FINANCIAL MARKETS
PCOA005
Module 6: Time Value of Money
Introduction
In this module, you will learn the concept of time value of money. Time value analysis has many
applications, including valuing stocks and bonds, setting up loan payment schedules, and making
corporate decisions regarding investing in new plants and equipment. At the end of this module, you
should be able to answer the exercise posted in your LMS.
Learning Outcomes
At the end of this module, you can:
1. Explain how the time value of money works and discuss why it is such an important concept in
finance.
2. Calculate the present value and future value of lump sums, annuities and uneven cash flow
streams.
Time Value of Money is the idea that money that is available at the present time is worth more than the
same amount in the future, due to its potential earning capacity. This core principle of finance holds
that provided money can earn interest, any amount of money is worth more the sooner it is received.
One of the most fundamental concepts in finance is that money has a time value attached to it. In simpler
terms, it would be safe to say that a peso was worth more yesterday than today and a peso today is worth
more than a peso tomorrow. The time value of money analysis (TVMA) is also called the discounted
cash flow analysis (DCFA).
An important time value of money analysis concept one must keep in mind is that in making financial
decisions that involves comparing two amounts of different time periods, both amounts must first be
converted into the same time value. This would mean that if we compare one amount from a current
and one amount from a future value, both amounts must first be valued at the same time period before
they can be correctly compared. There is no sense in comparing a present valued amount to a future
valued amount.
TIME LINES
An important tool used in time value analysis; it is a graphical representation used to show the timing
of cash flows. The first step in time value analysis is to set up a time line, which will help you visualize
what’s happening in a particular problem.
FUTURE VALUES
The amount to which a cash flow or series of cash flows will grow over a given period of time when
compounded at a given interest rate. A ₱1 in hand today is worth more than a ₱1 to be received in the
future because if you had it now, you could invest it, earn interest, and own more than a peso in the
future. The process of going to future value (FV) from present value (PV) is called compounding.
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Compounding is the arithmetic process of determining the final value of a cash flow or series of cash
flows when compound interest is applied.
Example:
Assume that you plan to deposit ₱100 for 3 years in a bank that pays a guaranteed 5% interest each
year. How much would you have at the end of Year 3? We first define some terms, and then we set up
a time line to show how the future value is calculated.
PV = Present Value (PV), the value today of a future cash flow or series of cash flows.
FVN = the value N periods into the future, after the interest earned has been added to the account.
CFt = Cash flow for a particular period.
i = Interest rate earned per year.
n = Number of periods involved in the analysis
PV = ₱100
n = 3 years
i = 5%
FV = ?
𝐹𝑉 = 𝑃𝑉(1 + 𝑖)
𝐹𝑉 = 100(1 + .05)
𝐹𝑉 = 115.76
Year 1
Year 2
Year 3
Principal
100.00
105.00
110.25
Interest (5%)
5.00
5.25
5.51
FV
105.00
110.25
115.76
The interest use on the above computation is compound interest.
Compound Interest - Occurs when interest is earned on prior periods’ interest. For the rest of the
examples, the interest that we will use is compound interest.
Simple Interest - Occurs when interest is not earned on interest. For simple interest future value:
𝐹𝑉 = 𝑃𝑉 + (𝑃𝑉 × 𝑖 × 𝑛)
𝐹𝑉 = ₱100 + (100 × .05 × 3)
𝐹𝑉 = 115
Year 1
Year 2
Year 3
Principal
100.00
100.00
100.00
Interest (5%)
5.00
5.00
5.00
FV
105.00
110.00
115.00
PRESENT VALUES
Present Value (PV) is the value today of a future cash flow or series of cash flows. Finding a present
value is the reverse of finding a future value.
𝐹𝑉 = 𝑃𝑉(1 + 𝑖)
𝑃𝑉 =
𝐹𝑉
(1 + 𝑖)
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Finding present values is called discounting; it is the reverse of compounding—if you know the PV,
you can compound to find the FV, while if you know the FV, you can discount to find the PV.
The fundamental goal of financial management is to maximize the firm’s value, and the value of a
business (or any asset, including stocks and bonds) is the present value of its expected future cash flows.
Example:
Assume that you plan to yield an amount of ₱100 for 3 years in a bank that pays a guaranteed 5%
interest each year. How much would you invest today to yield ₱100 at the end of Year 3?
