Econ173A_topic5

Econ173A_topic5 - Ec 173A-FINANCIAL MARKETS Foster, UCSD...

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Ec 173A—FINANCIAL MARKETS LECTURE NOTES Foster, UCSD 12-Jul-08 TOPIC 5 – MODERN PORTFOLIO THEORY A. Introduction to Modern Portfolio Theory (MPT) 1. Background: a) In 1952, Harry Markowitz [Nobel, 1990] revolutionized the field of portfolio selection and management. 1) The old theory of portfolios focused on single assets. Investors determine a reserva- tion price based on their assessment of an asset's expected return and risk. If market price P 0 < the reservation price, they buy the asset at P 0 . 2) Markowitz recognized that wise investors hold portfolios of several assets. Hence, the risk and return of any single asset is relevant only to the extent that it affects the overall risk and return of the portfolio to which it is added. 3) MPT is complicated, but worth the trouble, because the end result is an extremely simple rule for portfolio management! b) The assumptions of MPT. #1 Investors are risk averse and maximize E[U(W)], the expected utility of end-of- period wealth, over the same time horizon. #2 Investors make asset choices based on risk and expected return [σ(r p ), μ(r p )] of port- folios of assets. #3 Investors have homogeneous expectations and agree on μ(r) and σ(r) for all assets. #4 Information is free and simultaneously available to all investors. #5 There is a riskless asset with yield r f , and investors can borrow and lend at this rate. 2. Portfolio Rates of Return: a) Consider a portfolio of one stock and one bond. NOTATION -- for j { s, b } Symbol Definition Notes P j Beginning price of stock or bond n j Number of units of asset in portfolio V j Total value of asset in portfolio V j = n j P j CF j Cash flows per unit of asset r j Ex ante rate of return on asset r j = (CF j + P j )/P j V p Beginning portfolio price (value) V p = V s + V b ω j Proportion of portfolio value in asset ω j = V j /V p ; Σω j = 1
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Ec 173A MPT p. 2 b) Portfolio ex ante rate of return. b b s s p r r r ϖ + = . Proof: c) Portfolio returns and expected returns. 1) We saw above that portfolio return r p = ω s r s + ω b r b , a value-weighted linear function of the individual asset returns. 2) Therefore, r p is a linear function of random variables r s and r b , from which we deduce the portfolio expected return: . 3. Portfolio Risk: a) Covariance and correlation of ex ante rates of return. 1) Asset ex ante rates of return are random variables distributed over states of the world (SW). This implies that all asset rates of return are jointly distributed. 2) For assets S and B and states of the world SW j , j = 1. ..K, with probabilities Pr(j): b) Portfolio risk will be measured as the standard deviation σ(r p ) of ex ante rate of return r p . 1) For our stock-bond portfolio, r
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Econ173A_topic5 - Ec 173A-FINANCIAL MARKETS Foster, UCSD...

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