Time Value of Money

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What is Time Value of Money?
TIME allows one the opportunity to postpone consumption and
earn INTEREST.
This core principle of Time Value of Money is that, A dollar (taka)
on hand today is worth more than a dollar to be received in the
future because the dollar on hand today can be invested to earn
interest to yield more than a dollar in the future.
The time value of money mathematics quantify the value of a dollar
through time.

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Example: You have own a cash prize in lottery! You are given two
options to receive money.
a) Receive $100 now OR
b) Receive $100 after 1 year
What option would you choose?
For example, assuming a 5% interest rate on your savings account,
$100 invested or deposited today will be worth $105 in one year
($100+100X0.05X1)).
Conversely, $100 received one year from now is only worth $95.24
today ($100 divided by 1.05), assuming a 5%annual interest rate.
Future Value: describes the process of finding what an investment today
will grow to in the future.
Present Value: describes the process of determining what a cash flow to
be received in the future is worth in today’s dollars.

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What is Interest?
Interest is a rate which is charged or paid for the use of money.
The rate of interest usually denoted by ‘i’
Thus, the rate of interest represents the time value of the money
invested.
Simple Interest
Simple Interest is calculated on the original principal only.
Simple Interest = p X i X n
Where, p = Principal amount (original amount borrowed or loaned)
i = Interest rate for one period
n = number of periods
Compound Interest
When interest is paid on not only the principal amount invested but
also on any previous interest earned, this is called Compound
Interest.
Compounding means earning interest on interest

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Example: Simple Interest : Suppose, you lend someone $100, he
agrees to give 5% interest on this $100 every year. Then every year
you will receive how much as simple interest?
Solution: Simple Interest = p X i X n = $100 X0.05 X 1 = $5 as simple
interest amount
Now, if you continue to receive 5% interest on the original $100
amount you lent, How much you will receive over 5 years?
Year 1: 5% of $100 = $5+$100 =$105
Year 2: 5% of $100 = $5+$105 =$110
Year 3: 5% of $100 = $5+$110 =$115
Year 4: 5% of $100 = $5+$115 =$120
Year 5: 5% of $100 = $5+$120 =$125
Alternative Calculation Method
Simple Interest = p X i X n = $100 X0.05 X 5 = $25
After 5 years you will receive = ($100+$25) = $125

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QS 1: You borrow $10,000 for 3 years at 5% simple annual
interest. Calculate the amount of simple interest.
Solution: Simple Interest = p X i X n
= $10,000 X0.05 X 3 =$1500 (for 3 years)
QS 2: You borrow $10,000 for 60 days at 5% simple annual
interest (assume 365 days a year). Calculate the amount of
simple interest.
Solution: Simple Interest = p X i X n= $10,000 X0.05 X (60/365)
=$82.2 (for 60 days)
QS 3: Rahim has borrowed $1000 from Karim for 2 years at
8% simple annual interest. How much Rahim will pay karim in
2 years?.
Solution: Simple Interest = p X i X n= $1000 X0.09 X 2 =$160
Total amount will paid back to Karim in 2 years =(Original

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Example: Compound Interest : You have lent someone $100
for 5 years at the compound interest rate of 5% per year. How
much you will receive at the end of 5 years?
Solution:
Year 1: 5% of $100 = $5+$100 =$105
Year 2: 5% of $105 = $5.25 +$105 =$110.25
Year 3: 5% of $110.25 = $5.51+$110.25 =$115.76
Year 4: 5% of $115.76 = $5.79+$115.76 =$121.55
Year 5: 5% of $121.55 = $6.08+$121.55 =$127.63
After 5 years you will receive = $127.63
Alternative Calculation Method
Year 1: 5% of $100 = $5
Year 2: 5% of $105 = $5.25
Year 3: 5% of $110.25 = $5.51
Year 4: 5% of $115.76 = $5.79
Year 5: 5% of $121.55 = $6.08
Compound Interest = $5+ $5.25+ $5.51+ $5.79+ $6.08= $27.63
After 5 years you will receive = ($100+$27.63) = $127.63
Note that, in comparing the growth of simple and compound interest,
investments with simple interest grow in linear fashion and compound
interest results in geometric growth.

