5.5: Growing Perpetuities and Annuities

Question 1

What is the formula for the present value of a growing perpetuity?

Answer 1

PV₀ = PMT₀ * (1 + g) / (k - g)

Question 2

When does the relationship for the present value of a growing perpetuity hold true?

Answer 2

1. When k > g, otherwise the answer is negative.
2. Only future estimated cash flows and growth in these cash flows are relevant.
3. The relationship holds only when growth in payments is expected to take place indefinitely.

Question 3

How do you calculate the present value of a growing perpetuity?

Answer 3

PV₀ = PMT₀ * (1 + g) / (k - g)

Question 4

Provide an example calculation for the present value of a growing perpetuity.

Given: PMT₀ = $100,000; g = 4%; k = 15%

Answer 4

PV₀ = ($100,000 * (1 + 0.04)) / (0.15 - 0.04) = $945,454.55

Question 5

What is a growing annuity?

Answer 5

A growing annuity is a stream of cash flows that grow (or shrink) at a constant rate per period over a given period of time, ending at some terminal point (n).

Question 6

What is the formula for the present value of a growing annuity?

Answer 6

PV₀ = PMT₀ * (1 + g) / (k - g) * [1 - (1 + g / 1 + k)ⁿ]

Question 7

Provide an example calculation for the present value of a growing annuity.

Given: PMT₁ = $200,000; g = -10%; k = 20%; n = 10

Answer 7

PV₀ = ($200,000 / (0.20 - (-0.10))) * [1 - ((1 - 0.10) / (1 + 0.20))¹⁰] = $629,124.32

Question 8

Explain the steps to evaluate a growing perpetuity.

Answer 8

1. Identify the initial payment (PMT₀).
2. Determine the growth rate (g) and discount rate (k).
3. Apply the formula: PV₀ = PMT₀ * (1 + g) / (k - g).

Question 9

Explain how to calculate the present value of a growing annuity.

Answer 9

1. Identify the initial payment (PMT₁) and the growth rate (g).
2. Determine the discount rate (k) and the number of periods (n).
3. Apply the formula: PV₀ = PMT₁ * (1 + g) / (k - g) * [1 - (1 + g / 1 + k)ⁿ].