# TVM

Document Sample

```					  Time Value of Money

Future Value and Compounding
Present Value and Discounting
Annuity and Perpetuity
Amortization Loans
Different Compounding Periods
1
Principle of Time Value
Time is money.
A dollar today is more valuable than a
dollar tomorrow
Why?
Money can earn a positive rate of return
at no risk.

2
Time Line

0           1           2           3
i%

CF0          CF1        CF2         CF3

Time lines show the timing of cash flows.
Tick marks are at the end of periods. Time 0 is
today; Time 1 is placed at the end of Period 1
or at the beginning of Period 2.

3
Time line for a \$100 cash flow
received at the end of Year 2

0           1           2 Year
i%

100

4
Time line for equal payments of
\$100 for 3 years

0         1         2             3
i%

-100     -100        -100

5
Time line for an uneven cash flow
stream

0           1       2          3
i%

-150        100      75         50

6
Time Value of Money

One of the most important
fundamental concepts in finance.
A dollar in hand today is worth
more than a dollar to be received
in the future.

FV = PV  (1 + i )n
7
What’s the value of an initial \$100
deposit after 3 years if i = 10%?

0            1          2           3
10%

100                              FV = ?

The process of finding FVs is
called compounding.

8
After 1 year:
FV1 = PV + INT1 = PV + PV  i
= PV  (1 + i )
= \$100  (1.10)
= \$110.00

9
After 2 years:
FV2 = FV1  (1 + i )
= 110  (1.10)
= \$100  (1.10)2
= PV  (1 + i )2
= \$121.00

10
After 3 years:
FV3 =121(1 +0.10 )
=100(1 +0.10 )2 (1+0.10)
=100(1.10)3
=PV(1 + i )3
=\$133.10
In general:

FVn = PV  (1 + i )n
11
Future value equation

FVn = PV  (1 + i )n
There are 4 variables in the
equation: FVn , PV, i and n. If any 3
variables are known, the calculator
will solve for the 4th.

12
The setup to find FV:

INPUTS   3     10    100    0
N    I/YR   PV    PMT     FV

OUTPUT                           -133.10

13
How much should you save now to
have \$100 in 3 years if i = 10%?

0           1        2           3
10%

PV = ?                             100

Finding PVs is discounting, and it’s
the reverse of compounding.
14
Present Value Equation

PV = FVn  (1 + i )n

3
 1   
PV = \$100  
 1.10 
= \$100  0.7513 = \$75.13

15
Financial Calculator Solution

INPUTS   3    10              0    100
N   I/YR    PV      PMT   FV

OUTPUT              -75.13

16
What interest rate would cause \$100 to
grow to \$125.97 in 3 years?

Solve for i :

100 (1 + i )3 = \$125.97

INPUTS    3           -100      0   125.97
N    I/YR   PV     PMT     FV

OUTPUT         8.0
17
If sales grow at 20% per year, how
long before sales double?

Solve for n:
FVn = PV  (1 + i )n
2 = 1  (1.20)n

INPUTS          20     -1      0    2
N    I/YR   PV      PMT   FV

OUTPUT 3.8

18
Definition of Annuity

Annuity: a series of cash flows of an
equal amount at fixed intervals for a
specified number of periods.
Ordinary Annuity: an annuity whose
payments occur at the end of each
period.
Annuity Due: an annuity whose
payments occur at the beginning of
each period.
19
What’s 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
20
What’s the FV of a 3-year ordinary
annuity of \$100 at 10%?

0          1       2          3
10%

100     100        100
110
121
FV = 331

21
Financial Calculator Solution

INPUTS   3    10    0    -100
N   I/YR   PV   PMT      FV

OUTPUT                          331.00

22
What’s the PV of this ordinary
annuity?

0            1     2             3
10%

100   100        100
90.91
82.64
75.13
248.68 = PV

23
Financial Calculator Solution

INPUTS   3    10              100   0
N   I/YR     PV      PMT   FV

OUTPUT              -248.69

24
Loan Analysis
Loan amount=PV of all payments
discounted at the contractual interest rate

25
Amortization Loans

Amortized Loan: a loan that is
repaid in equal payments over its
life.