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Time value of money calculations always use compound interest
You must adjust the interest rate and the number of periods to
be consistent with compounding periods.
A 6% interest rate compounded annually for 5 years should be
entered as (i) =6% for number of periods (n) =5
A 6% interest rate compounded semi-annually for 5 years should
be entered as (i)= 3%= (6 / 2) for number of periods (n) =5 X 2 =10
A 6% interest rate compounded quarterly for 5 years should be
entered as (i)= 1.5%= (6 / 4) for number of periods (n) =5 X 4 = 20

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FUTURE VALUE
How much what you got now grows to in future when compounded
at a given rate?
Future Value of a Single Amount Future Value is the amount of
money than an investment made today (the present value) will
grow to by some future date. The difference between PV and FV
depends on the number of compounding periods involved and the
interest rate.
.
The relationship between the future value and present value can be
expressed as:
Formula: FV = PV(1+i)
n
Alternative, FV = PV (FVIFi,n) [Here, (1+i)
n
and (FVIFi,n) means
same]
Where, FV = Future Value, PV = Present Value, i= Interest rate
per period, n =number of compounding periods; FVIFi,n = Future
Value Interest Factor for interest rate ‘i’ & number of periods n.
(this value given at the table)

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NOTE: the exponential n
indicates earning interest on interest,
called compounding. Compounding could be daily, monthly,
quarterly, semi-annually or yearly. Unless otherwise specified
compounding is done usually on annual basis.
FVIFi,n = Future Value Interest Factor is the calculation of this
equation (1+i)n
The value of FVIF is found in appropriate Table which shows
interest factor of $1 at i% for ‘n’ periods.
Important Notes Regarding FVIF Tables
The FVIFi,n assume beginning of period cash flows
As i ↑, the FVIFi,n ↑ also
As n ↑, the FVIFi,n ↑ also

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Future Value
QS: What is the future value of $34 in 5 years if the interest rate is
5%?
Solution: FV = PV (FVIFi,n)
= 34 (FVIF5%, 5)
=34 X 1.27628
=$43.39
FV = PV(1+i)n

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QS: A financial institution offers to pay 6% interest
compounded semi-annually. How much will your $10,000
grow to in 5 years at this rate?
SOLUTION: Here, PV = 10,000 , i = 0.06/2 =0.03
N = 5 X 2 =10
FV = 10,000(1 +0.03)
10
=10,000 X1.3439163 =$13,439.16
QS: Suppose you deposit $1,000 in an account that pays
12% interest, compounded quarterly. How much will be in
your account after 8 years if there are no withdrawals?
SOLUTION: Here, PV = 1,000 , i = 0.12/4 =0.03
N = 8 X 4 =32
FV = 1,000(1 +0.03)32
=$2575.10

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PRESENT VALUE
Present value concerns the current dollar value (today’s value) of a
future amount of money.
Present Value of a Single Amount: Present value is an amount today
that is equivalent to a future payment that has been discounted by an
appropriate interest rate (i).
Discounting is the process of translating a future value or a set of
future cash flows into a present value.
The amount of future value received in ‘n’ years from now is
discounted to its present value.

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The relationship between the Present Value and
Future Value can be expressed as:
Here, PVIFi,n and means same thing.
Alternative Tabular Formula, PV = FV (PVIFi,n)
Where, FV = Future Value, PV = Present Value
i= Interest rate per period, n =number of discounting periods
PVIFi,n = Present Value Interest Factor for interest rate ‘i’ &
number of periods n. (this value given at the table)
It should be noted that the present value interest factor
is the inverse of the future value interest
factor(1+i)
n .
For this reason discounting is considered as the reverse of the
compounding.

