Chapter 5
Time Value of Money
Principles Used in this Chapter
Principle 1: Money Has a Time Value.
The concept of time value of money – a dollar received today,
other things being the same, is worth more than a dollar
received a year from now, underlies many financial decisions
faced in business.
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Using Timelines to Visualize Cashflows
A timeline identifies the timing and amount of a stream of
cash flows along with the interest rate.
A timeline is typically expressed in years, but it can also be
expressed in months, days, or any other unit of time.
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Time Line Example
i=10%
Years 0 1 2 3 4
Cash flow -$100 $30 $20 -$10 $50
The 4-year timeline illustrates the following:
The interest rate is 10%.
A cash outflow of $100 occurs at the beginning of the first year
(at time 0), followed by cash inflows of $30 and $20 in years 1
and 2, a cash outflow of $10 in year 3 and cash inflow of $50 in
year 4.
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Checkpoint 5.1
Creating a Timeline
Suppose you lend a friend $10,000 today to help him
finance a new Jimmy John’s Sub Shop franchise and in
return he promises to give you $12,155 at the end of the
fourth year. How can one represent this as a timeline?
Note that the interest rate is 5%.
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Checkpoint 5.1: Check yourself
Draw a timeline for an investment of $40,000 today that
returns nothing in one year, $20,000 at the end of year 2,
nothing in year 3, and $40,000 at the end of year 4.
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Timeline
i=interest rate; not given
Time Period 0 1 2 3 4
Cash flow -$40,000 $0 $20,000 $0 $40,000
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Simple Interest and Compound Interest
What is the difference between simple interest and
compound interest?
Simple interest: Interest is earned only on the principal amount.
Compound interest: Interest is earned on both the principal and
accumulated interest of prior periods.
Example 5.1: Suppose that you deposit $500 in your savings
account that earns 5% annual interest. How much will you
have in your account after two years using (a) simple interest
and (b) compound interest?
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Example 5.1
Simple Interest
Interest earned = 5% of $500 = .05×500 = $25 per year
Total interest earned = $25×2 = $50
Balance in your savings account:
= Principal + accumulated interest
= $500 + $50 = $550
Compound interest (assuming compounding once a year)
Interest earned in Year 1 = 5% of $500 = $25
Interest earned in Year 2 = 5% of ($500 + accumulated interest)
= 5% of ($500 + 25) = .05×525 = $26.25
Balance in your savings account:
= Principal + interest earned = $500 + $25 + $26.25 = $551.25
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Simple Interest
Simple interest is charged only on the principal amount. The
following formula can be used to calculate simple interest:
Simple Interest (Is) = P × i × t
Where,
P is the principle amount;
i is the interest rate per period;
t is the time for which the money is borrowed or lent.
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Compound Interest
Compound interest is charged on the principal plus any interest
accrued till the point of time at which interest is being calculated. In
other words, compound interest system works as follows:
Interest for the first period charged on principle amount.
For the second period, its charged on the sum of principle amount
and interest charged during the first period.
For the third period, it is charged on the sum of principle amount
and interest charged during first and second period, and so on ...
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Compound Interest
It can be proved mathematically, that the interest calculated
as per above procedure is given by the following formula:
Compound Interest (Ic) = P × (1 + i) n – P
Where,
P is the principle amount;
i is the compound interest rate per period;
n are the number of periods.
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Present Value and Future Value
Time value of money calculations involve Present value (what a cash
flow would be worth to you today) and Future value (what a cash flow
will be worth in the future).
In example 5.1, Present value is $500 and Future value is $551.25 (if the
yearly compounding rate is 5%).
The linkage between present value and future value is:
Future Value = Present Value x (1+Interest Rate per period)Number of periods
For annual compounding (compounding once a year),
Future Value = Present Value x (1+Annual Interest Rate)Number of years
If nothing is said, assume annual compounding.
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Example:
Example 5.1: The future value of $500 in 2 years with annual
compounding interest rate of 5% can be computed directly
from the formula:
FV2=PV(1+i)2 = 500(1+0.05)2=500(1.05)2=551.25.
Continue example 5.1 where you deposit $500 in savings
account earning 5% annual interest. Show the amount of
interest earned for the first five years and the value of your
savings at the end of five years.
You can do the calculation year by year
or use the formula for future value: FVn= PV(1+i)n
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Year by year compounding
YEAR PV or Interest Earned FV or
Beginning (5%) Ending Value
Value
1 $500.00 $500*.05 = $25 $525
2 $525.00 $525*.05 = $551.25
$26.25
3 $551.25 $551.25*.05 $578.81
=$27.56
4 $578.81 $578.81*.05=$28 $607.75
.94
5 $607.75 $607.75*.05=$30 $638.14
.39
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Use the Future Value Equation
We will obtain the same answer using the future value
equation: FV = PV(1+i)n
= 500(1.05)5 = $638.14
So the balance in savings account at the end of 5 years
will equal $638.14. The total interest earned on the
original principal amount of $500 will equal $138.14 (i.e.
$638.14- $500.00).
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Annuity
Let’s say you want to save money to go on a vacation, or you want to
save money now for your baby’s college education.
A strategy for saving a little bit of money in the present and having a
big payoff in the future is called an annuity.
An annuity is an account in which equal regular payments are made.
There are two basic questions with annuities:
Determine how much money will accumulate over time given that equal
payments are made.
Determine what periodic payments will be necessary to obtain a specific
amount in a given time period.
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Ordinary Annuity and Annuity Due
There are two types of annuity formulas.
One formula is based on the payments being made at the
end of the payment period. This called ordinary annuity.
The annuity due is when payments are made at the
beginning of the payment period.
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Future Value of Ordinary and Due
Annuity
ORDINARY ANNUITY
F pmt
1 nr 1
nt
r
n
ANNUITY DUE
F pmt
1 r nt
n 1
1 nr
r
n
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The Present and Future Values of
an Ordinary Annuity
It is important that ‘n’ and ‘I’ match. If periods are
expressed in terms of number of monthly payments, the
interest rates must be expressed in terms of interest rate
per month
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The Present and Future Values of
Due Annuity
Annuity due is an annuity in which all the cash flows occur at
the beginning of the period. For example, rent payments on
apartments are typically annuity due as rent is paid at the
beginning of the month.
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Sinking Funds
A sinking fund is when we know the future value of the
annuity and we wish to compute the monthly payment
For an ordinary unity this formula is
r
pmt F n
1
r nt
n 1
For an annuity due the formula is
r
1 1
pmt F n
r
n
r nt
n 1
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Perpetuity
A stream of equal payments expected to continue forever.
PV (Perpetuity) = Payment = PMT
Interest i
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