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# Arbitrage Pricing Theory, Risk, Cost of Capital, and Capital

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Factor Beta factor Expected value Actual value
Growth in GNP 2.04 3.5% 4.8
Interest rate -1.90 14.0 15.2
Stock return 10.0
Suppose a factor model is appropriate to describe the returns on a stock. Information about those factors in the following chart
a. What is the systematic risk of the stock return?
b. B. the firm announced that its market share had unexpectedly increased 23 percent to 27 percent. Investors know from their past experience that the stock return will increase by 0.36 percent per an increase of I percent in its market share. What is the systematic risk of the stock?
c. C. what is the total return on this stock?

Q2
Assume that the following market model adequetly describes the return-generating behavior of risky assets

Where Rit= the return for the ith asset at time t
And Rmt= the return on a portofolio containing all risky assets in some proportion, at time t
Rmt and Eit are statistically independent
Suppose the following data are true
Asset (beta)Bi E(ri) Var(ei)
A 0.7 8.41% 1.00%
B 1.2 12.06 1.44
C 1.5 13.95 2.25
Var (Rmt)=1.21%

a. Calculate the standard deviation of returns for each asset
b. Assume short selling is allowed
i. Calculate the variance of return of three portfolios containing an infinite number of asset type A,B, C respectively.
Ii. Assume Rf=3.3% and Rm=10.6% which assets will not be held by rational investors?
Iii. What equilibrium state will emerge such that no arbitrage opportunities exist? why?
Q3
The following table lists possible rates of return on [Company (C)] stock and debt and on the market portfolio. The probability of each state is also listed

State Probability Return on stock Return on debt Return on the market

1 0.1 3% 8% 5%
2 0.3 8 8 10
3 0.4 20 10 15
4 0.2 15 10 20
a. What is the beta of (C) debt?
b. What is the beta of (C) stock?
c. If the debt -to-equity ratio of (C) is 0.5, what is the asset beta of (C)? assume no taxes.
Q4
Calculate the weighted average cost of capital for the (B) Company. The book value of (B)'s outstanding debt is \$60 million. Currently, the debt is trading at 120 percent of book value and is priced to yield 12 percent . the 5 million outstanding shares of stock are trading at \$20 per share. The required return on (B)'s stock is 18 percent. The tax rate is 25%

#### Solution Preview

Arbitrage Pricing Theory, Risk, Cost of Capital, and Capital Budgeting questions
Q1
Factor Beta factor Expected value Actual value
Growth in GNP 2.04 3.5% 4.8
Interest rate -1.90 14.0 15.2
Stock return 10.0
Suppose a factor model is appropriate to describe the returns on a stock. Information about those factors in the following chart
a. What is the systematic risk of the stock return?

It is the undiversifiable risk associated with unanticipated factor movements:
It is calculated as (4.8%-3.5%)X2.04-1.90X(15.2-14) = 0.372%

b. B. the firm announced that its market share had unexpectedly increased 23 percent to 27 percent. Investors know from their past experience that the stock return will increase by 0.36 percent per an increase of I percent in its market share. What is the unsystematic risk of the stock?
Unsystematic risk is (27-23) X 0.36=1.44%
c. C. what is the total return on this stock?
Total return = Expected Return + Systematic Risk + Unsystematic risk
Total Return = 10+0.372+1.44=11.812%

Q2
Assume that the following market model adequetly describes the return-generating behavior of risky assets

Where Rit= the return for the ith asset at time t
And Rmt= the return on a portofolio containing all risky assets in some proportion, at time t
Rmt and Eit are statistically independent
Suppose the following data are true
Asset (beta)Bi E(ri) Var(ei)
A 0.7 8.41% 1.00%
B 1.2 12.06 1.44
C 1.5 13.95 2.25
Var (Rmt)=1.21%

a. Calculate the standard deviation of returns for each asset

The variance of returns on asset i is given by

substituting the values, we get
For A - variance = (0.7)^2X(1.21%)+1%
Variance = 0.015929
Standard deviation of A is Square root (0.015929)=0.1262
Doing in the same way, we get
For B, standard deviation is square root (0.031824)=0.1783
For C, standard deviation is square root (0.049725)=0.2229

b. Assume short selling is ...

#### Solution Summary

The solution explains calculations involving Arbitrage Pricing Theory, standard deviation and WACC

\$2.19

## The cost of equity capital and the CAPM

Review the background material on the capital asset pricing model, the material on the dividend growth model, and arbitrage pricing theory and do some of your own research using internet. These models provide some insights and tools to estimate the rate of return that investors in ST. JUDE MEDICAL 'require' in the sense that if they don't see the possibility that they'll earn that rate of return they'll sell the shares and that of course will lower the market price per share.

These models use a set of assumptions that are not necessarily tenable.

You are asked by your board of directors to write a report explaining the challenge of estimating or coming with a good 'feel' for the "cost of equity capital" or the rate of return that you feel your company investors require as the minimum rate of return that they expect of require your company to earn on their investment in the shares of the company. There are several asset pricing models used to estimate the cost of equity capital that you have read about for this module in the background materials. After reading through the background materials, write a report to the board of directors of ST JUDE MEDICAL in a FIVE page paper addressing the following issue:

Which of the three models (dividend growth, CAPM, or APT) is the best one for estimating the required rate of return (or discount rate) of your company?

In your paper include discussion of the following issues:

1. Ease of use of these three models

2. Accuracy of each of these three models

3. How realistic the assumptions of each model are

For this discussion you need to take a clear stand and pick one of these three models to defend to the Board of Directors. You cannot tell the Board of Directors that "I like all three models", they want you to come to them with a decisive choice of just one model.

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