Transcript Time Value of Money - Georgia College & State University

Chapter 4
Time Value of Money
1
Time Value Topics




Future value
Present value
Rates of return
Amortization
2
Determinants of Intrinsic Value:
The Present Value Equation
Net operating
profit after taxes
Free cash flow
(FCF)
Value =
Required investments
in operating capital
−
=
FCF1
FCF2
FCF∞
... +
+
+
(1 + WACC)1
(1 + WACC)2
(1 + WACC)∞
Weighted average
cost of capital
(WACC)
Cost of debt
Cost of equity
Why is timing important?
You are asked to choose from the
following options:
1. Receive $1 million today
2. Receive $1 million 10 years from now
Would you choose 1 or 2?
4
Money has time value

Most people prefer to receive it sooner
rather than later because they place a
higher value on the cash received
earlier.
5
Time value of money:
Practical relevance
Examples
 Retirement plan
 Mortgage payment
 Pricing a financial securities
 Helping your company to decide which
project to undertake
6
Time lines show timing of cash
flows.
0
1
2
3
CF1
CF2
CF3
I%
CF0
Tick marks at ends of periods, so Time 0
is today; Time 1 is the end of Period 1; or
the beginning of Period 2.
7
Time line for a $100 lump sum
due at the end of Year 2.
0
I%
1
2 Year
100
8
Time line for an ordinary
annuity of $100 for 3 years
0
I%
1
2
3
100
100
100
9
Time line for uneven CFs
0
-50
I%
1
2
3
100
75
50
10
Preparing BAII Plus for use



Press ‘2nd’ and [Format]. The screen will
display the number of decimal places that the
calculator will display. If it is not eight, press
‘8’ and then press ‘Enter’.
Press ‘2nd’ and then press [P/Y]. If the
display does not show one, press ‘1’ and then
‘Enter’.
Press ‘2nd’ and [BGN]. If the display is not
END, that is, if it says BGN, press ‘2nd’ and
then [SET], the display will read END.
11
FV of an initial $100 after
3 years (I = 10%)
0
1
2
3
10%
100
FV = ?
Finding FVs (moving to the right
on a time line) is called compounding.
12
After 1 year
FV1 =
=
=
=
PV + INT1 = PV + PV (I)
PV(1 + I)
$100(1.10)
$110.00
13
After 2 years
FV2 =
=
=
=
FV1(1+I) = PV(1 + I)(1+I)
PV(1+I)2
$100(1.10)2
$121.00
14
After 3 years
FV3 =
=
=
=
FV2(1+I)=PV(1 + I)2(1+I)
PV(1+I)3
$100(1.10)3
$133.10
In general,
FVN = PV(1 + I)N
15
Four Ways to Find FVs




Step-by-step approach using time line
(as shown in Slides 12-15).
Solve the equation with a regular
calculator (formula approach).
Use a financial calculator.
Use a spreadsheet.
16
Financial Calculator Solution
Financial calculators solve this
equation:
FVN = PV (1+I)N
There are 5 variables. If 4 are
known, the calculator will solve for
the 4th.
17
Here’s the setup to find FV
INPUTS
OUTPUT
3
N
10
-100
I/YR PV
0
PMT
FV
133.10
18
Spreadsheet Solution



Use the FV function
= FV(I, N, PMT, PV)
= FV(0.10, 3, 0, -100) = 133.10
19
What’s the PV of $100 due in
3 years if I/YR = 10%?
Finding PVs is discounting, and it’s the
reverse of compounding.
0
PV = ?
10%
1
2
3
100
20
Solve FVN = PV(1 + I )N for PV
PV =
FVN
(1+I)N
= FVN
1
PV = $100
1.10
1
1+I
N
3
= $100(0.7513) = $75.13
21
Financial Calculator Solution
INPUTS
OUTPUT
3
N
10
I/YR
PV
-75.13
0
PMT
100
FV
Either PV or FV must be negative. Here
PV = -75.13. Put in $75.13 today, take
out $100 after 3 years.
22
Spreadsheet Solution



