Chapter 6: Discounted Cash Flow Valuation
1.
Present Value and Multiple Cash Flows: Seaborn Co. has identified an
investment project with the following cash flows. If the discount rate is 10
percent, what is the present value of these cash flows? What is the present value at
18 percent? At 24 percent?
Year
1
2
3
4
Cash Flow
$ 950
1,040
1,130
1,075
Using formulas and a 10% discount rate:
PV 
FV
(1  r )t
PV 
$950
$1,040
$1,130
$1,075



 $3,306.365685
1
2
3
(1  0.10) (1  0.10) (1  0.10) (1  0.10) 4
We can also use the ‘stack register’ on a financial calculator:
0 gCF0 ;950 gCFj ;1,040 gCFj ;1,130 gCFj ;1,075 gCFj ;10i; fNPV  $3,306.365685
Using formulas and an 18% discount rate:
PV 
FV
(1  r )t
PV 
$950
$1,040
$1,130
$1,075



 $2,794.222480
1
2
3
(1  0.18) (1  0.18) (1  0.18) (1  0.18) 4
We can also use the ‘stack register’ on financial calculators:
0 gCF0 ;950 gCFj ;1,040 gCFj ;1,130 gCFj ;1,075 gCFj ;18i; fNPV  $2,794.222480
Using formulas and a 24% discount rate:
PV 
FV
(1  r )t
PV 
$950
$1,040
$1,130
$1,075



 $2,489.875027
1
2
3
(1  0.24) (1  0.24)
(1  0.24)
(1  0.24) 4
We can also use the ‘stack register’ on our financial calculators:
0 gCF0 ;950 gCF j ;1,040 gCF j ;1,130 gCF j ;1,075 gCF j ;24i; fNPV  $2,489.875027
2.
Present Value and Multiple Cash Flows: Investment X offers to pay you $6,000
per year for nine years, whereas Investment Y offers to pay you $8,000 per year
for six years. Which of these cash flow streams has the higher present value if the
discount rate is 5 percent? If the discount rate is 15 percent?
To find the present value of Investment X, an annuity, using a 5% discount rate
we use the following equation:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
9 
(
1

0
.
05
)

   $42,646.93005
PVAX  $6,000  


0.05




Using a financial calculator:
9n;5i;6,000 PMT ;0 FV ; PV  $42,646.93005
To find the present value of Investment Y, an annuity, using a 5% discount rate we
use the following equation:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
6 
(
1

0
.
05
)

   $40,605.53654
PVAY  $8,000  


0.05




Using a financial calculator:
6n;5i;8,000 PMT ;0 FV ; PV  $40,605.53654
Using a 5% discount rate, Investment X has a larger PV cash flow stream.
To find the present value of Investment X, an annuity, using a 15% discount rate
we use the following equation:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
9 
(1  0.15)  


PVAX  $6,000 
 $28,629.50352


0.15




Using a financial calculator:
9n;15i;6,000 PMT ;0 FV ; PV  $28,629.50352
To find the present value of Investment Y, an annuity, using a 15% discount rate
we use the following equation:
  1
1  
(1  r )t


PVA  C 

r



 




 

1

1  
6 
(1  0.15)  


