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Unformatted text preview: Economics 106V Investments : Lecture Note4 Daisuke Miyakawa UCLA Department of Economics June 27, 2008 4th lecture covers the following items: (1) Practical di culties on the optimal portfolio approach, (2) the basic idea and the structure of an index (a single factor) model, (3) the diversi&cation e/ect, (4) active and passive portfolio management, and (5) the Security Characteristic Line (SCL). After discussing those items, we solve some exercise problems. 1 Practical Di culties on the Optimal Portfolio Approach On the previous lecture, we studied how to construct an optimal portfolio. It is a theoretically concrete idea to (i) formulate an optimal risky portfolio and (ii) combine it with the riskfree asset. However, there are several practical di culties to use this method. First, the optimal portfolio approach may require a large size of information as its input. As one example, consider the situation where 50 securities are available for our portfolio choice. How may inputs do we need? E [ r i ] : 50 estimates of expected returns + V ar ( r i ) or & i : 50 estimates of Var or Std + Cov ( r i ;r j ) : 50 & (501) 2 estimates of Cov (why?) = Total : 1,325 estimates Table1: Inputs for the Optimal Portfolio Approach Obviously, 1,325 is not a small number. Moreover, as a second problem, it is extremely di cult to estimate such a large number of objects from a limited number of observations. This provides a good motivation to develop a more convenient and less estimationinteisive method for a portfolio construction. 2 The Basic Idea and the Structure of Index Model 2.1 A Key Equation The idea behind the index model is simple. The model assumes that (i) one big random variable a/ects all stock returns to a greater or lesser extent (this is the reason why we can also call this model as a "single factor" model) and (ii) for each stock, a completely idiosyncratic random variable a/ects its return. This is 1 simply expressed as the following key equation: r i ( t ) & r f ( t ) = & i + i m ( t ) + " i ( t ) where r i ( t ) : The return on stock i during date t r f ( t ) : The riskfree return during date t & i ; i : Stock& i specic constants m t : The market premium during date t (i.e., r m ( t ) & r f ( t ) ) " i ( t ) : Stock& i specic randomness The equation states that the excess return of the stock i consists of (i) a stock i specic constant term & i , (ii) a stock i specic sensitivity coe cient i times the market premium m ( t ) , and (iii) a stock i specic randomness " i ( t ) . One remark is that r f ( t ) is known ex ante (i.e., at the beginning of date t ) since it is not a random variable. Another remark is that " i ( t ) is always unobservable while r i ( t ) and m t are observable ex post (i.e., at the end of the date t ). Given this relationship among those variables, our intermediate goal is to get the estimates for the constants & b & i ; b i . Once we get this information and the forecast for....
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 Summer '08
 Miyakawa
 Economics

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