# L10: Discount Factors, beta, and Mean-variance Frontier1 Lecture 10: Discount Factor, Beta, Mean-variance Frontier The following topics will be covered:

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L10: Discount Factors, beta, and Mean-variance Frontier1 Lecture 10: Discount Factor, Beta, Mean-variance Frontier The following topics will be covered: –Discount factor (Existence Theorem –Beta Representations –Mean-variance Frontier –Relation between Discount Factors, Betas, and Mean-variance Frontiers (Equivalence Theorem) –Implications Materials are from chapters 4, 5, & 6, JC.

L10: Discount Factors, beta, and Mean-variance Frontier2 Law of One Price Payoff space –The payoff space X is the set of all the payoffs that investors can purchase, or it is a subset of the tradable payoffs that is used in a particular study. –Payoff space includes some set of primitive assets, but investors can form new payoffs by forming portfolios of the primitive assets. This leads to: Portfolio formation (A1) –x 1, x 2 є X  ax 1 + bx 2 є X for any real a, b. –i.e., \$1 to get R, \$2 to get 2R, -\$1 to get -R –This assumption rules out short sell constraint, bid/ask spread, leverage limitation, etc. –We can trade nonlinear functions of a basis payoff, e.g., call option

L10: Discount Factors, beta, and Mean-variance Frontier3 Law of One Price Law of one price (A2): p(ax 1 +bx 2 )=ap(x 1 ) + bp(x 2 ) –linearity –The law of one price says that investors cannot make instantaneous profits by repackaging portfolios. –The law is meant to describe a market that has already reached equilibrium. If there are any violations of the law of one price, trader will quickly eliminate them so they cannot survive in equilibrium

L10: Discount Factors, beta, and Mean-variance Frontier4 Theorem on the Existence of a Discount Factor Theorem: Given free portfolio formation A1, and the law of one price A2, there exists a unique payoff x* є X such that p(x)=E(x*x) for all x. –x* is a discount factor I.e., m=x* -- a special m p(x) = E(mx) = E[proj(m|X)x] = E(x*x) –x* = proj(m|X) On the existence of m (the discount factor). Unique in payoff space X. See Figure 4.2 It can be shown that:

L10: Discount Factors, beta, and Mean-variance Frontier5 What the Theorem Does Not Say

L10: Discount Factors, beta, and Mean-variance Frontier6 No Arbitrage and Positive Discount Factors No Arbitrage (absence of arbitrage) –Positive payoff implies positive price. I.e., x>0 then p>0. m>0 implies no arbitrage No arbitrage and the law of one price imply m>0.

L10: Discount Factors, beta, and Mean-variance Frontier7 What Does It Not Say? Discount factor m>0 exists, but it does not say that m>0 is unique. Does not say every m>0 We can extend the pricing function defined on X to all possible payoffs R s

L10: Discount Factors, beta, and Mean-variance Frontier8 Explicit Formula for x*, the discount factor Assuming the discount factor x* is the linear function of the shocks to payoffs: x* = E(x*) + (x – E(x))’b Finding b to ensure that x* prices the asset x: p = E(xx*) = E(x*)E(x) + E[x(x – E(x))’]b We have: The convenient alternative formula:

L10: Discount Factors, beta, and Mean-variance Frontier9 Expected Return-Beta Representation Linear factor pricing model: where β terms are defined as the coefficients in a multiple regression of returns on factors Factors f are proxies for marginal utility growth. The regression is not about predicting return from variables seen ahead of time. Its objective is to measure contemporaneous relations or risk exposure

L10: Discount Factors, beta, and Mean-variance Frontier10 Expected Return-Beta Representation Beta is the amount of exposure of asset i to factor a risks Expected return-beta relationship should be tested via cross-sectional regression Test the constraint:

L10: Discount Factors, beta, and Mean-variance Frontier11 Some Common Special Cases If there is a risk-free rate, we have If there is no risk-free rate, then γ must be estimated in the cross-sectional regression. It is called zero-beta rate. In the form of excess returns, we have: Also, it is often the case that the factors are returns or excess returns.

L10: Discount Factors, beta, and Mean-variance Frontier12 Mean-Variance Frontier Mean –variance frontier of a given set of assets is the boundary of the set of means and variances of the returns on all portfolios of the given assets.