PVAY  $8,000 
 $30,275.86155


0.15




Using a financial calculator:
6n;15i;8,000 PMT ;0 FV ; PV  $30,275.86155
Using a 15% discount rate, Investment Y has a larger PV cash flow stream.
3.
Future Value and Multiple Cash Flows: Paradise, Inc., has identified an
investment project with the following cash flows. If the discount rate is 8 percent,
what is the future value of these cash flows in year 4? What is the future value at a
discount rate of 11 percent? At 24 percent?
Year
1
2
3
4
Cash Flow
$ 940
1,090
1,340
1,405
Using formulas and an 8% discount rate:
FV  PV (1  r )t
FV  $940(1.08)3  $1,090(1.08) 2  $1,340(1.08)1  $1,405  $5,307.70528
Using a financial calculator:
3n;8i;0 PMT ;940 PV ; FV  $1,184.12928
2n;8i;0 PMT ;1,090 PV ; FV  $1,271.376
1n;8i;0 PMT ;1,340 PV ; FV  $1,447.20
$1,405.00
Sum = $5,307.70528
Using formulas and an 11% discount rate:
FV  PV (1  r )t
FV  $940(1.11)3  $1,090(1.11)2  $1,340(1.11)1  $1,405  $5,520.96214
Using a financial calculator:
3n;11i;0 PMT ;940 PV ; FV  $1,285.57314
2n;11i;0 PMT ;1,090 PV ; FV  $1,342.989
1n;11i;0 PMT ;1,340 PV ; FV  $1,487.40
$1,405.00
Sum = $5,520.96214
Using formulas and a 24% discount rate:
FV  PV (1  r )t
FV  $940(1.24)3  $1,090(1.24)2  $1,340(1.24)1  $1,405  $6,534.81056
Using a financial calculator:
3n;24i;0 PMT ;940 PV ; FV  $1,792.22656
2n;24i;0 PMT ;1,090 PV ; FV  $1,675.984
1n;24i;0 PMT ;1,340 PV ; FV  $1,661.60
$1,405.00
Sum = $6,534.81056
4.
Calculating Annuity Present Value: An investment offers $5,300 per year for
15 years, with the first payment occurring one year from now. If the required
return is 7 percent, what is the value of the investment? What would the value be
if the payments occurred for 40 years? For 75 years? Forever?
To find the present value of a 15-year annuity using an 7% discount rate we use
the following equation:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
15  
(
1

0
.
07
)

   $48,271.94423
PVA  $5,300  


0.07




Using a financial calculator:
15n;7i;5,300 PMT ;0 FV ; PV  $48,271.94423
To find the present value of a 40-year annuity using an 7% discount rate we use
the following equation:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
40  
(
1

0
.
07
)

   $70,658.05687
PVA  $5,300  


0.07




Using a financial calculator:
40n;7i;5,300 PMT ;0 FV ; PV  $70,658.05687
To find the present value of a 75-year annuity using an 7% discount rate we use
the following equation:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
75  
(
1

0
.
07
)

   $75,240.70446
PVA  $5,300  


0.07




Using a financial calculator:
75n;7i;5,300 PMT ;0 FV ; PV  $75,240.70446
To find the present value of a perpetuity using an 7% discount rate we use the
following equation:
PV 
C
r
PV 
19.
$5,300
 $75,714.28571
0.07
EAR versus APR: Big Dom’s Pawn Shop charges an interest rate of 30 percent
per month on loans to its customers. Like all lenders, Big Dom must report an
APR to consumers. What rate should the shop report? What is the effective annual
rate?
The APR is simply the interest rate per period multiplied times the number of
periods per year. Since the monthly interest rate is 30%, and there are 12 months
per year, the APR is:
APR  30% 12  360%
To find the EAR, we use the following equation:
m
APR 

EAR  1 
 1
m 

12
 3.6 
EAR  1 
  1  2,229.808512%
 12 
While such an interest rate would be considered above the usury rate for any
bank, such rates are common with pay-day loans!
20.
Calculating Loan Payments: You want to buy a new sports coupe for $68,500,
and the finance office at the dealership has quoted you a 6.9 percent APR loan for
60 months to buy the car. What will the monthly payments be? What is the
effective annual rate on the loan?
To find the monthly payment of a 60-month annuity, we use the following
equation:
  1
1  
(1  r )t


PVA  C 

r



 




 


 
1

1  
   0.069  60  

  1 
  
12

   $68,500
PVA  C   


0.069


12








C  $1,353.152587
Using a financial calculator:
60n;
6.9
i;68,500 PV ;0 FV ; PMT  $1,353.152587
12
To find the EAR, we use the following equation:
m
APR 

EAR  1 
 1
m 

12
 0.069 
EAR  1 
  1  7.12245%
12 

22.
Calculating EAR: Friendly’s Quick Loans, Inc., offers you “three for four or I
knock on your door.” This means you get $3 today and repay $4 when you get
your paycheck in one week (or else). What is the effective annual rate Friendly’s
earns on this lending business? If you were brave enough to ask, what APR would
Friendly’s say you were paying?
First, we need to determine the weekly interest rate using the following future
value (or present value) of a lump sum equation:
FV  PV (1  r )t
$4  $3(1  r )1W eek
r
$4
 1  33.33333333%PerWeek
$3
Using the 33.33333333% weekly interest rate, the APR would be:
APR  33.33333%  52  1,733.333333%
To find the EAR, we use the following equation:
m
APR 

