PART I: Present and Future Values of Lump Sum Payments
1. Important terminologies
FV
Future Value
(1+i)t Future Value Interest Factor [FVIF]
PV
Present Value
1/(1+i)t Present Value Interest Factor [PVIF]
i
n
Rate per period
# of time periods
1. Future Value
FVn (future value) of a lump sum payment is the ending amount in an account.
2. Present Value
PV (present value) is the value today of a future payment, or stream of payments, discounted at
the appropriate rate of interest. PV is also the beginning amount that will grow to some future
value.
3. Interest Rate, i
The parameter i is the periodic interest rate that an account pays.
4. No. of periods, n
‘n’ is the number of periods the money is left in the account.
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2. Future Value and Present Value of a Lump Sum Payment (Detailed
Discussion)
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3. Present vs. Future Value
What is the difference between future value and present value?


Present value (PV) is the current value of future cash flow.
Future value (FV) is the value of cash flow after a specified period.
A simple way to classify whether cash flows are present value or future value is to remember
that:


PV is the value at the beginning of a time period that you are considering.
FV is the value at the end of the time period that you are considering.
Recall the example from the beginning of this section:
John is an acquaintance of yours. He is a business major and has made the dean's list every
semester. John wants to start up a small consulting company during his senior year. He has
approached you and asked you to lend him $10,000, which he will pay back after three years.
For the three-year period that you consider lending, $10,000 is the value at the beginning and
hence called PV, and the amount that you will get back from John (i.e., $13,000) is the value at
the end and hence called FV. (Going interest rate is 9.139%)
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4. Conceptual Questions and Answers
Question 1
John is an acquaintance of yours. He is a marketing major and has made the dean's list every
semester. John decides to start up a small consulting company during his senior year. He has
approached you and asked you to lend him $10,000, which he will pay back after three years.


If you decide to keep the money, you can use it to pay your bills, take a vacation, add to
your savings, etc.
If you decide to lend him the money, you will have to go without the things that this
$10,000 can buy.
After some careful consideration, you decide to lend him the money. How much should you
expect back after three years? The same amount, or more or less than $10,000? Why?
Answer
In this example, you should expect to get more than $10,000 back from your acquaintance, John.
Why should you (as well as others) expect to get back more than the amount that you lend i.e.,
$10,000? Because if you do not lend the money, you can use it to do other things. By lending,
you are giving up using it for the next three years, and hence you require some returns to
compensate for what you will give up. This is a tradeoff between using money today and saving
for future use, and hence time has value.
The underlying basis of the Time Value of Money (TVOM) is that time has value. That is, a
dollar today is worth more than a dollar tomorrow.
Question 2
Why are (1+i) and (1+i)t called interest factors?
Answer:
1. Start with simple arithmetic problem on interest:
How much will $10,000 placed in a bank account paying 5% per year be worth compounded
annually?
Principal + Interest
$10,000 + $10,000 x .05 = $10,500
2. Factor out the $10,000.
10,000 x (1.05) = $10,500
3. This leaves (1.05) as the factor.
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Question 3
What is the value of any financial asset?
Answer:
The present value of its expected cash flows.
Question 4
1. What is an opportunity cost?
2. How is this concept used in Time value of money (TVM) analysis, and where is it shown on a
time line?
3. Is a single number used in all situations? Explain.
Answer
1. The opportunity cost rate is the rate of interest one could earn on an alternative investment
with a risk equal to the risk of the investment in question.
2. This is the value of i in the TVM equations, and it is shown on the top of a time line, between
the first and second tick marks.
3. It is not a single rate--the opportunity cost rate, i.e. interest rate varies depending on the
riskiness and maturity of an investment, and it also varies from year to year depending on
inflationary expectations.
Question 5
If a firm’s earnings per share grew from $1 to $2 over a 10-year period, the total growth would
be 100%, but the annual growth rate would be less than 10%. True or false? Explain. (Hint: If
you aren’t sure, plug in some numbers and check it out.)
Answer
True, because of compounding effects--growth on growth. The following example demonstrates
the point. The annual growth rate is i in the following equation:
$1(1 + i)10 = $2.
The term (1 + i)10 is the FVIF for i percent, 10 years. We can find i as:
Using the FVIF table: (1 + i)10 = FVIF = $2/$1 = 2. Look across the row for 10 periods until
you find a factor approximately equal to 2.0. The closest is 1.9672, found in the 7 percent
column. Thus, the correct growth rate is about 7 percent, not 10 percent. Viewed another way,
if earnings had grown at the rate of 10 percent per year for 10 years, then EPS would have
increased from $1.00 to $2.59, found as follows:
FV10 = PV(1.10)10 = PV(FVIF10%,10) = $1(2.5937) = $2.59.
This formulation recognizes the “interest on interest” phenomenon.
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A. Four important concepts in Part I
Introduction
The lump sum has the following characteristics:


