Topic C07: Time Value of Money Topic Overview p. 1

BAFS Learning and Teaching Example Updated in 2022

**Learning Objectives: **

1. To understand the fundamental concepts of time value of money;

2. To calculate future value and present value of a single and a series of cash flows;

3. To distinguish between nominal and effective rate of return; and

4. To apply the concepts and calculations of time value of money in financial management.

**Overview of Contents: **

Lesson 1 Future Value and Compounding Lesson 2 Present Value and Discounting

Lesson 3 Present Value and Future Value of Annuity

Lesson 4 Uneven Cash Flows, Nominal and Effective Interest Rate
**Extended Learning -- Rule of 72 (End of Lesson 2) **

**Resources: **

Topic Overview and Teaching Plan

Power Point Presentation

Student Worksheet

**Suggested Activity: **

Problem Solving

**Topic Overview**

**Topic ** BAFS Compulsory Part - Basics of Personal Financial Management
C07: Fundamentals of Financial Management – Time Value of Money

**Level ** S4

**Duration ** 4 lessons (40 minutes per lesson)

Topic C07: Time Value of Money Topic Overview p. 2

BAFS Learning and Teaching Example Updated in 2022

**Lesson 1**

**Theme ** Future Value and Compounding
**Duration ** 40 minutes

**Expected Learning Outcomes: **

Upon completion of this lesson, students will be able to:

1. Explain the difference between simple and compound interest;

2. Calculate the future value of a single cash flow for one year; and

3. Calculate the future value of a single cash flow for multi-year periods using compounding.

**Teaching Sequence and Time Allocation: **

**Activities ** **Reference ** **Time **

**Allocation **
**Part I: Introduction **

Teacher starts the lesson by taking out a $1,000 bill and asks the class “Who wants this $1,000?” in order to lead to the topic of the time value of money.

PPT#2 5 minutes

**Part II: Content **

Teacher introduces the concept of future value and explains how to calculate future value of a single cash flow for one year.

**Activity 1 – Problem Solving (Problem 1) **

Students are asked to solve Problem 1 and teacher invites a volunteer to show the calculation to the class.

*(Hint: Make sure students are able to perform this basic *
*calculation before introducing the next concept. *

*Teacher can provide more exercises to the students if *
**needed.) **

PPT #3-5

PPT #6-7 Student Worksheet

p.1

12 minutes

Teacher extends the calculation of future value to 2 years and explains the concepts of simple and compound interests.

Teacher demonstrates how to calculate the future value of single cash flow for multi-year periods using simple and compound interests.

PPT#8-10

PPT#11-15 20 minutes

Topic C07: Time Value of Money Topic Overview p. 3

BAFS Learning and Teaching Example Updated in 2022

**Activity 2 – Problem Solving (Problems 2 and 3) **

Students are asked to solve Problem 2 and 3 and teacher invites volunteers to show the calculations to the class.

*(Hint: Make sure students are able to perform this *
*calculation before moving on. Teacher can provide *
*more exercises to the students if needed.) *

PPT#16-19 Student Worksheet

p. 2

**Part III: Conclusion **

Teacher concludes the lesson by recapping different

formulas in calculating present value. PPT#20 3 minutes
**Preparation for the next lesson: **

Students are asked to review the examples and problems discussed during the class to have a thorough understanding of the concepts and calculations of future value.

Topic C07: Time Value of Money Topic Overview p. 4

BAFS Learning and Teaching Example Updated in 2022

**Lesson 2 **
**Theme ** Present Value and and Discounting
**Duration ** 40 minutes

**Expected Learning Outcomes: **

Upon completion of this lesson, students will be able to:

1. Explain the concept of discounting;

2. Calculate the present value of a single future cash flow for one year;

3. Calculate the present value of a single future cash flow for multi-year period using discounting;

**Teaching Sequence and Time Allocation: **

**Activities ** **Reference ** **Time **

**Allocation **
**Part I: Introduction **

Teacher posts a problem related to the calculation of

present value to the class. PPT#21 5 minutes

**Part II: Content **

Teacher illustrates the way to calculate present value of a single cash flow for one year and introduce the concept of discounting.

Teacher further explains the calculation of present value of a single cash flow for multi-year period using

discounting.

PPT#22-24 PPT#25-27

15 minutes

**Activity 3 – Problem Solving (Problems 4a & 4b) **

Students are asked to solve Problems 4a & 4b and teacher invites volunteers to show the calculation to the class.

Teacher reminds that if there are

multi-compounding-period in a year, the interest rate per annum should be divided by the number of periods when discounting to present value

*(Hint: Teacher can provide more exercises to students if *
*needed.) *

Teacher recaps the formulas involving present value calculations.

PPT#28-31 Student Worksheet

p.3

PPT#32

15 minutes

Topic C07: Time Value of Money Topic Overview p. 5

BAFS Learning and Teaching Example Updated in 2022

**Extended Learning - Rule of 72 (Optional) **

Teacher introduces Rule of 72 to estimate the time in years needed for an investment to double in value.

