CHAPTER 5

TIME VALUE OF MONEY

Chapter Outline

• Introduction• Future value• Present value• Multiple cash flow• Annuities• Perpetuities• Amortization

Introduction

• Is that the value of one Ringgit today is the same as the value of one Ringgit ten years ago or ten year to come?

• What will happen to the money if we invest or we save it in the bank or financial institutions?

Future Value

• Refer to the amount of money an investment will grow to over some period of time at some given interest rate.

• Cash value of an investment at some time in future.

Future Values• Suppose you invest $1,000 for one year at 5% per year.

What is the future value in one year?

• Suppose you leave the money in for another year. How much will you have two years from now?

Future Values: General Formula

• FV = PV(1 + r)t

– FV = future value– PV = present value– r = period interest rate, expressed as a

decimal– t = number of periods

Example:

• What will be the future value of RM 10, 000 in 5 years, given the rate of return of 15%?

• You plan to save in an account that pays 10% return per year. You put in RM 20,000 in the first year, RM 30,000 in the second year and another RM 50,000 in the third year . How much will you have by the end of 10 years?

Present Value

• Value today of a sum of money to be received in future.

• Value that we should invest or deposit, if we want some specific value in the future.

• How much do I have to invest today to have some amount in the future?

Present Values

– FV = PV(1 + r)t

– Rearrange to solve for PV = FV / (1 + r)t

• When we talk about discounting, we mean finding the present value of some future amount.

• When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

Present Value – One Period Example

• Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?

Present Values – Example 2

• You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?

Present Values – Example 3

• Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest?

Discount Rate

• Often we will want to know what the implied interest rate is on an investment

• Rearrange the basic PV equation and solve for r– FV = PV(1 + r)t

– r = (FV / PV)1/t – 1

Discount Rate – Example 1

• You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest?■r = (FV / PV )1/t – 1

Discount Rate – Example 2

• Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?

Discount Rate – Example 3• Suppose you have a 1-year old son and

you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?

Finding the Number of Periods

• Start with basic equation and solve for t (remember your logs)– FV = PV(1 + r)t

– t = ln(FV / PV) / ln(1 + r)• You can use the financial keys on the

calculator as well; just remember the sign convention.

Number of Periods – Example 1

• You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?

Multiple Cash Flows – Present Value Example 6.3

• Find the PV of each cash flows and add them– Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = -178.57

– Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = -318.88

– Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV = -427.07

– Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV = - 508.41

– Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93

Example 6.3 Timeline0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

Multiple Cash Flows – PV Another Example

• You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?– N = 1; I/Y = 10; FV = 1,000; CPT PV = -909.09– N = 2; I/Y = 10; FV = 2,000; CPT PV = -1,652.89– N = 3; I/Y = 10; FV = 3,000; CPT PV = -2,253.94– PV = 909.09 + 1,652.89 + 2,253.94 = 4,815.93

Annuities and Perpetuities Defined

• Annuity – finite series of equal payments that occur at regular intervals– If the first payment occurs at the end of the period, it

is called an ordinary annuity– If the first payment occurs at the beginning of the

period, it is called an annuity due• Perpetuity – infinite series of equal payments

Annuities and Perpetuities – Basic Formulas

• Perpetuity: PV = C / r• Annuities:

rrCFV

rrCPV

t

t

1)1(

)1(11

Annuity – Sweepstakes Example

• Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual end-of-year installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?– 30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29

Amortized Loan with Fixed Principal Payment - Example

• Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.

• Click on the Excel icon to see the amortization table

YearBeginning

BalanceInterestPayment

PrincipalPayment

Total Payment

EndingBalance

1 50,000 4,000 5,000 9,000 45,0002 45,000 3,600 5,000 8,600 40,0003 40,000 3,200 5,000 8,200 35,0004 35,000 2,800 5,000 7,800 30,0005 30,000 2,400 5,000 7,400 25,0006 25,000 2,000 5,000 7,000 20,0007 20,000 1,600 5,000 6,600 15,0008 15,000 1,200 5,000 6,200 10,0009 10,000 800 5,000 5,800 5,000

10 5,000 400 5,000 5,400 0

Amortized Loan with Fixed Payment - Example

• Each payment covers the interest expense plus reduces principal

• Consider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5,000.– What is the annual payment?

• 4 N• 8 I/Y• 5,000 PV• CPT PMT = -1,509.60

• Click on the Excel icon to see the amortization table

Year BeginningBalance

TotalPayment

Interest Paid

Principal Paid

Ending Balance

1 5,000.00 1,509.60 400.00 1,109.60 3,890.40

2 3,890.40 1,509.60 311.23 1,198.37 2,692.03

3 2,692.03 1,509.60 215.36 1,294.24 1,397.79

4 1,397.79 1,509.60 111.82 1,397.78 0.01

Totals 6,038.40 1,038.41 4,999.99