5.4: Annuities and Perpetuities

Question 1

What is the learning objective of section 5.4?

Answer 1

A: Differentiate between an ordinary annuity and an annuity due, and explain how special constant payment problems can be valued as annuities and, in special cases, as perpetuities.

Question 2

What does the example of the twins investing early illustrate?

Answer 2

The power of compound interest as time passes. Twin 1 starts investing $2,000 per year at age 21 for six years and stops, accumulating $1.2 million by age 65.

Twin 2 starts at age 27 and invests $2,000 per year for 38 years, also accumulating $1.2 million by age 65.

Question 3

What is an annuity?

Answer 3

A series of payments or receipts of the same amount made at regular intervals over a given period.

Question 4

What is an ordinary annuity?

Answer 4

Equal payments that are made at the end of each period of time.

Question 5

What is the manual formula to calculate the future value (FV) of an ordinary annuity?

Answer 5

A: FV_n = PMT * [(1 + k)^n - 1] / k

Question 6

How do you calculate the FV of the ordinary annuity in Example 5.8 using a financial calculator?

Answer 6

A:
1. Enter 164,020 and press PMT
2. Enter 6 and press N
3. Enter 0 and press PV
4. Enter 8 and press I/Y
5. Press CPT and then FV
The answer will be 1,203,239.

Question 7

What is the manual formula to calculate the present value (PV) of an ordinary annuity?

Answer 7

A: PV_0 = PMT * [1 - 1 / (1 + k)^n] / k

Question 8

What is an annuity due?

Answer 8

A: An annuity for which the payments are made at the beginning of each period.

Question 9

How does an annuity due differ from an ordinary annuity?

Answer 9

A: In an annuity due, payments are made at the beginning of each period, while in an ordinary annuity, payments are made at the end of each period.

Question 10

In Example 5.9, how do you calculate the future value (FV) of an annuity due using the given values: $164,020 annual payment, 8% interest rate, and 6 years?

Answer 10

A:
FV_6 = $164,020 (1.08)^6 + $164,020 (1.08)^5 + $164,020 (1.08)^4 + $164,020 (1.08)^3 + $164,020 (1.08)^2 + $164,020 (1.08)
= $1,299,498

Question 11

What is the formula to find the FV of an annuity due?

Answer 11

FV_n = PMT [(1 + k)^n - 1] / k * (1 + k)

Question 12

How do you calculate the future value (FV) of an annuity due in Example 5.9 using a financial calculator?

Answer 12

A:
1. Set the calculator to ‘Begin’ mode by pressing 2ND BGN 2ND SET
2. Enter 164,020 and press PMT
3. Enter 6 and press N
4. Enter 0 and press PV
5. Enter 8 and press I/Y
6. Press CPT and then FV
The answer will be 1,299,498.

Question 13

What is the formula to find the PV of an annuity due?

Answer 13

A: PV_0 = PMT [1 - 1 / (1 + k)^n] / k * (1 + k)

Question 14

How do you calculate the present value (PV) of an annuity due in Example 5.9 using a financial calculator?

Answer 14

1. Set the calculator to ‘Begin’ mode by pressing 2ND BGN 2ND SET
2. Enter 164,020 and press PMT
3. Enter 6 and press N
4. Enter 0 and press FV
5. Enter 8 and press I/Y
6. Press CPT and then PV
   The answer will be -818,904.

Question 15

Why do we multiply the FV of an ordinary annuity by (1 + k) to get the FV of an annuity due?

Answer 15

Because each payment in an annuity due receives one extra period of compounding interest compared to an ordinary annuity.

Question 16

What is a perpetuity?

Answer 16

A special annuity that provides payments forever.

Question 17

What is the formula to calculate the present value (PV) of a perpetuity?

Answer 17

PV_0 = PMT / k

Question 18

In Example 5.10, how do you calculate the present value (PV) of a $3,000 per year annuity that earns 12% annually for 30 years?

Answer 18

PV_0 = $3,000 * [(1 - 1 / (1.12)^30) / 0.12]
= $3,000 * 8.05518
= $24,165.55

Question 19

In Example 5.10, what is the present value (PV) of a $3,000 per year annuity that goes on forever with a 12% annual return?

Answer 19

PV_0 = $3,000 / 0.12
= $25,000

Question 20

How do you calculate the present value (PV) of an annuity for 30 years in Example 5.10 using a financial calculator?

Answer 20

A:
1. Enter 3,000 and press FV
2. Enter 30 and press N
3. Enter 12 and press I/Y
4. Press CPT and then PV
The answer will be -24,165.55.

Question 21

How do you calculate the future value (FV) for Twin 1 who invests $2,000 per year for 6 years at 12% annual return?

Answer 21

FV_0 = PMT * [(1 + k)^n - 1] / k
= $2,000 * [(1.12)^6 - 1] / 0.12
= $2,000 * 8.11519
= $16,230.38

Question 22

How do you calculate the future value (FV) of the accumulated savings for Twin 1 after 38 years (from age 27 to age 65) in Example 5.11?

Answer 22

FV_38 = PV_0 (1 + k)^n
= $16,230.38 (1.12)^38
= $16,230.38 * 74.17966
= $1,203,964.13

Question 23

How do you calculate the future value (FV) for Twin 2 who invests $2,000 per year for 38 years at 12% annual return in Example 5.11?

Answer 23

A:
FV_38 = PMT * [(1 + k)^n - 1] / k
= $2,000 * [(1.12)^38 - 1] / 0.12
= $2,000 * 609.83053
= $1,219,661.07

Question 24

Explain how to calculate the present value (PV) and future value (FV) of an ordinary annuity.

Answer 24

• PV of an ordinary annuity: PV_0 = PMT * [1 - 1 / (1 + k)^n] / k
• FV of an ordinary annuity: FV_n = PMT * [(1 + k)^n - 1] / k

Question 25

Explain how to calculate the present value (PV) and future value (FV) of an annuity due.

Answer 25

• PV of an annuity due: PV_0 = PMT * [1 - 1 / (1 + k)^n] / k * (1 + k)
• FV of an annuity due: FV_n = PMT * [(1 + k)^n - 1] / k * (1 + k)

Question 26

Define “perpetuity.”

Answer 26

A special annuity that provides payments forever.

Question 27

What is the formula to calculate the present value (PV) of a perpetuity?

Answer 27

PV_0 = PMT / k