Time Value of Money
Chapter 5
Important Concepts to be discussed
1. Opportunity cost and TVOM
2. Simple vs. Compound Interest
3. Lump sum payments
4. Annuities
5. Perpetuities
6. Complex Cash Flow Streams
7. Compounding Frequency
8. EAR and APR
Introduction
• ‘A dollar in hand today is worth more than a
dollar to be received in future’.
Example 1
Today
30 Days from now
Option 1
$100
$100
Option 2
$100
$105
• In case of Option 1 the seller would go for $100 today.
• In case of Option 2 the seller would be indifferent between
receiving $100 today or going for $105, 30 days from now.
• $5 thus is seller’s opportunity cost which can be defined as:
‘the cost of selecting an alternative is the benefit foregone
from the next best alternative’.
1. Opportunity Cost
• The Economist.com defines opportunity cost
as,
– “the true cost of something is what we give up to
get it”.
• Opportunity cost of an investment is the ROR
available on the best alternative investment of
similar risk.
Opportunity Cost (contd.)
• Personal Opportunity Cost
– Personal opportunity cost refers to the time,
health or energy.
– For example, time spent on studying usually
means lost time for leisure or working.
Opportunity Cost (contd.)
• Financial Opportunity Cost
– Financial opportunity cost involve monetary value
of decisions made.
– For example, the purchase of an item from your
savings means you can no longer earn interest on
these funds.
• As we know the true cost of something is what we give
up to get it so by purchasing an item we give up the
chance to earn interest on these savings.
Measuring Opportunity Cost using
Time Value of Money
• TVOM can be used to measure financial
opportunity cost using interest calculations.
– For example spending $1000 from a savings
account that was giving 4% a year means an
opportunity cost of $40 in lost interest.
$1000 x 0.04 = $40
2. Future Value of Lump Sum
Payments
• Future Value refers to the ‘amount of money
an investment will grow into over a period of
time at some given interest rate’.
• Defined in another way, ‘Future value refers to
the cash value of an investment at sometime
in future’.
2. Future Value of Lump Sum
• Example 2
• Investing for a Single Period (e.g. 1 year)
– Suppose we invested $100 in an account that pays
10% interest rate per year. By the end of the year
we’ll have 110 in our account.
2. Future Value of Lump Sum
• Calculation
• 100
x 10% =
• Principal
i
10
interest pmt.
• At the end of the year we’ll have $ 100
+$ 10
$ 110
2. Future Value of Lump Sum
• Formula
• FV = PV + (PV x i)
• 110 = 100 + (100 x 10%)
110 = 100 + 10
$110 = $110
FV = PV + (PV x i)
FV = PV (1 + i)
FV = 100 (1 + 0.10)
FV = 100 x 1.10
FV = $110
2. Future Value of Lump Sum
• Example 3
• Investing for more than period (e.g. 2 years)
– Going back to our $100 investment what we will have
after 2 years?
– During 1st year
FV = PV (1+i) (1+i)
• FV = PV (1+ i)
• FV = 100 (1.10) = $110
– During 2nd year
• FV = PV (1+ i)
• FV = 110 (1.10) = $121
FV = 100 (1.10) (1.10)
FV = $121
or
FV = PV (1+i)n
FV = PV (1+i)2
FV = 100 (1+0.10)n
FV = 100 (1.10)2
FV = $121
This 121 is the future value of $100, two years from now at 10% ROI.
Simple vs. Compound Interest
Example 4
Compounding
• The process of leaving your money and any
accumulated interest in an investment for
more than one period, thereby reinvesting the
interest is called ‘compounding’.
• Compounding the interest means earning
interest on interest so we call the result
‘compound interest’.
Future Value of a Single Amount
We can generalize this as . . .
FV = PV (1 + i)n
Future
Value
Present Value
Number
of Compounding
Periods
Interest
Rate
16
Numericals
Example 5
• Find the following future values:
• a. An initial £ 500 compounded for 1 year at 6 percent.
• b. An initial £ 500 compounded for 2 years at 6 percent.
Example 6
• What’s the future value of $100 after 3 years if it earns 10%, annual
compounding?
Example 7
• For example, you earn $500 from your summer job and want to save for
European trip in the next three years. How much will you have when you
go for the trip if you deposit the money in a savings account that earns
10% interest? Using the time line for this problem, complete the equation:
Revision of Last Session
• Future Value of a Lump Sum
• The future value of a lump sum usually is the easiest time value
concept to understand.