Teacher demonstrates how to use Rule of 72 and leads students to check the accuracy of the rule.

Teacher explains to students that the rule could be used to estimate the required rate of return for an investment to double in value within a certain number of years.

**Challenging Problem **

Students are asked to solve the Challenging

Problem. Then teacher asks volunteers to show the calculations to the class.

*(Hint: Teacher can provide more exercises to students if *
*needed.) *

PPT#33

PPT#34-36 PPT#37-38

PPT#39-40 Student Worksheet

p. 4

15 minutes

**Part III: Conclusion **

Teacher concludes the lesson with a review of the key

concepts covered. 5 minutes

**Preparation for the next lesson: **

Students are asked to review the examples and problems discussed during the class in order to have a thorough understanding of the concepts and calculations of present value.

Topic C07: Time Value of Money Topic Overview p. 6

BAFS Learning and Teaching Example Updated in 2022

**Lesson 3 **

**Theme ** Present Value and Future Value of Annuity
**Duration ** 40 minutes

**Expected Learning Outcomes: **

Upon completion of this lesson, students will be able to:

1. Explain the features of an annuity;

2. Calculate the present value of an annuity; and 3. Calculate the future value of an annuity.

**Teaching Sequence and Time Allocation: **

**Activities ** **Reference ** **Time **

**Allocation **
**Part I: Introduction **

Teacher posts a problem involving annuity to the class

to introduce the concept of annuity. PPT#41 5 minutes
**Part II: Content **

Teacher explains the present value of an annuity can be found by calculating the present values of all cash flows first and then add the present values together to arrive at the answer.

Teacher demonstrates to the class the steps to find the present value of an annuity.

**Activity 4 – Problem Solving (Problem 5) **

Students are asked to solve Problem 5 and teacher invites a volunteer to show the calculation to the class.

*(Hint: Teacher can provide more exercises to students if *
*needed.) *

PPT#42-45

PPT#46-47 PPT#48-49

Student Worksheet

p.5

15 minutes

Teacher explains how to calculate the future value of an annuity and works through the example with students.

**Activity 5 – Problem Solving (Problem 6) **

Students are asked to solve Problem 6 and teacher invites a volunteer to show the calculation to the class.

PPT#50-53

PPT#54-55 Student Worksheet

15 minutes

Topic C07: Time Value of Money Topic Overview p. 7

BAFS Learning and Teaching Example Updated in 2022

*(Hint: Teacher can provide more exercises to students if *
*needed.) *

p.6
**Part III: Conclusion **

Teacher concludes the lesson by reviewing the key

concepts covered. 5 minutes

**Preparation for the next lesson: **

Students are asked to review the examples and problems discussed during the class in order to have a thorough understanding of the concepts and calculations of annuity.

Topic C07: Time Value of Money Topic Overview p. 8

BAFS Learning and Teaching Example Updated in 2022

**Lesson 4 **

**Theme ** Uneven Cash Flows, Nominal and Effective Interest Rate
**Duration ** 40 minutes

**Expected Learning Outcomes: **

Upon completion of this lesson, students will be able to:

1. Calculate the present value of a series of uneven cash flows;

2. Calculate the future value of a series of uneven cash flows;

3. Distinguish between nominal and effective rates of return;

**Teaching Sequence and Time Allocation: **

**Activities ** **Reference ** **Time **

**Allocation **
**Part I: Introduction **

Teacher extends the discussion of a series of cash flows to uneven cash flows and explains the calculations of future and present value of a series of uneven cash flows.

PPT#56 5 minutes

**Part II: Content **

With an example, teacher explains how to calculate the present value of a series of uneven cash flows.

**Activity 6 – Problem Solving (Problem 7) **

Students are asked to solve Problem 7 and teacher invites a volunteer to show the calculation to the class.

PPT#57-58 PPT#59-60

Student Worksheet

p. 7

5 minutes

With an example, teacher explains how to calculate the future value of a series of uneven cash flows.

**Activity 7 – Problem Solving (Problem 8) **

Students are asked to solve Problem 8 and teacher invites a volunteer to show the calculation to the class.

PPT#61-62 PPT#63-64

Student Worksheet

p.8

5 minutes

Teacher explains what is nominal rate of return (NRR) and how it is different from effective rate of return (ERR).

PPT#65-66

10 minutes

Topic C07: Time Value of Money Topic Overview p. 9

BAFS Learning and Teaching Example Updated in 2022

Teacher demonstrates how to use ERR to compare the
actual returns of different investment plans. **Activity **
**8 – Problem Solving (Problem 9) **

Students are asked to solve Problem 9 and teacher invites a volunteer to show the calculation to the class.

PPT#67-68 PPT#69-70

Student Worksheet

p.9

10 minutes

**Part III: Conclusion **

Teacher reviews concepts and calculations of time value

of money and concludes the lesson. 5 minutes

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?

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?

**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?

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?

**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?

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?

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.

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

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

BAFS Learning and Teaching Example Updated in 2022