Practical financial manaegment lasher 7th ed chapter 06

Chapter 6 - Time Value of Money


Time Value of Money
A sum of money in hand today is worth more than the
same sum promised with certainty in the future.
Think in terms of money in the bank
The value today of a sum promised in a year is the amount
you'd have to put in the bank today to have that sum in a
year.
Example:

Future Value (FV) = $1,000
k = 5%
Then Present Value (PV) = $952.38 because
$952.38 x .05 = $47.62
and $952.38 + $47.62 = $1,000.00


Time Value of Money
Present Value
– The amount that must be deposited today to
have a future sum at a certain interest rate

Terminology
– The discounted value of a future sum is its
present value

3

Outline of Approach
Four different types of problem
– Amounts

– Annuities

Present value

Present value

Future value

Future value

4


Outline of Approach
Develop an equation for each
Time lines - Graphic portrayals
Place information on the time line

5


The Future Value of an Amount
How much will a sum deposited at interest rate k
grow into over some period of time
If the time period is one year:
FV1 = PV(1 + k)

If leave in bank for a second year:
FV2 = PV(1 + k)(1 ─ k)
FV2 = PV(1 + k)2
Generalized:
FVn = PV(1 + k)n

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The Future Value of an Amount
(1 + k)n depends only on k and n
Define Future Value Factor for k,n as:
FVFk,n = (1 + k)n
Substitute for:
FVn = PV[FVFk,n]
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The Future Value of an Amount
Problem-Solving Techniques
– All time value equations contain four
variables
In this case PV, FVn, k, and n

Every problem will give you three and
ask for the fourth.
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Concept Connection Example 6-1
Future Value of an Amount
How much will $850 be worth in three years at 5% interest?
Write Equation 6.4 and substitute the amounts given.
FVn = PV [FVFk,n ]
FV3 = $850 [FVF5,3]


Concept Connection Example 6-1
Future Value of an Amount
Look up FVF5,3 in the three-year row under the 5%
column of Table 6-1, getting 1.1576


Concept Connection Example 6-1
Future Value of an Amount
Substitute the future value factor of 1.1576 for FVF5,3

FV3 = $850 [FVF5,3]
FV3 = $850 [1.1576]
= $983.96


Financial Calculators
Work directly with equations
How to use a typical financial calculator
– Five time value keys
Use either four or five keys

– Some calculators require inflows and

outflows to be of different signs
If PV is entered as positive the computed FV is
negative
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Financial Calculators
Basic Calculator functions


Financial Calculators
What is the present value of $5,000 to be received in one
year if the interest rate is 6%?
Input the following values on the calculator and compute
the PV:

N

1

I/Y

6

FV

5000

PMT

0

PV

4,716.98

Answer

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The Present Value of an Amount
FVn = PV ( 1+k )

n

Solve for PV


1
PV = FVn 

n
1 +k ) 

(

1 4 2 43
Interest Factor

FVFk,n

1
=
PVFk,n

PV= FVn [PVFk,n ]
Future and present value factors are reciprocals
– Use either equation to solve any amount problems
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Concept Connection Example 6-3
Finding the Interest Rate
Finding the Interest Rate
what interest rate will grow $850 into $983.96 in three
years. Here we have FV3, PV, and n, but not k.
Use Equation 6.7
PV= FVn [PVFk,n ]

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Concept Connector Example 6-3
PV= FVn [PVFk,n ]
Substitute for what’s known
$850= $983.96 [PVFk,n ]
Solve for [PVFk,n ]
[PVFk,n ] = $850/ $983.96
[PVFk,n ] = .8639

Find .8639 in Appendix A (Table A-2). Since n=3 search only row 3,
and find the answer to the problem is (5% ) at top of column.


Concept Connection Example 6-3
Finding the Interest Rate


Annuity Problems
Annuities
– A finite series of equal payments separated
by equal time intervals
Ordinary annuities
Annuities due

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Figure 6-1 Future Value: Ordinary
Annuity

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Figure 6-2 Future Value: Annuity Due

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The Future Value of an Annuity—

Developing a Formula
Future value of an annuity
– The sum, at its end, of all payments and all
interest if each payment is deposited when
received
– Figure 6-3 Time Line Portrayal of an
Ordinary Annuity

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Figure 6-4 Future Value of a Three-Year
Ordinary Annuity

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For a 3-year annuity, the formula is:
FVA = PMT ( 1+k ) + PMT ( 1+k ) + PMT ( 1+k )
0

1

2

Generalizing the Expression:
FVA n = PMT ( 1+k ) + PMT ( 1+k ) + PMT ( 1+k ) + L + PMT ( 1+k )
0

1

2

which can be written more conveniently as:
n

FVA n = ∑ PMT ( 1+k )

n−i

i=1

Factoring PMT outside the summation, we obtain:
FVA n = PMT

n

∑ ( 1+k )
i=1

n−i

FVFAk,n

n -1


The Future Value of an Annuity—
Solving Problems
Four variables in the future value of an

annuity equation
– FVAn

future value of the annuity

– PMT

payment

–k

interest rate

–n

number of periods
Helps to draw a time line

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