• The term “lump sum” refers to a single-sum payment or receipt at one
point in time.
• The future value of a single sum is the future amount of an initial
deposit when it is compounded for a given number of periods and at a
given interest rate.
• Compounding is the process whereby interest is earned upon interest.
• When a deposit is made, interest is earned on the deposit in the first
period; in subsequent periods, interest is earned not only on the
original deposit but also on the interest earned in each of the
previous compounding periods. Thus, interest is earned on increasing
amounts over time.
Simple Interest
• Interest is the fee paid to use someone else’s money.
• Interest on loans of a year or less is frequently calculated as simple
interest, which is paid only on the amount borrowed or invested and not
on past interest.
• The amount borrowed or deposited is called the principal.
• The rate of interest is given as a percent per year, expressed as a decimal.
For example, 6% = .06 and 11 1/2 % = .115.
• The time during which the money is accruing interest is calculated in
years. (6 months mean 6/12 = 0.5 year, 9 months mean 9/12 = 0.75 year)
• Simple interest is the product of the principal, rate, and time.
• Simple interest is normally used only for loans with a term of a year or less
and for bonds (A typical bond pays simple interest twice a year ).
Simple Interest (Example 8)
Simple Interest (Example 9)
Compound Interest
• With annual simple interest, you earn interest each year on
your original investment.
• With annual compound interest, however, you earn interest
both on your original investment and on any previously
earned interest.
• To see how this process works, suppose you deposit $1000 at
5% annual interest. The following chart shows how your
account would grow with both simple and compound
interest:
(Example 10)
Compound Interest
• As the chart shows, simple interest is computed
each year on the original investment, but
compound interest is computed on the entire
balance at the end of the preceding year.
• So simple interest always produces $50 per year in
interest, whereas compound interest produces $50
interest in the first year and increasingly larger
amounts in later years (because you earn interest on
your interest).
Question: Simple vs. Compound Interest
Example 11
• What is the future value of $100 after 5 years
– at 10% compound interest?
– At 10% simple interest?
Future Value of a Single Amount when
Interest is Compounded
FV = PV (1 + i)n
Future
Value
Present Value
Number
of Compounding
Periods
Interest
Rate
25
Future Value of Lump Sum Amount when
Interest is Compounded
(Example 12 & 13)
Present Value of a Lump Sum Amount
• The process of determining the present value of a
payment or a stream of payments that is to be received
in the future.
• The formula for future value of lump sum payment, FV
= PV(1 + i)n, has four variables:
FV , PV , i , and n .
• Given the values of any three of these variables, the
value of the fourth can be found. In particular, if the
future value, i , and n are known, then PV can be
found.
• Here, PV is the amount that should be deposited
today to produce FV dollars in n periods.
Present Value of a Lump Sum Amount
(Example 14)
Interest Rate of a Lump Sum Amount
(Example 15)
Summing Up: Lump Sum Payments
• FV = PV (1 + i)n
• PV = FV/ (1 + i)n
• Similarly we can also calculate i and n from the
above formula, if any of the 3 variables are given.
Time Line
• Graphical representation to show timing of the cash flows.
• Number above tick mark represent end of year values.
• For instance 1 means end of year 1 and beginning of year 2. 2
means end of year 2 and beginning of year 3.
• Interest rate is placed directly above the time line.
• Cash flows are placed directly below the time line.
Annuities
• So far, only lump-sum deposits and payments
have been discussed. Many financial
situations, however, involve a sequence of
payments at regular intervals, such as weekly
deposits in a savings account or monthly
payments on a car loan.
• Such periodic payments are now the subject
of our discussion.
Annuities
• Definition: A stream of equal payments occurring
at fixed intervals for a specified time.
• For a payment to be classified as annuity certain
conditions must be met:
1. Payment amount should be same.
2. Time interval between occurrence of any two
periods should be same.
3. Payment last for a certain time period.
– For example: $1500 (1) deposited at the end of each
year (2) for the next 6 years (3) in an account paying
8% interest compounded annually.
Ordinary Annuities
• Characteristics
– In an ordinary annuity payment occurs at the end
of each period.
– The first payment starts one period from the
beginning of the timeline. (because for ordinary annuity
payments occurs at end of each of period, so for example 1st period
ends at 1 on timeline, not 0 which is the beginning of timeline)
– Last payment is at the end of the timeline.