Use the PV function:
= PV(I, N, PMT, FV)
= PV(0.10, 3, 0, 100) = -75.13
23
Finding the Time to Double
0
-1
20%
1
2
?
2
Q: if deposit $1 today, and i=20%, when will it double?
24
Time to Double (Continued)
$2
(1.2)N
N LN(1.2)
N
N
=
=
=
=
=
$1(1 + 0.20)N
$2/$1 = 2
LN(2)
LN(2)/LN(1.2)
0.693/0.182 = 3.8
25
Financial Calculator Solution
INPUTS
N
OUTPUT 3.8
20
I/YR
-1
PV
0
PMT
2
FV
26
Spreadsheet Solution



Use the NPER function:
= NPER(I, PMT, PV, FV)
= NPER(0.10, 0, -1, 2) = 3.8
27
Finding the interest rate
0
-1
?%
1
FV =
$2 =
(2)(1/3) =
1.2599 =
I=
2
I)N
PV(1 +
$1(1 + I)3
(1 + I)
(1 + I)
0.2599 = 25.99%
3
2
28
Financial Calculator
INPUTS
OUTPUT
3
N
I/YR
25.99
-1
PV
0
PMT
2
FV
29
Spreadsheet Solution

Use the RATE function:

= RATE(N, PMT, PV, FV)

= RATE(3, 0, -1, 2) = 0.2599
30
Exercises



Suppose you deposit $150 in an account today and the interest
rate is 6 percent p.a.. How much will you have in the account
at the end of 33 years?
You deposited $15,000 in an account 22 years ago and now the
account has $50,000 in it. What was the annual rate of return
that you received on this investment?
You currently have $38,000 in an account that has been paying
5.75 percent p.a.. You remember that you had opened this
account quite some years ago with an initial deposit of $19,000.
You forget when the initial deposit was made. How many years
(in fractions) ago did you make the initial deposit?
31
Perpetuity 1
Perpetuity: a stream of equal cash flows ( C )
that occur at the end of each period and go on
forever.
PV of perpetuity =
C
r
32
Perpetuity 2

We use the idea of a perpetuity to
determine the value of


A preferred stock
A perpetual debt
33
Perpetuity questions

Suppose the value of a perpetuity is $38,900
and the discount rate is 12 percent p.a..
What must be the annual cash flow from this
perpetuity?
Verify that C = $4,668.

An asset that generates $890 per year
forever is priced at $6,000. What is the
required rate of return?
Verify that r = 14.833 %
34
Ordinary Annuity

Ordinary annuity: a cash flow stream where a
fixed amount is received at the end of every
period for a fixed number of periods.
35
What’s the FV of a 3-year
ordinary annuity of $100 at 10%?
0
10%
1
2
100
100
3
FV
100
110
121
= 331
36
Financial Calculator Solution
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
331.00
Have payments but no lump sum PV, so
enter 0 for present value.
37
Spreadsheet Solution



Use the FV function:
= FV(I, N, PMT, PV)
= FV(0.10, 3, -100, 0) = 331.00
38
What’s the PV of this ordinary
annuity?
0
1
2
3
100
100
100
10%
90.91
82.64
75.13
248.69 = PV
39
Financial Calculator Solution
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-248.69
Have payments but no lump sum FV, so
enter 0 for future value.
40
Spreadsheet Solution



Use the PV function:
= PV(I, N, PMT, FV)
= PV(0.10, 3, 100, 0) = -248.69
41
Annuity, find FV
You open an account today with $20,000
and at the end of each of the next 15
years, you deposit $2,500 in it. At the
end of 15 years, what will be the balance
in the account if the interest rate is 7
percent p.a.?
PV=-20000, PMT=-2500, N=15, I/Y=7, FV=?
42
Annuity, find I/Y
You lend your friend $100,000. He will
pay you $12,000 per year for the ten
years and a balloon payment at t = 10
of $50,000. What is the interest rate
that you are charging your friend?
PV=-100,000, FV=50,000, PMT=12,000, N = 10, I/Y=?
43
Annuity, find PMT
Next year, you will start to make 35 deposits of $3,000 per
year in your Individual Retirement Account (so you will
contribute from t=1 to t=35). With the money accumulated
at t=35, you will then buy a retirement annuity of 20 years
with equal yearly payments from a life insurance company
(payments from t=36 to t=55). If the annual rate of return
over the entire period is 8%, what will be the annual
payment of the annuity?
44
Annuity Due

