nth risk factor Definition 4 Idiosyncratic risk ε k is the idiosyncratic risk specific to asset k A E fn = 0 n = N Normalization - mean = 0 [ A E fn f ] m = 0 n m = N and n m Risk factors are mutually independent A3 E f n = n = N Normalization - variance = A4 E [ ε k ] = 0 k = K A5 E [ ε k ε j ] = 0 k j = K and k j A6 E [ ε k f ] n = 0 k = K and n = N A7 E [ ε k] < k = K Normalizations A-A3 can be weakened as we can always transform by linear combination the risk factors to make them satisfy such conditions If we define the expected return of asset k as a k then we have the linear projection 9 Rk = a k + N b kn fn + ε k k = K and n = N n=
LECTURE NOTES 3 6 Definition 5 Asymptotic arbitrage Let a portfolio of K assets be described by the vector of weights w K = w K w K wk K Consider the increasing sequence K = 3 Let σ kj be the covariance between the return of asset k and return of asset j An asymptotic arbitrage exists if the following conditions hold: A K k= wk k B lim K K k= = 0 zero net investment K j= wk k wk j σ kj = 0 Portfolio s return becomes certain C K k= wk k a k > 0 portfolio s return bounded above zero Theorem 6 APT If asymptotic arbitrages then 0 a k = λ 0 constant + N n= b kn λ n + risk premium for f n v k expected return deviation where i K k= v k = 0 ii K k= b knv k = 0 n = N K iii lim K K k= v k = 0 Condition iii says that the average squared error of the pricing rule 0 goes to zero as K becomes large So expected returns on average become closely approximated by 0 Notice that if the economy contains a risk-free asset then λ 0 = R f Proof Back page 5 Claim A multibeta generalization of the CAPM with weaker assumptions as an economy with N priced sources of risk can always be described by an economy with only one priced source of risk see Back exercise #65 page 8 Remark The APT gives no guidance about candidates for the economy s multiple underlying risks For example Chen Roll and Ross choose macroeconomic factors; Fama and French choose empirically driven firm-specific factors that best fit crosssection of returns; Carhart added a fourth empirically driven momentum factor V Hansen-Jagannathan Lower Bound on Risk Premia Recall from the binomial model that S k0 = E φ Sk E φ Rk = Also cov φ Rk + R k = by cov xỹ = E [ xỹ] xȳ If a riskless asset with risk-free return R f then R f = = R f Substituting cov φ Rk + R Rk f = R f cov Rk R f = R f cov Let φ = SDF and R k R f = cov φ R k φ Rk φ Rk + R k = R f Thus = ρ k φ σ kσ φ Thus σ kσ φ ρ k φ σ kσ φ σ kσ φ
LECTURE NOTES 3 7 as ρ k φ [ ] Making substitutions Rearranging terms σ kσ φ σ φ R k R f σ kσ φ R k R f σ k Sharpe s ratio σ φ Sharpe ratio σ φ Remark For the mean-variance efficient portfolio the relation in equation is a strict equality V Mehra-Prescott Equity Risk Premium Puzzle Let φ = βu C U C 0 where C is aggregate consumption A lnc = lnc 0 + ε C = C 0 e ε where ε N µ ε σε continuously compounded growth rate of consumption A RA with power utility function U C = θ C θ where θ= coefficient of absolute risk aversion Hence θ C φ = β C 0 taking logs and exponentials gives = e lnβ C θ C 0 = e lnβ θlnc lnc0 = βe θε where θε N θµ ε θ σε E φ = βe θµε+ θ σ ε and σ = E φ φ by definition of variance with φ = β e θε Let θε N θµ ε 4θ σε Then E φ = β e θµε+θ σ ε and = β e θµε+θ σ ε So E φ = e θ σ ε and σ = φ e θ σ ε σ φ = e θ σε Taking Taylor series expansion around 0 st e x = e 0 + e 0 x + higher order terms = + x + o For small x e x + x σ φ + θ σε = θσ ε Thus 3 Sharpe s ratio θσ by H&J lower bound For the S&P 500 Sharpe s ratio = 049 pa and 00 σ 004 Then if σ = 00 θ 49 σ = 004 θ Too high! shouldn t be more than 5! References [Black 97 Journal of Business] [Markowitz 95 Journal of Finance] [Roll 977 Journal of Financial Economics ] [Reisman 988 Econometrica] [Hansen-Jagannathan 99 Journal of Political Economy]