Problem Set: Time Value of Money II
Annuities, Perpetuities, Story Problems
(Solutions can be found at the end. They show how to do these problems
using formulae, but you may also use a calculator.)
Annuities
1. Value an annuity of $40.00 per year for ten years (r = 13%).
2. Value an annuity of $30.78 per year for fifty years (r = 11.8%).
3. Value an annuity due of $121.00 per year for six years (r = 6.7%).
4. Value an annuity due of $300.00 per year for eight years (r = 16.5%).
5. Value an annuity of $40.00 per year for ten years (r = 13.1%) that begins in six
years.
6. Value an annuity of $56.06 per year for 3 years (r = 5.5%) that begins in ten years.
7. Value an annuity of $600.00 per month for 3 years (r = 11.5%).
8. Value an annuity of $600.00 per week for 5 years (r = 7.8%).
9. Value an annuity of $300.00 per month for 7 years (r = 12.3%) that begins in 3
years.
10. Value an annuity that pays $300.00 per year for 7 years, then $500.00 for five years
(r = 12.3%).
11. If a five year annuity is worth $500.00 and r = 10.2%, what is the yearly cash flow?
12. If a ten year annuity is worth $100.00 and r = 11.4%, what is the yearly cash flow?
13. If a two year weekly annuity is worth $5000.00 and r = 9.8%, what is the weekly
cash flow?
Perpetuities
14. Value a perpetuity of $400.00 per year (r = 14.9%).
15. Value a perpetuity of $30.66 per year (r = 8.9%).
16. Value a perpetuity of $456.09 per year growing at 3% (r = 17.8%).
17. Value a perpetuity of $121.66 per year declining at 6% (r = 10.2%).
18. Value perpetuity of $300.00 per year growing at 1% that begins in five years. (r =
4.5%).
19. Value a perpetuity of $300.00 per month (r = 9.9%).
20. Value a perpetuity of $10.44 per week (r = 11.6%).
21. Value a perpetuity of $1,011.22 per month growing at 2% annually (r = 14.4%).
22. Value a perpetuity of $5.00 per day (365 days/year) (r = 9.8%).
23. Value perpetuity of $300.00 per week that begins in two years. (r = 14.5%).
24. If a perpetuity is worth $1,000 and r = 15.5%, what is the cash flow?
25. If a perpetuity is worth $5,556 and r = 2.3%, what is the cash flow?
26. If a growing perpetuity is worth $3,224.55 (r = 12.7% and g = 4%), what is the cash
flow?
27. If a perpetuity beginning in year three is worth $455.67 (r = 14.0%), what is the cash
flow?
28. If a growing perpetuity that pays $1,000 next year is worth $8,000.00 (r = 10.1%),
what is the growth rate?
Story Problems
29. You expect to go to graduate school in the Fall of 2000, and the tuition will be
$15,000 per year for the two-year M. B. A. program. If the interest rate is 8%, how
much do you need to save each academic year (1996/97, 1997/98, 1998/99,
1999/2000) to accumulate enough money to pay for all of your tuition by the time
you enter the program?
30. A late night TV ad offers the following: send in $100 today, and you will receive
payments for three years (beginning next year). The first payment will be $50 and
each subsequent year the payment will be increased by $10. What is the net
present value of this ‘deal’ if the appropriate discount rate is 22%?
31. You have borrowed $35,000 at an interest rate of 9%. If you plan to pay the loan off
in annual installments of $1,000 (beginning next year), when can you pay back the
loan?
32. The type of house you would like to buy requires a down-payment of $50,000. You
plan to make that down-payment five years from now. How much do you need to
save per year (beginning next year), if your money gets 7% (annually), and the
annual inflation rate for housing prices is 15%?
33. You hope to go to graduate school, and the tuition will be $10,000 per year for the
two-year M.B.A. program. If can only afford to save $3,000/year and the interest
rate is 9%, how long will you need to save to accumulate enough money to pay for
all of your tuition by the time you enter the program?
34. The house you plan to buy will require a down-payment of $40,000 in two years.
How much do you need to save per month (beginning next month), if your savings
gets 8% (annually), and is compounded monthly?
35. You have borrowed $10,000 at an interest rate of 8.7%. If you plan to pay the loan
off in quarterly installments of $1,000 (beginning next quarter), how long will it take
you to pay back the loan?
Problem Set 3: Time Value of Money II (Solutions)
Annuities
1.
PV 

40 
1
1
 = $217.05
0.13  1.13 10 


2.
PV 

30.78 
1
1
 = $259.86
0.118  1.118 50 


3.
PV  121 

121 
1
1
 = $621.13
0.067  1.067 5 


4.
PV  300 

300 
1
1 
 = $1,493.94
0.165  1.165 7 


5.
PV 

40 
1
1
1

= $116.82
10
0.131  1.131  1.1315


6.
PV 

56.06 
1
1
1

= $93.41
3
0.055  1.055   1.055 9






600 
1
 = $18,195.05
1
7. PV 
0.115   0.115 312 
12   1  12  






600 
1
 = $129,098.10
1
8. PV 
0.078   0.078 552 
52   1  52  






300 
1
1

1
= $13,184.93
9. PV 
712
0.123   0.123    0.123 212
12   1  12    1  12 


10.

