Time Value of Money Practice Problems
Solutions to each
problem follows problem number 51.
1. An investor deposited
$10,000 in a savings account paying 5% converted quarterly. At the
end of 5 years what is the value of the account?
2. A depositor planned to leave $2,000 in a savings account paying
5% converted semiannually for 5 years. However, at the end of 2 1/2
years the depositor had to withdraw $1,000. What amount will be in
the account at the end of the original 5 year period?
3. Find the value of $1,000 invested at 8% for 10 years with
interest compounded annually.
4. Find the amount of $6,000 invested at 12% for 5 years,
compounded -- Annually, Semi-Annually,
Quarterly, Monthly, Daily.
5. Find the present value of $5,000 due in 4 years if money is
worth 4% compounded semi-annually.
6. What is the present value of a certificate of deposit with a
maturity value of $1,000 due in 3 years, if money is worth 6%
compounded semi-annually?
7. A person can buy a piece of property for $4500 cash OR for
$2000 down and $3000 in 3 years. If money is earning 6% compounded
semi-annually, which is the better purchase plan and by how much?
8. A piece of property can be purchased for $2850 cash OR for
$3000 in 12 months. Which is the better plan if money is worth 7%
compounded quarterly?
9. Find the amount of an annuity of $5,000 per year for 10 years
at 6% and 7% with interest compounded annually.
10. What is the value of an annuity of $100 paid monthly for 6
years if money is worth 6% compounded monthly?
11. An investor wants to provide for a $3000 scholarship every
year for 10 years. If the school can get a 5.5% return on its
investment, how much money should the investor give now?
12. Wilson agrees to pay Smith $1000 each year for 5 years. If
money is worth 7% what is the cash equivalent of this debt?
13. If money is worth 9% converted semi-annually, what is the
present value of $145.50 due every 6 months for 2 years?
14. An investor makes a $2000 annual deposit into a mutual fund
that produces a return of 12% annually for 3 years. How much will
the investor have at the end of the three year term?
15. What is the annual yield on: a) a 3% account compounded
monthly b) a 6 1/8% account compounded daily c)
a 9% account compounded semi-annually
16. An investor saves $500 per quarter in an IRA account for 30
years at 7 3/4% interest compounded quarterly. At the end of 30
years, she wants to pay herself equal (annual) payments for the next
25 years. If money continues to be worth 7.75% (now compounded
annually), how much will the annual payments be?
17. A city’s population is expected to increase at a rate of 4.95%
per year for the next ten years. If the current population is
322,000, what is the expected population at the end of the next ten
years?
18. Find the amount of $6000 for 8 years at 8% compounded a)
annually, b) semiannually, c) quarterly, d) monthly.
19. $2000 is deposited into an account earning 6.75%. What is the
balance of this account in 6 years, 8 months if interest is
compounded monthly?
20. An investment of $4000 is made for 12 years. During the first
5 years the interest rate is 7% compounded semiannually; the rate
then drops to 6% for the remainder of the time. What is the final
amount?
21. The University is given a gift of $400,000 for construction of
a science building. The University receives 8% on the money for 9
years then the rate drops to 7%. If the building is constructed 25
years after the gift was received, how much is in the fund at that
time?
22. What principal is needed to accumulate $3000 in 8 years at
4.5% interest compounded semiannually?
23. What is the present value at 4% compounded quarterly of
$12,000 due in 18 months?
24. A person owns a note for $2500 due in 5 years. What should a
buyer pay for the note if money is worth 5% compounded quarterly?
25. Find the present value of $2000 due in 15 months if money is
worth 7% compounded semiannually?
26. A University alumni wanted to set aside an amount such that
$250,000 will accumulate in 50 years to give to the University. What
should she set aside now to provide for the gift if money is worth
6% compounded monthly?
27. An investor deposits $1000 per year into an account earning
8.125% compounded annually for 14 years. At that time, she stops
contributing, the balance in the account continues to grow for
another 11 years (at the same interest rate). What is the balance at
the end of the term?
28. An investor’s IRA grows with $2,000 annual deposits and earns
9% for 30 years. The sum is used to live off of for the next 30
years. What is the value of the annuity before the withdrawals and
what is the amount of the annual withdrawals?
29. $500 rent payments are paid to a landlord. What is the value
of 5 years of monthly payments if money is worth 6.5% compounded
monthly?
30. $200 monthly deposits are made into a mutual fund that earns
12.25% compounded monthly. What is the value of the account at the
end of 7.25 years?
31. What is the monthly payment of a $28,000 car loan at 9.62%
interest for 5 years? How much is paid to the principal? How
much is paid to interest?
32. The landlord holding a $400 per month, four year lease wants
to sell the obligation to a bank. With money worth 7.15%,
(compounded monthly), how much will the bank pay for the lease?
What does the bank earn from this transaction?
33. To retire with $1,000,000 in 40 years, how much would have to
be deposited monthly to meet the goal if money is worth 7.375%? How much principal is paid into the account?
How much interest was earned?
34. What is the current cash value of $2500 annual payments for 8
years with money worth 6.175% compounded annually?
35. In 5 years, your annual salary will be $47,500 if you receive
8.2% annual raises. What is your current salary?
36. How long would it take to turn $2500 into $10,000 if a 15%
return (compounded monthly) was available?
