# Chapter 6 Time Value of Money - PowerPoint

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```					CHAPTER 6
Time Value of Money
   Future value
   Present value
   Annuities
   Rates of return
   Amortization
6-1
A Fundamental Principle of
Finance: The Time Value of \$
   Would you be indifferent between
receiving \$1 today or receiving the \$1 a
year from now?
   Text: “A dollar today is worth more
than a dollar tomorrow.”
   Why? The time value of money: A
dollar today can be invested at interest
and be worth more tomorrow.
6-2
Future Value of an Amount
Invested Today
PV  Present Value : an amount today
FVn  Future Value : an amount n periods in the future
i  The interest rate (or rate of return) per period

FVn  PV  accumulated interest
FV1  PV  iPV
FV1  PV (1  i )1
FV2  FV1 (1  i )1
 PV (1  i )1 (1  i )1
 PV (1  i ) 2
FV3  FV2 (1  i )1
 PV (1  i ) 2 (1  i )1
 PV (1  i ) 3
Generalizing :
FVn  PV (1  i ) n

6-3
Time lines
0                1            2             3
i%

CF0             CF1         CF2            CF3

    Show the timing of cash flows.
    Tick marks occur at the end of periods, so
Time 0 is today; Time 1 is the end of the
first period (year, month, etc.) or the
beginning of the second period.
6-4
Drawing time lines:
\$100 lump sum due in 2 years;
3-year \$100 ordinary annuity

\$100 lump sum due in 2 years
0                 1             2
i%

100
3 year \$100 ordinary annuity
0             1           2           3
i%

100        100          100
6-5
Drawing time lines:
Uneven cash flow stream; CF0 = -\$50,
CF1 = \$100, CF2 = \$75, and CF3 = \$50

Uneven cash flow stream
0           1            2       3
i%

-50         100           75      50

6-6
What is the future value (FV) of an initial
\$100 after 3 years, if I/YR = 10%?

   Finding the FV of a cash flow or series of
cash flows when compound interest is
applied is called compounding.
   FV can be solved by using the arithmetic,
methods.
0                 1            2              3
10%

100                                        FV = ?
6-7
Solving for FV:
The arithmetic method
   After 1 year:
 FV1 = PV ( 1 + i ) = \$100 (1.10)
= \$110.00
   After 2 years:
 FV2 = PV ( 1 + i ) = \$100 (1.10)
2              2
=\$121.00
   After 3 years:
 FV3 = PV ( 1 + i ) = \$100 (1.10)
3              3
=\$133.10
   After n years (general case):
 FVn = PV ( 1 + i )
n
6-8
The “Future Value Interest
Factor”

In the equation, FVn  PV (1  i ) n
The term,(1  i ) n is the Future Value in n periods of \$1 invested today
at an interest rate of i per period.

We call this term the " Future Value Interest Factor"( FVIF, n )
i

Values for FVIF, n can be found in a table.
i

6-9
Future Value: Tabular Method

FVn  PV (1  i )   n

 PV  FVIF,n
i

 \$100 * FVIF %, 3
10

 \$100 *1.3310
 \$133.10
6-10
Compound Interest
18
16           10% Simple
14
12           10% Compound
FV of \$1

10
8
6
4
2
0
12

15

18

21

24

27

30
0

3

6

9

Number of Years
6-11
The Present Value of a Future
Amount
The future value at the end of period n of an amount today is :
FVn  PV (1  i ) n
If wesolve for PV , we get :
FVn
PV 
(1  i ) n
1
 FVn
(1  i ) n
1
The term,             , is the PV of \$1 received n periods in the future
(1  i ) n
We designate this term as the " Present Value Interest Factor"( PVIF, n ).
i

Tables are available to determine the values of PVIF, n
i

6-12
What is the present value (PV) of \$100
due in 3 years, if I/YR = 10%?
   Finding the PV of a cash flow or series of
cash flows when compound interest is
applied is called discounting (the reverse of
compounding).
   The PV shows the value of cash flows in
0               1            2              3
10%

PV = ?                                      100
6-13
Solving for PV:
The arithmetic method
   Solve the general FV equation for PV:
   PV = FVn / ( 1 + i )n

