Exercise 1: Long Term Discounting (30 points) Assume a project will result in benefits of $1 million each year for 150 years (i.e. annual benefits) by building a hydroelectric dam that will provide clean energy to a city. (a) (10 pts) Compute the present value of these benefits using a time-constant discount rate of 3. (b) (15 pts) Compute the present value of these benefits using the following time-declining discount rate schedule: 3.5 percent for years 1-50; 2.5 percent for years, 51-100; 1 percent for years 101-150. (c) (5 pts) How does your answer in (a) compares to (b)? is the difference significant enough as to be important? Note: Use discrete time interest rates for this exercise (instead of continuous time interest rates), i.e. to discount "X" for a year at an interest rate of 5% we would use PV (X) = 145% (instead of PV (X) = X - e-5%)

(a)

The present value of these benefits using a time-constant discount rate of 3% will be $32,937,698.03.

 

(b)

The present value of these benefits using the time-declining discount rate schedule is $46,198,655.58.

 

(c)

The present value of the benefits using the time-declining discount rate is higher than the present value of the benefits using the time-constant discount rate and the difference of $13,260,957.55 between them is significant enough to be considered important.

Step-by-step explanation

Exercise 1:

(a)

PV= P(1-(1+r)^-n)/r

PV is the present value=?

P is the size of each annual benefit= $1,000,000

r is the discount rate= 3% or 0.03

n is the number of years that the benefits will be received= 150

PV= $1,000,000(1-(1+0.03)^-150)/0.03

PV= $1,000,000(1-1.03)^-150)/0.03

PV= $32,937,698.03

Therefore, the present value of these benefits using a time-constant discount rate of 3% will be $32,937,698.03.

 

(b)

Step 1: Calculate the present value of the benefits that will be received in years 1-50.

PV₁= P(1-(1+r₁)^-n)/r₁

PV₁ is the present value of the benefits that will be received in years 1 to 50=?

P is the size of each annual benefit= $1,000,000

r₁ is the discount rate applicable in years 1-50= 3.5% or 0.035

n is the number of years that the benefits will be received= 50

PV₁= $1,000,000(1-(1+0.035)^-50)/0.035

PV₁= $1,000,000(1-1.03)^-50)/0.03

PV₁= $23,455,617.87

 

Step 2: Calculate the value of the benefits that will be received in years 51-100 at the end of the 50^th year.

PV₂= P(1-(1+r₂)^-n)/r₂

PV₂ is the present value of the benefits that will be received in years 51 to 100 at the end of 50^th year=?

P is the size of each annual benefit= $1,000,000

r₂ is the discount rate applicable in years 51-100= 2.5% or 0.025

n is the number of years that the benefits will be received= 50

PV₂= $1,000,000(1-(1+0.025)^-50)/0.025

PV₂= $1,000,000(1-1.025)^-50)/0.025

PV₂= $28,362,311.68

 

Step 3: Calculate the value of the benefits that will be received in years 101-150 at the end of the 100^th year.

PV₃= P(1-(1+r₃)^-n)/r₃

PV₃ is the present value of the benefits that will be received in years 101 to 150 at the end of 100^th year=?

P is the size of each annual benefit= $1,000,000

r₃ is the discount rate applicable in years 101-150= 1% or 0.01

n is the number of years that the benefits will be received= 50

PV₃= $1,000,000(1-(1+0.01)^-50)/0.01

PV₃= $1,000,000(1-1.01)^-50)/0.01

PV₃= $39,196,117.53

 

Step 4: Calculate the present value of these benefits using the time-declining discount rate schedule

PV= PV₁ + PV₂/(1+r₂)⁵⁰ + PV₃/(1+r₃)¹⁰⁰

PV₁ is the present value of the benefits that will be received in years 1 to 50= $23,455,617.87

PV₂ is the present value of the benefits that will be received in years 51 to 100 at the end of 50^th year= $28,362,311.68

r₂ is the discount rate applicable in years 51-100= 2.5% or 0.025

PV₃ is the present value of the benefits that will be received in years 101 to 150 at the end of 100^th year= $39,196,117.53

r₃ is the discount rate applicable in years 101-150= 1% or 0.01

PV= $23,455,617.87 + $28,362,311.68/(1+0.025)⁵⁰ + $39,196,117.53/(1+0.01)¹⁰⁰

PV= $23,455,617.87 + $28,362,311.68/1.025⁵⁰ + $39,196,117.53/1.01¹⁰⁰

PV= $23,455,617.87 + $8,251,793.584 + $14,491,244.13

PV= $46,198,655.584

PV≈ $46,198,655.58

Therefore, the present value of these benefits using the time-declining discount rate schedule is $46,198,655.58.

 

(c)

Calculate the difference:

The difference= The present value of the benefits using the time-declining discount rate schedule - The present value of the benefits using the time-constant discount rate

The present value of the benefits using the time-declining discount rate schedule= $46,198,655.58

The present value of the benefits using the time-constant discount rate= $32,937,698.03

The difference= $46,198,655.58 - $32,937,698.03

The difference= $13,260,957.55

Therefore, the present value of the benefits using the time-declining discount rate is higher than the present value of the benefits using the time-constant discount rate and the difference of $13,260,957.55 between them is significant enough to be considered important.