𝑃𝑉 =
𝐹𝑉
(1 + 𝑖)
𝑃𝑉 =
100
(1 + .05)
𝑃𝑉 =
100
= 30.21
3.31
ANNUITIES
Annuity is a series of equal payments at fixed intervals for a specified number of periods. The above
discussion dealt with single payments, or “lump sums.” However, many assets provide a series of cash
inflows over time; and many obligations, such as auto, student, and mortgage loans, require a series of
payments. When the payments are equal and are made at fixed intervals, the series is an annuity.
For example, ₱100 paid at the end of each of the next 3 years is a 3-year annuity. If the payments occur
at the end of each year, the annuity is an ordinary (or deferred) annuity. If the payments are made at
the beginning of each year, the annuity is an annuity due. Ordinary annuities are more common in
finance.
Here are the time lines for a ₱100, 3-year, 5% ordinary annuity and for the same annuity on an annuity
due basis. With the annuity due, each payment is shifted to the left by one year. A ₱100 deposit will be
made each year, so we show the payments with minus signs:
Ordinary Annuity
Period
0
1
2
3
-100
-100
-100
0
1
2
3
-100
-100
-100
5%
Payment
Annuity Due
Period
Payment
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Keep in mind that annuities must have constant payments and a fixed number of periods. If these
conditions don’t hold, then the payments do not constitute an annuity.
FUTURE VALUE OF AN ORDINARY ANNUITY
Future value of an ordinary annuity is computed when the payment is at the of each year with equal
amounts.
𝐹𝑉 𝑜𝑓 𝑂𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 (𝐹𝑉 ) = 𝑃𝑀𝑇 ×
(1 + 𝑖) − 1
𝑖
*PMT - Payment each year or cash flow each period
FUTURE VALUE OF AN ANNUITY DUE
Because each payment occurs one period earlier with an annuity due, all of the payments earn interest
for one additional period. Therefore, the FV of an annuity due will be greater than that of a similar
ordinary annuity.
𝐹𝑉 𝑜𝑓 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝐷𝑢𝑒 (𝐹𝑉 ) = 𝑃𝑀𝑇 ×
(1 + 𝑖) − 1
× (1 + 𝑖)
𝑖
Example:
Enzo Company is to make an annual investment of ₱200,000 for four years. The interest for this
investment was 9%.
Required:
A. Compute the future value if payment is made every year end.
B. Compute the future value if payment is made every beginning of the year.
Solution
A. Ordinary Annuity
(1 + 𝑖) − 1
𝐹𝑉 = 𝑃𝑀𝑇 ×
𝑖
(1 + .09) − 1
. 09
𝐹𝑉
= 200,000 ×
𝐹𝑉
= 200,000 × 4.5731 = 𝟗𝟏𝟒, 𝟔𝟐𝟎
B. Annuity Due
(1 + 𝑖) − 1
× (1 + 𝑖)
𝑖
𝐹𝑉
= 𝑃𝑀𝑇 ×
𝐹𝑉
= 200,000 ×
𝐹𝑉
= 200,000 × 4.5731 × 1.09 = 𝟗𝟗𝟔, 𝟗40
(1 + .09) − 1
× (1 + .09)
. 09
Future Value using Excel Formula
= 𝐹𝑉(𝑟𝑎𝑡𝑒, 𝑛𝑝𝑒𝑟, 𝑝𝑚𝑡, [𝑃𝑉], 𝑡𝑦𝑝𝑒)
rate = interest rate
nper = number of years
pmt = cash flow each period
[PV] = present value
Type = 0 for Ordinary annuity; 1 for Annuity due
Self-Test: Try to compute the above example using the excel formula.
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PRESENT VALUE OF AN ORDINARY ANNUITY
𝑃𝑉 𝑜𝑓 𝑂𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝐴𝑛𝑛𝑢𝑖𝑡𝑦(𝑃𝑉 ) = 𝑃𝑀𝑇 ×
1 − (1 + 𝑖)
𝑖
PRESENT VALUE OF AN ANNUITY DUE
1 − (1 + 𝑖)
𝑃𝑉 𝑜𝑓 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝐷𝑢𝑒(𝑃𝑉 ) = 𝑃𝑀𝑇 ×
𝑖
× (1 + 𝑖)
Example:
On January 1 of the current year, Keith Corporation sold its equipment costing ₱500,000 for ₱800,000
to Arvin Corporation. Arvin paid ₱200,000 as down payment and the balance will be paid with a
noninterest-bearing note for ₱600,000. The note shall be paid in equal annual installments (this is the
series of amounts that Keith will receive in the future) every year amounting to ₱200,000/year. The
prevailing interest rate for this type of note is 10%. You have been tasked by Keith Corporation on the
present value of the note receivable to be recognized by the company.
