TOPIC 3

TIME VALUE OF MONEY

CONTENTS

1. Future Value2. Present Value3. Annuity4. Perpetuity5. Effective Annual Rate (EAR)

Time Value of Money

Basic Principle: A dollar received today is worth more than a dollar received in the future.

This is due to opportunity cost. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner.

This concept is so important in understanding financial management

Time Value of Money Translate $1 today into its equivalent in the future

(compounding) – Future Value (FV)

today future

Translate $1 in the future into its equivalent today (discounting) – Present Value (PV)

today future

$1 $?

$1$?

1. FUTURE VALUE (FV)

Compound interest occurs when interest paid on the investment during the first period is added to the principal; then, during the second period, interest is earned on this new sum.

Compounding is the process of determining the Future Value (FV) of cash flow.

The compounded amount (FV) is equal to the beginning amount plus interest earned.

Example: If we place RM1,000 in savings account paying 5% interest compounded annually. How much will it be worth at the end of each year?

RM1000 i = 5%

0 1 2 3 4 n..

Year 1: RM1000 (1.05) = RM1050.00Year 2: RM1050 (1.05) = RM1102.50Year 3: RM1102.50 (1.05) = RM1157.63Year 4: RM1157.63 (1.05) = RM1215.51

Formula of FV

or

where;FVn = the FV of the investment at the end of n year

n = the number of yearsi = the annual interest ratePV = original amount invested at beginning of the first year(1 + i) is also known as compounding factor

FVn = PV (1 + i)n FVn = PV (FVIF i,n )

Example:

If we place RM1,000 in a savings account paying 5% interestcompounded annually. How much will our account accrue in 4 years?PV = RM1,000 i = 5% n = 4 years

FVn = PV (1 + i)n FVn = PV (FVIF i,n )

= 1,000(1 + 0.05)4 = 1,000(FVIF 5%,4 )

= 1,000 (1.2155) = 1,000 (1.2155) = RM1,215.50 = RM1,215.50

Example:If Anuar invests RM10,000 in a bank where it will earn 6%interest compounded annually. How much will it be worth at theend of a) 1 year and b) 5 years

a) Compounded for 1 yearFV1 = RM10,000 (1 + 0.06)1 FV1 = RM10,000 (FVIF 6%,1 )

= RM10,000 (1.06)1 = RM10,000 (1.0600) = RM10,600 = RM10,600

b) Compounded for 5 yearsFV5 = RM10,000 (1 + 0.06)5 FV1 = RM10,000 (FVIF 6%,5 )

= RM10,000 (1.06)5 = RM10,000 (1.3382) = RM13,380 = RM13,382

Exercise:1. If Danial invests RM20,000 in a bank where it will earn 8%

interest compounded annually. How much will it be worth at the end of a) 5 years and b) 15 years

Exercise:2. If the interest rate increases to 10%, how much will the

Danial’s savings grow?

Finding iAt what annual rate would the following have to be invested; $500 to grow to RM1183.70 in 10 years.

FVn = PV (FVIF i,n )

1183.70 = 500 (FVIF i,10 )

1183.70/500 = (FVIF i,10 )

2.3674 = (FVIF i,10 ) refer to FVIF table

i = 9%

Finding nHow many years will the following investment takes? $100 togrow to $672.75 if invested at 10% compounded annually

FVn = PV (FVIF i,n )

672.75 = 100 (FVIF 10%,n )

672.75/100 = (FVIF 10%,n )

6.7272 = (FVIF 10%,n ) refer to FVIF table

n = 20 years

Exercise:1. How many years will the following investments take:

$100 to grow to $298.60 if invested at 20% compounded annually

$550 to grow to $1044.05 if invested at 6% compounded annually

2. At what annual rate would the following investments have to be invested: $200 to grow to $497.65 in 5 years

$180 to grow to $485.93 in 6 years

Compound Interest with Non-annual Periods

Non-annual periods occurs semiannually, quarterly ormonthly

if semiannually compounding:FV = PV (1 + i/2)nx2 or FV = PV (FVIF i/2,nx2 )

if quarterly compounding: FV = PV (1 + i/4)nx4 or FV = PV (FVIF i/4,nx4 )

if monthly compounding: FV = PV (1 + i/12)nx12 or FV = PV (FVIF i/12,nx12 )

Example:If you deposit $100 in an account earning 6% with semiannually compounding, how much would you have in the account after 5 years?

FV = PV (1 + i/2)nx2 FV = PV (FVIF i/2,nx2 )

= 100 (1 + 6%/2)5x2 = 100 (FVIF 6%/2,5x2 )

= 100 (1 + 0.03)10 = 100 (FVIF 3%,10 )

= 100 (1.3439) = 100 (1.3439)= $134.39 = $134.39

Example:

If you deposit $1000 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?

FV = PV (1 + i/4)nx4

= 1000 (1 + 6%/4)5x4

= 1000 (1 + 0.015)20

= $1,346.86

*6% /4 = 1.5% (can’t use FVIF table)

Exercise:1. If you deposit $1,000 in an account earning 8% with

quarterly compounding, how much would you have in the account after 3 years?