   PV = FV3 / ( 1 + i )3
= \$100 / ( 1.10 )3
= \$75.13

6-14
Solving for PV: The Tabular
Method
FVn
PV 
(1  i ) n
1
PV  FVn *
(1  i ) n
PV  FVn * PVIF, ni

PV  FV3 * PVIF %,3
10

PV  \$100 * .7513
 \$75.13
6-15
Lessons of the PVIF Table
   As the discount rate, i, goes up, the PV of \$1
goes down.
   As the discount rate, i, goes down, the PV of
\$1 goes up.
   As n increases, the PV of \$1 decreases.
   As n decreases, the PV \$1 increases.
   Overall conclusion: The “value” - what you’d
pay today - of a future \$1 is affected by i and
n.
6-16
Solving for n:
If sales grow at 20% per year, how long
before sales double?

Tabular Method:
FVn  PV (1  i ) n
FVn  PV * FVIFi , n
\$2    \$1 * FVIF20% , n
FVIF20%, n  2.0
Look in the FVIF table for 2.0 in the 20% column :
n  slightly less than 4.0 years (3.8 years with interpolation)

6-17
Annuities
   An annuity is a series of equal
payments made at fixed intervals for a
specified number of periods.
   Most common type is an ordinary or
at the end of each period.
   An annuity due has payments made at
the beginning of each period.

6-18
What is the difference between an
ordinary annuity and an annuity due?
Ordinary Annuity
0            1      2          3
i%

PMT    PMT        PMT
Annuity Due
0            1      2          3
i%

PMT          PMT    PMT
6-19
Solving for FVA:
3-year ordinary annuity of \$100 at 10%

Tabular Method:
n
FVAn  PMT (1  i ) n t
t 1
n
The term,  (1  i) n - t is the FV of an n period ordinary annuity of \$1 at an
t 1

interest rate of i per period.

We designate this term FVIFA, n and the values are available in tables.
i
n
(1  i ) n  1
FVIFA, n   (1  i )
i
n t

t 1                        i
FVAn  PMT * FVIFA, n
i

FVA3  \$100 * FVIFA %, 3
10

 \$100 * 3.3100
 \$331.00
6-20
Solving for PVA:
3-year ordinary annuity of \$100 at 10%

Tabular Method:
n
1
PVAn  PMT
t 1   (1  i ) t
1
1
n
1                   (1  i ) n       1      1
PVIFA, n                                                  
(1  i )                                i i (1  i ) n
i                       t
t 1                            i
Therefore:
PVAn  PMT * PVIFA, n
i

PVA3  \$100 * PVIFA %, 3
10

 \$100 * 2.4869
 \$248.69
6-21
What is the PV of this uneven
cash flow (CF) stream?

0         1   2     3         4
10%

100     300   300       -50
90.91
247.93
225.39
-34.15
530.08 = PV
6-22
PV of an Uneven Cash Flow
Stream of n periods: The Math

CF1         CF2               CFn
PV                         
(1  i) 1
(1  i) 2
(1  i ) n

n
CFt

t 1 (1  i) t

6-23
“The 4-column setup” to calculate PV of
Uneven Cash Flow Stream

Year (t)      CFt    PVIF @   PV
10%
1           100     .9091      90.91
2           300     .8264     247.93
3           300     .7513     225.39
4           -50     .6830     -34.15
Total                     530.08
6-24
Solving for i:
What interest rate would cause \$100 to
grow to \$125.97 in 3 years?

Tabular Method:

FVn  PV (1  i ) n
FVn  PV * FVIF, n
i

125.97  100 * FVIF, 3
i

FVIF, 3  1.2597
i

Look in the FVIF table for 1.2597 in the n  3 row :
i  8%

6-25
The Present Value of a
Perpetuity
   If you have an ordinary annuity where n is
infinite or very large, you have a “perpetuity.”
   British Consul bonds are perpetuities.
Perpetuities are also used to value
commercial real estate projects and corporate
acquisitions.

