Ff topic 3_time_value_of_money

Topic 3

Time Value of Money

Learning Objectives

Define the time value of money.

Explain the significance of time value of money in financial

management.

Define the meaning of compounding and discounting.

Calculate the future value and present value.

Calculate future and present value, ordinary annuity or

annuity due.

Define the meaning of perpetuity and how to calculate it.

Generally, receiving $1 today is worth more than $1 in the future. This is due to opportunity costs.

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

Today Future

If we can measure this opportunity cost, we can:

Translate $1 today into its equivalent in the future (compounding).

Translate $1 in the future into its equivalent today (discounting).

?Today Future

Today

?Future

Significance of the time value of money

Time value of money is important in understanding financial management.

It should be considered for making financial decisions.

It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities.

Simple Interest

Interest is earned only on principal.

Example: Compute simple interest on $100 invested at 6% per year for three years. 1st year interest is $6.00

2nd year interest is $6.00

3rd year interest is $6.00

Total interest earned: $18.00

Compound Interest

Compounding is when interest paid on an investment during the first period is added to the principal; then, during the second period, interest is earned on the new sum (that includes the principal and interest earned so far).

Is the amount a sum will grow to in a certain number of years when compounded at a specific rate.

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

Compounded amount = Future Value (beginning amount plus interest earned. )

Compound Interest

Example: Compute compound interest on $100 invested at 6% for three years with annual compounding. 1st year interest is $6.00 Principal now is $106.00

2nd year interest is $6.36 Principal now is $112.36

3rd year interest is $6.74 Principal now is $119.11

Total interest earned: $19.10

Future Value Future Value is the amount a sum will grow to in a certain number of years

when compounded at a specific rate.

Two ways to calculate Future Value (FV): by using Manual Formula or Using Table.

Manual Formula Table

FVn = PV (1 + r)n FVn = PV (FVIFi,n)n

Where :

FVn = the future of the investment at the end of “n” years

r = the annual interest (or discount) rate

n = number of years

PV= the present value, or original amount invested at the beginning of the first year

FVIF=Futurevalueinterestfactororthecompoundsum$1

Future Value - single sums

If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?

Mathematical Solution:FV = PV (FVIF i, n )

FV = 100 (FVIF .06, 1 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)1 = $106

0 1

PV = -100 FV = ???

Future Value - single sums

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

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .06, 5 ) (use FVIF table, or)FV = PV (1 + i)n

FV = 100 (1.06)5 = $133.82

0 5

PV = -100 FV = ???

Compound Interest With Non-annual Periods

Non-annual periods : not annual compounding but occur semiannually, quarterly, monthly or daily…

If semiannually compounding : FV = PV (1 + i/2)n x 2 or FVn= PV (FVIFi/2,nx2)

If quarterly compounding : FV = PV (1 + i/4)n x 4 or FVn= PV (FVIFi/4,nx4)

If monthly compounding : FV = PV (1 + i/12)n x 12 or FVn= PV (FVIFi/12,nx12)

If daily compounding : FV = PV (1 + i/365)n x 365 or FVn= PV (FVIFi/365,nx365)

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .015, 20 ) (can’t use FVIF table)FV = PV (1 + i/m) m x n

FV = 100 (1.015)20 = $134.68

0 20

PV = -100 FV = 134.68

Future Value - single sumsIf you deposit $100 in an account earning 6% with

quarterly compounding, how much would you have in the account after 5 years?

Example:If you invest RM10,000 in a bank where it will earn 6% interest compounded

annually. How much will it be worth at the end of a) 1 year and b) 5 years

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

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

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .005, 60 ) (can’t use FVIF table)FV = PV (1 + i/m) m x n

FV = 100 (1.005)60 = $134.89

0 60

PV = -100 FV = 134.89

Future Value - single sumsIf you deposit $100 in an account earning 6% with

monthly compounding, how much would you have in the account after 5 years?

Present Value Present value reflects the current value of a future payment or

receipt. How much do I have to invest today to have some amount in the

future? Finding Present Values(PVs)= discountingManual Formula Table PVn = FV/ (1 + r)n PVn = FV (PVIFi,n)n

Where :FVn = the future of the investment at the end of “n” yearsr = the annual interest (or discount) rate n = number of yearsPV= the present value, or original amount invested at the beginning of

the first yearPVIF=Present Value Interest Factor or the discount sum$1

Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .06, 1 ) (use PVIF table, or)PV = FV / (1 + i)n

PV = 100 / (1.06)1 = $94.34

Present Value - single sumsIf you receive $100 one year from now, what is the PV

of that $100 if your opportunity cost is 6%?


PV = 100 / (1.06)5 = $74.73

Present Value - single sumsIf you receive $100 five years from now, what is the

PV of that $100 if your opportunity cost is 6%?


PV = 1000 / (1.07)15 = $362.45

Present Value - single sumsWhat is the PV of $1,000 to be received 15 years from

now if your opportunity cost is 7%?

