Question

Un chat the mill would be willing to accept? Exercise 3: Benefit/cost analysis, NPV, option value The town of Dryville is considering building a dam in order to provide irrigation water during droughts. If they build the dam this year (year 0) they expect it to provide benefits of $800 in years 0, 1, 2, 3, 4, and 5. After year 5, the dam will be worn out and need to be replaced. 1. What is the net present value of the stream of benefits in year 0, assuming a discount rate of .05? nu = 70.05) 2. The dam costs $3500 to build. What is the net benefit of the project (benefits - costs)? Should Dryville build the damn? 3. Researchers are currently developing new climate change models, and in year 1 we expect to have new information about the benefits of building the dam. There is a 50% chance of a "wet" model, in which the annual benefits will be $400, and a 50% chance of a "dry" model, in which the annual benefits will be $1200. What is the NPV in year 0 of building the dam in year 1, assuming a "wet" model? What is the net benefit of the project? (Hint: be sure to find the net present value of the cost in year 0.) Should Dryville build the damn in this case? Note that if the dam is built in year 1, Dryville receives 0 benefit in year 0, followed by $400 in years 1, 2, 3, 4, 5, and 6.

Answer 1

Discount rate r=0.05

1. Net present value of the stream of benefits of 800$ for year 0,1,2,3,4,5 and 6

=800+800/(1+r)+800/(1+r)²+800/(1+r)³ + 800/(1+r)⁴ +800/(1+r)⁵

= 4263.58$

2. Since the dam costs 3500$ to build,the net benefits will be4263.58-3500= 763.58$ and Dryville should build the dam

3. NPV in year 0 of building the dam in Year 1 assuming a wet model and starting from year 1 to year 6

= 400/(1+r)+400/(1+r)²+400/(1+r)³ + 400/(1+r)⁴ +400/(1+r)^{5 +}400/(1+r)⁶= 2030.28$

NPV in year 0 Of the cost of 3500$ in year 1= 3500/1.05=3333.33$

No,Dryville should not build the dam in this case as the NPV of the benefits will be less than that of the costs.

4. NPV in year 0 of building the dam in Year 1 assuming a dry model and starting from year 1 to year 6

= 1200/(1+r)+1200/(1+r)²+1200/(1+r)³ + 1200/(1+r)⁴ +1200/(1+r)^{5 +}1200/(1+r)⁶= $6090.83

So Dryville should build the dam in this case as the NPV of the benefits are greater than that of the cost (NPV Of the cost it the same 3333.33$)

5. 1. If you do not build the dam at all, your benefits and cost will be 0 so your expected net benefit=0

2. For waiting and building in year 1, There is 50% chance of it being dry model,so your net benefits in NPV will be approximately 6090.83-3333.33=2757.5 if you wait and build in year 1

And there is a 50% chance of it being wet model, so net losses will be approximately 2030.28-3333.33=-1303.05$ if you wait and build the dam in year 1, so if you wait and it turns out to be a wet model, you will obviously not build the dam and face the losses, so the net benefit in this case will be 0

So there is a 50% chance of dry model and a profit of 2757.5 and a 50% chance of wet model and a profit of 0 if you wait and build in year 1(Since dam is not built and losses not faced), so your expected benefit in NPV of waiting for a year and then deciding= 0.5*2757.5+0=1378.75

3. For building the dam in year 0

The net benefit in case it turns out a dry model after year 1= 800(Assuming no dry or wet model in year 0 as before) +1200/(1+r)+1200/(1+r)²+1200/(1+r)³ + 1200/(1+r)⁴ +1200/(1+r)⁵-3500=$5995.37-3500=2495.37$

The net benefit in case it turns out a wet model after a year of building= 800+400/(1+r)+400/(1+r)²+400/(1+r)³ + 400/(1+r)⁴ +400/(1+r)⁵-3500=2531.79-3500=-968.2$

So the expected net benefit if you build the dam in year 0 without waiting= 0.5*2495.37-0.5*968.2=$763.585 (Since the dam was already build and you will have to face the losses in case it turns out a wet model and both profit and loss have a 50% chance of happening)

So the option for waiting and building the dam in year 1 has the expected net benefit =1378.75(As calculated in 2)

for building the dam in year 0 is 763.59 (As calculated in 3.) And not building it is 0

Value of the option for being able to build it in year 1 should be 1378.75-763.59=$615.16 (In a sense You would be willing to pay upto 615 $ to have an option of building the dam after a year)

Hope it's clear. Do ask for any clarifications required.