There are 2 cash flows on the time line:
o Present value (PV) at the beginning of the time line
o Future value (FV) at the end of the time line
There is no cash flow in between PV and FV.
Four important concepts within the broad concept of lump sum payments.
1. Finding the future value, FV, of a lump sum payment.
2. Finding the present value, PV, of a lump sum payment.
3. Finding the interest rate, i, when FV, PV of a lump sum payment along with number of
years are given.
4. Finding the number of years, n, when FV, PV of a lump sum payment along with interest
rate, i, is given.
B. Other Important Topics for Discussion
1. Time line
2. Simple vs. Compound interest rate
3. PV and FV of a lump sum investment (different interest rate each period)
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A. Four important concepts in Part I
1. Finding the future value, FV, of a lump sum payment
Formula for finding future value of a lump sum payment
Finding future values (moving to the right along the time line) is called compounding.
The equation for the future value of a lump sum payment is:
FVn = PV(1+i)n
Where:




FV = future value of lump sum
PV = present value of lump sum
i = interest rate per period
n = number of compounding periods
Solved Examples
Example 1
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.
Solution
a.
Given,
Present value (PV) = £. 500
Interest rate (i) = 6%
0
6%
1
- 500
b.
FV = ?
FVn = PV(1 + i)n
FV1 = PV (1 + i)1
= £ 500 (1 + 0.06)1 = £. 530
Present value (PV) = £. 500
Interest rate (i) = 6%
0
- 500
6%
1
2
FV = ?
FVn = PV(1 + i)n
FV2 = PV (1 + i)2 = £. 500 (1 + 0.06)2 = £. 561.80
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Example 2
What’s the future value of $100 after 3 years if it earns 10%, annual compounding?
Solution
FVn = PV(1+i)n
So FV3 = $100(1.10)3 = $100(1.3310) = $133.10.
Example 3
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:
FVn = PV(1+i)n




FV = ?
PV = 500
r = .10
t=3
FV3 = 500(1+.10)3 = $665.5
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2. Finding the present value, PV, of a lump sum payment.
Formula for finding future value of a lump sum payment
Finding present values, or discounting (moving to the left along the time line), is the reverse of
compounding, and the basic present value equation is the reciprocal of the compounding
equation.
The equation for the future value of a lump sum payment is:
PV = Fn /(1+i)n
Where:




FV = future value of lump sum
PV = present value of lump sum
i = interest rate per period
n = number of compounding periods
Solved Examples
Example 1
Find the present values:
a. The present value of £ 500 due in 2 years at a discount rate of 6 percent.
Solution
a.
Future value (FV) = £. 500
Interest rate (i) = 6%
No. of periods (n) = 2
Present value (PV) = ?
0
6%
1
PV = ?
PV
2
FV = 500
FVn
FV2
= (1 + i)n = (1 + i)2
= £ 445
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Example 2
What’s the present value of $100 to be received in 3 years if the interest rate is 10%, annual
compounding?
Solution
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3. Finding the interest rate, i, when FV, PV of a lump sum payment along with number
of years is given.
Solved Examples
Example 1
An investor deposits $10,000. Ten years later it is worth $17,910. What rate of return did the
investor earn on the investment?
Solution
$17,910 = $10,000 x (1+i)10
(1+i)10 = $17,910/10,000 = 1.7910
(1+i) = (1.7910) 1/10 = 1.060
i = .060 = 6.0%
Example 2
What annual interest rate would cause $100 to grow to $125.97 in 3 years?
Solution
FV = PV (1 + i)n
125.97 = 100 (1 + i )3
i = 7.999%
4. Finding the number of years, n, when FV, PV of a lump sum payment along with
interest rate, i, is given.
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B. Other important topics for discussion
1. Timeline
Introduction
A time line is a graphical representation that is used to show the timing of cash flows. It is
simply a straight line that shows cash flow, its timing, and interest rate. The tick marks represent
end of periods (often years), so time 0 is today; Time 1 is the end of the first year, or 1 year from
today; and so on. The time line is a very useful tool for an analysis of the time value of money
because it provides a visual for setting up the problem.
A time line consists of the following components:



