Question

Suppose that the rate of return on risky assets is given by the following single factor model: where F is the factor affecting returns on all securities and e, is a firm-specific dis- turbance. The risk-free rate is 4% and borrowing at this rate is possible. Two well- diversified portfolios P and Q are found to have the following expected returns and betas: E(rp) 12% ßP : 0.8 (a) Explain why these data are inconsistent with Arbitrage Pricing Theory (b) Construct a zero investment portfolio that yields a sure profit and does not involve borrowing. What are the weights of P and Q in your portfolio? Are you investing in bills?

Answer 1

Answer:

Part a)

Arbitrage pricing theory APT pricing theory was basically developed to overcome the limitations of CAPM,CAPM only considers the market risk in order to price any asset however there no. Of risk involved with an asset like liquidity risk, operating risk, interest rate risk, credit risk,economy risk, industry risk,GDP etc.While APT says that the return of any asset for the purpose of pricing has a linear relationship with N no. Of factors which represent different types of risk stated above APT equation is given by

Er= Rf+ B1× F1+B2×F2+B3×F3+.............Bn×Fn + e

Where,

Er= return expected from an asset

Bi= sensitivity of an asset with different factors

Fi= factors

e= asset specific risk

Although APT has two fundamental assumptions

1) there should be a perfectly competitive market

2) no. Of factors should not be more than the no. Assets in the market.

Now as we can see that in this question we are only given Er and Beta i.e. one sensitivity and that too risk factor associated with this beta is also not given hence this data is inconsistent with APT

Part b)

If we want to create a risk risk free portfolio with risky assets only (i.e. no borrowing or investing at Risk free rate) then we need to solve the following two equations

Rf= Erp×Wp+ Erq× Wq

0= Wp+ Wq

First equation says that porfotlio should provide risk less profit

Second equation says that net investment should be Zero

We have

4= 12×Wp+ 15×Wq

0=Wp+Wq

Solving for Wp we get Wp= 4/(-3)= -1.33333

And Wq= 1.3333333

Which means that we should shortsell asset P with133% and buy asset with 133% weight this way we will achieve both of our goal.

Thakyou!