Lecture 10: Discount Factor, Beta, Mean-variance Frontier

1. 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 Frontier

2. 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) • x1, x2є X ax1 + bx2є 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 Frontier

3. Law of One Price • Law of one price (A2): p(ax1+bx2)=ap(x1) + bp(x2) • 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 Frontier

4. 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* єX 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 Frontier

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

6. 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 Frontier

7. 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 Rs L10: Discount Factors, beta, and Mean-variance Frontier

8. 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 Frontier

9. 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 Frontier

10. 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 Frontier

11. 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 Frontier

12. 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 Frontier

13. 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 Frontier

14. 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 Frontier

15. 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 Re* L10: Discount Factors, beta, and Mean-variance Frontier

16. Orthogonal Decomposition and Mean-Variance Frontier • Re* is an excess return that represents means on Re 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(Re) = E[proj(1|Re) Re] = E(Re*Re) • Theorem: Every Ri can be expressed as Ri = R* + wiRe* + ni where ni is an excess return with property E(ni) = 0 and the components are orthogonal. • Theorem: Rmv is on the mean-variance frontier if and only if Rmv = R* + wiRe* for some real number w. L10: Discount Factors, beta, and Mean-variance Frontier

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

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

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

20. 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 Frontier

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

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

23. 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)=Rf 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 Frontier

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

25. P=E(mx)  β Beta representation using m Beta representation using x* and R* Theorem: 1=E(mRi) 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 Frontier

26. 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 Frontier

27. 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 L10: Discount Factors, beta, and Mean-variance Frontier