Annuity due: a cash flow stream where a fixed
amount is received at the beginning of every
period for a fixed number of periods.
45
Ordinary Annuity vs. Annuity Due
Ordinary Annuity
0
I%
1
2
3
PMT
PMT
PMT
1
2
3
PMT
PMT
Annuity Due
0
PMT
I%
46
Ordinary Annuity vs. Annuity Due
$300
$300
T=0
T=1
T=2
$300
$300
$300
T=0
T=1
T=2
$300
T=3
T=3
47
a relationship between ordinary
annuity and annuity due?
PV of annuity due
= (PV of ordinary annuity) x (1 + r)
FV of annuity due
= (FV of ordinary annuity) x (1 + r)
48
Find the FV and PV if the
annuity were an annuity due.
0
1
2
100
100
3
10%
100
49
PV and FV of Annuity Due
vs. Ordinary Annuity
PV of annuity due:
= (PV of ordinary annuity) (1+I)
= ($248.69) (1+ 0.10) = $273.56

FV of annuity due:
= (FV of ordinary annuity) (1+I)
= ($331.00) (1+ 0.10) = $364.10

50
PV of Annuity Due: Switch
from “End” to “Begin”
BEGIN Mode
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-273.55
51
FV of Annuity Due: Switch
from “End” to “Begin”
BEGIN Mode
INPUTS
OUTPUT
3
10
N
I/YR
0
100
PV
PMT
FV
-364.10
52
Excel Function for Annuities
Due




Change the formula to:
=PV(0.10,3,-100,0,1)
The fourth term, 0, tells the function there
are no other cash flows. The fifth term tells
the function that it is an annuity due. A
similar function gives the future value of an
annuity due:
=FV(0.10,3,-100,0,1)
53
What is the PV of this
uneven cash flow stream?
0
90.91
247.93
225.39
-34.15
1
2
3
4
100
300
300
-50
10%
530.08 = PV
54
Solving with Calculator

Input cash flows in the calculator’s “CF” register:
Press CF key
 CF0 = 0, ENTER,
 C01 = 100, ENTER,
F01=1, ENTER,
 C02 = 300, ENTER,
F02=2, ENTER,
 C03= 50, +/- key, ENTER,
F03=1, ENTER,

Press NPV key,

I = 10, ENTER,
press CPT key to get
NPV = 530.087. (Here NPV = PV.)
55
Excel Formula in cell A3:
=NPV(10%,B2:E2)
56
In-class group project

You will need to pay for your son’s private school tuition (first
grade through 12th grade) a sum of $8,000 per year for Years 1
through 5, $10,000 per year for Years 6 through 8, and $12,500
per year for Years 9 through 12. Assume that all payments are
made at the beginning of the year, that is, tuition for Year 1 is
paid now (i.e., at t = 0), tuition for Year 2 is paid one year from
now, and so on. In addition to the tuition payments you expect
to incur graduation expenses of $2,500 at the end of Year 12. If
a bank account can provide a certain 10 percent p.a. rate of
return, how much money do you need to deposit today to be
able to pay for the above expenses?
57
Nominal rate (INOM)




Stated in contracts, and quoted by
banks and brokers.
Not used in calculations or shown on
time lines
Periods per year (M) must be given.
Examples:


8%; Quarterly
8%, Daily interest (365 days)
58
Periodic rate (IPER )

IPER = INOM/M, where M is number of compounding
periods per year. M = 4 for quarterly, 12 for monthly,
and 360 or 365 for daily compounding.

