Ff 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 yearinterest 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 = ???

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

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
If 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 year
FV1 = 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 years
FV5 = 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

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
If 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)= discounting
Manual Formula Table
PVn = FV/ (1 + r)n PVn = FV (PVIFi,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
PVIF=Present Value Interest Factor or the discount sum$1

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
PV = ??? FV = 100
0 1
Present Value - single sums
If you receive $100 one year from now, what is the PV
of that $100 if your opportunity cost is 6%?

PV = FV / (1 + i)n
PV = 100 / (1.06)5 = $74.73
If you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
0 5
PV = ??? FV = 100

PV = FV / (1 + i)n
PV = 1000 / (1.07)15 = $362.45
What is the PV of $1,000 to be received 15 years from
now if your opportunity cost is 7%?
0 15
PV = -362.45 FV = 1000

Finding i
1. 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?
11,439 = 5,000 (FVIF ?, 5 )
11,439/ 5,000= (FVIF ?, 5 )
2.3866 = (FVIF ?, 5 )
i = .18

Finding n
1. How many years will the following investment takes? $100 to
grow 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
2. 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 Discounting
Cash Flow Streams
0 1 2 3 4

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

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 - annuity
If you invest $1,000 each year at 8%, how much
would you have after 3 years?

PVA = PMT (PVIFA i, n )
PVA = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)
1
PVA = PMT 1 - (1 + i)n
i
1
PV A= 1000 1 - (1.08 )3 = $2,577.10
.08
Present Value - annuity
What 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.

PMT
i
PV =
 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
1000 1000 1000
4 5 6 7 8
year year year
5 6 7
PV
in
END
Mode
FV
in
END
Mode

Begin Mode vs. End Mode
1000 1000 1000
4 5 6 7 8
year year year
6 7 8
PV
in
BEGIN
Mode
FV
in
BEGIN
Mode

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.
0 1 2 3
1000 1000 1000

What about this annuity?
Same 3-year time line,
Same 3 $1000 cash flows, but
The cash flows occur at the beginning
of each year, rather than at the end
of each year.
This is an “annuity due.”
0 1 2 3
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
(1 + i)
(1.08)

Present Value - annuity due
Mathematical 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)
1
PVA due = PMT 1 - (1 + i)n
i
1
PVA 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 rate
m
APY = ( 1 + ) 4 - 1
APY = .0808, or 8.08%
.0785
4

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 (PVIFAr,t)
PV = $3,000 (PVIFA8%,10)
PV = $3,000 (6.7101)
PV = $20,130
b. PV = PMT (PVIFAr,t)
PV = $50 (PVIFA3%,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

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