There is firm B, whose dividend payments next year will be $102 if the economy i

Question

There is firm B, whose dividend payments next year will be $102 if the economy is strong and $52 if the economy is weak. The beta of firm B is 0.95. Firm B has no leverage.

There is another firm C, whose dividend payments next year will be $102 if the economy is strong and fire does not break out in its factory, $82 if the economy is strong but fire breaks out in its factory, $52 if the economy is weak and fire does not break out, and $32 if the economy is weak but fire breaks out. Fire will break out with probability 10% but this fire risk is firm-specific risk, which will occur independently of the fundamental state of the economy. Firm C has no leverage.

Now, suppose there are 1M firms which have the same payoff structure as that of firm C. That is, they will pay $102 when the economy is strong and $52 when the economy is weak. But when fire breaks out, the dividends will decrease by $20. Again, fire risk is firm-specific risk. All these firms have no leverage, either. In this circumstance, suppose you hold 1/1M unit of the share of each of these companies.

1) How much do you expect to get in total when the economy is strong and when the economy is weak, respectively?

2) Calculate the beta and required expected return for firm C’s equity.

3) Calculate the present value of firm C’s equity. Tell me how the idiosyncratic risk affects its stock price.

4) We can express firm C’s return as

is an idiosyncratic component. Compute  for each of the above four scenarios. What’s the probability that the economy will be strong and fire will not break out? What’s the probability of other three scenarios? Check the expected value of  is 0.

Explanation / Answer

HI there

When the economy is strong :-

Number of firms = 1M

Number of shares = 1/1M

Expected value of each share

E(S) = E(S|F) * P(F) + E(S|NF) * P(NF)

where , E(S) indicates expected returns when the economy is strong

E(S|F) indicates the expected returns when there is a fire and economy is strong

E(S|NF) denotes the expected returns when there there is no fire and the economy is strong

P(F) denotes the probability of fire

P(NF) = probability of NO fire

There fore , E(S) = 82 * 0.1 + 102 * 0.9 = 100

Similarly , During a weak economy ,

E(W) = E(W|F) * P(F) + E(W|NF) * P(NF)

= 32 * 0.1 + 52 * 0.9 = 50

Expected value of the portfolio = 1M * 1/1M * 100 = $100 (strong economy)

= 1M * 1/1M * 50 = $50 (weak economy)

2)For Firm B ,

Beta = 0.95

unlevered company

Considering that the firm is US based , so Risk free rate(rf) = 3.06 % (considering 10 year T ill rate as risk free)

Expected Return from B = P(S) * E(S) + P(W) * E(W)

As nothing is mentioned , we consider that P(S) = P(W) = 1/2

E(B) = 1/2 * 102 + 1/2 * 52 = 77

Applying CAPM MODEL :-

E(R) = Rf + Beta(Rm-Rf)

77 = 0.036 + 0.95 * (Rm - 0.036)

Rm = $ 81.05

For Firm C ,

E(R) = 1/2 * 100 + 1/2* 50 (100 is E(R) for strong , 50 is for weak and probability of weak and strong is 1/2 each)

E(R) = $75

75 = 0.036 + Beta ( 81.05-0.036)

Beta = 0.9253

Expected Return = $75

3)It is given that we will be recieving dividends after an year

intrest rate = 6.25%

PV = 75/(1 + 0.0625) = $70.58

Idiosyncratic risk is asset specific or class specific

For example , in the above problem all those 1 M firms had this risk

It can reduced by diversification

I.e by investing a part of the income in firm B in the above question.

4)Probability that Economy is strong and Fire will break out i.e P(S^F) = 1/2* 0.1 = 0.05

P(S ^ NF) = 1/2 * 0.9 = 0.45

P(W ^ F ) = 1/2 * 0.1 = 0.05

P(W ^ NF ) = 1/2 * 0.9 = 0.45

I hope this helps.If you have any doubts or eed further clarification , put in the comments.

Have a nice day :)

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