Note-3 - Optimal Portfolio Problem

Note-3 - Optimal Portfolio Problem - Economics 106V...

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Economics 106V Investments : Lecture Note-3 Daisuke Miyakawa UCLA Department of Economics June 27, 2008 3rd lecture covers the following items: (1) The structure of optimal portfolio choice problem, (2) a Line (CML), (4) multiple asset model, (5) optimal risky portfolio, (6) global minimum variance portfolio, (7) e¢ cient frontier, and (8) diversi±cation e/ect. After discussing those items, we solve some exercise problems. 1 The Structure of Optimal Portfolio Choice Problem In the previous two lecture, we have studied the tools necessary to discuss the optimal portfolio construction problem. 1.1 Overview Suppose we are asked to construct a portfolio, which is a set of risky assets and risk-free asset. For simplicity, assume that there are only two risky assets and one risk-free asset. Figure-1 summarizes the structure of this problem. Risky Asset-A r A Risky Asset-B r B Risky-Free Asset r f A σ B 0 = f Step-1: Optimal "Risky" Portfolio Construction Step-2: Optimal Portfolio Construction Figure-1: The Structure of the Problem In order to construct an optimal portfolio, we need to follow the two steps. First, an optimal "risky" portfolio, which consists of only the risky assets, is constructed. Then, we allocates our funds to the risk-free asset and the optimal risky portfolio. Figure-2 is the road map for the material on this lecture. Note that there 1
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are three steps including "Step-0", which corresponds to the preparation for constructing an optimal risky portfolio. i σ [ ] i r E Risky Asset-A Risky Asset-B Risk-Free Asset Efficient Frontier i [ ] i r E Step-1 : Optimal Risky Portfolio Sharpe Ratio Capital Allocation Line (CAL) i [ ] i r E Step-2 : Optimal Portfolio Indifference Curve <Step-0> <Step-1> <Step-2> Figure-2: Road map 1.2 Some Important Concepts in Each Step 1.2.1 Step-0: Drawing an E¢ cient Frontier On this step, we derive an "e¢ cient frontier". This is a set of ( E [ r i ] i ) which can be achieved by combining the risky asset-A and -B. We will see why the frontier takes the shape later. 1.2.2 Step-1: Optimal Risky Portfolio the e¢ cient frontier. This straight line is called as a "Capital Allocation Line (CAL)" and has an important role on the next step. 1.2.3 Step-2: Optimal Portfolio optimal portfolio can be obtained as a point at which the CAL and the indi∕erence curve is tangent. From the next section, we will discuss those three steps in detail.
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This note was uploaded on 09/23/2008 for the course ECON 106v taught by Professor Miyakawa during the Summer '08 term at UCLA.

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Note-3 - Optimal Portfolio Problem - Economics 106V...

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