Future Value and Compounding

1
Time Value of Money

• Future Value and Compounding
• Present Value and Discounting
• Annuity and Perpetuity
• Amortization Loans
• Different Compounding Periods

2
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.

3
Time Line
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.
4
Time line for a 100 cash flow received at the
end of Year 2
100
5
Time line for equal payments of 100 for 3 years
6
Time line for an uneven cash flow stream
0
1
2
3
i
100
50
75
-150
7
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
8
Whats the value of an initial 100 deposit after
3 years if i 10?
The process of finding FVs is called compounding.
9
After 1 year
FV1 PV INT1 PV PV ? i PV ? (1 i )
100 ? (1.10) 110.00
10
(No Transcript)
11
After 3 years
FV3 121?(1 0.10 ) 100?(1 0.10 )2
?(10.10) 100?(1.10)3 PV?(1 i )3 133.10
12
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.
13
The setup to find FV
3 10 100 0
-133.10
14
How much should you save now to have 100 in 3
years if i 10?
Finding PVs is discounting, and its the reverse
of compounding.
15
Present Value Equation
PV FVn ? (1 i )n
16
Financial Calculator Solution
3 10 0 100
-75.13
17
What interest rate would cause 100 to grow to
125.97 in 3 years?
3 -100 0 125.97
8.0
18
If sales grow at 20 per year, how long before
sales double?
20 -1 0 2
3.8
19
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.

20
Whats the difference between an ordinary annuity
and an annuity due?
21
Whats the FV of a 3-year ordinary annuity of
100 at 10?
22
Financial Calculator Solution
3 10 0 -100
331.00
23
Whats the PV of this ordinary annuity?
24
Financial Calculator Solution
3 10 100 0
-248.69
25
Loan Analysis

• Loan amountPV of all payments discounted at the
  contractual interest rate

26
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.

27
If you borrow a 3-year, 1,000, 10 amortized
loan, how much do you need to pay every year?
3 10 1000 0
-402.11
28
Loan 1,000 PV of all payments discounted at
10Loan of ,1000
3 10 1000 0
-402.11
29
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.

30
Find interest charge for Year 1
Interest charge Initial balance ? interest
rate 1,000 ? (0.10) 100
31
Find principal repayment in Year 1
Principal repayment Total payment interest
charge 402.11 - 100 302.11
32
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.
33
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
34

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.
35
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.
36
Annual vs. Semiannual Compounding
37
Annual vs. Semiannual Compounding
6 5 100 0
-134.01
38
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.

39
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.

40
Calculating Effective Annual Rate
41
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
42
Calculating annual nominal rate (inom) from
effective compounding period rate (keff)

• inomkeffxm
• Also,
• keffinom/m

43
Perpetuities

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

44
Growing Perpetuities

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