# 4 - 1 FINS 5513 Investments and Portfolio Selection...

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1 FINS 5513 Investments and Portfolio Selection University of New South Wales Semester 1 2012 Week 4 Russell Jame

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Optimal Risky portfolios BKM 7.1 – 7.2 2
Portfolio of two risky assets Last week we attempted to create optimal portfolios when choosing between one risky asset and one risk free asset. We determined that when borrowing and lending rates were the same the optimal solution was: Choose the risky asset with the highest Sharpe ratio Allocation between the risky and the risk free asset according to your risk aversion 3

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Portfolios of Two Risky Assets Let’s suppose you are considering investing in two stocks: MSFT and IBM. Suppose there is no risk free asset. E[MSFT] = 8% E[IBM]=8% Correl(MSFT, IBM) = 45% σ(MSFT) =12% σ (IBM) =15% You estimate the Sharpe ratio for both stocks: SharpeMSFT =8/12 = 0.67 4
Portfolio of Two Risky Assets Can you somehow combine MSFT and IBM to earn a higher Sharpe ratio? Let’s try a simple strategy of investing 50% of your wealth in MSFT and 50% in IBM. E[Rp]=.5*8+.5*8=8% 5 2 2 2 2 2 2 2 2 2 2 2 ( , ) .5 *.12 .5 *.15 2*.5*.5*.45*.12*.15 0.0132 0.115 p MSFT MSFT IBM IBM MSFT IBM MSFT IBM p p W W W W Cov r r σ = + + = + + = =

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Diversification Why does a combined portfolio work better than either asset by itself? Diversification! There are two ways to think about diversification: Economic Intuition Statistically 6
The Economics of Diversification The return of a givens tock consists of different components Market factors This cannot be diversified away. This type of risk if often caller market risk, systematic risk, or non-diversifiable risk. Industry factors Firm factors This risk can be diversified away. This type of risk is called firm-specific risk, idiosyncratic risk, unique risk, or diversifiable risk. 7

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The statistics of diversification BKM 7.3 -7.4 8
Concept Check: Diversification Consider n asset, each of which has an E(r)=m and a variance of σ2. Assume the asset returns are uncorrelated and that you invest the same weight in each stock (i.e. wi = 1/n.) What does this assumption of uncorrelated returns mean economically? What is the mean and variance of this portfolio? 9

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Diversification In general, asset returns are correlated. Does this positive correlation reduce the benefits of diversification? Does this positive correlation eliminate the benefits of diversification? To answer this question, let’s consider the same n assets, except now they have a correlation of 0.9. 10
Diversification The mean and variance of this portfolio are: Even in this case, we can see that the 11 ( 29 , 2 2 2 2 2 2 2 2 1 2 2 2 2 , 2 1, 1 .1 1 1 .9( ) .9 1 .9 1 1 ( ) ( ) j ij j i j i j n i i n n n n n i j i j i i i i i i j i j i j i j i j w w n n n n n n n E R E r nm m n n Var R w w w w w w ρ σ σ σ = = = = = = + = + = + - = + - = + = = = =

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Mean-Std Dev Diagram of a portfolio with Two Assets ρ-=-1 ρ-=-1 ρ=1 -1<ρ<1 12
Optimization and the global minimum variance portfolio 13 BKM 7.3 -7.4

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Optimal Risky Portfolios Although we have established that
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## This note was uploaded on 03/19/2012 for the course FINS 5513 taught by Professor Wangjianxin during the Three '10 term at University of New South Wales.

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4 - 1 FINS 5513 Investments and Portfolio Selection...

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