The Two Key Concepts in Finance
Chapter Two
The Two Key Concepts in Finance
ANSWERS TO QUESTIONS
1.
With simple interest the effective rate of interest will equal the nominal
rate. Any compounding will cause the effective rate to exceed the nominal
rate.
PV  Ct  Ct
1
1

R R1  R N 1
2.
3.
A future value can equal a present value only if there is no time value of
money (R = 0). Present values exceed future values only when interest rates
are negative (R < 0).
4.
If the first payment occurs immediately, it can be invested to earn interest
over the first period. If it occurs at the end of the period, it cannot earn
interest during that first period.
5.
This is best seen via a mathematical example. There is little difference
between daily compounding and continuous compounding (for reasonable
levels of interest rates).
Monthly versus annual compounding results in the greater difference.
6.
The statement is true. Consider the extremes. If R = 0, continuous
compounding yields the same as simple interest. As R increases, the
difference between simple interest and compound interest increases, too.
7.
False. Utility measures the combined influences of expected return and
risk. A small sum of money to be received for certain has very little utility
associated with it, whereas a small investment in a very risky venture, such
as a lottery ticket, has considerable utility to some people.
8.
The answer depends on the individual, but many people will change their
selection if the game can be played repeatedly.
The Two Key Concepts in Finance
9.
The answer depends on the individual. Because you incur the $50 cost
despite the choice, it should not necessarily cause a person to change their
selection.
10.
Yes. Set equations 2-9 and 2-11 equal to each other, cancel out the initial
cash flow “C,” assume some initial value for “N” or for “g” and solve for
the other variable.
11.
Mathematically, no, but practically speaking, yes, if the time period is long
enough. Depending on the interest rate used, the present value of an annuity
approaches some limit as the period increases. If the period is long enough,
there is no appreciable difference in the two values.
ANSWERS TO PROBLEMS
1.
Not answered here.
2.
After the last payment to the custodian, the fund will have a zero balance.
This means( PV payments in) – (PV payments out) = 0, or, equivalently,
PV of payments in = PV of payments out
Payments out:
PV 
5000
5000(1.04) 5000(1.04) 2
5000(1.04)14



...

(1.08) 26
(1.08) 27
(1.08) 28
(1.08) 40
Multiply both sides of the equation by (1.08)26:
5000(1.04) 5000(1.04) 2 5000(1.04) 3
5000(1.04)14
(1.08) PV  5000 


... 
(1.08)
(1.08) 2
(1.08) 3
(1.08)14
26
 5000 
5000
5000
5000

 ... 
2
1.03846 1.03846
1.0384614
1.0826 PV  5000  53356.66
PV 
Payments in:
58356.66
 7889.92
(1.08) 26
The Two Key Concepts in Finance
Let x = the first payment
PV  x 
x
x(1.04) x(1.04) 2
x(1.04) 25


...

1.08
(1.08) 2
(1.08) 25
x
x
x

 ... 
2
1.03846 (1.03846)
(1.03846) 25
PV  x  15.8795x  16.8795x
Payments out = Payments in
16.8795x  7889.92
x =$467.43


20
100  1.05
 = $1,035.63
1
1
.
12


.12  .05 
3.
PV 
4.
FV  PV 1  R 
5.
PV 1  R   FV
 $1,035.631.12
20
20
 $9,989.99
t
1  R t  FV PV
1  R   FV PV t
1

R  FV

1
t
PV
1
1
 7  1  10.00%
R   3,898,000
2,000,000 

6.
C1 = 200
PV= 2500
g = 0.03
PV 
C1
Rg
The Two Key Concepts in Finance
C1
g
PV
200

 .03  11.00%
2500
R
1

1
PV  C  
N 
 R R1  R  
7.
1

1
1  R 10
FV  C  
N 
 R R1  R  
C
C
8. PV = 1500
FV
1

1
1  R 10
 
N 
 R R1  R  
50,000
 1

1
1.0810


10 
 .08 .081.08 
N = 18
C = 100
 $3,451.47
R=?
1
1 
PV  C  
N 
 R 1  R  

1500  1
1
 
18 
100  R R1  R  
R = 0.0199/month
.0199 per month x 12 months per year = 23.88% per year
9. C = 1000
R = .06
PV4 
C $1000

 $16,667.67 (This is the value of the perpetuity at
R
.06
time 4.)
PV0 
$16,667.67
 $13,201.56
(1.06) 4
The Two Key Concepts in Finance
10. PV = 200,000
R = .08
N = 20
annuity due
 1

1
$200,000  C  C 

19 
 .08 .081.08 
C = $18,861.52
11. PV 
12.
$50
$50
$50


 $39.86  $35.59  $31.78  $107.22
2
3
(1.12)
(1.12)
(1.12) 4
C1
$11.035

 $9.86  9 7/8
R  g .14  .035
PV 
13. FV = 35000
N = 24
PV 
R = .06/12 = .005
C=?
FV
$35,000

 $31,051.50
24
(1  R )
(1.005) 24
1

1
PV  C  
N 
 R R(1  R) 
C
14.
15.
$31,051.50
 1

1


24 
 .005 .005(1.005) 

$31,051.50
 $1,376.22
22.56
PV9 
C10
$1000

 $33,333.33
R  g .07  .04
PV 
PV9
$33,333.33

 $18,131.12
9
(1  R)
(1.07) 9
PV of $50,000 annuity:
N = 20
R = 8%
PV of growing perpetuity:
PV = $490,907.37
The Two Key Concepts in Finance
g = .04
PV 
C1
Rg
C1 = PV(R – g) = $490,907.37 (.08 - .04) = $19,636.29
16.
This is potentially a complicated problem, depending on how you view it.
Cost = construction cost + maintenance
 $25,000 
$500
 $32,142.86
.12  .05
500 crypts:
Return 
X
benefit
500 X

 .12
cost
32142.86
.12($32142.86)
 $7.71
500
To recover costs:
$32,142.86
 $64.29 per crypt
500
To earn a 12% return:
$62.29 + $7.71 = $72.00 per crypt
17. R = 9%
N = 10
C = 2500
PV = 16,044.14
annuity due:
1

1
PV  C  C  
N 1 
 R R(1  R) 
 1

1
$16,044.14  C  C 

9 
 .09 .09(1.09) 
The Two Key Concepts in Finance
C = $2,293.66
18. 264,000 miles = 264000 miles x 5280 ft/mi. x 12 in/ft = 1.672704 x 1010
inches
Let D = number of doublings
.004 x 2D = 1.672704 x 1010
2D 
D
19.
1.672704 x1010
 4.18176 x1012
3
4.00 x10
ln( 4.18176 x1012 )
= 41.92  42 doublings
ln( 2)
Let x = 11.9999
(1)
Then 10x = 119.9999
(2)
Subtract (1) from (2):
9x = 108.0000
x = 12.0000 Q.E.D
20. Calculate the after-tax present value of the annuity due and compare it to the
after-tax value of the lump sum, which is $1,026,100:
The after tax annuity payment would be $250,000 - $75,305 - $10,000 =
$164,695.
 1

1
PV  $164,695  $164,695

 $1,746,371.36
19 
 .08 .08(1.08) 
The annuity is more valuable.
For a good elaboration on the role of taxes with lottery winnings, see “The
Lotto Jackpot: The Lump Sum Versus the Annuity,” by Allen Atkins and
Edward Dyl. Financial Practice and Education, Fall/Winter 1995, pp. 107 –
111.
The Two Key Concepts in Finance