EAR  1 
 1
m 

52
 17.333333333 
EAR  1 
  1  313,916,511.6%
52


Loan sharks territory!
24.
Calculating Annuity Future Values: You are planning to make monthly
deposits of $300 into a retirement account that pays 10 percent interest
compounded monthly. If your first deposit will be made one month from now,
how large will your retirement account be in 30 years?
To determine the future value of an annuity, we use the following equation:
FVA  C 
(1  r )  1
t
r
 0.10 360 
  1
1 
12 


FVA  $300 
 $678,146.3774
0.10
12
Using a financial calculator:
360n;
10
i;300 PMT ;0 PV ; FV  $678,146.3774
12
26.
Calculating Annuity Present Values: Beginning three months from now, you
want to be able to withdraw $2,300 each quarter from your bank account to cover
college expenses over the next four years. If the account pays 0.65 percent interest
per quarter, how much do you need to have in your bank account today to meet
your expense needs over the next four years?
To find the present value of an annuity, we use the following equation:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
16  


1

0
.
0065

   $34,843.70881
PVA  $2,300  


0.0065




Using a financial calculator:
16n;0.65i;2,300 PMT ;0 FV ; PV  $34,843.70881
32.
Calculating Annuities: You are planning to save for retirement over the next 30
years. To do this, you will invest $700 a month in a stock account and $300 a
month in a bond account. The return on the stock account is expected to be 11
percent, and the bond account will pay 6 percent. When you retire, you will
combine your money into an account with a 9 percent return. How much can you
withdraw each month from your account assuming a 25-year withdrawal period?
First, we need to determine the future value of the two (stock and bond) annuities:
FVA  C 
(1  r )  1
t
r
FVAStock
 0.11 360 
  1
1 
12 


 $700 
 $1,963,163.816
0.11
12
Using a financial calculator:
360n;
11
i;700 PMT ;0 PV ; FV  $1,963,163.816
12
FVABond
 0.06 360 
  1
1 
12 


  $301,354.5127
 $300 
0.06
12
Using a financial calculator:
360n;
6
i;300 PMT ;0 PV ; FV  $301,354.5127
12
The expected value of the retirement accounts in 30 years is:
$1,963,163.816  $301,354.5127  $2,264,518.329
Now, we need to determine the monthly withdrawal amount using the present
value annuity equation and solving for C:
  1
1  
(1  r )t
PVA  C   

r



 




 


 
1

1  
   0.09  300  

  1 
12   



PVA  C 
 $2,264,518.329


0.09


12








C  $19,003.75547
Using a financial calculator:
300n;
9
i;2,264,518.329 PV ;0 FV ; PMT  $19,003.75547
12
You would be able to withdraw $19,003.76 each month for the next 25 years, after
which time your account balance will be almost zero.
34.
Calculating Annuity Payments: You want to be a millionaire when you retire in
40 years. How much do you have to save each month if you can earn an 12
percent annual return? How much do you have to save if you wait 10 years before
you begin your deposits? 20 years?
To determine the monthly payment, C, we need to save at 12% for 40 years in
order to be a millionaire we use the following equation:
FVA  C 
(1  r )  1
t
r
 0.12  480 
  1
1 
12 


$1,000,000  C 
0.12
12
C  $84.99951862
Using a financial calculator:
480n;
12
i;0 PV ;1,000,000 FV ; PMT  $84.99951862
12
To determine the monthly payment, C, we need to save at 12% for 30 years in
order to be a millionaire we use the following equation:
FVA  C 
(1  r )  1
t
r
 0.12 360 
  1
1 
12 


$1,000,000  C 
0.12
12
C  $286.1259693
Using a financial calculator:
360n;
12
i;0 PV ;1,000,000 FV ; PMT  $286.1259693
12
To determine the monthly payment, C, we need to save at 12% for 20 years in
order to be a millionaire we use the following equation:
FVA  C 
(1  r )  1
t
r
 0.12  240 
  1
1 
12 


