1. 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.

2. 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

3. 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

4. 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:

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

6. 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.

7. 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

8. 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:

9. 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

10. 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:

11. 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.

12. 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.

13. 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:

14. 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

15. 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 *

16. 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.

17. 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

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

19. 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.

20. 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

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

22. 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)

23. 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

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

25. 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,

26. 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)

27. 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