Question

5) Suppose a project is dependent on whether there is a good economy and whether a com- petitor enters the market. If there is no competitor, the project will produce $50 million in a good economy and $15 million in a poor economy. If there is a competitor, the project will produce $30 million in a good economy and $10 million in a poor economy. The probability of a good economy is 60%, versus 40% for a poor economy; and the probability for a competitor to enter the market is 30%, versus 70% for no competitor.

a) Find the expected value of the cash flow from the project.

b) Assume the project cannot begin for three years and has a risk-adjusted dis- count rate of 18%. Calculate the present value of the project’s expected cash flow.

c) Using the calculation from part b, calculate the expected three-year holding- period return, the variance of the three-year holding-period return, and the standard deviation of the three-year holding-period return.

d) Adjust the variance and the standard deviation in part c to annual terms.

6) Assume that the project in Problem 5 is valuable if it can generate a cash flow above
$25 million. A decision on whether to develop the project at a cost of $6.5 million needs to be made today if the project is to be implemented in three years. Given a 5% annual risk-free rate and the calculations from Problem 5, generate a binomial tree (each stage of the tree being one year) to value the development option and make a decision.

Answer 1

Expected cash flow = Expected pay off with competitor * 30% + Exp. without competitor * 70%

E(CF) = (50*.6+15*.4) * 0.3 + (30*0.6+10*.0.4) = $29.20 million

		Good Economy	Poor Economy	Expected payoff
	prob	0.6	0.4
With No Competitor	0.3	50.00	15.00	36.00
With competitor	0.7	30.00	10.00	22.00
Expected /Mean Payoff		48.00	10.00	26.20

Question (b)

It is assumed that the investment is required at beginning of year 1.

Expected value of project after four years = $26.20 mn (assumed that project begins after 3 years and gives cash flows at the end of fourth year)

Present value of the project = 26.20 / (1+ discount rate)^4 = 26.20 / (1+18%)^4 = $13.51 mn

Question c)

The present value of the project that gives cashflows at the end of 4th year is $13.51 million

It is assumed that we invest $13,51million in the project at time 0 but project begins after three years and produces cash flows at the end of 4th year - however the question says 3- year holding period return which would mean cash flows are received at the end of 3rd year. So, If the investment is made at time 0, then holding period is 4 years.

In any case, question c refers to only holding period.

holding period return = (Cash received at end of holding period - initial investment) / Initial investment

Holding period return = ( 26.2- 13.51) / 13.51 * 100 = 93.88%

Note: Annualized holding period return = Discount rate = 18% = (1+93.88%)^(1/3) -1

(Note also that if the borrowing cost of initial investment of $13.51 is more than 18%, then the net return made by the company will be negative. In the problem, we are ignoring the borrowing cost and evaluating only the project return)

Variance of the return:

Variance = Summation of (return - mean return)^2 * probability

The return table is calculated and given below:

		Investment	13.51
		Expected payoff	Exp Return
	prob
With No Competitor	0.3	36.00	166.40%
With competitor	0.7	22.00	62.80%
Expected /Mean Payoff		26.20	93.88%

Variance = (166.40%-93.88%)^2*0.3 + (62.80%-93.88%)^2*0.7 = 22.54%

Standard deviation = SQRT(22.54%) = 47.48%

c) Variance and std in annual terms

Annualized standard deviation = Holding period std / SQRT(periods)

Annualized std = 47.48% / SQRT(3) = 47.48 / SQRT(3) = 27.41%

Annualized variance = (27.41%)^2 = 7.51%

Note: If cash is invested in time t=0, project begins after 3 years and gives cash flows at the end of fourth year, then annualized return should refer to four year holding period. However, problem asks annualized variance for three year holding period return.

If cash is invested at time=0 and cash flows received at the end of 4th year, then

Annualized std = 47.48% / SQRT(3) = 47.48 / SQRT(4) = 27.73%

Annualized variance = (27.41%)^2 = 5.63%

Question (6)

The question is not clear. The states of economy or events are not interdependent and there are no cash flows involved in initial three years. Hence, the intermediate states does not affect the outcomes calculated in problem 5. The pay off will remain same irrespective of the intermediate states and it is not clear why a binomial tree is required. Even at the end of three year, the uncertainty will exist at the same level as the uncertainty at time=0.

As per problem (5), the project gives expected mean pay-off of 26.20 million at the end of fourth year and requires investment of $13.51 million at time=0

The present value of the project is assumed to be its investment required in problem (5). Hence, the given estimate of $6.5 mn is less than present value. If the required invsetment is only $6.5mn at time=0, then the decision should be to go ahead with the project.

Assuming that the project requires investment only after three years when the project begins, the present value of the pay off of $26.20 mn at time t=0 can be calculated as follows

Investment required after three years at discount rate of 18% = 26.20/ (1.18) = $22.20

Present value of investment at time=0 , Investment = 22.20 / (1.18)^3 = $13.51 mn

Hence, if the initial investment required is only $6.5 million for the project, then the actual return will be greater than the discount rate of 18%, this can be noticed from the NPV

Net Present Value = 13.51 - 6.5 = $7.014 million

Hence the decision should be to go ahead with the project as long as the borrowing cost of the firm is also less than 18%.

If the borrowing cost is less than 18%, the project will give a positive NPV over investment requirement of $13.51 calculated above.