Each periodic payment includes
not only interest but also a portion
of principal.

26
If you borrow a 3-year, \$1,000, 10%
amortized loan, how much do you need
to pay every year?

0                1               2                  3
10%

1,000             PMT         PMT                PMT

INPUTS      3         10    1000                0
N        I/YR   PV        PMT      FV

OUTPUT                               -402.11
27
Loan= \$1,000
PV of all payments discounted at 10%=Loan of \$,1000

0               1               2                  3
10%

1,000             PMT         PMT                PMT

INPUTS      3         10    1000                0
N        I/YR   PV        PMT      FV

OUTPUT                               -402.11
28
Amortization Schedule

 Amortization  Schedule is a table
that shows precisely how a loan
will be repaid.
 Construct a amortization
schedule for the 3-year, \$1,000,
10% amortized loan.

29
Find interest charge for Year 1

Interest charge
= Initial balance  interest rate
= \$1,000  (0.10) = \$100

30
Find principal repayment in Year 1

Principal repayment
= Total payment – interest charge
= \$402.11 - \$100 = \$302.11

31
Find end balance for Year 1

End balance
= Initial balance - principal repayment
= \$1,000 - \$302.11 = \$697.89

Repeat these steps for Years 2 and 3
to complete the amortization table.

32
Amortization Schedule

BEG                     PRIN    END
YR   BAL      PMT     INT    PMT     BAL

1    \$1,000   \$402    \$100   \$302    \$698
2       698     402     70     332    366
3       366     402     36     366      0
1,206    206   1,000

33
\$
402.11
Interest charge

302.11

Principal repayments

0        1            2      3
Level payments. Interest declines because
outstanding balance declines. Lender earns
10% on loan outstanding, which is falling.
34
Different Compounding Periods

Will the FV of a lump sum be larger or
smaller if we compound more often,
holding the stated i% constant?

LARGER! If compounding is more
frequent than once a year--for
example, semiannually, quarterly, or
daily--interest is earned on interest
more often.
35
Annual vs. Semiannual
Compounding
0             1       2            3
10%

100                                133.10

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

100                                ?
36
Annual vs. Semiannual
Compounding

INPUTS   6    5     100    0
N   I/YR   PV    PMT     FV

OUTPUT                          -134.01

37
Effective Annual Rate

EAR is the interest rate which
causes PV to grow to the same FV
as under multi-period compounding.
EAR is the actual rate of return
investors earn, or the actual rate of
interest borrowers pay.

38
Effective Annual Rate

The return on an investment with
monthly payments is different from
one with the same nominal value
but quarterly payments.
We must convert both into EAR
basis to compare the rates of
return.

39
Calculating Effective Annual Rate

m
    iNom 
EAR   = 1 +       -1
     m 
    0 . 10  2
= 1 +         - 1 .0
       2 
= ( . 05 )2 - 1 . 0
1
= 0 . 1025 = 10 . 25 %
40
Calculating Effective Annual Rate

Nominal Rate                        = 10%

EARAnnual                           = 10%

EARQ        = (1 + 0.10/4)4 - 1     = 10.38%

EARM        = (1 + 0.10/12)12 - 1   = 10.47%

EARD(365) = (1 + 0.10/365)365 - 1 = 10.52%

41
Calculating annual nominal rate (inom) from
effective compounding period rate (keff)

inom=keffxm
Also,
keff=inom/m

42
Perpetuities

A perpetuity is an annuity that
continues forever.
The present value of a perpetuity is:

PMT
PV =
i
43
Growing Perpetuities

The cash flows of a growing perpetuity
grow at a constant rate forever.
The present value of a growing
perpetuity is:

PMT1
PV =
i-g
44

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 5 posted: 6/30/2011 language: English pages: 44
How are you planning on using Docstoc?