Lecture5.pdf - Investments and Portfolio Analysis Lecture 5 — Portfolio Allocation II Dr Jianan Liu Fall 2019 Today’s Agenda I Review I New topics 1

Lecture5.pdf - Investments and Portfolio Analysis Lecture 5...

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Investments and Portfolio Analysis Lecture 5 — Portfolio Allocation II Dr. Jianan Liu Fall, 2019
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Today’s Agenda I Review I New topics: 1. Combining risky assets 2. N risky assets and one risk free assets 3. Book Chapter 7 I The midterm
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Review of last lecture I Book Chapter 6
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Utility function (value) I A common functional form of utility function: U = E ( R p ) - 1 2 A σ 2 ( R p ) I A is the risk aversion parameter I Investors with A > 0 and finite are mean-variance investors I Mean-variance investor’s preference: I Given expected return, assets that have lower risk give investors higher utility value I Given level of risks, assets that have higher expected return give investors higher utility value
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Calculation to pin down A I Calculation to pin down A —one way is to find the exact ( E ( R p ) , σ 2 ( R p )) that makes investor just accepts the bet I Just indicates that the investors are indifferent between accepting the bets and not accepting the bets; that is, U = 0
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Portfolios and weights A combination of two risky assets— A and B —with weights ω A and 1 - ω A is denoted as portfolio c I expected return on c : E ( R c ) = ω A E ( R A ) + (1 - ω A ) E ( R B ) I variance of portfolio c : Var ( R c ) = ω 2 A Var ( R A )+(1 - ω A ) 2 Var ( R B )+2 ω A (1 - ω A ) Cov ( R A , R B ) I covariance of A and B Cov ( R A , R B ): Cov ( R A , R B ) = Corr ( R A , R B ) × p Var ( R A ) × p Var ( R B )
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Portfolio weights and balance sheet There is a mapping between balance sheet and portfolio weights. An investor has initial capital $ V , and she invests in assets A and B I when 0 ω A 1,
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Portfolio weights and balance sheet I when ω A > 1, she is shorting B and buying A with shorting proceeds
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Portfolio weights and buying on margin I when ω A > 1 and B is cash, it’s buying A on margin case I Recall, when initial margin is 50%, the investor buying A on margin has balance sheet: I Her portfolio is consisted: asset A and cash(margin loans) denoted as B , with weights: ω A = $2 V / $ V = 2 ω B = - ($ V ) / $ V = - 1
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Portfolio weights and short-selling I when ω A < 0 and B is cash, it’s short selling A case I Recall, when initial margin is 50%, the investor short selling A has balance sheet: I Her portfolio is consisted of asset A and cash (proceeds and deposits) denoted as B , with weights: ω A = - $2 V / $ V = - 2 ω B = (3$ V ) / $ V = 3
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Capital allocation line—combining one risky asset and one risk-free asset I Risky asset has expected return— E ( R p ) and standard deviation— σ ( R p ), and the riskfree rate is R f I Opportunity sets—all possible combinations of p and riskfree asset, which are on Capital Allocation Line (CAL)
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Golden formulas of CAL The weight on the risky asset is y . A combined portfolio c has: I Risk premium: E ( R c ) - R f = y ( E ( R p ) - R f )
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Golden formulas of CAL The weight on the risky asset is y . A combined portfolio c has: I Risk premium: E ( R c ) - R f = y ( E ( R p ) - R f ) I standard deviation: σ ( R c ) = y σ ( R p )
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