FIN 40153: Intermediate Financial
Management
INTRODUCTION TO VALUATION
(TIME VALUE OF MONEY)
(BASED ON RWJ CHAPTER 4)
Money has “time value”
 Why?
 Because interest rates (and expected returns)
are positive.
 Implies that:
 A dollar today is worth more than a dollar
tomorrow.
 Cash flows at different points in time cannot
be added together.
Basic Tools for Valuation: Future and
Present Values (Discounting)
Terminology:
r is the interest rate or as is also referred to as the discount rate
PV will be the value today or the present value of a future payment
FVn will be the future value of money invested today n years from now
 If you invest PV today then n years from now it will be worth
FVn = PV(1+r)n
This same basic formula can be used to determine the current amount
that is equivalent to (i.e. can grow to) a stated future amount, since
PV = FVn/(1+r)n
Example
You need $20 million five years from now to
fund a capital investment. If r = 6%, what
amount can be set aside now to fund the
investment?
PV = 20.0/(1.06)5 = 20.0/1.33823 = $14.945
million.
What if you can’t really earn r = 6%?
Example
 My Uncle Pete wants me to loan him $15,000 today so he can buy
some frozen fruit for his new smoothie stand. He says that he is
willing to get the full $15,000 back to me in exactly three years, and
is willing to give me $3,500 in three years on top of that for interest.
What is the annual rate of interest that I am receiving on my loan to
Pete under his proposal?
 18,500 = 15,000(1+r)3 = 18,500
 (18,500/15,000) = (1+r)3
 r = (18,500/15,000)1/3 -1 = 0.0724 = 7.24%
On HP10BII:
-15,000
•Amber C All
PV
18,500
FV
3
N
0
PMT
Push the I/YR key to get 7.24%
Note: Must be sure that P/YR is set = 1 !!!
Present Value Of A Series of Future
Cash Flows
 The present value of a stream of cash flows can be found
using the following general valuation formula:
C3
CN
C1
C2
PV 


 ... 
2
3
(1  r1 ) (1  r2 ) (1  r3 )
(1  rN ) N
N
Ct
=
t
(
1

r
)
t 1
t
 In words: (1) discount each cash flow back to the present using
the appropriate discount rate, and (2) sum the present values.
The latter step is OK, because each future cash flow has been
restated as its equivalent amount at a common point in time.
Valuing Streams of Structured Future
Cash Flows
 Perpetuity A stream of equal payments, made at
equal time intervals, forever.
 The present value of a perpetuity that pays the
amount C at the end of each period, when the
discount rate is r, is:
C
C
C
…
0
1
2
3
C
PV 
r
 Note that this gives the value as of time 0, one
period before the first payment arrives.
Example
You wish to endow a Permanent Chair in
finance at TCU. The chair needs $150,000/yr
forever (in perpetuity).
 The trustees of TCU can invest your money at
6%/yr through guaranteed investment contracts
with the NeverBankrupt Insurance Co.
 How much money do you need (what is the PV of the
perpetuity) ?
 PV = C / r = 150,000 / 0.06 = $2,500,000
Valuing Streams of Cash Flows
 Growing Perpetuity
 A growing perpetuity is a stream of periodic payments
that grow at a constant rate and continue forever.
 The present value of a perpetuity that pays the amount
C1 next period, growing at the rate g indefinitely when
the discount rate is r is
C1
C1(1+g)
C1(1+g)2
…
0
1
2
C1
PV 
rg
3
Examples
Suppose that the Finance professor to be hired for the
Chair wants 5% raises per year guaranteed:
Growing perpetuity: $150,000 received at time t = 1, growing at
5% per period forever and discounted at
6% per period
0
C
C(1 + g)
C(1 + g) 2
1
2
3
…
PV = C/(r –g ) = $150,000/(0.06 – 0.05) = $15,000,000
Sidebar: Valuation by
Earnings or Cash Flow “Multiples”
 Analysts and investors sometimes take a
shortcut approach to valuation:
 multiply earnings or cash flow for the current year
by an appropriate “multiple”.
 The best known of these is the “price/earnings
ratio”.
 Valuation by multiple in the case of infinitelived, constant growth, cash flows:
PV  M * C1 ,
1
where M 
rg
Multiples Appropriate for Valuing
Infinite-Lived Cash Flows with Constant
Growth
growth rate (g) -2% 0% 2% 4% 6% 8% 10% 12% 14%
interest rate
6%
12.5 16.7 25.0 50.0
8%
10.0 12.5 16.7 25.0 50.0
10%
8.3 10.0 12.5 16.7 25.0 50.0
12%
7.1 8.3 10.0 12.5 16.7 25.0 50.0
14%
6.3 7.1 8.3 10.0 12.5 16.7 25.0 50.0
16%
5.6 6.3 7.1 8.3 10.0 12.5 16.7 25.0 50.0
18%
5.0 5.6 6.3 7.1 8.3 10.0 12.5 16.7 25.0
20%
4.5 5.0 5.6 6.3 7.1 8.3 10.0 12.5 16.7
22%
4.2 4.5 5.0 5.6 6.3 7.1 8.3 10.0 12.5
24%
3.8 4.2 4.5 5.0 5.6 6.3 7.1 8.3 10.0
Larger multiples are justified by higher cash flow growth,
or by lower interest (discount) rates
Annuities
 An annuity is a series of equal payments made at
fixed intervals for a specified number of periods
 e.g., $100 at the end of each of the next three
years
 If payments occur at the end of each period it is an
Ordinary Annuity.
 If payments occur at the beginning of each period
it is an Annuity Due.
0
1
2
3
100
100
100
Ordinary Annuity
Valuing Annuities
 The present value of a T period annuity paying a periodic
cash flow of C when the discount rate is r is:
1
1
PV  C  
T
r
r
(
1

