# Future Value of an Annuity

In document Topic Overview (Page 50-65)

Kitty, a rising pop music artist is negotiating with EBI Music Company for the terms of her new 3-year contract. EBI offers two alternatives for her remuneration:

(a) An upfront payment of \$6,000,000; or

(b) Payment of \$2,200,000 at the end of each year for the next three years.

Suppose the appropriate interest rate is 5%, advise Kitty which package to take.

Topic C07: Time Value of Money Student Worksheet p.6

BAFS Learning and Teaching Example Updated in 2022

Problem 6: Future Value of an Annuity

Daniel plans to take a backpack trip to Europe after his university graduation, and he is expected to save up \$20,000 at the end of each year in the coming 3 years. If he can earn 3.5% interest per year from his savings, how much would he have at the end of the third year?

Topic C07: Time Value of Money Student Worksheet p.7

BAFS Learning and Teaching Example Updated in 2022

Problem 7: Present Value of Uneven Cash Flows

Using a discount rate of 7%, what is the total present value of the following series of cash flows?

End of first year \$12,000 End of second year \$22,000 End of third year \$ 5,000

Topic C07: Time Value of Money Student Worksheet p.8

BAFS Learning and Teaching Example Updated in 2022

Problem 8: Future Value of Uneven Cash Flows

Based on an interest rate of 7%, what is the future value of the following series of cash flows?

End of first year \$12,000 End of second year \$22,000 End of third year \$ 5,000

Topic C07: Time Value of Money Student Worksheet p.9

BAFS Learning and Teaching Example Updated in 2022

Problem 9: NRR Vs ERR

Ken received a bonus worth \$100,000. He plans to deposit the money into a savings account for 2 years. Which of the following savings plans should he choose and how much will he obtain at the maturity date?

Plan I Plan II

Interest rate 5.5% 6.5%

Frequency of interest compounding

Half-yearly Yearly

Topic C07: Time Value of Money Student Worksheet p.10

BAFS Learning and Teaching Example Updated in 2022

Topic C07: Time Value of Money Student Worksheet p.1

BAFS Learning and Teaching Example Updated in 2022

BAFS Compulsory Part - Basics of Personal Financial Management

Topic C07: Fundamentals of Financial Management - Time Value of Money

## Problem Solving

Problem 1: Future Value of a Single Cash Flow

Your mother has bought \$4,500 Australian dollars and placed it in the bank on a fixed deposit term of 1 year. If the deposit interest rate is 5.19% per year, how much will she have at the maturity of the deposit?

Solution:

At the maturity of the deposit, your mother will have:

FV = PV × (1 + r)

= A\$4,500 × (1 + 0.0519) = A\$4,733.55

Topic C07: Time Value of Money Student Worksheet p.2

BAFS Learning and Teaching Example Updated in 2022

Problem 2: Future Value of a Single Cash Flow Using Compound Interest

An investment product promises to pay an 8% return for each year, compounded annually. How much will an initial investment of

\$100,000 become at the end of the 3rd year?

Solution:

FV = PV × (1 + r)n

= \$100,000 × (1 + 0.08)3

= \$125,971.20

Problem 3: Future Value of a Single Cash Flow Using Compounding

You have made a 3-year fixed New Zealand dollar deposit today.

The principal amount is NZ\$10,000 and interest rate is 6.75%, compounded yearly. How much would you get back at the maturity of your deposit?

Solution:

FV = PV × (1 + r)n

= NZ\$10,000 × (1 + 0.0675)3 = NZ\$12,164.76

Topic C07: Time Value of Money Student Worksheet p.3

BAFS Learning and Teaching Example Updated in 2022

Problem 4a: Present Value of a Single Cash Flow (multi-year periods)

How much would you be willing to lend to Thomson with a current interest rate of 3%, if he promises to pay you \$1,000 in 1 year time?

What if Thomson pays you 3 years later?

Solution:

 For 1 year:

PV = FV / (1 + r)

= \$1,000 / 1.03

= \$970.87

 For 3 years:

PV = FV / (1 + r)n

= \$1,000 / 1.033

= \$915.14

Problem 4b: Present Value of a Single Cash Flow (multi-compounding periods in a year)

Jason wants to renovate his house at \$200,000 a year later. If the interest rate is 6% per annum, compounded semi-annually, how much does he need to deposit into his bank account now to save up for the renovation work?