2. To what amount will the following investments accumulate: $5,000 invested for 5 years at 10% with quarterly compounding

$4,000 invested for 6 years at 6% with semiannually compounding

2. PRESENT VALUE (PV) PV is the current value of futures sum Finding PV is called discounting and can be

calculated by using this equation: or

[1/ (1 + i)n ] is also known as discounting factor

PV = [ FVn / (1 + i)n ] PVn = FV (PVIF i,n )

Example:

What is the PV of $800 to be received 10 years from today if ourdiscount rate is 10%?

PV = 800 / (1.10)10 PV = 800 (PVIF10%,10 )

= $308.43 = 800 (0.3855) = $308.40

Exercise:

Find the PV of $10,000 to be received 10 years from today if our discount rates:a) 5%

b) 10%

c) 20%

PV with Multiple, Uneven Cash Flows

What is the PV of an investment that yields $300 to be received in 2 years and $450 to be received in 8 years if the discount rate is 5%?

PV = 300 (PVIF5%,2 ) + 450 (PVIF5%,8 )

= 300 (0.907) + 450 (0.677)= 272.10 + 304.65= $576.75

Exercise:

1. What is the PV of an investment that yields $500 to be received in 3 years and $750 to be received in 5 years if the discount rate is 5%?

2. What is the PV of an investment that yields $1,000 to be received in 2 years and $2,500 to be received in 4 years if the discount rate is 6%?

3. ANNUITYAn annuity is a series of equal payments for a specifiednumber of years.

100 100 100 100 100 0 1 2 3 4

There are 2 type of annuities: Ordinary annuity Annuity due

*in finance, ordinary annuities are used much more frequent compared to annuities due

ORDINARY ANNUITY

Ordinary annuity is an annuity which the payments occur at theend of each period

a. Present Value Annuity (PVA) or

b. Future Value Annuity (FVA)

PVAn = PMT / (1+i)nPVAn = PMT (PVIFA i,

n )

FVAn = PMT (1 +i)n FVAn = PMT (FVIFA i,n )

Example:

Find the PV of $500 received at the end of each year of the next3 years discounted back to the present at 10%?

PVA3 = 500/1.101 + 500/1.102 + 500/1.103

= 454.55 + 413.22 +375.66= $1,243.43

OR

PVA3 = 500 (PVIFA 10%, 3 )

= 500 (2.487)= $1,243.50

Example:We are going to deposit $15,000 at the end of each year for the next 5 years in a bank where it will earn 9% interest. How muchwill we get at the end of 5 years?

FVA5 = 15000 (1.09)4 + 15000 (1.09)3 + 15000 (1.09)2 +

15000 (1.09)1 + 15000= $89,770.66

OR

FVA5 = 15000(FVIFA 9%,5 )

= 15000 (5.9847)= $89,770.50

Exercise:

1. What is the accumulated sum of each of the following streams of payments $500 a year for 15 years compounded annually at 5%

$850 a year for 10 years compounded annually at 7%

2. What is the PV of the following annuities $2500 ay ear for 15 years discounted back to the present at 8%

$280 a year for 5 years discounted back to the present at 9%

ANNUITY DUE

Annuity due is an annuity in which the payments occur at thebeginning of each period

a. Present Value of Annuity Due(PVAD)

b. Future Value of Annuity Due(FVAD)

PVADn = PMT (PVIFA i, n )(1 + i)

FVADn = PMT (FVIFA i,n )(1 + i)

Example:Find the PV of $500 at the beginning of each year of the next 5 years discounted back to the present at 6%?PVAD = 500 (PVIFA 6%,5 ) (1 + 0.06)

= 500 (4.212) (1.06)= $2,232.36

We are going to deposit $1,000 at the beginning of each year for the next 5 years in a bank where it will earn 5% interest. How much will we get at the end of 5 years?FVAD = 1000 (FVIFA 5%,5 ) (1 + 0.05)

= 1000 (5.526) (1.05)= $5,802.30

4. PERPETUITY Perpetuity is an annuity that continues forever The equation representing the PV of annuity

PV = PP / i

Example: What is PV of $1,000 perpetuity discounted back to the present at 8%?

PV = PP / i = 1000 / 0.08 = $12,500

Exercise:

What is the present value of the following: A $100 perpetuity discounted back to the present at 12%

A $95 perpetuity discounted back to the present at 5%

EFFECTIVE ANNUAL RATE (EAR)

Making Interest Rates Comparable We cannot compare rates with different

compounding periods. For example, 5% compounded annually is not the same as 5% percent compounded quarterly.

To make the rates comparable, we compute the annual percentage yield (APY) or effective annual rate (EAR).

EFFECTIVE ANNUAL RATE (EAR)

Quoted rate could be very different from the effective rate if compounding is not done annually.

Example: $1 invested at 1% per month will grow to $1.126825 (= $1.00(1.01)12) in one year. Thus even though the interest rate may be quoted as 12% compounded monthly, the effective annual rate (EAR) or APY is 12.68%.

EFFECTIVE ANNUAL RATE (EAR)

APY = (1 + quoted rate/m)m – 1

Where m = number of compounding periods

= (1 + .12/12)12 – 1 = (1.01)12 – 1= .126825 or 12.6825%