B.3 Portfolio Theory II-1-1.pptx

# B.3 Portfolio Theory II-1-1.pptx - 1 Outline Portfolio...

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1 Outline: Portfolio Theory II Add a risk-free asset to the two risky asset model Identify portfolios that combine risky and risk-free assets The new frontier is linear Slope is the Sharpe ratio Identify the optimal portfolio of risky assets – the tangency portfolio Determining the optimal complete portfolio Generalize to N assets (Markowitz model) Efficient portfolios Optimal complete portfolio All investors hold same portfolio of risky assets Mutual fund theorem Asset allocation in the real world Assessing risk preferences Venti Portfolio Theory II

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Review: The efficient frontier with two risky securities Venti Portfolio Theory II 2 . Portfolio B Frontier shows all possible combinations of risk and return available to investors holding varying amounts of risky portfolios B and S (without short-selling) Portfolio S . . Efficient frontier
Next: Two Extensions Venti Portfolio Theory II 3 1. Add the risk-free asset to the mix What is the optimal portfolio of risky assets? What is the optimal complete portfolio (including both risky and risk-free assets)? 2. Generalize from 2 risky securities (a stock fund and a bond fund) to N risky securities (many stocks and many bonds)

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Venti Portfolio Theory II 4 Add a Risk-Free Asset to the Portfolio Risk-Free Asset : Assume return is r f =5%, f =0.0 Can borrow and lend at this rate We can combine the risk-free asset with any of the portfolios we have constructed from stocks and bonds. Consider an arbitrarily chosen portfolio of risky assets P Let r c denote the return on a portfolio that combines the risk-free asset and portfolio P. Then: (1) E(r c ) = E(r P ) + (1- )r f (2) c = P where is the share held in the portfolio of risky assets.
Possible Capital Allocations with Stocks, Bonds and T-bills Venti Portfolio Theory II 5 CAL P . Portfolio P r f =5% CAL: Capital Allocation Line: Opportunity set with risky portfolio P and risk-free asset

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Substitute (2) into (1) and rearrange to yield: What is the slope of the CAL? Venti Portfolio Theory II 6 P f C C f P ff C P P E r r σ E r =r E r r = σ σ P f P E r -r Slope= Sharpe Ratio σ
Optimal Portfolio of Risky Assets Venti Portfolio Theory II 7 Which portfolio of risky assets is optimal? Maximize the slope of the CAL (the Sharpe ratio) for any possible portfolio, P Ans. When it is just tangent to the efficient frontier. The optimal portfolio of risky assets (stocks and bonds) is the tangency portfolio T When is At a maximum? p f p p E r -r S = σ

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Opportunity Set with Stocks, Bonds and a Risk-Free Asset Venti Portfolio Theory II 8 CAL T . Portfolio T (r T =11%, s T =14.2%)
Let w B be the share in bonds and (1-w B ) in stocks Substitute the following into the Sharpe ratio: E(r P ) = w B E(r B ) + w S E(r S ) p 2 = w B 2 B 2 + w S 2 S 2 + 2w B w S BS B S Maximize S P w/r/t w B to obtain: Aside: Calculating the Optimal Risky Portfolio Venti Portfolio Theory II 9 2 B f S s

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