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Ch.7 The Capital Asset Pricing Model: Another View About Risk. 7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Using the CAPM to value risky cash flows 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset

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## Ch.7 The Capital Asset Pricing Model: Another View About Risk

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**Ch.7 The Capital Asset Pricing Model: Another View About**Risk • 7.1 Introduction • 7.2 The Traditional Approach to the CAPM • 7.3 Using the CAPM to value risky cash flows • 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset • 7.5 Characterizing Efficient Portfolios -- (No Risk-Free Assets) • 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio • 7.7 The Zero-Beta Capital Asset Pricing Model • 7.8 The Standard CAPM • 7.9 Conclusion**7.1 Introduction**• Equilibrium theory (in search of appropriate risk premium) • Exchange economy • Supply = demand: for all asset j, • Implications for returns**7.2 The Traditional Approach**• All agents are mean-variance maximizers • Same beliefs (expected returns and covariance matrix) • There exists a risk free asset • Common linear efficient frontier • Separation/ Two fund theorem • T = M**(7.1)**7.2 The Traditional Approach to the CAPM • a. The market portfolio is efficient since it is on the efficient frontier. • b. All individual optimal portfolios are located on the half-line originating at point . • c. The slope of the CML is .**Proof of the CAPM relationship (Appendix 7.1)**• Refer to Figure 7.1. Consider a portfolio with a fraction 1- a of wealth invested in an arbitrary security j and a fraction a in the market portfolio As a varies we trace a locus which - passes through M (- and through j) - cannot cross the CML (why?) - hence must be tangent to the CML at M Tangency = = slope of the locus at M = slope of CML =**(7.2)**(7.3) Define Only a portion of total risk is remunerated = Systematic risk**(7.4)**(7.5) A portfolio is efficient iff no diversifiable risk (7.6) b is the only factor; SML is linear**7.4 The Mathematics of the Portfolio Frontier: Many Risky**Assets and No Risk-Free Asset Goal: understand better what the CAPM is really about – In the process: generalize. • No risk free asset • Vector of expected returns e • Returns are linearly independent • Vij = sij**Definition 7.1: A frontier portfolio is one which displays**minimum variance among all feasible portfolios with the same . since**(vector)**(vector) l (scalar) g (scalar) (7.15) (vector) (vector) (scalar) If E = 0, wp = g If E = 1, wp = g + h**Proposition 7.1: The entire set of frontier portfolios can**be generated by ("are convex combinations" of) g and g+h. • Proof: To see this let q be an arbitrary frontier portfolio with as its expected return. Consider portfolio weights (proportions) ; then, as asserted, **.**• Proposition 7.2. The portfolio frontier can be described as convex combinations of anytwo frontier portfolios, not just the frontier portfolios g and g+h. • Proof: To see this, let be any two distinct frontier portfolios; since the frontier portfolios are different, Let q be an arbitrary frontier portfolio, with expected return equal to . Since , there must exist a unique number such that (7.16) Now consider a portfolio of with weights a, 1- a, respectively, as determined by (7.16). We must show that .**(7.17)**• For any portfolio on the frontier, , with g and h as defined earlier. Multiplying all this out yields:**(i) the expected return of the minimum variance portfolio is**A/C; • (ii) the variance of the minimum variance portfolio is given by 1/C; • (iii) equation (7.17) is the equation of a parabola with vertex (1/C, A/C) in the expected return/variance space and of a hyperbola in the expected return/standard deviation space. See Figures 7.3 and 7.4.**Figure 7-3 The Set of Frontier Portfolios: Mean/Variance**Space**Figure 7-5 The Set of Frontier Portfolios: Short Selling**Allowed**7.4 Characterizing Efficient Portfolios -- (No Risk-Free**Assets) • Definition 7.2: Efficient portfolios are those frontier portfolios for which the expected return exceeds A/C, the expected return of the minimum variance portfolio.**Proposition 7.3 : Any convex combination of frontier**portfolios is also a frontier portfolio. • Proof : Let , define N frontier portfolios ( represents the vector defining the composition of the ith portfolios) and let , i = 1, ..., N be real numbers such that . Lastly, let denote the expected return of portfolio with weights . The weights corresponding to a linear combination of the above N portfolios are : Thus is a frontier portfolio with . **Proposition 7.4 : The set of efficient portfolios is a**convex set . • This does not mean, however, that the frontier of this set is convex-shaped in the risk-return space. • Proof : Suppose each of the N portfolios considered above was efficient; then , for every portfolio i. However ; thus, the convex combination is efficient as well. So the set of efficient porfolios, as characterized by their portfolio weights, is a convex set. **7.5 Background for Deriving the Zero-Beta CAPM: Notion of a**Zero Covariance Portfolio • Proposition 7.5: For any frontier portfolio p, except the minimum variance portfolio, there exists a unique frontier portfolio with which p has zero covariance. We will call this portfolio the "zero covariance portfolio relative to p", and denote its vector of portfolio weights by . • Proof: by construction.**E ( r )**p A/C MVP ZC(p) E[r ( ZC(p)] (1/C) Var ( r ) Figure 7-6 The Set of Frontier Portfolios: Location of the Zero-Covariance Portfolio**Let q be any portfolio (not necessary on the frontier) and**let p be any frontier portfolio. (7.27)**7.6 The Zero-Beta Capital Asset Pricing Model (Equilibrium)**(i) agents maximize expected utility with increasing and strictly concave utility of money functions and asset returns are multivariate normally distributed, or (ii) each agent chooses a portfolio with the objective of maximizing a derived utility function of the form , W concave. (iii) common time horizon, (iv) homogeneous beliefs about e and V**(7.28)**(7.29) • All investors hold mean-variance efficient portfolios • the market portfolio is convex combination of efficient portfolios is efficient. • (7.21) .**(7.30)**n x 1 n x n a number n x 1 7.7 The Standard CAPM Solving this problem gives**(7.31)**(7.32) (7.33)**(7.34)**or (7.35) or**What have we accomplished?**• The pure mathematics of the mean-variance portfolio frontier goes a long way • In particular in producing a SML-like relationship where any frontier portfolio and its zero-covariance kin are the heroes! • The CAPM = a set of hypotheses guaranteeing that the efficient frontier is relevant (mean-variance optimizing) and the same for everyone (homogeneous expectations and identical horizons)**What have we accomplished?**• The implication: every investor holds a mean-variance efficient portfolio • Since the efficient frontier is a convex set, this implies that the market portfolio is efficient. This is the key lesson of the CAPM. It does not rely on the existence of a risk free asset. • The mathematics of the efficient frontier then produces the SML**What have we accomplished?**• In the process, we have obtained easily workable formulas permitting to compute efficient portfolio weights with or without risk-free asset**Conclusions**• The asset management implications of the CAPM • The testability of the CAPM: what is M? the fragility of betas to its definition (The Roll critique) • The market may not be the only factor (Fama-French) • Remains: do not bear diversifiable risk!

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