UNIT 3
Time Value of Money
The price then that the borrower has to
pay for the loan of capital, and which he
regards as interest, is from the point of
view of the lender more properly to be
regarded as profits: for it includes
insurance against risks which are often
very heavy, and earnings of arrangement
for the task, which is often very arduous,
of keeping those risks as small as possible.
Variations in the nature of these risks and
of the task of management will of course
occasion corresponding variations in the
gross interest—so called that is paid of the
use of money”.
—Alfred Marshall, Principles of Economics: Volume 2 (London: MacMillan & Co., 1890), p. 623
Overview
Unit 3 marks the beginning of the second major area of study of this course, namely valuing
financial assets which we discussed briefly in Unit 1. In that Unit, we observed that many firms
trade their securities in financial markets, and that as a result, the financial executive should
focus on raising the market values of the firm’s securities.
Against this background, it is important that we understand how securities are valued and
traded in capital markets. Therefore, in this Unit, we will examine risk-free assets such as
government securities whose future cash flows are essentially known. We will also demonstrate
the use of both conceptual knowledge and calculations to work out financial problems. Because
of the importance of this unit, I highly recommend that you practice as many problems as
possible.
It should be noted that for this course, you should be familiar with the use of your financial
calculator as this will not be taught in the unit. While the formulas are available to show the
variables which make up the inputs, the financial calculator workings are not showed in this
unit. Such workings may be accessed from any standard finance text or you tube videos.
MGMT2023 Financial Management_UNIT 3 – VERSION 1 1 Unit 3 Learning Objectives
By the end of this unit you should be able to:
1. Calculate the future value of an investment made today. 2. Calculate the present value of cash to be received today. 3. Determine the future value and present values of an investment with multiple cash flows. 4. Distinguish between nominal and effective annual rates of interest. 5. Describe the procedures involved in (1) determining deposits needed to accumulate a future sum, (2) loan amortization, (3) finding interest or growth rates and (4) finding an unknown number of period. This Unit is divided into three sessions as follows:
Session 3.1: A Single Amount
Session 3.2: Annuities
Session 3.3: Uneven Cash Flow
Readings & Resources
Note to Student:
Some hyperlinks to resources may not open on clicking. If any link fails to open, please copy
and paste the link to your browser to view/download the resource.
Required Reading
Drakes, P.P. & Fabozzi. F.J. (2010). The basics of Finance: An Introduction to Financial
Markets, Business Finance, and Portfolio Management. Chapter 10. The Math of Finance.
Pages 203- 229. Available at:
http://onlinelibrary.wiley.com.library.open.uwi.edu/doi/10.1002/9781118267790.ch2/
pdf
Use of a Financial Calculator is required for this Unit. Texas Instruments- BAII Plus is highly
recommended. Students may however purchase the brand of their choice.
MGMT2023 Financial Management_UNIT 3 – VERSION 1 2 Suggested Reading [or Other Resources]
Sections 1 to 6 of The Time Value of Money. Available at:
https://www.boundless.com/finance/textbooks/boundless-finance-textbook/thetime-value-of-money-5/
MGMT2023 Financial Management_UNIT 3 – VERSION 1 3 Session 3.1
A Single Amount
Introduction
Time value is based on the belief that a dollar today is worth more than a dollar that will be
received at some future date. The techniques employed in compounding and moving money
through various time periods are critical to almost every business decision made by both the
financial executive and financial analysts since they use them to help understand how bonds
and stocks are valued (which you will learn in Units 4 and 5 respectively) and how a project
value is determined (which you will learn in Unit 7). Our emphasis is in this session is on the
movement of a single money value through the present and future periods. To perform this
task, we will employ the future value and present value concepts. Future Value (FV)
Future Value of an investment is what the investment will be worth after earning interest for
one or more time periods. Compounding is the term used to define the process of converting the
initial amount into the future value. The best way to understand the concept of future value is
by an example.
REFLECTION
Suppose you place $100 in a bank savings account that pays interest at 10 percent a year.
How much money will you have in one year?
MGMT2023 Financial Management_UNIT 3 – VERSION 1 4 Most of you can intuitively come up with the correct answer, $110, (principal + interest earned)
without the aid of a formula. However, we can use our intuitive calculation to generate a more
general formula as follows:
FV = PV + (PV x i)
FV = P x (1 i)
FV = $100 x (1+ 0.10)
= $100 x 1.10
=$110
1
1
1
0
0
0
+
n
n
1
Where:
FVn = Future value or ending amount of your account after N periods
PVn = Present value or the beginning amount
I = Interest earned per year. The interest rate is sometimes written as i and we often assume
that interest is paid at the end of a year
N = Number of periods involved in the analysis
Present Value (PV)
In our discussion of future value above, we asked the question “If you put $100 in a bank that paid 10 percent annual interest, how much money would accumulate in one year? Another type of question that arises frequently in finance concerns present value. This question asks, “What is the value today of a cash flow promised in the future? Discounting is the process by which the present value of future cash flows is obtained. Let’s take a look at an example. REFLECTION
Suppose your aunt gives you a bank note that matures in one year and pays $110. The note
pays 10 percent interest annually and cannot be redeemed before the maturity. You however
need the money now and would like to sell the note to your friend. How much should your friend
pay for the note?
