FBE441_04_Diversification

# FBE441_04_Diversification - FBE 441 Investments Prof. David...

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FBE 441 Investments Prof. David Solomon Lecture 4: Portfolio Diversification Efficient Frontier of Risky Assets Diversifiable vs. Non-Diversifiable risk Adding Risk-Free Asset: An Optimal Risky Portfolio Readings: BKM chapters 7 and 8

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2 - “Don’t put all your eggs in one basket” . Portfolio diversification with several risky assets - Decomposing risk: . Diversifiable vs. non-diversifiable - Optimal Portfolios: . Adding the risk-free asset: one optimal risky portfolio for all tastes - Estimation risk . From theory to practice Lecture Outline:
3 Portfolio Diversification Portfolios of 2 risky assets: Suppose we invest in US and JP, with means and covariances of returns being given by: Excel: FBE441_Lect_04_a_2Assets_MSCI.xls E[r] sigma[r] US 13.6% 15.4% JP 15.0% 23.0% correl[US,JP) 0.27

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4 Portfolio Diversification Ex (Cont.): If an investor holds 60% in the US and 40% in JP what is the mean and volatility (standard deviation) of the portfolio? - mean: E(R p ) = 0.6*0.136 + 0.4*0.15 = 0.141 - variance: var(R p ) = 0.6 2 *0.154 + 0.4 2 *0.23 + 2*0.4*0.6*0.27*0.154*0.23 =0.02144 s.d.(R p ) = σ p =0.146 This portfolio has higher expected return and lower risk than the US market alone!
5 Portfolio Diversification We say that the 60/40 portfolio ‘dominates’ the 100% US portfolio in mean-variance terms This means that portfolio A dominates portfolio B If: E(R a ) >= E(R b ) Std Dev(R a ) <= Std Dev (R b ) At least one of these inequalities is strict i.e. either E(R a ) > E(R b ) and/or Std Dev(R a ) < Std Dev (R b ) This is called ‘stochastic dominance’, because portfolio A is better than portfolio B in expectation.

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6 Portfolio Diversification Stochastic dominance implies that an investor who cares about only mean and variance will always pick portfolio A over portfolio B. Stochastic dominance (stochastic = pertaining to a random variable) is weaker than strict dominance, which means that A beats B in every state of the world
7 Portfolio Diversification 2 Given Assets, Reducing risk by varying weights Let ϖ be the weight in the US, and 1- ϖ the weight in JP The expected return of the portfolio is: E(R p ) = ϖ *0.136 + (1- ϖ )*0.15 The variance of the portfolio return is: var(R p ) = ϖ 2 *(0.154) 2 + (1- ϖ ) 2 *(0.23) 2 +2* ϖ *(1- ϖ )*0.154*0.23*0.27 What happens when we vary ϖ ?

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8 Portfolio Diversification Ex (Cont.): varying weights 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.136 0.138 0.14 0.142 0.144 0.146 0.148 0.152 US JP s tandard deviation ex pec ted return w=0.1 w=0.9 w=0.8 w=0.7 w=0.6 w=0.5 w=0.2 w=0.3 w=0.4 w E[Rp] Stdev[Rp] -1 16.4% 44.4% -0.9 16.2% 42.1% -0.8 16.1% 39.9% -0.7 16.0% 37.7% -0.6 15.8% 35.4% -0.5 15.7% 33.3% -0.4 15.5% 31.1% -0.3 15.4% 29.0% -0.2 15.3% 26.9% -0.1 15.1% 24.9% 0 15.0% 23.0% 0.1 14.8% 21.1% 0.2 14.7% 19.4% 0.3 14.5% 17.9% 0.4 14.4% 16.5% 0.5 14.3% 15.4% 0.6 14.1% 14.6% 0.7 14.0% 14.2% 0.8 13.8% 14.2% 0.9 13.7% 14.6% 1 13.6% 15.4% 1.1 13.4% 16.4% 1.2 13.3% 17.8% 1.3 13.1% 19.3% 1.4 13.0% 21.0% 1.5 12.8% 22.8% 1.6 12.7% 24.8% 1.7 12.6% 26.8% 1.8 12.4% 28.8% 1.9 12.3% 30.9% 2 12.1% 33.1%
9 Terminology 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 Mean-standard deviation frontier for US and Japan σ Er US JP Efficient Frontier Minimum Variance Portfolio short US short JP -> Your job:

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10 Terminology Feasible portfolio all portfolios in the frontier Efficient portfolio is the set of portfolio that has the highest E(r) for a given standard deviation The minimum variance portfolio is the portfolios that
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## FBE441_04_Diversification - FBE 441 Investments Prof. David...

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