Assignment for Chapter 4宋逢明金融工程习题

Assignment for Chapter4:

1. Assuming a one-factor model of the form: i i i e F b r ++=%4

Consider three well-diversified portfolios (zero unsystematic risk). The expected value of the

Is one of the portfolio ’s expected return not in line with the factor model relationship? Which one? Can you construct a combination of the other two portfolios that has the same factor sensitivity as the “out-of-line ” portfolio? What is the expected return of that combination? What action would you expect investors to take with respect to these three portfolios?

2. Assume that security returns are generated by a one-factor model. Hap Morse portfolio whose

Specify an arbitrage portfolio in which Hap might invest. (Remember there are an infinite number of possibilities; choose one.) Demonstrate that this portfolio satisfies the conditions of an arbitrage portfolio.

3. One factor arbitrage model: r i =E(r i )+βi F+e i ，risk-free rate is 5%. Now there are two portfolios

A and B, full diversification. Their β values and expected yields as following:

Asset A B Expected yield 8% 6% β value

0.45

0.3

（1）Can you use A and risk-free bond to replicate B? If there is an arbitrage opportunity, how to

do?

（2）Extend the model to the two-factor model: r i =E(r i )+βi1F 1+βi2F 2+e i ，risk-free rate is 5%. Now

there are two portfolios A and B, full diversification. Their β values and expected yields as

following: Asset A B Expected yield

8% 6% βi1 0.45 0.3 βi2

1.5

1.0

Can you use A and risk-free bond to replicate B? If there is an arbitrage opportunity, how to do?