Question

Zuti has a capital investment project that could start immediately. The project will require a machine costing $2.4 million. The total discounted value now of the cash inflows from the project will be either $2.6 million or $1.9 million with equal probability. The risk-free rate is 3%.

Instead of starting immediately the project could be delayed until one year from now to gain more market information. Its total discounted cash inflows at that time will be known as either $2.6 million, or $1.9 million, with certainty.

(i) What is the present value of the option to delay?

(ii) The supplier of the machine has offered to deliver it (if required) in one year’s time at a price of only $2 million, if Zuti pays a non-refundable deposit now. What is the maximum the firm should pay as a deposit now? What type of real option does this represent for Zuti? Identify the specific components of the option contract.

Answer 1

Present value of the option to delay

Value of a call option = P_a*N(d_1) - P_e*N(d_2)*e^{-rt}

where,

d_1 = \frac{ln(P_a/P_e)+(r+0.50s^2)*t}{s\sqrt{t}}

d_2 = d_1-{s\sqrt{t}}

Pa = $2.6M*50%+$1.9M*50% = $2.25M]

here, cost of asset, Pe = $2.40M

d1 = [ln($2,250,000/$2,400,000)+(0.30+0.50*(0.50)2)*1]/(0.50*1)]

= [ln(0.9375)+0.425]/0.50

= [-0.0280+0.425]/0.50

= 0.794

= 0.79

d2 = 0.794- (0.50*1) = 0.294 = 0.29

N(d1) = 0.50+0.2852 = 0.7852 ;

N(d2) = 0.50+0.1141 = 0.6141

Value of call option = [$2,250,000*0.7852]-[$2,400,000*0.6141*2.7183(-0.03*1)]

= $1,766,700 - $1,473,840*0.9705

= $1,766,700 - $1,430,362 = $336,338

We first compute the values for N(d1) and N(d2) as below -

d1 = [ln($2,000,000/$2,400,000)+(0.30+0.50*(0.50)2)*1]/(0.50*1)]

= [ln(0.8333)+0.425]/0.50

= [-0.0792+0.425]/0.50

= 0.6916

= 0.69

d2 = 0.6916- (0.50*1)

= 0.1916

= 0.19

N(d1) = 0.50+0.2549 = 0.7549 ;

N(d2) = 0.50+0.0753 = 0.5753

Then, value of call option can be computed as

Value of call option =

[$2,000,000*0.7549-[$2,400,000*0.5753*2.7183(-0.03*1)]

= $1,509,800 - $1,380,720*0.9705

= $1,509,800 - $1,339,989

= $169,811

Thus, Zuti should pay a maximum security deposit of $169,811.

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