Required:
A. Compute the present value if payment is made every year end.
B. Compute the present value if payment is made every beginning of the year.
Solution:
A. Ordinary Annuity
1 − (1 + 𝑖)
𝑃𝑉 = 𝑃𝑀𝑇 ×
𝑖
1 − (1 + .10)
. 10
𝑃𝑉
= 200,000 ×
𝑃𝑉
= 200,000 × 2.4869
𝑃𝑉
= 𝟒𝟗𝟕, 𝟑𝟕𝟎. 𝟒𝟎
B. Annuity Due
𝑃𝑉
= 𝑃𝑀𝑇 ×
1 − (1 + 𝑖)
𝑖
× (1 + 𝑖)
𝑃𝑉
= 200,000 × 𝑃𝑀𝑇 ×
1 − (1 + 𝑖)
𝑖
𝑃𝑉
= 200,000 × 2.4869 × 1.10
𝑃𝑉
= 𝟓𝟒𝟕, 𝟏𝟎𝟕. 𝟒𝟒
× (1 + .10)
Present Value using Excel Formula
= 𝑃𝑉(𝑟𝑎𝑡𝑒, 𝑛𝑝𝑒𝑟, 𝑝𝑚𝑡, [𝐹𝑉], 𝑡𝑦𝑝𝑒)
rate = interest rate
nper = number of years
pmt = cash flow each period
[FV] = future value
Type = 0 for Ordinary annuity; 1 for Annuity due
Self-Test: Try to compute the above present value using the excel formula.
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DISTRIBUTION, UPLOADING, OR POSTING ONLINE IN ANY FORM OR BY ANY MEANS WITHOUT THE WRITTEN PERMISSION OF
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PERPETUITIES
A perpetuity is simply an annuity with an extended life. In computing for annuities, the payments or
receipts are to be made over some predetermined time frame. This can be seen in the examples we have
studied. However, there are annuities that may go on indefinitely or perpetually. This type of annuity is
referred to as perpetuities.
Formula:
𝑃𝑉 𝑜𝑓 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 =
𝑃𝑎𝑦𝑚𝑒𝑛𝑡
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒
Example:
Assume that the Philippine government sold small-denominated bonds in 1980. Assume further that
these bonds remain floating up to the present. In 2010, the government sold huge amount of bonds to
pay off the smaller bond issues they made in the '80s. The purpose for issuing the huge amount of bonds
is to consolidate the government's past debts due to the smaller bonds issued. Assume that each
consolidation promised to pay ₱100,000 per year in perpetuity. How much would each bond be worth
if the discount rate is a. 10%? b.15%? c.20%?
a. 10% 𝑃𝑉 𝑜𝑓 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 =
₱
,
%
= ₱𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎
b. 15% 𝑃𝑉 𝑜𝑓 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 =
₱
,
%
= ₱𝟔𝟔𝟔, 𝟔𝟔𝟕
c. 20% 𝑃𝑉 𝑜𝑓 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 =
₱
,
%
= ₱𝟓𝟎𝟎, 𝟎𝟎𝟎
It can be observed that the value of the perpetuity significantly varies when the interest rate is changed.
There is an indirect proportional relationship between the interest rate and the PV perpetuity.
UNEVEN CASH FLOWS
The definition of an annuity includes the words constant payment—in other words, annuities involve
payments that are equal in every period. Although many financial decisions involve constant payments,
many others involve uneven, or nonconstant, cash flows. For example, the dividends on common stocks
typically increase over time, and investments in capital equipment almost always generate uneven cash
flows. Throughout the module, we reserve the term payment (PMT) for annuities with their equal
payments in each period and use the term cash flow (CFt) to denote uneven cash flows, where t
designates the period in which the cash flow occurs.
There are two important classes of uneven cash flows:
(1) a stream that consists of a series of annuity payments plus an additional final lump sum and
(2) all other uneven streams.