$1,000,000  C 
0.12
12
C  $1,010.861336
Using a financial calculator:
240n;
36.
12
i;0 PV ;1,000,000 FV ; PMT  $1,010.861336
12
Comparing Cash Flow Streams: You’ve just joined the investment banking firm
of Dewey, Cheatum, and Howe. They’ve offered you two different salary
arrangements. You can have $95,000 per year for the next two years, or you can
have $70,000 per year for the next two years, along with a $45,000 signing bonus
today. The bonus is paid immediately, and the salary is paid at the end of each
year. If the interest rate is 10 percent compounded monthly, which do you prefer?
Since the interest rate is compounded monthly, we need to first determine the
EAR:
m
APR 

EAR  1 
 1
m 

12
 0.10 
EAR  1 
  1  10.4713063%
12 

The present value of the $95,000 per year at the end of each of two years is:
  1
1  
(1  r )t


PVA  C 

r



 




 

1

1  
2 


1

0
.
104713063

   $163,839.0884
PVA  $95,000  


0.104713063




Using a financial calculator:
2n;10.4713063i;95,000 PMT ;0 FV ; PV  $163,839.0884
The present value of the $70,000 per year at the end of each of two years is:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
2 

1  0.104713063  


PVA  $70,000 
 $120,723.5388


0.104713063




Adding the $45,000 signing bonus received immediately, the value of this contract
is:
$120,723.5388  $45,000  $165,723.5388
Using a financial calculator:
2n;10.4713063i;70,000 PMT ;0 FV ; PV  $120,723.5388
$120,723.5388  $45,000  $165,723.5388
The salary contract with $70,000 payable at the end of each of the next two years
with an immediate $45,000 signing bonus is more valuable!
38.
Growing Annuity: You job pays you only once a year for all the work you did
over the previous 12 months. Today, December 31, you just received your salary
of $50,000 and you plan to spend all of it. However, you want to start saving for
retirement beginning next year. You have decided that one year from today you
will begin depositing 5 percent of your annual salary in an account that will earn
11 percent per year. Your salary will increase at 4 percent per year throughout
your career. How much money will you have on the date of your retirement 40
years from today?
Since your salary grows at 4% per year, your next year’s salary is:
NextYear' sSalary  $50,000(1  0.04)  $52,000
Therefore, your deposit into your retirement account next year will be:
NextYear' sDeposit  $52,000  0.05  $2,600
Since your salary grows at a constant 4% per year and you put in a constant 5%
of your salary into your retirement account, your annual deposit into the
retirement account grows at 4% per year. Therefore, we can use the present value
of a growing perpetuity equation to find the present value of your deposits today:
 1   1   1  g t 
  
  
PVGP  C  

r

g
r

g



  1  r  

40

1
1
 
  1  0.04  
PVGP  $2,600  


   $34,399.4534
 0.11  0.04   0.11  0.04   1  0.11  
Now, we can find the value of this lump sum in 40 years:
FV  PV (1  r )t
FV  $34,399.4534(1  0.11) 40  $2,235,994.306
With your amount of your deposit increasing at 4% per year for the next 40 years,
and earning a 10% annual rate of return, your retirement account would grow to
$2,235,994.306
54.
Calculating Annuities Due: You want to buy a new sports car from Muscle
Motors for $68,000. The contract is in the form of a 60-month annuity due at an
7.85 percent APR. What will your monthly payment be?
With an annuity due, we simply multiply the value of an ordinary annuity by 1+r.
Using the present value of an annuity equation adjusted for an annuity due, we
have:
  1
1  
(1  r )t
PVADue  (1  r )  C   

r



 




 


 
1


1 
60 


 0.0785   
  1 

12   
 0.0785 



$68,000  1 
 $1,364.989126
C 


0.0785
12 



12








We can also use our financial calculators to solve for the monthly payment of this
annuity due. In order to do so, we need to set our financial calculator to the
annuity due mode. This requires that we change the annuity from END to BEG
(for begin). Once you change the mode, you should see the word BEGIN on the
register. Using the annuity due mode we get:
60n;
7.85
i;0 FV ;68,000 PV ; PMT  $1,364.989126
12
We can also use our financial calculators in the ordinary annuity mode to solve
this problem. Once we have obtained the monthly payment for an ordinary
annuity, we can simply discount this payment back one period:
60n;
7.85
i;0 FV ;68,000 PV ; PMT  $1,373.91843
12




1
  $1,364.989126
$1,373.91843  
 1  0.0785 
12 

55.
Amortization with Equal Payments: Prepare an amortization schedule for a
five-year loan of $42,000. The interest rate is 8 percent per year, and the loan calls
for equal principal and interest payments. How much interest is paid in the third
year? How much total interest is paid over the life of the loan?
Equal P&I payments can be determined using the present value of an annuity
equation:
  1
1  
(1  r )t
PVA  C   

r



 