Finding i1. At 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%2. If you sold land for $11,439 that you bought 5 years ago for $5,000,

what is your annual rate of return? FV = PV (FVIF i, n )11,439 = 5,000 (FVIF ?, 5 ) 11,439/ 5,000= (FVIF ?, 5 ) 2.3866 = (FVIF ?, 5 )i = .18

Finding n1. How many years will the following investment takes? $100 togrow to $672.75 if invested at 10% compounded annuallyFVn = 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 years2. Suppose you placed $100 in an account that pays 9% interest,

compounded annually. How long will it take for your account to grow to $514?

FV = PV (1 + i)n

514 = 100 (1+ .09)N

514/100 = (FVIF 9%,n )5.14 = (FVIF 9%,n ) refer to FVIF table

n = 19 years

Hint for single sum problems: In every single sum present value and future

value problem, there are four variables:FV, PV, i and n.

When doing problems, you will be given three variables and you will solve for the fourth variable.

Keeping this in mind makes solving time value problems much easier!

Compounding and DiscountingCash Flow Streams

Two types of annuity: ordinary annuity and annuity due.ordinary annuity: a sequence of equal cash flows, occurring

at the end of each period. Annuity due: annuity payment occurs at the beginning of

the period rather than at the end of the period.

0 1 2 3 4

Annuities

Mathematical Solution:FVA = PMT (FVIFA i, n )FVA = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or)

FVA = PMT (1 + i)n - 1 i

FVA = 1,000 (1.08)3 - 1 = $3246.40 .08

Future Value - annuityIf you invest $1,000 each year at 8%, how much

would you have after 3 years?

Mathematical Solution:PVA = PMT (PVIFA i, n )PVA = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)

1PVA = PMT 1 - (1 + i)n

i

1PV A= 1000 1 - (1.08 )3 = $2,577.10

.08

Present Value - annuityWhat is the PV of $1,000 at the end of each of the

next 3 years, if the opportunity cost is 8%?

Perpetuities

Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.

You can think of a perpetuity as an annuity that goes on forever.

So, the PV of a perpetuity is very simple to find:

Present Value of a Perpetuity

What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

PMT $10,000 i .08

= $125,000

PV = =

Ordinary Annuity vs.

Annuity Due

$1000 $1000 $1000

4 5 6 7 8

Begin Mode vs. End Mode

year year year 5 6 7

Begin Mode vs. End Mode

year year year 6 7 8

Earlier, we examined this “ordinary” annuity:

Using an interest rate of 8%, we find that:

The Future Value (at 3) is $3,246.40.The Present Value (at 0) is $2,577.10.

1000 1000 1000

What about this annuity?

Same 3-year time line,Same 3 $1000 cash flows, butThe cash flows occur at the beginning

of each year, rather than at the end of each year.

This is an “annuity due.”

1000 1000 1000

Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the

end of year 3?

Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

FVA due = PMT (FVIFA i, n ) (1 + i) FVA due = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or)

FVA due = PMT (1 + i)n - 1 i

FVA due = 1,000 (1.08)3 - 1 = $3,506.11 .08

Present Value - annuity dueMathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

PVA due = PMT (PVIFA i, n ) (1 + i) PVA due = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or)

1PVA due = PMT 1 - (1 + i)n

i

1PVA due = 1000 1 - (1.08 )3 = $2,783.26

.08

(1 + i)

(1.08)

Annual Percentage Yield (APY)

Which is the better loan: 8% compounded annually, or 7.85% compounded quarterly? We can’t compare these nominal (quoted) interest rates,

because they don’t include the same number of compounding periods per year!

We need to calculate the APY.

Note: APY can be called as the Effective Annual rate (EAR)

Annual Percentage Yield (APY)

Find the APY for the quarterly loan:

The quarterly loan is more expensive than the 8% loan with annual compounding!

APY = ( 1 + ) m - 1quoted ratem

APY = ( 1 + ) 4 - 1

APY = .0808, or 8.08%

.07854

Practice Problems

1. To what amount will the following investments accumulate?

a. $4,000 invested for 11 years at 9% compounded annually

b. $8,000 invested for 10 years at 8% compounded annually

2. How many years will the following take?

a. $550 to grow to $1,043.90 if invested at 6% compounded

annually

b. $40 to grow to $88.44 if invested at 12% compounded annually

3. At what annual rate would the following have to be invested?

a. $550 to grow to $1,898.60 in 13 years

b. $275 to grow to $406.18 in 8 years

4. What is the present value of the following annuities?

a. $3,000 a year for 10 years discounted back to the present at 8%

b. $50 a year for 3 years discounted back to the present at 3%

a. PV = $3,000 (PVIFA r,t) PV = $3,000 (PVIFA 8%,10)

PV = $3,000 (6.7101)PV = $20,130

b. PV = PMT (PVIFA r,t)PV = $50 (PVIFA 3%,3)PV = $50 (2.8286)PV = $141.43

To pay for your child’s education, you wish to have

accumulated $25,000 at the end of 15 years. To do this, you

plan on depositing an equal amount in the bank at the end

of each year. If the bank is willing to pay 7% compounded

annually, how much must you deposit each year to obtain

your goal?

FVA = PMT (FVIFA i, n )$25,000 = PMT (PVIFA .07, 15 )

$25,000 = PMT (25.129) Thus, PMT = $994.87

Ff topic 3_time_value_of_money

Economy & Finance

Transcript of Ff topic 3_time_value_of_money