Question

Use the following alternative discount rate values (0.01; 0.03; 0.04; 0.06; 0.07) to investigate the sensitivity of the present value of net benefits of the dam in exercise (1) to the assumed value of the real discount rate. Compute the "breakeven" value of the discount rate, dBE.

Exercise 1 was as follows:

The initial cost of constructing a permanent dam (i.e., a dam that is expected to last forever) is $425 million. The annual net benefits will depend on the amount of rainfall: $18 million in a “dry” year, $29 million in a “wet” year and $52 million in a “flood” year. Meteorological records indicate that over the last 100 years there have been 86 “dry” years, 12 “wet” years, and 2 “flood” years. Assume the annual benefits, measured in real dollars, begin to accrue at the end of the first year. Using the meteorological records as a basis for prediction, what are the net benefits of the dam if the real discount rate is 5 percent?

Answer 1

Here, we are going to use the basic formula for calculating net present value (NPV), which is as follows:

NPV= -C₀ +C₁/(1+r) + C₂/(1+r)² + .......+ C_n/(1+r)ⁿ, where

C₀=initial cash flow (sunk cost hence the negative sign in the equation above)

C=cash flow

r=discount rate

n=time

Now, according to the given question,

C₀=$425 million

C= $18 million in a dry year, @29 million in a wet year, $52 million in a flood year (3 types of cash flows)

r =0.05

n=infinity

Probability of a dry year =0.86, wet year=0.12, flood year=0.02

So, the net benefits of the dam can be calculated as follows:

Net benefit = -425 + 0.86 * 18 * (1/(1+0.05) +1/(1+0.05)² +..............) +

0.12 * 29 *(1/(1+0.05) +1/(1+0.05)²+...............) +

0.02* 52 *(1/(1+0.05) +1/(1+0.05)²+.........)

= -425 + 0.86* 18* (1/1.05)/(1-1/1.05) + 0.12* 29 * (1/1.05)/(1-1/1.05) +0.02*52*(1/1.05)/(1-1/1.05)

= -425 + (15.48 * 20) + (3.48 * 20) + (3.61* 20)

= -425 + 451.4 = 26.4 ($ million)

Now for investigating the sensitivity we may compute the net present benefit by using the discount rate as 1%, 3%, 4%, 6%, 7% as calculated above.

At 1%, the net present value is  -425 + 0.86 * 18 * (1/(1+0.01) +1/(1+0.01)² +..............) +

0.12 * 29 *(1/(1+0.01) +1/(1+0.01)²+...............) +

0.02* 52 *(1/(1+0.01) +1/(1+0.01)²+.........)

which is equal to 1832

Similarly, you can calculate for other values of discount rate. As discount rate increases, the net present benefits decreases.

Now for the breakeven discount rate, it is the discount rate at which net present value is 0.

0=-425 + (15.48+3.48+3.61) * ((1/1+r)/((1- (1/(1+r)))

This gives a break even discount rate, r=0.0531