300 
1
1

0.123  1.123 7 



500 
1
1
1

5
0.123  1.123   1.123 7


= 1,356.20 + 794.27
= $2,150.47
PV 
11.

C 
1
1 

0.102  1.102 5 




1

500  0.102  C  1 
 1.102 5 


500  0.102
C


1
1

 1.102 5 


C = $132.57
500 
12.

C 
1
1 

0.114  1.114 10 




1

100  0.114  C  1 
 1.114 10 


100  0.114
C


1
1

 1.114 10 


C = $17.27
100 
13.




C 
1

5000 
1
0.098   0.098 252 
52   1  52  






0.098
1


5000 
 C 1
252
  0.098
52
 
114  
  1 
52
 
 
0.098
5000 
52
C




1
1

  0.098 252 
 1

52  
 
C = $52.99
Perpetuities
14.
PV 
400
= $2, 684.56
0.149
15.
PV 
30.66
= $344.49
0.089
16.
PV 
456.09
= $3,081.69
0.178  0.03
17.
PV 
121.66
= $750.99
0.102   0.06 
18.
PV 
300
1
= $7,187.67
0.045  0.01 1.045 4
19.
PV 
300
= $36, 363.64
0.099
12
20.
PV 
10.44
= $4,680.00
0.116
52
21.
PV 
1,011.22
= $97,860.00
0.144  0.02
12
22.
PV 
5
= $18, 622.45
0.098
365
23.
PV 
300
0.145
52
1
 0.145 
 1+ 52 


252 1
24.
C
0.155
1,000  0.155  C
1,000 
C = $155.00
25.
C
0.023
5,556  0.023  C
5,556 
C = $127.79
26.
3,224.55 
C
 0.127  0.04 
3,224.55  0.127  0.04   C
C = $280.54
27.
455.67 
C
1
0.14 1.14 2
455.67 1.14  
2
C
0.14
455.67 1.14   0.14  C
2
= $82.91
28.
= $80,759.90
8,000 
1,000
 0.101  g 
0.101  g 
1,000
8,000
1,000
 0.101
8,000
1,000
g
 0.101
8,000
g = -2.4%
g 
Story Problems
29. Value needed at matriculation:
15,000 
15,000
 $28,888.89
1.08
Savings annuity (FV annuity formula solved for C):
28,888.89
1.08 
4
1
0.08
 $6,411.05
30.
100 
50
60
70


 $19.84
2
3
1.22 1.22 
1.22 
31. You could never pay it back at this discount rate. The most you could ever pay
back would be the present value of a perpetuity of $1,000 per year, i.e.,
1,000
 $1,111.11
0.09
32. In five years the down-payment will be:
50,000 1.15   $100,567.86
5
Thus you will need to save (per year):
100,567.86
1.07 
5
0.07
1
 $17,487.81
33. You need to have the following by matriculation:
10,000 
10,000
 $19,174.32
1.09
Thus, you must save for:
 19,174.32  0.09 
ln 
 1
3,000

  $5.27 years
ln 1.09 
34.
  0.08 212

 1
 1

12 
  $40,000
C


0.08


12


C = $1,542.42
35.


1,000 
1
1
0.087   0.087 4T
4   1  4 

T = 2.85 years


  10,000



You could solve the entire formula for T, but it is easier to do all the mathematical
operations first. That is 0.087/4 = 0.0218, etc. When you have done this, you get:

1
45,977.01 1 
 1.0218 4T

1
1
 0.2175
4T
1.0218 
1  0.2175 
0.7825 
1.0218 

  10,000


1
1.0218 
4T
1
1.0218 
4T

4T
1
0.7825
1.0218   1.2780
4T ln 1.0218   ln 1.2780 
ln 1.2780 
0.2453
T 

 2.8436
4ln 1.0218  0.0863
4T
The slight difference is due to rounding error, since in this calculation I used a
calculator and rounded to 2 decimals points, but used Excel for the first
calculation.