r
)




textbook uses the shorthand notation ATr% to denote the factor
in brackets.
If we have an annuity due instead, the net effect is that every
payment occurs one period sooner, so the value of each payment is
higher by the factor (1+r).
To obtain the PV with beginning-of-period payments, simply
multiply the answer obtained based on end-of-period payments by
the factor (1+r).
The
Annuity Example: A five year annuity
paying $2000 per year, with r = 5%.
 Valuing the payments individually we get:
1
2
3
4
5

2,000.00
2,000.00
2,000.00
2,000.00
2,000.00
1,904.76
1,814.06
1,727.68
1,645.40
1,567.05
_________
8,658.95
Using the annuity formula we get:
 1

1
 = $8658.95
PV = 2000
5
0.05
0.05(1.05
)



Do on HP: PMT=2000, N=5, I/YR=5, FV=0, push PV
and get 8658.95 (note it displays as a negative number)
 Alternatively, suppose you were given
$8,658.95 today instead of the annuity
year
1
2
3
4
5
principal
interest
PMT
$ 8,658.95 $ 432.95 $ (2,000.00)
$ 7,091.90 $ 354.60 $ (2,000.00)
$ 5,446.50 $ 272.32 $ (2,000.00)
$ 3,718.82 $ 185.94 $ (2,000.00)
$ 1,904.76 $ 95.24 $ (2,000.00)
Ending Bal
$ 7,091.90
$ 5,446.50
$ 3,718.82
$ 1,904.76
$
0.00
 Notice that you can replicate the annuity cash flows by
investing the present value to earn 5%.
 This demonstrates that present value calculations provide
a literal equality, in that the future cash flows can be
converted into the present value and vice versa, if (and
only if) the selected discount rate is representative of
actual capital market conditions.
Growing Annuities. A stream of regular
payments that grows at a constant rate, g.
• The present value of a T period growing annuity that
pays C1 next period, with subsequent payments
growing at rate g, when the discount rate is r is:
0
C1
C1(1+g)
…
C1(1+g)T-2 C1(1+g)T-1
1
2
T-1
T
 1
1
1

g


PV  C1 


 r  g r  g  1  r 

T




T

 c1 
1

g


1  
PV  

 
 r  g 
 1 r  



or
Example (growing annuity)
 A new venture is expected to generate $1
million in free cash flow during its first year.
Subsequent cash flows will grow by 4% per
year due to market growth and cost savings.
Cash flows will continue for 20 years. The
appropriate discount rate is 14%.
20