Solution:

 The amount to be deposited into bank now:

PV = FV / (1 + r)n

= \$200,000 / (1.03)2

= \$188,501

Topic C07: Time Value of Money Student Worksheet p.4

BAFS Learning and Teaching Example Updated in 2022

## Extended Learning (Lesson 2)

Challenging Problem: Rule of 72

Today is Jacky’s 30th birthday and he is working on a retirement plan. Currently, he has a saving of \$100,000 and plan to invest the money in a mutual fund that is expected to have an annual rate of return of 12%. By means of the Rule of 72, estimate:

(a) How long will it take for the investment to double?

(b) How much would the investment be worth when Jacky is 60 years old?

Solution:

(a) By the Rule of 72, it takes about 6 years (i.e. 72/12) for the investment to double.

(b) The investment period equals 60 – 30 = 30 years, and given the investment will double every 6 years, the investment will double 30 / 6 = 5 times. So \$100,000 will grow to \$100,000

× 25 = \$3,200,000.

Topic C07: Time Value of Money Student Worksheet p.5

BAFS Learning and Teaching Example Updated in 2022

Problem 5: Future Value of an Annuity

Kitty, a rising pop music artist is negotiating with EBI Music Company for the terms of her new 3-year contract. EBI offers two alternatives for her remuneration:

(a) An upfront payment of \$6,000,000; or

(b) Payment of \$2,200,000 at the end of each year for the next three years.

Suppose the appropriate interest rate is 5%, advise Kitty which package to take.

Solution:

PV of first \$2,200,000 = \$2,200,000 / 1.05

= \$2,095,238.10 PV of second \$2,200,000 = \$2,200,000 / 1.052

= \$1,995,464.85 PV of third \$2,200,000 = \$2,200,000 / 1.053

= \$1,900,442.72 PV of annuity = Total PVs

= \$5,991,145.67 which is less than \$6,000,000.

Thus, Kitty is better off to be paid \$6,000,000 upfront.

Topic C07: Time Value of Money Student Worksheet p.6

BAFS Learning and Teaching Example Updated in 2022

Problem 6: Future Value of an Annuity

Daniel plans to take a backpack trip to Europe after his university graduation, and he is expected to save up \$20,000 at the end of each year in the coming 3 years. If he can earn 3.5% interest per year from his savings, how much would he have at the end of the third year?

Solution:

Total FVs:

= \$20,000 × (1+0.035)2

+ \$20,000 × (1+0.035) + \$20,000 = \$21,424.5 + \$20,700 + \$20,000 = \$62,124.5

Topic C07: Time Value of Money Student Worksheet p.7

BAFS Learning and Teaching Example Updated in 2022

Problem 7: Present Value of Uneven Cash Flows

Using a discount rate of 7%, what is the total present value of the following series of cash flows?

End of first year \$12,000 End of second year \$22,000 End of third year \$ 5,000

Solution:

PV of \$12,000 in the first year

= \$12,000 / 1.07

= \$11,214.95

PV of \$22,000 in the second year

= \$22,000 / 1.072

= \$19,215.65

PV of \$5,000 in the third year

= \$5,000 / 1.073

= \$4,081.49 Total PVs = \$34,512.09

Topic C07: Time Value of Money Student Worksheet p.8

BAFS Learning and Teaching Example Updated in 2022

Problem 8: Future Value of Uneven Cash Flows

Based on an interest rate of 7%, what is the future value of the following series of cash flows?

End of first year \$12,000 End of second year \$22,000 End of third year \$ 5,000

Solution:

FV of \$12,000 in Year 1

= \$12,000 × 1.072

= \$13,738.80 FV of \$22,000 in Year 2

= \$22,000 × 1.07

= \$23,540 FV of \$5,000 in Year 3

= \$5,000

Total FVs = \$42,278.80

Topic C07: Time Value of Money Student Worksheet p.9

BAFS Learning and Teaching Example Updated in 2022

Problem 9: NRR Vs ERR

Ken received a bonus worth \$100,000. He plans to deposit the money into a savings account for 2 years. Which of the following savings plans should he choose and how much will he obtain at the maturity date?

Plan I Plan II

Interest rate 5.5% 6.5%

Frequency of interest compounding

Half-yearly Yearly

Solution:

ERR of each plan are as follows:

ERR of Plan I ERR of Plan II

(1+0.055/2) 2 -1

= 5.6%

(1+0.065) 1 -1

= 6.5%

Plan II offers a higher return.

FV of Plan II:

\$100,000 x (1+6.5%) 2

= \$113,422.5

In document Topic Overview (Page 50-65)