Efficient set_etc

Efficient set_etc - Professor Reza (Bus 172A) Lecture...

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Professor Reza (Bus 172A) Lecture Notes: Portfolio Selection These notes are not a substitute for the text or class lectures; they complement them. Efficient Portfolios When considering investing in risky assets, investors limit themselves to those assets or portfolios that offer the maximum expected return for a given level of risk, or offer the minimum amount of risk for a given level of expected return. The portfolios that meet these conditions are known as the efficient se t (or efficient frontier). Depending on the correlations among the expected returns of the risky assets, the efficient set may look different. In general, the efficient set looks like Fig. 1. Fig. 1 Expected Return Q* S K . . P*-----P . . E* Q Standard Deviation (risk) Portfolios in the efficient set will be found on the border of the bullet-shaped diagram (the interior along with the border constitutes the feasible set). Points such as F and K are efficient portfolios whereas P and Q are not. Why? Because we can always find portfolios that offer higher return with the same risk (as in Q*) or lower risk with the same return (as in P*). Therefore, only portfolios lying on the boundary between E and S are considered efficient. Investors ignore inefficient portfolios. Risk-free Asset This is an asset that there is no uncertainty about its terminal value; it may or may not pay any dividend, interest or capital gain. The covariance between the risk-free asset and risky assets is zero. Statistics Concepts: Covariance reflects how one variable changes when another variable changes. Covariance is positive if the first variable rises when the second rises; it is negative if the first variable falls when the second rises. The covariance of a variable with itself is called variance; the square root of variance is called standard deviation. If we let σ ij = covariance between variables i and j, then σ ii = variance of i; we can use σ i = standard deviation of variable i. Also, σ ij = ρ ij * σ i * σ j , where ρ ij is the correlation between variables i and j. Notation: for brevity’s sake, “return” and “ rate of return” are used interchangeably. Furthermore, the risk-free asset’s expected return has standard deviation σ f = 0. In practice the only asset that qualifies as riskfree is a US Treasury security with the maturity that matches the length of the investor’s holding period . With the risk-free asset, investors can now construct additional portfolios by mixing the riskfree asset with the risky assets, as shown in Fig 2. Expected Retrun Fig. 2 V G R f E Std Deviation (Risk) Observe that we can now have portfolios such as R f (all our investment in the riskfree asset), points such as G (some investment in the riskfree asset and some in the risky portfolio V), or all the investment going to the risky portfolio (at point V). Any portfolio that lies on the line connecting R f to V is possible.
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Efficient set_etc - Professor Reza (Bus 172A) Lecture...

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