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 / 1.03^{3}

= $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

× 2^{5} = $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 / 1.05^{2}

= $1,995,464.85
PV of third $2,200,000 = $2,200,000 / 1.05^{3}

= $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 / 1.07^{2}

= $19,215.65

PV of $5,000 in the third year

= $5,000 / 1.07^{3}

= $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 × 1.07^{2}

= $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