Question

Suppose a local asteroid defense system will yield benefits of $1 billion in 300 years by averting large-scale destruction and loss of life. Write an expression for calculating the present discounted value of these benefits using the following time-declining rate schedule: 4 percent for years 1-50, 3 percent for years 51-100, 2 percent for years 101-200, 1 percent for years 201-300. DO NOT CALCULATE!

Answer 1

Answer: The present discounted value of the benefits is $541.5 million

Step-by-step explanation: To calculate the present discounted value of the benefits, we need to use the formula:

PV=FV/(1+r)n

where PV is the present value, FV is the future value, r is the interest rate or discount rate, and n is the number of periods.

Since the benefits are received in 300 years, we need to divide the time into four segments according to the given rate schedule:

Years 1-50: r = 0.04

Years 51-100: r = 0.03

Years 101-200: r = 0.02

Years 201-300: r = 0.01

We also need to assume that the benefits are received at the end of each segment, so we can use the following values for n:

Years 1-50: n = 50

Years 51-100: n = 50

Years 101-200: n = 100

Years 201-300: n = 100

Using these values, we can calculate the present value of the benefits for each segment as follows:

# Years 1-50

PV1 = FV / (1 + r)^n

PV1 = 1000000000 / (1 + 0.04)^50

PV1 = 14050761.24

# Years 51-100

PV2 = FV / (1 + r)^n

PV2 = 1000000000 / (1 + 0.03)^50

PV2 = 22809668.32

# Years 101-200

PV3 = FV / (1 + r)^n

PV3 = 1000000000 / (1 + 0.02)^100

PV3 = 135335283.24

# Years 201-300

PV4 = FV / (1 + r)^n

PV4 = 1000000000 / (1 + 0.01)^100

PV4 = 366032341.64

Finally, we can add up all the present values to get the total present discounted value of the benefits:

# Total present discounted value

PDV = PV1 + PV2 + PV3 + PV4

PDV = 14050761.24 + 22809668.32 + 135335283.24 + 366032341.64

PDV = 541518054.44

Therefore, the present discounted value of the benefits is $541.5 million. Hope this helps, and have a great day! =)