Time Value of Money CH 4

1
Time Value of Money (CH 4)
TIP If you do not understand something, ask
me!

• Future value
• Present value
• Annuities
• Interest rates

2
Last week

• Objective of the firm
• Business forms
• Agency conflicts
• Capital budgeting decision and capital structure
  decision

3
The plan of the lecture

• Time value of money concepts
• present value (PV)
• discount rate/interest rate (r)
• Formulae for calculating PV of
• perpetuity
• annuity
• Interest compounding
• How to use a financial calculator

4
Financial choices

• Which would you rather receive today?
• TRL 1,000,000,000 ( one billion Turkish lira )
• USD 652.72 ( U.S. dollars )
• Both payments are absolutely guaranteed.
• What do we do?

5
Financial choices

• We need to compare apples to apples - this
  means we need to get the TRLUSD exchange rate
• From www.bloomberg.com we can see
• USD 1 TRL 1,637,600
• Therefore TRL 1bn USD 610.64

6
Financial choices with time

• Which would you rather receive?
• 1000 today
• 1200 in one year
• Both payments have no risk, that is,
• there is 100 probability that you will be paid

7
Financial choices with time

• Why is it hard to compare ?
• 1000 today
• 1200 in one year
• This is not an apples to apples comparison.
  They have different units
• 1000 today is different from 1000 in one year
• Why?
• A cash flow is time-dated money
• It has a money unit such as USD or TRL
• It has a date indicating when to receive money

8
Present value

• To have an apple to apple comparison, we
• convert future payments to the present values
• or convert present payments to the future values
• This is like converting money in TRL to money in
  USD

9
Some terms

• Finding the present value of some future cash
  flows is called discounting.
• Finding the future value of some current cash
  flows is called compounding.

10
What is the future value (FV) of an initial 100
after 3 years, if i 10?

• Finding the FV of a cash flow or series of cash
  flows is called compounding.
• FV can be solved by using the arithmetic,
  financial calculator, and spreadsheet methods.

11
Solving for FVThe arithmetic method

• After 1 year
• FV1 c ( 1 i ) 100 (1.10) 110.00
• After 2 years
• FV2 c (1i)(1i) 100 (1.10)2 121.00
• After 3 years
• FV3 c ( 1 i )3 100 (1.10)3 133.10
• After n years (general case)
• FVn C ( 1 i )n

12
Set up the Texas instrument

• 2nd, FORMAT, set DEC9, ENTER
• 2nd, FORMAT, move ? several times, make sure
  you see AOS, not Chn.
• 2nd, P/Y, set to P/Y1
• 2nd, BGN, set to END
• P/Yperiods per year,
• ENDcashflow happens end of periods

13
Solving for FVThe calculator method

• Solves the general FV equation.
• Requires 4 inputs into calculator, and it will
  solve for the fifth.

3
10
0
-100
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
133.10
14
What is the present value (PV) of 100 received
in 3 years, if i 10?

• Finding the PV of a cash flow or series of cash
  flows is called discounting (the reverse of
  compounding).
• The PV shows the value of cash flows in terms of
  todays worth.

0
1
2
3
10
PV ?
100
15
Solving for PVThe arithmetic method

• i interest rate, or discount rate
• PV C / ( 1 i )n
• PV C / ( 1 i )3
• 100 / ( 1.10 )3
• 75.13

16
Solving for PVThe calculator method

• Exactly like solving for FV, except we have
  different input information and are solving for a
  different variable.

3
10
0
100
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
-75.13
17
Solving for NIf your investment earns interest
of 20 per year, how long before your investments
double?
20
0
2
-1
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
3.8
18
Solving for iWhat interest rate would cause
100 to grow to 125.97 in 3 years?
3
0
125.97
-100
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
8
19
Now lets study some interesting patterns of cash
flows

• Perpetuity
• Annuity

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ordinary annuity and annuity due
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Value an ordinary annuity

• Here C is each cash payment
• n is number of payments
• If youd like to know how to get the formula
  below, see me after class.

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Example

• you win the 1million dollar lottery! but wait,
  you will actually get paid 50,000 per year for
  the next 20 years if the discount rate is a
  constant 7 and the first payment will be in one
  year, how much have you actually won?

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Using the formula
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Solving for FV3-year ordinary annuity of 100
at 10

• 100 payments occur at the end of each period.
  Note that PV is set to 0 when you try to get FV.

3
10
-100
0
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
331
25
Solving for PV3-year ordinary annuity of 100
at 10

• 100 payments still occur at the end of each
  period. FV is now set to 0.

3
10
100
0
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
-248.69
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Solving for FV3-year annuity due of 100 at 10

• 100 payments occur at the beginning of each
  period.
• FVAdue FVAord(1i) 331(1.10) 364.10.
• Alternatively, set calculator to BEGIN mode and
  solve for the FV of the annuity

BEGIN
3
10
-100
0
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
364.10
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Solving for PV3-year annuity due of 100 at 10

• 100 payments occur at the beginning of each
  period.
• PVAdue PVAord(1I) 248.69(1.10) 273.55.
• Alternatively, set calculator to BEGIN mode and
  solve for the PV of the annuity

BEGIN
3
10
100
0
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
-273.55
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What is the present value of a 5-year 100
ordinary annuity at 10?

