$($REAL OPTIONS$)$

A company is considering investing in a project. The present value $($PV$)$ of future discounted expected cash flows is either Sh$.3$ million if the market goes up or Sh$.0\mathrm{.}5$ million if the market goes down next year. The objective probability the market will go up is $20\%.$ The appropriate risk$-$adjusted rate of return $($cost of capital$)$ is $25\%.$ The initial capital investment required at time $0$ is Sh$.1\mathrm{.}2$ million. The risk$-$free rate is $10\%$ per year.

a$.$ Determine the PV of the project at time $0.$

b$.$ Determine the NPV of the project at time $0.$

c$.$ Should the company invest in this project.

$2.$ DECISION TREE

Mirror Young Cycles Ltd$.$ wishes to design a new sports bicycle. The Company would have to invest Ksh$.10,000$ at the beginning of the first year for the design and model testing of the new bicycle. Mirror$$s managers believe that there is a $60\%$ probability that this phase will be successful and the project will continue. If phase $1$ is not successful, the project will be abandoned with zero salvage value.

The next phase, if undertaken would consist of making the molds and producing two prototype bicycles. This would cost Ksh$.500,000$ at the end of the first year. If the bicycles test well, Mirror would go into production. If they do not, the molds and prototypes could be sold for Ksh$.100,000.$ The managers estimate that the probability is $80$ percent that the bikes will pass testing and that Phase $3$ will be undertaken.

Phase $3$ consists of changing over one current production line to produce the new design. This would cost Ksh$\mathrm{.}1$ million in Year $2.$ If the economy is strong at this point, the net value of sales would be Ksh$\mathrm{.}3$ million, while if the economy is weak the net value would be Ksh$\mathrm{.}1\mathrm{.}5$ million. Both net values occur during Year $3,$ and the two states of the economy are equally likely. Mirror$$s cost of capital is $12$ percent.

Required:

$($a$)$ Construct a decision tree and determine the project$$s expected NPV assuming that the project has average risk.

$($b$)$ Calculate the project$$s standard deviation of NPV and coefficient of variation of NPV$.$

$($c$)$ If Mirror$$s average project had a coefficient of variation of between $1\mathrm{.}0$ and $2\mathrm{.}0,$ would this project be of high, low or average stand$-$alone risk?