 1,000,000 
 1.04  
PV  
 1  
   10,000,0001  .15943  $8,405,701
 .14  .04    1.14  
Annuities with Deferred Starts
 The standard formulas for level and growing
annuities assume that the first payment occurs
one period from now.
 If the first payment is deferred until later than
one period from now, the standard formulas can
still be used, but:
The answer denotes the value as of one period
before the first payment, not the value now.
A Valuation Problem
What is the value of a 10-year annuity that pays $300 a year at
the end of each year, if the first payment is deferred until 6
years from now, and if the discount rate is 10%?
0
1
2
3
4
5
6
7
8
300 300 300
9
•
10
11
12
13
14
15
•
•
•
•
•
300
The value of the annuity payments as of five years from now
is:
 1

1
PV5  300

  $1,843.37
10 
 0.10 0.10(1  0.10) 
Now discount this equivalent payment back 5 years to time
zero:
1843.37
PV0 
 $1144.58
5
(1  0.10)
Application of Time Value Analysis:
Planning for Retirement.
You have determined that you will require $2.5 million when you
retire 25 years from now. Assuming an interest rate of r = 7%, how
much should you have already saved? set aside each year from
now until retirement?
Determine the present equivalent of the targeted 2.5 million.
PV = 2,500,000/(1.07)25
PV = 2,500,000/5.42743 = $460,623.
Hmmm, you don’t have that much saved??
Determine the annuity payment that has an equivalent present
value:
 1

1
460623  C 

 .07 .07(1.07) 25 



460623  C 11.65358
39526  C
Retirement Planning Example, Continued.
 You expect your earnings to grow by 4% per year. Your
retirement goal is unchanged but, instead of making equal
payments, you wish to make payments that grow along with
your earnings. How large should the first payment be? How
large should the 25th payment be?
25

c1

  1.04  
460623  
 1  
 
 .07  .04   1.07  
 c 
460623   1 .50882185
 .03 
460623  C1 16.960728
$27158  C1 , and
C25  271581.04   $69614.
24
A college planning example
 You have determined that you will need $60,000 per
year for four years to send your daughter to college.
The first of the four payments will be made 18 years
from now and the last will be made 21 years from
now. You wish to fund this obligation by making
equal annual deposits over the 21 years. You expect
to earn r = 8% per year.
Time Line:
0
1
2 3
60 60 60 60
17 18 19 20 21 22
College planning, cont.
• Step 1: Determine the time 17 value of the obligation:
 1

1
PV17  60000

  600003.312127   $198,727.
4 
 0.08 0.08(1  .08) 
• Step 2: Determine the equivalent time zero amount:
PV 
198727
198727

 $53,710
17
3.70002
1.08
College planning example, continued.
• Step 3: Determine the 21-year annuity that is equivalent
to the stipulated present amount.
 1

1

53,710  C 

21 
 .08 .08(1.08) 
53,710  C 10.016803
$5,362  C
Verification of Answer Obtained in CollegePlanning Example
Year
BOY Bal.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0
5362
11153
17407
24162
31457
39335
47844
57034
66958
77677
89253
101755
115258
129840
145590
162599
180969
140808
97435
50592
Int. @8% Dep. (EOY) Pymt (EOY)
0
429
892
1393
1933
2517
3147
3828
4563
5357
6214
7140
8140
9221
10387
11647
13008
14477
11265
7795
4047
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
5362
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
60000
60000
60000
60000
EOY Bal.
5362
11153
17407
24162
31457
39335
47844
57034
66958
77677
89253
101755
115258
129840
145590
162599
180969
140808
97435
50592
1
Time Value Summary
 Discounted cash flow, or present value, analysis is the
foundation for valuing assets or comparing opportunities.
 To use DCF you need to know three things
 Forecasted cash flows
 Timing of cash flows
 Discount rate (reflects current capital market conditions
and risk).
 If there are no simple cash flow patterns, then each cash
flow must be valued individually, and the present values
summed. You can look for annuities or growing annuities
to allow short cuts.
 Different streams of cash flows can be meaningfully
compared (or equated) after they are each converted to
their equivalent present values.