MGMT2023 Financial Management_UNIT 3 – VERSION 1 5 The answer lies in the present value. To find the PV of a future cash flow or its value today we
“reverse” the compounding process and divide the FV ($110) by the value factor (1+ 0.10). Let
us write the formula.
n
n
PV = FV / (1+i)
Where:
FVn = Future value or ending amount of your account after N periods
PVn = Present value or the beginning amount
i
= Interest earned per year. The interest rate is sometimes written as i and we often
assume that interest is paid at the end of a year
n = Number of periods involved in the analysis
LEARNING ACTIVITY 3.1
1. Watch the following video presentations to help you get an appreciation of PV and
FV.
Available at: https://youtu.be/zR3L5mLTi7s
2. Read Sections 2 and 3 of The Time Value of Money. Available at:
https://www.boundless.com/finance/textbooks/boundless-finance-textbook/thetime-value-of-money-5/
3. Take the self-assessment questions on this section of the unit at this time.
Session 3.1 Summary
This Session examined the movement of money through time using both future value and
present value formulas.
MGMT2023 Financial Management_UNIT 3 – VERSION 1 6 Session 3.2
Annuities
Introduction
This Session begins by defining what an annuity is. It then identifies the different types of
annuities and examines the techniques used in calculating them.
Annuity
Many assets provide a stream of cash inflows over a time period and obligations, such as
student and mortgage loans, require a series of payments. If payments are equal and fixed, then
the series is referred to as an annuity. For example, a fixed amount of $550, paid at the end of
each year over a four-year period, is a four-year annuity. As long as the payments are made at
the end of each period, they are known as an ordinary annuity.
An annuity is therefore a fixed cash payment received at regular intervals for some limited
length of time. Conversely, if the payments are made at the beginning of each year, then we
have an annuity due. In finance, ordinary annuities are more common, and, in this course, we
will assume that payments are made at the end of the year unless otherwise stated.
Future Value of Annuities
Compounding and determining the present value of an annuity can be found by using the
methods we described previously. However, these methods can be time consuming, especially
for larger annuities. Thus, we have modified the formulas to deal directly with annuities.
Recall that an ordinary annuity involves depositing or investing an equal sum of money at the
end of each year for a certain number of years and allowing it to grow. Perhaps we are saving
for retirement, to purchase a home or take a holiday. The future value of an ordinary annuity
can be found using a financial calculator or a spreadsheet. However, in this Unit, we will use
the step-by-step approach or the formula to calculate the future value of an ordinary annuity.
Equation 4.5 below shows the step-by-step method used to calculate the future value of an
ordinary annuity.
MGMT2023 Financial Management_UNIT 3 – VERSION 1 7 FVAn = CF (1+ I)n −1 +CF (1+ I)n −2 +CF (1+ I)n −3 +CF (1+ I)n −4
Where:
FVn = the future value of the annuity at the end of the nth year
CF = Cash Flow deposited or received at the end of each year
i = the annual interest (or discount) rate
n = the number of years for which the annuity will last
Present Value of Annuities
Let us now examine how to calculate the present value of annuities by looking at an example.
What is the present value (PV) of a four-payment annuity of $120 per quarter, starting
immediately, if the effective annual interest rate is 8%?
PV of annuity = CF [(1-PVIFi,n) / i]
= CF[(1- (1 + i) -n ] / i
Where
CF = the Cash Flow deposited or received at the end of each year
i = the annual discount (or interest) rate
PV = the present value of the future annuity
n = the number of years for which the annuity will last
Required Reading
Read Section 4- Annuities of The Time Value of Money. Available at :
https://www.boundless.com/finance/textbooks/boundless-financetextbook/the-time-value-of-money-5/
LEARNING ACTIVITY 3.2
This Session focused on explaining annuities and the techniques used to calculate
them. The following time value of money techniques were used to perform the
calculations:
1. Future value of annuities
2.
