Note-4 - Index Model

# Note-4 - Index Model - Economics 106V Investments Lecture...

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Unformatted text preview: Economics 106V Investments : Lecture Note-4 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 risk-free 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 & (50-1) 2 estimates of Cov (why?) = Total : 1,325 estimates Table-1: 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 estimation-inteisive 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 risk-free return during date- t & i ;¡ i : Stock& i speci¡c constants m t : The market premium during date- t (i.e., r m ( t ) & r f ( t ) ) " i ( t ) : Stock& i speci¡c randomness The equation states that the excess return of the stock- i consists of (i) a stock- i speci¡c constant term & i , (ii) a stock- i speci¡c sensitivity coe¢ cient ¡ i times the market premium m ( t ) , and (iii) a stock- i speci¡c 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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Note-4 - Index Model - Economics 106V Investments Lecture...

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