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Important Notes Regarding PVIF Tables
The PVIFi,n assume end of period cash flows
As i ↑, the PVIFi,n also↓
As n ↑, the PVIFi,n also ↓
QS1: You want to buy a house of 5 years from now for
$1,50,000. Assuming a 6% annual interest rate, how much
should you invest today to yield $1,50,000 in 5 years?
Solution: FV =1,50,000, i= 0.06, n =5
=$1,12,088.73

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QS-2: What is most you would pay now for an opportunity to
receive $10,000 at the end of 10 years, if you can earn 9% on
similar investments?
Solution: FV =10,000, i= 0.09, n =10
PV = FV (PVIFi,n)
= 10,000 (PVIF 9%,10)
=10,000 X 0.422 =$4,220
Thus, given a rate of interest of 9%, you would not pay more
than $4,220 now for the opportunity to receive $10,000 in 10
years.

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QS3: A financial institution that offers an semi-
annual interest rate of 6% . How much money should
you deposit today to yield $1,50,000 in 5 years?
Solution: FV =1,50,000, i= 0.06/2 =0.03, n =5X2=10
=$1,11,614.09

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ANNUITY
An annuity is a series of equal payments or receipts
that occur at evenly spaced intervals for a specified
period. The annuity values are assumed to occur at
the end of each period. Examples of annuities
include:
Lease and rental payments
Student loan payments
Car loan Payments
Insurance premiums
Mortgage payments

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FUTURE VALUE OF AN ANNUITY
The Future Value of an Annuity is the value that a series of
expected or promised future payments (or cash flow) will
grow to after a given number of periods at a specific
compounded interest.
FV of Ordinary Annuity: If payments are made at the end of
each period. (e.g. at the end of each year)
FV of Annuity Due: If payments are made at the beginning
of each period. (e.g. at the beginning of each year)

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The future value of an annuity could be found by calculating
the future value of each individual payment in the series
using the future value formula and then summing the results.
FV of Ordinary Annuity :Formula
FVA= PMT (1+I)
n-1
+PMT(1+i)
n-2
+---------+PMT(1+i)
n-n
Alternative Formula: FVA =
Alternative Tabular Formula: FVA = PMT (FVIFAi,n)
Where, FVA = Future Value of an Annuity over n periods.
PMT = Amount of each payment
i= interest rate per period
n= number of periods or number of payments
FVIFAi,n= The future value interest factor of an annuity
FVIFA factors are the calculations of Σ(1+i)n-t

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FV of Annuity Due :Formula
FVA= [PMT (1+I)
n-1
+PMT(1+i)
n-2
+---------+PMT(1+i)
n-n
] X (1+i)
Alternative Formula: FVA =
Alternative Tabular Formula: FVA = PMT (FVIFAi,n)(1+i)

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QS1: If we deposit $1000 at the end of each year for 4 years in
a savings deposit account and our fund grows at 10%. What
is the future value of this annuity?
SOLUTION: FV of Ordinary Annuity
FVA= PMT (1+I)
n-1
+PMT(1+i)
n-2
+---------+PMT(1+i)
n-n
Here, n = 4, PMT =$1000, i =10%
FVA= 1000(1+0.10)4-1
+1000(1+0.10)4-2
+1000(1+0.10)4-3
+1000(1+0.10)4-4
=1331+1210+1100+1000 = $4641
ALTERNATIVE METHOD: ALTERNATIVE METHOD:
FVA = PMT (FVIFAi,n)
FVA = 1000 (FVIFA10%,4 )
FVA = 1000 x 4.64
= $4641 =1000X4.64
=$4641