PV  PMT
i
6-26
Semiannual and Other
Compounding Periods
   If interest is paid more frequently than
annually, how do you calculate FVn and PV?
   Let m = number of compounding periods per
year
   Let n = number of years until future pmt. is
i mn
FVn  PV (1  )
m

FVn                   1
PV                  FVn
i mn                  i mn
(1  )               (1  )
m                    m       6-27
What is the FV of \$100 after 3 years under
10% semiannual compounding? Quarterly
compounding?
iNOM mn
FVn  PV ( 1      )
m

0.10 23
FV3S  \$100 ( 1      )
2
6
FV3S  \$100 (1.05)  \$134.01
FV3Q  \$100 (1.025)12  \$134.49
6-28
Compounding Intervals
i        ii         iii       iv                     v
Periods Interest              Value                  Annually
per     per        APR        after                  compounded
year    period     (i x ii)   one year               interest rate

1         6%         6%       1.06                          6.000%

2         3          6        1.032   = 1.0609              6.090
4         1.5        6        1.0154 = 1.06136              6.136

12            .5     6        1.00512 = 1.06168             6.168
52      .1154        6        1.00115452 = 1.06180          6.180

365     .0164        6        1.000164365 = 1.06183         6.183

6-29
What’s the FV of a 3-year \$100
annuity, if the quoted interest rate is
10%, compounded semiannually?
1          2            3
0            1    2    3     4      5     6
5%

100        100         100

   Payments occur annually, but compounding
occurs every 6 months.
   Cannot use normal annuity valuation
techniques.

6-30
Compound each cash flow
1         2          3
0         1    2    3    4    5     6
5%

100       100       100
110.25
121.55
331.80

FV3 = \$100(1.05)4 + \$100(1.05)2 + \$100
FV3 = \$331.80
6-31
Will the FV of a lump sum be larger or
smaller if compounded more often,
holding the stated I% constant?
   LARGER, as the more frequently compounding
occurs, interest is earned on interest more often.
0                1             2                     3
10%

100                                                  133.10
Annually: FV3 = \$100(1.10)3 = \$133.10
0                    1              2                3
0            1       2      3       4      5         6
5%

100                                                 134.01
Semiannually: FV6 = \$100(1.05)6 = \$134.01
6-32
Loan amortization
   Amortization tables are widely used for
loans, retirement plans, etc.
   Financial calculators and spreadsheets are
great for setting up amortization tables.

   EXAMPLE: Construct an amortization
schedule for a \$1,000, 10% annual rate
loan with 3 equal payments.

6-33
Step 1: Find the Annual Payment (PMT)
Using the PVIFA Table

PVAn  PMT * PVIFA, n
i

\$1000  PMT * PVIFA %, 3
10

\$1000  PMT * 2.4869
PMT  \$402.11

6-34
Step 2:
Find the interest paid in Year 1
   The borrower will owe interest upon the
initial balance at the end of the first
year. Interest to be paid in the first
year can be found by multiplying the
beginning balance by the interest rate.

INTt = Beg balt (i)
INT1 = \$1,000 (0.10) = \$100
6-35
Step 3:
Find the principal repaid in Year 1
   If a payment of \$402.11 was made at
the end of the first year and \$100 was
paid toward interest, the remaining
value must represent the amount of
principal repaid.

PRIN= PMT – INT
= \$402.11 - \$100 = \$302.11
6-36
Step 4:
Find the ending balance after Year 1
   To find the balance at the end of the
period, subtract the amount paid
toward principal from the beginning
balance.

END BAL = BEG BAL – PRIN
= \$1,000 - \$302.11
= \$697.89
6-37
Step 5: Construct an amortization
table: Repeat steps 2 – 4 until end of
loan
Year    BEG BAL PMT          INT        PRIN   END
BAL
1        \$1,000      \$402      \$100       \$302    \$698
2           698       402          70      332    366
3           366       402          37      366       0
TOTAL             1,206.34    206.34      1,000      -

6-38
Illustrating an amortized payment:
Where does the money go?
\$
402.11
Interest

302.11

Principal Payments

0        1              2   3
   Constant payments.
   Declining interest payments.
   Declining balance.
6-39

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