Used in calculations, shown on time lines.

Examples:

8% quarterly: IPER = 8%/4 = 2%.

8% daily (365): IPER = 8%/365 = 0.021918%.
59
The Impact of Compounding


Will the FV of a lump sum be larger or
smaller if we compound more often,
holding the stated I% constant?
Why?
60
The Impact of Compounding
(Answer)


LARGER!
If compounding is more frequent than
once a year--for example, semiannually,
quarterly, or daily--interest is earned on
interest more often.
61
Common examples
Compounding
period
Six-months /
semiannual
Quarter
Compounding
frequency
2
4
Month
12
Day
365
62
When frequency of compounding
is more than once a year
PV 
‘n’ = number of years
FV
r 

1  
m

‘m’ = frequency of
compounding per year
mn
r 

FV  PV  1  
m

‘r’ = nominal rate
mn
m
r 

Effect iveint erestrat e  1    1
m

63
Effective Annual Rate (EAR =
EFF%)

The EAR is the annual rate that causes
PV to grow to the same FV as under
multi-period compounding.
64
Effective Annual Rate Example

Example: Invest $1 for one year at 12%,
semiannual:
FV = PV(1 + INOM/M)M

FV = $1 (1.06)2 = $1.1236.
EFF% = 12.36%, because $1 invested for
one year at 12% semiannual compounding
would grow to the same value as $1 invested
for one year at 12.36% annual compounding.
65
$100 at a 12% nominal rate with
semiannual compounding for 5 years
INOM
FVN = PV 1 +
M
FV5S
0.12
= $100 1 +
2
= $100(1.06)10
MN
2x5
= $179.08
66
FV of $100 at a 12% nominal rate for
5 years with different compounding
FV(Ann.)
FV(Semi.)
FV(Quar.)
FV(Daily)
=
=
=
=
$100(1.12)5
$100(1.06)10
$100(1.03)20
$100(1+(0.12/365))(5x365)
=
=
=
=
$176.23
$179.08
$180.61
$182.19
67
Comparing Rates


An investment with monthly payments
is different from one with quarterly
payments. Must put on EFF% basis to
compare rates of return. Use EFF%
only for comparisons.
Banks say “interest paid daily.” Same
as compounded daily.
68
EFF% for a nominal rate of 12%,
compounded semiannually
EFF% =
=
INOM
1 +
M
0.12
1 +
2
M
−1
2
−1
= (1.06)2 - 1.0
= 0.1236 = 12.36%.
69
EAR (or EFF%) for a Nominal
Rate of of 12%
EARAnnual
= 12%.
EARQ
= (1 + 0.12/4)4 - 1
= 12.55%.
EARM
= (1 + 0.12/12)12 - 1 = 12.68%.
EARD(365)
= (1 + 0.12/365)365 - 1= 12.75%.
70
Can the effective rate ever be
equal to the nominal rate?


Yes, but only if annual compounding is
used, i.e., if M = 1.
If M > 1, EFF% will always be greater
than the nominal rate.
71
When is each rate used?
INOM:
Written into contracts, quoted
by banks and brokers. Not used
in calculations or shown
on time lines.
72
When is each rate used?
(Continued)
IPER: Used in calculations, shown on
time lines.
If INOM has annual compounding,
then IPER = INOM/1 = INOM.
73
When is each rate used?
(Continued)


EAR (or EFF%): Used to compare
returns on investments with different
payments per year.
Used for calculations if and only if
dealing with annuities where payments
don’t match interest compounding
periods.
74
Annuity with semiannual
compounding

You would like to accumulate $16,500
over the next 8 years. How much must
you deposit every six months, starting
six months from now, given a 4 percent
per annum rate with semiannual
compounding?
75
Loan Amortization


Amortization is the process of separating a
payment into interest payment and
repayment of principal.
Amortization schedule is a table that shows
how each payment is split into principal
repayment and interest payment.
76
Amortization