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QS2: If one saves $100 at the beginning of every year for 3 years
in an account earning 7% interest, compounded annually, how
much will one have at the end of the third year?
SOLUTION: FV of Annuity Due
FVA=[ PMT (1+I)n-1
+PMT(1+i)n-2
+---------+PMT(1+i)n-n
] X (1+i)
Here, n = 3, PMT =$100, i =7%
FVA= [100(1+0.07)
3-1
+100(1+0.07)
3-2
+100(1+0.07)
3-3
]X(1+0.07)
=(114.49+107+100)X1.07 = $343.99
ALTERNATIVE METHOD: ALTERNATIVE METHOD:
FVA = PMT (FVIFAi,n)(1+i)
FVA = 100 (FVIFA7%,3 )(1+0.07)
FVA = 100 x 3.2149 X 1.07
= $343.99 =1000X3.2149X1.07
=$343.99

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PRESENT VALUE OF AN ANNUITY
The Present Value of an Annuity is the value of a series of equal amount
of expected or promised future payments (or cash flow) that have been
discounted to a single equivalent value today.
The present value of any asset (e.g. stock or bond or real asset) is based
on the present value of the future cash flows that asset is expected to
generate in the future.
Investors and security analysts examines the relationship between the
present value of an asset to its current market price or cost of investment
to make investment decision.
PV calculation is extremely useful for comparing two or multiple separate
cash flows e.g. received from investments, dividend, interest payment
etc.
PV of Ordinary Annuity: If payments are made at the end of each period.
(e.g. at the end of each year)
PV of Annuity Due: If payments are made at the beginning of each period.
(e.g. at the beginning of each year)

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PV of Ordinary Annuity: Formula
Alternative Tabular Formula: PVA = PMT (PVIFA i,n)
Here, PVA = Present Value of an Annuity
PMT = Amount of each payment
i = discount rate per period
n = number of periods
PVIFA i,n= Present Value Interest Factor of an Annuity
PVIFA are the calculations of ∑[1/(1+i)n]
Alternative Formula:

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PV of Ordinary Annuity Due: Formula
Alternative Tabular Formula: PVA = PMT (PVIFA i,n)(1+i)
Here, PVA = Present Value of an Annuity
PMT = Amount of each payment
i = discount rate per period
n = number of periods
PVIFA i,n= Present Value Interest Factor of an Annuity
PVIFA are the calculations of ∑[1/(1+i)n]
Alternative Formula:

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QS1: If Samual agrees to repay a loan by paying $ 1000 a year
at the end of every year for three years and the discount rate
is 7%, how much could Samual borrow today?
Solution: PV of Ordinary Annuity
Here, PMT = 1000, i =0.07, n = 3
= $2624.32
=
ALTERNATIVE METHOD: ALTERNATIVE METHOD:
PVA = PMT (PVIFA i,n)
PVA = 1000 (FVIFA7%,3 )
PVA = 1000 x 2.6243
= $2624.32
=1000X2.6243 =$2624.32

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QS2: If Robin agrees to repay a loan by paying $ 1000 a year
at the beginning of every year for three years and the
discount rate is 7%, how much could Robin borrow today?
Solution: PV of Annuity Due
Here, PMT = 1000, i =0.07, n = 3
= 2624.32 X 1.07 = $2808.02
=
ALTERNATIVE METHOD: ALTERNATIVE METHOD:
PVA = PMT (PVIFA i,n)(1+i)
PVA = 1000 (FVIFA7%,3 )(1+0.07)
PVA = 1000 x 2.6243x1.07
= $2808.02
= 2624.32X1.07=$2808.32

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Present Value of a Perpetuity
Perpetuities are stream of equal payment made regularly like
every month or every year and expected to continue forever.
Perpetuities are not very common in financial decision
making. There are few examples.
For example, in the case of irredeemable preference share
(i.e. preference share without a maturity), the company is
expected pay preference dividend perpetually.
By definition, in a perpetuity, the time period n is so large that
(mathematically n approached infinity) the expression (1+i)n
or this discounting factor tend to become zero and the
formula for a perpetuity becomes.
PV (of a perpetuity) = Payment / Interest Rate

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QS1: How large a deposit you should make today, If you want
to start a Merit Scholarship of $1000 per year forever in a
University, given that the University will always earn 3% on
such deposits?
Solution: PV (of a perpetuity) = Payment / Interest Rate
PV = $1000 /0.03
=$33,333
QS2: If you wanted to award a Finance Scholarship of $1000
per year forever, given that the University will always earn 5%
on such deposits. How large a deposit would you have to
make today?

