Practical financial managment 7e LASHER chapter 6

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

6


The Future Value of an Amount

n

(1 + k) depends only on k and n

Define Future Value Factor for k,n as:
n
FVFk,n = (1 + k)

Substitute for:
FVn = PV[FVFk,n]

7


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.

8


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

12


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

14


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

15


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 FV 3, PV, and n, but
not k.
Use Equation 6.7
PV= FVn [PVFk,n ]

16


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

19


Figure 6-1 Future Value: Ordinary Annuity

20


Figure 6-2 Future Value: Annuity Due

21


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

22


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

23


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
– PMT
–k
–n

future value of the annuity
payment
interest rate
number of periods

Helps to draw a time line

25