Construct an amortization schedule for
a $1,000, 10% annual rate loan with 3
equal payments.
77
Step 1: Find the required
payments.
0
10%
-1,000
INPUTS
OUTPUT
3
N
1
2
3
PMT
PMT
PMT
10
I/Y
R
-1000
PV
0
PMT
FV
402.11
78
Step 2: Find interest charge
for Year 1.
INTt = Beg balt (I)
INT1 = $1,000(0.10) = $100
79
Step 3: Find repayment of
principal in Year 1.
Repmt = PMT - INT
= $402.11 - $100
= $302.11
80
Step 4: Find ending balance
after Year 1.
End bal = Beg bal - Repmt
= $1,000 - $302.11 = $697.89
Repeat these steps for Years 2 and 3
to complete the amortization table.
81
Amortization Table
BEG
BAL
$1,000
PMT
$402
2
698
402
70
332
366
3
366
402
37
366
0
YEAR
1
TOT
1,206.34
PRIN
INT
PMT
$100 $302
END
BAL
$698
206.34 1,000
82
Interest declines because
outstanding balance declines.
$450
$400
$350
$300
$250
$200
$150
$100
$50
$0
Interest
Principal
PMT 1
PMT 2
PMT 3
83

Amortization tables are widely
used--for home mortgages, auto
loans, business loans, retirement
plans, and more. They are very
important!
84
Example of loan amortization 1
You have borrowed $8,000 from a bank and
have promised to repay the loan in five equal
yearly payments. The first payment is at the
end of the first year. The interest rate is 10
percent. Draw up the amortization schedule for
this loan.
85
Example of loan amortization 2
1) Compute periodic payment.
PV=8000, N=5, I/Y=10, FV=0, PMT=?
Verify that PMT = -2,110.38
86
Example of loan amortization 3
Suppose we want to work out the remaining
balance immediately after the 2nd payment.
 Press [2ND], [AMORT] to activate the
Amortization worksheet in BA II Plus.
 Press P1=2, [ENTER], ,
 Press P2=2, [ENTER], ,
 You will see BAL=5,248.20
87
Example of loan amortization 4



Press  again and you see the portion of the
year 2 payment going towards repaying
principal, PRN = -1,441.42
Press  again and you see the portion of year
2 payment going towards interest,
INT = -668.96
To get out of the Amortization schedule,
press [2ND], Quit.
88
Verify the amortization schedule
Year
Beg.
Balance
Payment
Interest
Principal
0
End.
Balance
8,000.00
1
8,000.00
2,110.38
800.00
1,310.38
6,689.62
2
6689.62
2,110.38
668.96
1,441.42
5,248.20
3
5248.20
2,110.38
524.82
1,585.56
3,662.64
4
3662.64
2,110.38
366.26
1,744.12
1,918.53
5
1918.53
2,110.38
191.85
1,918.53
0.00
89
Non-matching rates and periods

What’s the value at the end of Year 3 of
the following CF stream if the quoted
interest rate is 10%, compounded
semiannually?
90
Time line for non-matching
rates and periods
0
1
2
3
4
5%
100
100
5
6
6-mos.
periods
100
91
Non-matching rates and periods


Payments occur annually, but
compounding occurs each 6 months.
So we can’t use normal annuity
valuation techniques.
92
1st Method: Compound Each
CF
0
5%
1
2
100
3
4
100
5
6
100.00
110.25
121.55
331.80
FVA3 = $100(1.05)4 + $100(1.05)2 + $100
= $331.80
93
2nd Method: Treat as an
annuity, use financial calculator
Find the EFF% (EAR) for the quoted rate:
EFF% =
0.10
1 +
2
2
− 1 = 10.25%
94
Use EAR = 10.25% as the
annual rate in calculator.
INPUTS
OUTPUT
3
10.25
0
-100
N
I/YR
PV
PMT
FV
331.80
95
What’s the PV of this stream?
0
5%
90.70
82.27
74.62
247.59
1
2
3
100
100
100
96
After Chapter Homework

Problems: 1-30.
97