• Be sure your financial calculator is set back to
  END mode and solve for PV
• N 5, I/YR 10, PMT 100, FV 0.
• PV 379.08

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What if it were a 10-year annuity? A 25-year
annuity? A perpetuity?

• 10-year annuity
• N 10, I/YR 10, PMT 100, FV 0 solve for
  PV 614.46.
• 25-year annuity
• N 25, I/YR 10, PMT 100, FV 0 solve for
  PV 907.70.
• Perpetuity (Ninfinite)
• PV PMT / i 100/0.1 1,000.

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What is the present value of a four-year
annuity of 100 per year that makes its first
payment two years from today if the discount rate
is 9?  
100 100 100 100
323.97
297.22
0 1 2 3 4
5
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What is the PV of this uneven cash flow stream?
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Solving for PVUneven cash flow stream

• Input cash flows in the calculators CF
  register
• CF0 0
• CF1 100
• CF2 300
• CF3 300
• CF4 -50
• Enter I/YR 10, press NPV button to get NPV
  530.09. (Here NPV PV.)

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Detailed steps (Texas Instrument calculator)

• To clear historical data
• CF, 2nd ,CE/C
• To get PV
• CF , ?,100 , Enter , ?,? ,300 , Enter, ?,2,
• Enter, ?, 50, /-,Enter, ?,NPV,10,Enter, ?,CPT
• NPV530.0867427

34
The Power of Compound Interest

• A 20-year-old student wants to start saving for
  retirement. She plans to save 3 a day. Every
  day, she puts 3 in her drawer. At the end of
  the year, she invests the accumulated savings
  (1,0953365) in an online stock account. The
  stock account has an expected annual return of
  12.
• How much money will she have when she is 65 years
  old?

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Solving for FVSavings problem

• If she begins saving today, and sticks to her
  plan, she will have 1,487,261.89 when she is 65.

45
12
-1095
0
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
1,487,262
36
Solving for FVSavings problem, if you wait
until you are 40 years old to start

• If a 40-year-old investor begins saving today,
  and sticks to the plan, he or she will have
  146,000.59 at age 65. This is 1.3 million less
  than if starting at age 20.
• Lesson It pays to start saving early.

25
12
-1095
0
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
146,001
37
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.

Annually FV3 100(1.10)3 133.10
Semiannually FV6 100(1.05)6 134.01
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What is the FV of 100 after 3 years under 10
semiannual compounding? Quarterly compounding?
39
Classifications of interest rates

• 1. Nominal rate (iNOM) also called the APR,
  quoted rate, or stated rate. An annual rate that
  ignores compounding effects. Periods must also be
  given, e.g. 8 Quarterly.
• 2. Periodic rate (iPER) amount of interest
  charged each period, e.g. monthly or quarterly.
• iPER iNOM / m, where m is the number of
  compounding periods per year. e.g., m 12 for
  monthly compounding.

40
Classifications of interest rates

• 3. Effective (or equivalent) annual rate (EAR,
  also called EFF, APY) the annual rate of
  interest actually being earned, taking into
  account compounding.
• If the interest rate is compounded m times in a
  year, the effective annual interest rate is

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Example, EAR for 10 semiannual investment

• EAR ( 1 0.10 / 2 )2 1 10.25
• An investor would be indifferent between an
  investment offering a 10.25 annual return, and
  one offering a 10 return compounded
  semiannually.

42
EAR on a Financial Calculator
Texas Instruments BAII Plus
43
Why is it important to consider effective rates
of return?

• An investment with monthly payments is different
  from one with quarterly payments.
• Must use EAR for comparisons.
• If iNOM10, then EAR for different compounding
  frequency
• Annual 10.00
• Quarterly 10.38
• Monthly 10.47
• Daily 10.52

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If interest is compounded more than once a year

• EAR (EFF, APY) will be greater than the nominal
  rate (APR).

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47
Whats the FV of a 3-year 100 annuity, if the
quoted interest rate is 10, compounded
semiannually?

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

48
Method 1Compound each cash flow

• FV3 100(1.05)4 100(1.05)2 100
• FV3 331.80

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Method 2Financial calculator

• Find the EAR and treat as an annuity.
• EAR ( 1 0.10 / 2 )2 1 10.25.

3
10.25
-100
0
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
331.80
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When is periodic rate used?

• iPER is often useful if cash flows occur several
  times in a year.

51
Exercise
You agree to lease a car for 4 years at 300
per month. You are not required to pay any money
up front or at the end of your agreement. If
your discount rate is 0.5 per month, what is the
cost of the lease?
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Solution