Time period
Interest rate
Cash flow
We will go over each of these in the following screens.
Time period (t) can be any time interval such as year, half a year, and month. Each period should
have equal time interval. Zero represents a starting point, and a tick represents the end of one
period. From the example at the beginning of this section, there are three years or annual periods
and hence the time line has three ticks. Each tick represents a period of one year.
Interest rate (r) is the rate earned or paid on cash flow per period. It is labeled above the time
line.
If a 10% return per year is required, the time line should be:
Cash flow (CF) is amount of money. It is placed directly below a tick at time period that it
occurs. Cash flow can be known or unknown amount.
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Remember: Time line can help you simplify a complex problem.
Example 1
Draw time lines for a $100 lump sum cash flow at the end of Year 2.
Solution
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2. Simple v. Compound interest rate
What is compound interest rate? How is it different from simple interest rate?


For a simple interest rate, interest is earned on the original principal only.
For a compound interest rate, interest is earned on both the original principal and interests
reinvested from prior periods.
The difference is the amount of interest that is earned on the reinvested interests.
Example 1 (Simple interest rate)
For example, you invest $100 for 3 years in an investment company that provides a simple 10%
interest rate. What is your payback at the end of 3 years?
Solution
If the interest rate is a simple rate, the payback for:



The first year (X1) is $100 of original principal + $10 of interest:
X1 = $100 + (10% × $100) = $110.
The payback for the second year (X2) is $100 of the principal + $10 of interest from the
first year + $10 of interest from the second year:
X2 = $100 + (10% × $100) + (10% × $100) = $120.
The payback for the third year (X3) is $100 of the principal + $10 of interest from the first
year + $10 of interest from the second year + $10 of interest from the third year:
X3 = $100 + (10% × $100) + (10% × $100) + (10% × $100) = $130.
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Example 2 and Solution (Compound interest rate)
If the interest rate is an annual compound rate (i.e., computed once a year), the payback for:



The first year (X1) is $100 of original principal + $10 of interest:
X1 = $100 + (10% × $100) = $110
The payback for the second year (X2) is, $100 of the principal + $10 of interest from the
first year + $11 of interest from the second year:
X2 = $100 + (10% × $100) + (10% × $110) = $121
The payback for the third year (X3) is $100 of the principal + $10 of interest from the first
year + $11 of interest from the second year + $12.1 of interest from the third year:
X3 = $100 + (10% × $100) + (10% × $110) + (10% × 121) = $133.1
Note that the payback for the first year is the same for both simple and compound rate. However,
the paybacks after the first year are higher for the compound rate than for the simple rate because
interest is computed on the prior year's principal, not the original principal. The longer the time
period is, the larger the difference. TVOM assumes compound interest rate.
Example 3
Josh, your close friend, borrowed $500 from you three years ago. He has promised to return the
money to you today with 6% interest rate per year. If Josh returns $595 and some change to you,
is the interest rate a simple or compound rate?
Solution
For a simple interest rate of 6%, the amount of interest per year would be:
X1 = $500 + (.06 × $500) = $530
X2 = $500 + (.06 × $500) + (.06 × $500) = $560
X3 = $500 + (.06 × $500) + (.06 × $500) + (.06 × $500) = $590
As such, you should get back $590.
For a compound interest rate of 6%, the amount of interest per year would be:
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X1 = $500 + (.06 × $500) = $530
X2 = $500 + (.06 × $500) + (.06 × $530) = 561.80
X3 = $500 + (.06 × $500) + (.06 × $530) + (.06 × $561.80) = $595.51
Since Josh returns $595.51, $500 of principal and $95.51 of interest, the interest rate is a
compound rate. Under a compound rate, interest amount is greater than under a simple rate
because interest is earned on the prior year's interest.
3. PV and FV of a lump sum investment (different interest rate each
period)
Example 1
Suppose you make an investment of $1,000. This first year the investment returns 12%, the
second year it returns 6%, and the third year in returns 8%. How much would this investment
be worth, assuming no withdrawals are made?
Solution:
1000*(1.12) x (1.06) x (1.08)
= $1,282
Example 2
Suppose you make an investment worth $A. This first year the investment returns 10%, the
second year it returns 16%, and the third year in returns 2%. How much would this investment
be worth, assuming no withdrawals are made?
Solution:
A*(1.10) x (1.16) x (1.02)
Example 3
Suppose you make an investment of $1,000. This first year the investment returns 5%, the
second year it returns i. Write an expression, using i, that represents the future value of the
investment at the end of two years.
Solution:
FV=1,000 x (1.05) x (1+i)
Example 4
An investment is worth $50,000 today. This first year the investment returns 9%, the second
year it returns i. Write an expression using i that represents the original value of the investment.
Solution
PV=50,000÷[(1.09) x (1+i)]
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Self-Test Questions
Question 1
1. You lent $10,000 to John in 1997 and he returned $13,000 to you in 2000, which amount
should be called PV and FV?
2. You lend John $10,000 in 2006, and get $13,000 back three years after that, in 2009. Is
$10,000 PV or FV?
Question 2
A farmer can spend $60/acre to plant pine trees on some marginal land. The expected real risk
free rate of return is 4%, and the expected inflation rate is 6%. What is the expected value of the
timber after 20 years?
Answer:
First we’ll determine the risk free rate of return.
Krf = K* + IP
Krf = 0.04 + 0.06
Krf or i= 0.10 or 10%
FVn= FV20 = PV(1 + i)20, = $60(1.10)20= $403.65 per acre.
We could have asked: How long would it take $60 to grow to $403.65, given the real rate of
return of 4% and an inflation rate of 6%? Of course, the answer would be 20 years.
Question 3
Using the time line below, indicate the values for A, B and C for each of the three problems.
(The first one has been done for you)
1. The bank lends you $20,000 and requires a 10% return. To assist you in calculating the
amount of money you would pay back, you would label parts A, B, and C of the time line
as:
A. 10%
B. + 20,000 (inflow)
C. ? (FV is unknown)
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2. Your brother borrows $100 and has also told you that he will return $120 at the end of
three years. To assist you in calculating the return on this investment, you would label
parts A, B, and C of the time line as:
A. ? (interest is unknown)
B. ____________________
C. ____________________
3. Based on a 10% return, your roommate determines she will need $500 at the end of three
years to pay back a loan. To assist you in calculating the amount of money originally
borrowed by your roommate, you would label parts A, B, and C of the timeline as:
A. ____________________
B. ____________________
C. ____________________
Question 6
You’ve located an investment that pays 12 percent per year. That rate sounds good to you, so
you invest $400.
1. How much will you have in three years? ($561.97)
2. How much will you have in seven years? ($884.27)
3. At the end of seven years, how much interest will you have earned? (As you invested $400,
the interest in the $884.27 future value is $884.27 - 400 = $484.27.)
4. How much of that interest results from compounding?
Solution to part 4
At 12 percent, your $400 investment earns $400 x 0.12 = $48 in simple interest every year. Over
seven years, the simple interest over 7 years thus totals 7 x $48 = $336. However with the help of
compounded interest rate we earned $484.27, thus the other $484.27 - 336 = $148.27 is from
compounding
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Critical Thinking Questions
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Solutions to Critical Thinking Questions
Question 1
Question 2
Question 3
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Question 4
Question 5
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Question 9
Question 17
Question 18
Question 19
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Question 20
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Solved Examples
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