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FUTURE VALUE OF UNVEVEN CASH FLOW STREAM
The future value of an uneven cash flow stream is found by
compounding each payment to the end of the stream and
then summing the future values.
The FV is found by this equation:
FV= PMT (1+I)
n-1
+PMT(1+i)
n-2
+---------+PMT(1+i)
n-n
QS1: For the next 4 years, you will get $320 at the end of 1st
year, $400 at the end of 2nd
year, $650 at the end of 3rd
year and
$300 at the end of 4th
year. If you invest the money and get 6.5%
interest from it, how much will you have at the end of 4 years?
Solution: FV= PMT (1+I)
n-1
+PMT(1+i)
n-2
+---------+PMT(1+i)n-n
FV = 320(1+0.065)4-1
+ 400(1+0.065)4-2
+ 650(1+0.065)4-3
+ 300(1+0.065)4-4
= 386.54 + 453.69 +692.25 + 300
= $1832.48

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QS 2: For the next 5 years, you will get $100 at the
end of 1st
year, $400 at the end of 2nd
year, $400 at
the end of 3rd
year, $400 at the end of 4th year and
$300 at the end of 5th
year. If you invest the money
and get 8% interest from it, how much will you have
at the end of 5 years?

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PRESENT VALUE OF UNVEVEN CASH FLOW STREAM
The present value of an uneven series of payment or receipts
(i.e. future income) is the sum of the present value of the
individual payment of the cash flow stream.
The PV is found by this equation:
QS1: Suppose an investment promises a cash flow of $500 at
the end of 1st
year, $600 at the end of 2nd
year and $10,700 at
the end of 3rd
year. If the discount rate is 5%, what is the
value of this investment today?
Solution:
= 476.19 + 544.22 + 9243.26 =$10,263.67

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QS2: Suppose an investment promises a cash flow of $100 at
the end of 1st
year, $400 each at the end of 2nd
, 3rd
and 4th
year
and $300 at the end of 5th
year. If the discount rate is 8%, what
is the value of this investment today?

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Nominal Interest & Effective Interest Rate (EAR)
Nominal interest rate is also defined as a stated or quoted, annual interest
rate. This interest works according to the simple interest and does not
take into account the compounding periods. Note that when we talk about
a nominal (stated) interest rate we mean the annual rate (e.g., 10% annual
rate of return on an investment). Usually denoted by ‘i’.
Effective annual interest rate (EAR) Effective interest rate is the actual
interest rate when interest is compounded more than once a year. In this
case, interest is compounded on both the principal (initial investment)
and the interest that has already accrued. As the result, effective interest
rate differs and will be higher from the nominal (stated) interest rate when
compounding occurs more than once a year, and it depends on the
frequency of compounding. When we talk about the effective annual
interest rate, we mean the actual rate resulting from interest
compounding (e.g., 10.25% annual rate of return on the same investment).
It is used to compare the annual interest between loans with different
compounding periods like week, month, year etc. EAR depicts the true
picture of financial payments. Here,
i= Nominal or Stated annual rate
n= number of compounding periods

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Nominal Interest & Effective Interest Rate (EAR)
Example: Consider a stated annual rate of 10% offered on a $1000 on
deposit in a Bank.
Hence, nominal or simple interest= $1000X.10X1= $100
Hence, Compounded yearly, nominal rate will turn $1000 into $1100.
Example: Consider a stated annual rate of 10% offered on a $1000 on
deposit in a Bank compounded monthly.
=10.47%
Effective Interest= $1000X0.1047X1=$104.70
Hence, if compounding occurs monthly, $1000 would grow to $1104.70 by
the end of the year, rendering an effective annual interest rate of
10.47%. Basically the effective annual rate is the annual rate of interest
that accounts for the effect of compounding.

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