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Bonds represent the best example of the first type, while stocks and capital investments illustrate the
second type. Here are numerical examples of the two types of flows:
Annuity plus additional final payment
Period
0
1
12%
Cash flows
0
Irregular cash flows
Period
0
2
3
4
5
100
100
100
100
100
1000
1100
1
2
3
3
3
100
300
300
300
500
12%
Cash flows
0
PRESENT VALUE OF AN UNEVEN CASH FLOW STREAM
Example:
The following annual payments of notes payable of Dorothy Company with 8% discount rate. What is
the present value of the annual payment?
Series of future values of
Period
amounts to be paid
1
200,000
2
250,000
3
300,000
4
375,000
5
375,000
6
375,000
7
375,000
Solution:
Period
(A)
Series of future values
of amounts to be paid
(B)
PV Factor
1 − (1 + 𝑖)
𝑖
(C)
1
200,000
2
250,000
3
300,000
4
375,000
5
375,000
6
375,000
7
375,000
Present Value of Cash Flows
0.9259
1.7833
2.5771
3.3121
3.9927
4.6229
5.2064
PV Factor of
1 at 8%
(Current year
PV factor –
Previous year
PV factor)
(D)
0.9259
0.8573
0.7938
0.7350
0.6806
0.6302
0.5835
PV of Php1 for
each amount
E = (B x D)
185,185.19
214,334.71
238,149.67
275,636.19
255,218.70
236,313.61
218,808.90
1,623,646.96
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DISTRIBUTION, UPLOADING, OR POSTING ONLINE IN ANY FORM OR BY ANY MEANS WITHOUT THE WRITTEN PERMISSION OF
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FUTURE VALUE OF AN UNEVEN CASH FLOW STREAM
The future value of an uneven cash payment/receipt is sometimes referred to as terminal value. This
can be computed by compounding each payment/receipt each year and adding all the future values.
Example:
Assume that Rose Company is to make an investment of uneven cash payments for four years. The
interest for this investment was 9% and the investment is made every year-end. What is the future value
of this annuity? Consider the data below:
Period
1
2
3
4
Amount Invested
200,000
250,000
300,000
325,000
Solution
Period
(A)
Amount
invested
(B)
FV Factor at 9%
(1 + 𝑖) − 1
𝑖
(C)
1
200,000
2
250,000
3
300,000
4
325,000
Future Value of Cash Flows
1.0000
2.0900
3.2781
4.5731
FV Factor of 1
(Current year FV
factor – Previous year
FV factor)
(D)
1.0000
1.0900
1.1881
1.2950
Future Value
E = (B x D)
200,000.00
272,500.00
356,430.00
420,884.43
1,249,814.43
SEMIANNUAL AND OTHER COMPOUNDING PERIODS
The examples presented so far involved compounding or discounting interest rates annually. However,
there are financial contracts like acquisitions on installment basis or corporate bond contracts that may
require for semi-annual, quarterly, or monthly compounding periods.
Under these situations, we compute:
Period = number of years (n)
Interest = interest rate (i)
x
2 (semi-annual)
x
4 (quarterly)
x
12(monthly)
/
/
/
2 (semi-annual)
4 (quarterly)
12 (monthly)
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Example:
Find the future value of Sophia Corporation with a ₱100,000 investment. The investment is good for 5
years with 6% annual interest. Assume that the investment is compounded semi-annually.
Period = 5x2 = 10
Interest = 6% /2 = 3%
In computing for the future value factor, you will be using 3% rate and the period is not 5 periods but
10 periods. Based on the new period and interest rate, the future value factor is 11.4639.
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 𝐹𝑎𝑐𝑡𝑜𝑟 =
(1 + 𝑖) − 1
𝑖
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 𝐹𝑎𝑐𝑡𝑜𝑟 =
(1 + .03) − 1
= 11.4639
. 03
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 = ₱100,000 × 11.4639
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 = ₱1,146,390
References
Brigham, E. and Houston, J., n.d. Fundamentals of financial management. 8th ed. 5191
Natorp Blvd Mason, OH 45040 USA: Cengage Learning.
Anastacio, M. L., Dacanay, R. C., & Aliling, L. E. (2016). Fundamentals of Financial
Management (Revised ed.). REX Book Store.
THIS MODULE IS FOR THE EXCLUSIVE USE OF THE UNIVERSITY OF LA SALETTE, INC. ANY FORM OF REPRODUCTION,
DISTRIBUTION, UPLOADING, OR POSTING ONLINE IN ANY FORM OR BY ANY MEANS WITHOUT THE WRITTEN PERMISSION OF
THE UNIVERSITY IS STRICTLY PROHIBITED.
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