Time Value of Money

1. Chapter Five
Time Value of Money
Principles of Managerial Finance
First Canadian Edition
Lawrence J. Gitman and Sean Hennessey
© 2004 Pearson
5-1

2. Learning Goals
LG1 – Discuss role of time value of money in
finance, and use of computational aids.
LG2 – Understand concept of future value, its
calculation for single amount, and the
effect of compounding interest more
frequently.
LG3 – Find the future value of an ordinary
annuity and an annuity due, and compare
these two types of annuities.
© 2004 Pearson
5-2

3. Learning Goals (continued)
LG4 – Understand concept of present value, its
calculation for a single amount, and its
relationship to future value.
LG5 – Calculate present value of a mixed
stream of cash flows, an annuity, a
mixed stream with an embedded
annuity, and a perpetuity.
© 2004 Pearson
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4. Learning Goals (continued)
LG6 – Describe the procedures involved in
1) Determining the periodic investments
required to accumulate a future sum,
2) Loan amortization, and
3) Determining growth and interest rates.
© 2004 Pearson
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5. Role of Time Value in Finance
• The timing of cash flows has important
economic consequences that are recognized
as the Time Value of Money.
• 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.
© 2004 Pearson
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6. Future vs. Present Value
• Future Value is cash you will receive at a
given future date.
• Present Value is the equivalent of cash on
hand today.
• A time line can be used to depict the cash
flows associated with a given investment.
• Since financial managers make decisions at
time zero, they tend to rely on present value
techniques.
© 2004 Pearson
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7. Figure 5.2 Compounding and
Discounting
© 2004 Pearson
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8. Computational Aids
• Financial Tables are commonly used as
quick reference tools for determining
present and future values at various interest
or discount rates of a range of time periods.
• Modern calculators are programmed to
perform the complete computational
analysis using the underlying formulas for
present and future value.
© 2004 Pearson
5-8

9. Figure 5.3 Financial Tables
© 2004 Pearson
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10. Future Value of Single Amount
• Principal is the amount of money on which
interest is paid.
• Compound Interest is the interest earned on
a given deposit that becomes part of the
principal at the end of a specified period.
• Future Value of a present amount is found
by applying compound interest over a
specified period of time.
© 2004 Pearson
5-10

11. Equation of Future Value
FVn = future value at the end of period n.
PV = initial principal, or present value.
k = annual rate of return.
n = number of periods the money is left on deposit.
(5.4) FVn = PV * (1+k)n
© 2004 Pearson
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12. Using Tables & Calculators
• Table A-1 provide values for the Future
Value Interest Factor (FVIF) which
simplifies the process of calculating FV in
equation (5.4).
(5.5) FVIFk,n = (1+k)n
(5.6) FVn = PV * FVIFk,n
© 2004 Pearson
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13. Graphic View of Future Value
© 2004 Pearson
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14. Compounding More
Frequently Than Annually
• Semiannual Compounding involves two
compounding periods within the year.
• Quarterly Compounding involves four
compounding periods within the year.
(5.7) FVn = PV * (1+k/m)m*n
Where m is the number of compounding periods
within the year.
© 2004 Pearson
5-14

15. Continuous Compounding
• Continuous Compounding involves
compounding over every microsecond.
(5.8) FVn(continuous) = PV * (ek*n)
(5.9) FVIFk,n(continuous) = ek*n
© 2004 Pearson
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16. Nominal and Effective Annual
Rates of Interest
• The Nominal, or State, Annual Rate is that
charged by a lender or promised by a borrower.
• The Effective Annual Rate (EAR) is the interest
actually paid or earned due to compounding.
(5.10) EAR = (1+k/m)m - 1
© 2004 Pearson
5-16

17. Future Value of An Annuity
• An Annuity is a stream of equal annual cash
flows, either inflows or outflows.
• There are two basic types of annuities:
– Ordinary Annuity where the cash flow occurs at
the end of each period, and
– Annual Due Annuity where the cash flow
occurs at the beginning of each period.
© 2004 Pearson
5-17

18. FV of Ordinary Annuity
• The Future Value Interest Factor for an
Annuity (FVIFA) is:
(5.14)
= + n
t
(1 ) 1 ,
FVIFAk n k
å=
-
t
1
(5.15) FVAn = PMT * (FVIFAk,n)
Where PMT is the amount of each cash flow payment.
© 2004 Pearson
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19. FV of Annuity Due
• Since an Annuity Due requires the cash flow at
the beginning of the period only a simple
adjustment to the FVIFA is needed.
(5.16) FVIFAk,n(Annuity Due)=FVIFAk,n* (1+k)
© 2004 Pearson
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20. Present Value of Single Amount
• Present Value is the current dollar value of a future
amount; the amount of money that would have to be
invested today at a given rate of return over a
specified period to equal the future amount.
• The process of finding Present Value is often referred
to as Discounting Cash Flows.
• The annual rate of return used is referred to as the
discount rate, required return, cost of capital, or
opportunity cost.
© 2004 Pearson
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21. Equation for Present Value
(5.19) PV = FVn = FVn * ( 1 )
(1+k)n (1+k)n
• Tables may also be used to look up the
Present Value Interest Factor (PVIF).
(5.21) PVIFk,n = ( 1 )
(1+k)n
© 2004 Pearson
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22. Graphic View of Present Value
© 2004 Pearson
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23. PV of Mixed Stream
• A Mixed Stream is cash flows of different
amounts during the future periods.
• To determine the Present Value of a Mixed
Stream we must calculate the present value
of each future amount, then sum the total of
the present value calculations.
© 2004 Pearson
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24. PV of an Annuity
• An Annuity is a series of uniform future cash
flows. We may determine the Present Value
of an Annuity using a Present Value Interest
Factor for an Annuity (PVIFA).
n
(5.26)
å=
PVIFA
1
k , n (1 +
k
)
t =
t
1
(5.27) PVAn = PMT * (PVIFAk,n)
© 2004 Pearson
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25. PV of Mixed Stream with
Embedded Annuity
• Three steps to determine the Present Value of a
Mixed Stream with an Embedded Annuity.
– Find the present value of the annuity at specified
discount rate.
– Add the present value calculated to any other cash
flow occurring in the period just before the start of
the annuity to determine a revised cash flows.
– Discount the revised cash flows back to time zero in
the normal fashion at specified discount rate.
© 2004 Pearson
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26. PV of a Perpetuity
• A Perpetuity is an annuity with an infinite life.
• Adjusting the PVIFA where n=¥ we have:
(5.28) PVIFAk,¥ = 1
k
© 2004 Pearson
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27. Investments Required to
Accumulate a Future Sum
• To determine the Payments necessary to
accumulate a Future Sum, we simply
rearrange the formula for the future value of
an annuity (5.15):
(5.30) PMT = FVAn
FVIFAk,n
© 2004 Pearson
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28. Loan Amortization
• Loan Amortization is the determination of
the equal annual loan payments necessary
to provide a lender with a specified interest
return and to repay the loan principal over a
specified period.
• Rearranging the formula for PVA (5.27):
(5.32) PMT = PVAn
PVIFAk,n
© 2004 Pearson
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29. Growth or Interest Rates
• It is often necessary to calculate the compound
annual growth rate of a series of cash flows.
• Either future value or present value interest
factors can be used depending on the situation.
• Financial Calculators can determine the precise
annual interest rate.
• This rate is called the Internal Rate of Return
(IRR).
© 2004 Pearson
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