Ordinary Annuities
• Calculating FV Ordinary Annuity
• Ordinary Annuities —ones where the payments are made at the end of
each period.
• Example 16
FV of Ordinary Annuity
• Step by step method (we treat each pmt as a single amount)
• Formula method




FVn = FV of annuity at the end of nth period.
PMT = annuity payment deposited or received at the end of each period.
i = interest rate per period
n = number of periods for which annuity will last.
Example 17 (Future Value of an
Ordinary Annuity Stream)
Jill has been faithfully depositing $2,000 at the
end of each year over the past 5 years into an
account that pays a guaranteed 8% per year.
How much money has she have accumulated
in the account?
$11733.202
FV of Ordinary Annuity (Example 18)
Solving for PMT in
an Ordinary Annuity
(when FV is given in question)
Instead of figuring out how much money you will
accumulate (i.e. FV), you may like to know how much
you need to save each period (i.e. PMT) in order to
accumulate a certain amount at the end of n years.
In this case, we know the values of n, i, and FVn in the
formula and we need to determine the value of PMT.
39
Examples
 19 You would like to have $25,000 saved 6 years from now to pay
towards your down payment on a new house. If you are going to
make equal annual end-of-year payments to an investment
account that pays 7%, how big do these annual payments need
to be?
 20 How much must you deposit in a savings account each year,
earning 8% interest in order to accumulate $5,000 at the end of
10 years?
 21 If you can earn 12% on your investments, and you would like
to accumulate $100,000 for your child’s education at the end of
18 years, how much must you invest annually to reach your goal?
Verify the answers: 3494.89; 345.15;1793.73
40
Solving for ‘n’
Taking data from Example 19:
 FV =25000, i = 7%, PMT = $3494.895, n = ?
n = 6 years
41
Solving for ‘i’
Taking data from Example 19:
 FV =25000, i = ?, PMT = $3494.895, n = 6
42
The Present Value of an Ordinary Annuity
 The present value of an ordinary annuity measures the
value today of a stream of cash flows occurring in the
future.
43
Example 20: We compute the PV of each single cash flow and sum them up.
• Also verify through formula.
44
The Present Value of an Ordinary Annuity
 Example 21:
 (a) What is the value today or lump sum equivalent of
receiving $3,000 every year for the next 30 years if the
interest rate is 5%
 (b) what will be the value of annuity after 30 years?
For the example, FV=199,316.54. PV=46,117.35.
45
Formulas for the Present and Future Values of
an Ordinary Annuity
46
Checkpoint 5.1: Check Yourself
The Present Value of an Ordinary Annuity
 Your grandmother has offered to give you $1,000 per year for the next 10
years. What is the present value of this 10-year, $1,000 annuity discounted
back to the present at 5%?
47
Checkpoint 5.1
 Verify the answer:7721.73;
48
Checkpoint 5.2: Check Yourself
What is the present value of an annuity of $10,000 to be
received at the end of each year for 10 years given a 10
percent discount rate?
Verify the Answer: 61,445.67
49
Annuities Due
 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.
 Computation of future value of an annuity due requires compounding the
cash flows for one additional period, beyond an ordinary annuity.
 Computation of present value of an annuity due requires discounting the cash
flows for one less period than an ordinary annuity.
 Formula Adjustment :FV or PV (annuity due) = (FV or PV (ordinary annuity)x(1+i)
50
Example
 Example 22
 In example 21, we calculated the future value of 30-year
ordinary annuity of $3,000 earning 5% to be $199,316.54.
What will be the future value if the deposits of $3,000 were
made at the beginning of the year i.e. the cash flows were
annuity due?
FVAD=199,316.54 x 1.05= $ 209282.367
51
Checkpoint 5.3
 Checkpoint 5.2 where we computed the PV of 10-year ordinary
annuity of $10,000 at a 10% discount rate to be equal to
$61,446. What will be the present value if $10,000 is received
at the beginning of each year i.e. the cash flows were annuity
due?
PVAD=61446x1.1= $67590.6
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Class Practice and Homework
• Attempt Question # 5-1, 5-2, 5-3, 5-4, 5-6, 5-9,
5-10, 5-12, 5-13, 5-14, 5-15 (of your Textbook)
for homework.
Assignment Questions
• Please start solving Assignment 4, Part I.