Present value of annuities
MGMT2023 Financial Management_UNIT 3 – VERSION 1 8 Session 3.2 Summary
This Session focused on explaining annuities and the techniques used to calculate them. The
following time value of money techniques were used to perform the calculations:
1. Future value of annuities
2. Present value of annuities
MGMT2023 Financial Management_UNIT 3 – VERSION 1 9 Session 3.3
Uneven Cash Flows
Introduction
This Session focuses on demonstrating the techniques used to calculate the present value and
future value of uneven cash flows. An uneven or mixed cash flow stream is one where the
amounts to be received vary each period, as oppose to an annuity where the cash flows
are equal across periods. There is often a need to find the present value or future value
of a stream of cash flows that varies. You should find the techniques used in this session
rather useful in every day decision making.
Present Value of Uneven Cash Flow
An annuity means that payments are constant in every period. Typically, a number of financial
decisions involve fixed amounts. However, some include payments of uneven cash flows.
Examples of uneven cash flows include payment of dividends on common shares and
investments made on capital equipment. Both usually generate such cash flows.
In this Session, we concentrate on two important classes of uneven cash flows namely: 1) a series
of annuity payments plus an additional lump sum (for example, bonds) and, 2) uneven streams
of cash flows which include stocks and capital investments.
Future Value of Uneven Cash Flow
Now, let us consider uneven or irregular cash flows based on the information in the previous
section. Using the future value concept to calculate uneven cash flows, we compound the cash
flows rather than discount them. We discounted those cash flows to find the present value, but
now we need to compound them to find the future value.
Required Reading
Read Section 5- Valuing Multiple Cash Flows of The Time Value of Money.
Available at: https://www.boundless.com/finance/textbooks/boundless-financetextbook/the-time-value-of-money-5/
MGMT2023 Financial Management_UNIT 3 – VERSION 1 10 Take the self-­‐assessment questions on this section of the unit at this time. Compounding Interest More Frequently
We may sometimes need to work with cash flows that are received more frequently than once
per year or are paid more frequently than once per year. For example, savings institutions
compound interest monthly, daily or even continuously. We are therefore made to believe we
are getting a better deal from one institution over the other when the interest rate is quoted on a
monthly or daily basis. This may not be so, and we will not be certain until we calculate what
the true effective interest rate of the offer. To compare loan costs or investment returns over
different compounding periods, nominal and effective interest rates must be distinguished.
View the following video, which will explain the difference between nominal and
effective annual rates of interest. Available at:
https://youtu.be/qVfVQXkT-RU
Special Applications of Time Value
There are four applications of time value that are of interest to us in this course:
1. Payments to accumulate a Future Balance
2. Loan Amortization
3. Interest Growth Rate
4. Finding an unknown number of periods
Required Reading
Read Section 6 of The Time Value of Money. Available at:
https://www.boundless.com/finance/textbooks/boundless-finance-textbook/the-time-value-ofmoney-5/
MGMT2023 Financial Management_UNIT 3 – VERSION 1 11 LEARNING ACTIVITY 3.3
Attempt the “Try It” question on page 212 of the recommended text by Drakes & Fabozzi
(2010). Available at:
http://onlinelibrary.wiley.com.library.open.uwi.edu/doi/10.1002/9781118267790.ch10/pdf
Check page 239 of the text to see if you got those correct.
Self-Assessment: As a wrap-up to this Unit attempt the following quiz on Time Value of
Money. https://www.boundless.com/quizzes/time-value-money-quiz-175342/
Email your answers to your tutor. (Login is required)
Session 3.3 Summary
This Session examined the present value and future value of uneven cash flows. We then
distinguished nominal interest rate from effective interest rate and highlighted that fact that
consumers should pay attention to the frequency of compounding before investing or
borrowing to avoid being bluffed by financial institutions. We finally examined four typical
cases of time value which affect our daily lives.
Unit 3 Summary
In this Unit, we illustrated the movement of money through different time periods through the
use of the future value equation which calculates the future value of a present given amount and
then showed how the equation can be transformed to compute the present value of a given future
amount. We then focused on single payments, ordinary annuities, annuities due and uneven cash flow
streams. Finally, we used the present and future value formulas to calculate uneven cash flows
and highlighted four special applications of time value and how it affects our personal financial
decisions.
The time value of money concept illustrated in this Unit is important in finance because it is
used to find the values of bonds, stocks and capital budgeting projects. It is also important for you
to note that the various techniques used in this Unit will be used in Units 4 to 8.
MGMT2023 Financial Management_UNIT 3 – VERSION 1 12 References
Gitman, J. L. & Zutter, J. C. (2011). Principles of Managerial Finance. (13 Edition).
Chapter 5. Time Value of Money. Pages 164 – 187. Retrieved from
http://homes.ieu.edu.tr/hbaklaci/xmba510/Managerial%20Finance_13e.pdf
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MGMT2023 Financial Management_UNIT 3 – VERSION 1 13