Real option analysis can $be$ used $to$ alter the timing, scale, $or$ other aspects $ofan$ investment $in$ response $to$ market conditions. Businesses face the dilemma $of$ whether $to$ invest $in$ a project $or$ abandon $itifit$ does not add value $to$ the firm. Real option analysis allow financial managers $to$ determine the financial consequences $of$ this flexibility and the value $of$ the option.

Consider the case $of$ Fat Sheep Media Company:

Fat Sheep Media Company, a social networking company, has seen triple$-$digit growth $in$ its website$s$ registrations over the past two years. Most $of$ the website$s$ subscribers live outside the United States, and the company $is$ seeing a significant increase $in$ the number $of$ users from Brazil. $As$ a result, Fat Sheep $is$ considering opening a marketing office $in$ Brazil $to$ expand its marketing efforts there. Management, however, $is$ not sure $if$ the Brazilian expansion via the opening $of$ a subsidiary office will necessarily help the company grow and increase its value. Management$s$ uncertainty $is$ the result $of$ the possibility that Brazil$s$ Internet connectivity will $be$ insufficient $to$ support all $of$ Fat Sheep$s$ forecasted growth.

One $of$ Fat Sheep$s$ employees, Luana, who $is$ originally from Brazil, conducted some preliminary market research and submitted the following details about the potential five$-$year project:

$*$ Opening the new marketing office $in$ Brazil will require $an$ initial investment $of$ $$4\mathrm{.}00$ million.

$*$ According $to$ research $on$ Brazil$s$ mobile technology infrastructure, Luana noted there $isa60\%$ probability that the country$s$ mobile connectivity will $be$ sufficient $to$ generate additional advertising cash flows $of$ $$6\mathrm{.}00$ million per year for the company for the next five years.

$*$ Alternatively, there $isa40\%$ chance that Brazil$s$ mobile Internet connectivity will $be$ insufficient $to$ support Fat Sheep$s$ desired growth $in$ Brazil. $In$ this case, the company expects $to$ generate additional net advertising$-$related annual cash flows $of$ only $$2\mathrm{.}00$ million for the next five years.

$*$ The project$s$ expected cost $of$ capital $is11\mathrm{.}00\%,$ and the risk$-$free rate $is4\%.$ The project$s$ WACC should $be$ used $to$ discount all cash flows.

Note: For all questions, $do$ not round intermediate calculations, then round your final answer $to$ two decimal places unless noted otherwise.

Given this information, the project$s$ expected net present value $(NPV)$ without the consideration $of$ the growth option $is.$

After further research, Luana added a few more details $to$ her proposal:

$*If$ Brazil$s$ Internet connectivity $is$ good, then $at$ the end $of$ Year $3,$ Fat Sheep should consider investing $$3\mathrm{.}00$ million $to$ purchase $an$ existing Brazilian marketing firm and creating a new subsidiary.

$*$ The new subsidiary $is$ expected $to$ generate $$2\mathrm{.}40$ million $of$ additional annual cash flows $in$ years $4$ and year $5.$

$*$ However, $if$ the Internet connectivity $in$ Brazil $is$ inadequate $to$ support Fat Sheep$s$ desired customer growth, then the company will not invest the additional funds $in$ year $3or$ earn the expected additional advertising$-$related cash flows.

Based $on$ Luana$s$ additional information, use the decision tree analysis $to$ calculate the $NPVof$ the project including the growth option. Then, calculate the value $of$ the growth option $by$ itself, and select the correct answers from the choices available $in$ the following table. Remember $to$ use the project$scostof$ capital $to$ discount all cash flows.

Value

$NPVof$ the project with growth option $$12\mathrm{.}75$ million

Growth option value

Lastly, Luana wants $to$ use the the Black$-$Scholes option pricing model $(OPM)to$ determine the value $of$ the growth option. $Todo$ this, she has collected and computed the values for several additional variables, and has given you the Black$-$Scholes OPM equation for the valuation $ofan$ option $(V)$:

$V=(PxN(d))-(Xx$ erRFt

$xN(d))$

where,

$P=$ the current, $or$ a proxy, price $of$ the value $of$ the underlying asset $(P)-$which equals the present value $of$ the delayed project$s$ forecasted future cash flows

$N(d)$ and $N(d)=$ estimates $of$ the variance $of$ the project$s$ expected return

$X=$ the option$s$ strike price, which $is$ the cost $of$ purchasing the Brazilian firm that will become the Fat Sheep$s$ subsidiary

$e=$ the mathematical constant equal $to2\mathrm{.}718281828459045235360...,$ which can $be$ truncated and rounded $to2\mathrm{.}7183$

$rRF$

$=$ the market$s$ risk$-$free rate

$t=$ the time until the option expires, which, $in$ this situation, $is$ assumed $tobe$ the end $of$ third year, when the potential purchase $of$ the subsidiary would take place

According $to$ Luana, these variables should assume the following values:

Variable

Value

Project$scostof$ capital $11\mathrm{.}00\%$

Current value $of$ the delayed investment $(P)$ $$4\mathrm{.}48$ million

$N(d),as$ estimated $by$ Luana $0\mathrm{.}7573$

$N(d),as$ estimated $by$ Luana $0\mathrm{.}7082$

Delayed investment$s$ strike price $(X)$ $$2\mathrm{.}40$ million

Mathematical constant $e2\mathrm{.}7183$

Risk$-$free rate $(rRF$

$)0\mathrm{.}04\%$

Time until the option expires $(t)3$ years

Give