Time value of money

1. SF & AFM
LECTURE 03
BY;
DR. S. M. ALI TIRMIZI
TIME VALUE
OF MONEY
FOUNDATION UNIVERSITY ISLAMABAD
RAWALPINDI CAMPUS
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2. OBJECTIVES
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.
3. Identify the different types of annuities, calculate the present value and
future value of both an ordinary annuity and an annuity due, and
calculate the relevant annuity payments.
4. Calculate the present value and future value of an uneven cash flow
stream.
5. Explain the difference between nominal, periodic, and effective interest
rates.
6. Discuss the basics of loan amortization and develop a loan amortization
schedule that you might use when considering an auto loan or home
mortgage loan. 2

3. INTRODUCTION
 Time value analysis has many applications, including planning for
retirement, valuing stocks and bonds, setting up loan payment
schedules, and making corporate decisions regarding investing in
new plants and equipment.
 In fact, of all financial concepts, time value of money is the single
most important concept.
 This chapter is heavy on calculations.
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4. TIME LINE
 The first step in time value analysis is to set up a time line, which
will help to visualize what’s happening in a particular problem.
 Cash flows are shown directly below the tick marks, and the
relevant interest rate is shown just above the time line.
 Unknown cash flows, which you are trying to find, are indicated
by question marks.
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5. FUTURE VALUES
 A dollar in hand today is worth more than a dollar to be received in
the future because if you had it now, you could invest it, earn
interest, and own more than a dollar in the future.
 The process of going to future value (FV) from present value
(PV) is called compounding.
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6. 6
FUTURE VALUES

7. PRESENT VALUES
 Finding a present value is the reverse of finding a future value.
 Example: A broker offers to sell you a Treasury bond that will pay
$115.76 three years from now. Banks are currently offering a
guaranteed 5% interest on 3-year certificates of deposit (CDs); and
if you don’t buy the bond, you will buy a CD. The 5% rate paid on
the CDs is defined as your opportunity cost, or the rate of return
you could earn on an alternative investment of similar risk.
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8. FINDING THE INTEREST RATE I
 Suppose we know PV, FV, and N and want to find I.
 For example, suppose we know that a given bond has a cost of
$100 and that it will return $150 after 10 years.
 Thus, we know PV, FV, and N, and we want to find the rate of
return we will earn if we buy the bond.
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9. FINDING THE NUMBER OF YEARS N
 We sometimes need to know how long it will take to accumulate a
certain sum of money, given our beginning funds and the rate we
will earn on those funds.
 For example, suppose we believe that we could retire comfortably
if we had $1 million.
 We want to find how long it will take us to acquire $1 million,
assuming we now have $500,000 invested at 4.5%.
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10. ANNUITIES
 Normally, 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.
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Cont….

11. ANNUITIES
 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.
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12. FUTURE VALUE
OF ORDINARY ANNUITIES
 Suppose you deposit $100 at the end of each year for 3 years and
earn 5% per year. How much will you have at the end of the third
year?
 The answer, $315.25, is defined as the future value of the annuity,
FVAN;
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13. FUTURE VALUE
OF ANNUITIES 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.
 The earlier calculations of ordinary annuity used for annuity
due will become FV of $331.01 versus $315.25 for the
ordinary annuity using the following formula.
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14. PRESENT VALUE
OF ORDINARY ANNUITIES
 To find the PV, we discount PMT calculations & dividing each
payment by (1 + I)t
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15. FINDING ANNUITY PAYMENTS, PMT
 Suppose we need to accumulate $10,000 and have it available 5
years from now.
 Suppose further that we can earn a return of 6% on our savings,
which are currently zero.
 Thus, we know that FV = 10,000, PV = 0, N = 5, and I YR = 6%.
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16. FINDING THE NUMBER
OF PERIODS, N
 Suppose you decide to make end-of-year deposits, but you can
save only $1,200 per year.
 Again assuming that you would earn 6%, how long would it take
to reach your $10,000 goal?
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17. FINDING THE INTEREST RATE, I
 Suppose you can save only $1,200 annually, but you still need the
$10,000 in 5 years. What rate of return would enable you to
achieve your goal?
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18. PERPETUITIES
 A perpetuity is simply an annuity with an extended life.
 It’s easy to find the PV of a perpetuity with a formula;
 For example, that you buy preferred stock in a company that pays
you a fixed dividend of $2.50 each year the company is in
business. If we assume that the company will go on indefinitely,
the preferred stock can be valued as a perpetuity. If the discount
rate on the preferred stock is 10%, the present value of the
perpetuity, the preferred stock, is $25.
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19. 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 non-constant, cash flows.
 For annuities with their equal payments in each period and use the
term cash flow (CFt) to denote uneven cash flows.
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Cont…

20. UNEVEN CASH FLOWS
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21. UNEVEN CASH FLOWS
 We can find the PV of either stream by using the following
equation;
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Cont…

22. UNEVEN CASH FLOWS
 In order to find the PV of the uneven cash flows, NPV command
in MS Excel is applied;
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Cont…

23. UNEVEN CASH FLOWS
 The future value of an uneven cash flow is calculated by first
calculating NPV and than calculating the FV of NPV.
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Cont…

24. UNEVEN CASH FLOWS – SOLVING FOR I
 Finding the rate requires a trial-and-error process, which
means that a financial calculator or a spreadsheet is
needed.
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25. SEMIANNUAL
& OTHER COMPUNDING PERIODS
 Suppose, however, that you deposit $100 in a bank that pays a 5%
annual interest rate but credits interest each 6 months.
 So in the second 6-month period, you earn interest on your original
$100 plus interest on the interest earned during the first 6 months.
This is called semiannual compounding.
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26. COMPARING INTEREST RATES
 Different compounding periods are used for different types of
investments.
 For example, bank accounts generally pay interest daily; most
bonds pay interest semiannually; stocks pay dividends quarterly;
and mortgages, auto loans, and other instruments require monthly
payments.
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27. AMORTIZED LOAN
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28. AMORTIZED LOAN
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29. END OF LECTURE 3
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HARDWORK IS KEY TO SUCCESS