 

1

1  
5 
(
1

0
.
08
)


$42,000  C  


0.08




C  $10,519.17108
Using a financial calculator:
5n;8i;0 FV ;42,000 PV ; PMT  $10,519.17109
The interest payment each year is the beginning balance for that year times the
interest rate. The ending balance for each year is the beginning balance for that
year minus the principle payment for that year. The principle payment each year
is the total P&I payment minus the interest payment for that year. The ending
balance for one year is the beginning balance for the next year. The amortization
schedule is:
Year
1
2
3
4
5
Beginning
Balance
$42,000.00
34,840.83
27,108.93
18,758.47
9,739.98
Total
Payment
$10,519.17
10,519.17
10,519.17
10,519.17
10,519.17
Interest
Payment
$3,360.00
2,787.27
2,168.71
1,500.68
779.19
Principle
Payment
$7,159.17
7,731.90
8,350.46
9,018.49
9,739.98
Ending
Balance
$34,840.83
27,108.93
18,758.47
9,739.98
0
Interest in year 3 is $2,168.71
Total Interest is the sum of the interest payments for all 5 years, which is
$10,595.85
56.
Amortization with Equal Principal Payments: Rework Problem 55 assuming
that the loan agreement calls for a principal reduction of $8,400 (i.e., $42,000/5)
every year instead of equal annual payments.
Again, the interest payment each year is the beginning balance for that year times
the interest rate. Total payment for each year is the interest payment plus $8,400,
the principal payment. The ending balance for each year is the beginning balance
for that year minus the principle payment of $8,400. The ending balance for one
year is the beginning balance for the next year. The amortization schedule is:
Year
1
2
3
4
5
Beginning
Balance
$42,000.00
33,600.00
25,200.00
16,800.00
8,400.00
Total
Payment
$11,760.00
11,088.00
10,416.00
9,744.00
9,072.00
Interest
Payment
$3,360.00
2,688.00
2,016.00
1,344.00
672.00
Principle
Payment
$8,400.00
8,400.00
8,400.00
8,400.00
8,400.00
Ending
Balance
$33,600.00
25,200.00
16,800.00
8,400.00
0
Interest in year 3 is $2,016.00
Total Interest is the sum of the interest payments for all 5 years, which is
$10,080.00
58.
Calculating Annuity Values: After deciding to buy a new car, you can either
lease the car or purchase it on a three-year loan. The car you wish to buy costs
$32,000. The dealer has a special leasing arrangement where you pay $99 today
and $450 per month for the next three years. If you purchase the car, you will pay
it off in monthly payments over the next three years at an 7 percent APR. You
believe you will be able to sell the car for $23,000 in three years. Should you buy
or lease the car? What break-even resale price in three years would make you
indifferent between buying and leasing?
We can find the present value of each of these two options (buying versus leasing)
and compare. Since we are looking at the present value of expenses, the least
expensive option is the best.
The present value of leasing is the present value of the leased payments:
  1
1  
(1  r )t


$99  PVA  $99  C 

r



 




 


 
1


1 
   0.07  36  

  1 
12   



$99  PVA  $99  $450 
 $14,672.90826


0.07


12








The present value of buying the car is the cost of purchasing the car minus the
present value of resale value of the car:
PVRe sale 
$23,000
 0.07 
1 

12 

36
 $18,654.81624
Using the financial calculator:
36n;
7
i;23,000 FV ;0 PMT ; PV  $18,654.81602
12
The present value of the buying option is:
$32,000  $18,654.82  $13,345.18
Comparing the two options:
PVLease  $14,672.91
PVBuy  $13,345.18
Since the buy option is less expensive, it is the best option.
To determine the resale value that would make us indifferent between the two
options, we set the PV of the lease option equal to the purchase price of the car
minus the PV of the resale:
$14,672.91  $32,000  PVRe sale
PVRe sale  $17,327.09
Now, we need to determine what the resale price would be 3 years from now (i.e.,
the future value):
FV  PVRe sale (1  r )t  $17,327.09
36
 0.07 
FV  $17,327.091 
  $21,363.01237
12 