L10: Discount Factors, beta, and Mean-variance Frontier13 Lagrangian Approach to Get Mean-Variance Frontier Start with a vector of asset return R. mean return: E= E(R) variance-covariance matrix: ∑=E[(R-E)(R-E)’] we construct an optional portfolio and choose weight w Objective: choose a portfolio to minimize variance for a given mean:

L10: Discount Factors, beta, and Mean-variance Frontier14 Solving the System Introduce Lagrange multiplier on the constrains, we have: Note w is a vector We have the expression for the variance of the minimum variance portfolio specified in 5.7 (p82) –The variance is a quadratic function of the mean. Minimum variance portfolio has minimum variance, weight specified on page 83

L10: Discount Factors, beta, and Mean-variance Frontier15 Orthogonal Decomposition and Mean-Variance Frontier Alternative approach to derive mean-variance frontier Hansen and Richard (1987) approach We can describe any return by a 3-way orthogonal decomposition, then the problem is solved. Define R* -- the return corresponding the payoff x* Define R e *

L10: Discount Factors, beta, and Mean-variance Frontier16 Orthogonal Decomposition and Mean-Variance Frontier R e * is an excess return that represents means on R e space with an inner product in the same way that x* is a payoff in X space that represents prices with an inner product. E(R e ) = E[proj(1|R e ) R e ] = E(R e *R e ) Theorem: Every R i can be expressed as R i = R * + w i R e* + n i where n i is an excess return with property E(n i ) = 0 and the components are orthogonal. Theorem: R mv is on the mean-variance frontier if and only if R mv = R * + w i R e* for some real number w.

L10: Discount Factors, beta, and Mean-variance Frontier17 Mean-Variance Frontier in Payoff Space Note: R e space is the space of excess returns, thus p=0

L10: Discount Factors, beta, and Mean-variance Frontier18 Orthogonal Decomposition in Mean Standard Deviation Space

L10: Discount Factors, beta, and Mean-variance Frontier19 Algebraic Proof With decomposition, E(n i )=0 and that the three components are orthogonal, we have: For each desired value of the mean return, there is a unique w i that minimize variance for each mean.

L10: Discount Factors, beta, and Mean-variance Frontier20 Spanning the Mean-Variance Frontier You can span the mean-variance frontier with any two portfolios that on the frontier Use risk free rate or its kind to span the space –Zero-beta return –Constant-mimicking portfolio return –Minimum variance return Span and diversification

L10: Discount Factors, beta, and Mean-variance Frontier21 R*, R e *, x* See page 89 through 93.

L10: Discount Factors, beta, and Mean-variance Frontier22 R*, R e *, x* (8) R* is the minimum second moment return (9) E(R e* ) = E(R e*2 ) (10) If there is risk-free rate, R e* =1 - (1/R f )*R* (11) R f = R* + R f R e* (12) If there is no risk-free rate, Proj(1|X)=proj(1|R e )+proj(1|R*) (13) (14)

L10: Discount Factors, beta, and Mean-variance Frontier23 Hansen-Jagannathan Bounds We have, Hansen and Jagannathan (1991) read this as a restriction on the set of discount factors that can price a given set of returns We need very volatile discount factors with mean near 1 (E(m)=R f when risk free rate exists) to understand stock returns. The higher the Sharpe ratio, the tighter the bound on the volatility of discount factor

L10: Discount Factors, beta, and Mean-variance Frontier24 Explicit Form

L10: Discount Factors, beta, and Mean-variance Frontier25 P=E(mx)  β Beta representation using m Beta representation using x* and R* Theorem: 1=E(mR i ) implies an expected return-beta model x*=proj(m|X) or R*=x*/E(x* 2 ) as factors,

L10: Discount Factors, beta, and Mean-variance Frontier26 More … MV Frontier  p=E(mx) and β (page 103) Discount factors and beta models to mean-variance frontier (page 110) (Page 105)

L10: Discount Factors, beta, and Mean-variance Frontier27 Factor-Mimicking Portfolios When factors are not already returns or excess returns, it is convenient to express a beta pricing model in terms of its factor-mimicking portfolios rather than factors themselves. Defining f*=proj(f|X) m=b’f carries the same pricing implication on X as does m=b’f p=E(mx)=E(b’fx)=E[b’(proj(f|X)x]=E[b’f*x] Factor mimicking return

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