37. How long would it take for $1800 to grow to $5000 with money
worth 9 1/2%, compounded daily?
38. How long would it take for money to double at 11% interest
compounded quarterly.
39. In 1998, a mutual fund investing in computer stocks turned a
$1000 investment into $1993.30. What was the rate of return if money
was compounded monthly?
40. $11,500 invested in America On Line (AOL) in 1992 would now
have $5,000,000.00 (that’s right, five million). What is the rate of
return assuming a 7 year term and annual compounding?
41. The DOW (Stock Market) rose from 7539 to 9181 in 1998. What is
the rate of return assuming annual compounding?
42. A $2000 IRA deposit is made annually into an account earning
16%. How long would it take to raise $500,000?
43. How long would it take to save $100,000 if $200 is deposited
in an account earning 8 1/2% compounded monthly?
44. How long would it take to save $10,000 with $100 monthly
deposits earning 9 ¼% compounded monthly?
45. What is the future value of $5000 invested for 10 years at 8%
compounded continuously?
46. What is the future value of $1500 invested for 6 ½ years at 7
¼% compounded continuously?
47. What is the present value of $10,000 at 8 1/8% compounded
continuously for 7 ¾ years?
48. What is the present value of $1000 at 11 3/8% interest
compounded continuously for 5 years 3 months?
49. How long would it take money to double at 15% compounded
continuously?
50. $1000 is invested for 6 years producing $2500. What is the
rate of return if money was compounded continuously?
51. $10,000 turns into $20,000 in 4 ½ years. What was the rate of
return under continuous compounding?
SOLUTIONS TO THE ABOVE PROBLEMS
1. An investor deposited
$10,000 in a savings account paying 5% converted quarterly. At the
end of 5 years what is the value of the account? [This is a future
value of a single sum problem. i = .05/4, n = 4x5. the answer
is $12,820.37]
2. A depositor planned to leave $2,000 in a savings account paying
5% converted semiannually for 5 years. However, at the end of 2
1/2 years the depositor had to withdraw $1,000. What amount will
be in the account at the end of the original 5 year period?
[Calculate this problem by using the future value of a single sum
for half of the term (2 1/2 years), then subtract $1000, then
calculate the remaining balance for the 2 1/2 years that is left. $1,428.76]
3. Find the value of $1,000 invested at 8% for 10 years with
interest compounded annually. [Future value of a single sum. i =
.08/1, n = 10x1. $2,158.93]
4. Find the amount of $6,000 invested at 12% for 5 years,
compounded -- Annually [$10,574.05] Semi-Annually [$10,745.09]
Quarterly [$10,836.67] Monthly [$10,900.18] Daily [$10,931.63]
Use the future value of a single sum for each of these. This
is good practice using the different compounding frequencies.
5. Find the present value of $5,000 due in 4 years if money is
worth 4% compounded semi-annually. [This is a simple present value
of a single sum. i = .04/2. n = -4x2. $4,267.45]
6. What is the present value of a certificate of deposit with a
maturity value of $1,000 due in 3 years, if money is worth 6%
compounded semi-annually? [Present value of a single sum. i =
.06/2, n = -3x2. $837.48]
7. A person can buy a piece of property for $4500 cash OR for
$2000 down and $3000 in 3 years. If money is earning 6% compounded
semi-annually, which is the better purchase plan and by how much?
[pay in cash $4500., you would be better off by $12.45][Think of
this problem this way, the $4500 cash is in present value terms, the
$2000 down is in present value terms, the $3000 is due in 3 years,
that is in future value terms. Put everything in present value
terms which requires taking the present value of the $3000 and
adding it to the $2000.]
8. A piece of property can be purchased for $2850 cash OR for
$3000 in 12 months. Which is the better plan if money is worth 7%
compounded quarterly? [pay in payments, you would be better off by
$51.12][Same thing as the problem above, put all amounts in
present value terms.]
9. Find the amount of an annuity of $5,000 per year for 10 years
at 6% and 7% with interest compounded annually. [Use the future
value of an annuity formula for each of these. i = .06/1 and .07/1
respectively. n = 10 for each. 6%=$65,903.98;
7%=$69,082.24]
10. What is the value of an annuity of $100 paid monthly for 6
years if money is worth 6% compounded monthly? [Simple future
value of an annuity. i = .06/12, n = 6x12 $8640.89]
11. An investor wants to provide for a $3000 scholarship every
year for 10 years. If the school can get a 5.5% return on its
investment, how much money should the investor give now?
[Tricky. Imagine that the school has to receive an amount today to
afford to give the annual scholarship. The amount given
would be invested in at the given interest rate. Use the
present value of an annuity formula. i = .055, n = -10. $22,612.88]
12. Wilson agrees to pay Smith $1000 each year for 5 years. If
money is worth 7% what is the cash equivalent of this debt?
[Take the present value of an annutity. i = .07, n = -5. $4100.20]
13. If money is worth 9% converted semi-annually, what is the
present value of $145.50 due every 6 months for 2 years? [Present
value of an annuity. i = .09/2, n = -2x2. $521.99]
14. An investor makes a $2000 annual deposit into a mutual fund
that produces a return of 12% annually for 3 years. How much will
the investor have at the end of the three year term? [Future value
of an annuity. i = .12, n = 3. $6748.80]
15. What is the annual yield on: a) a 3% account compounded
monthly [3.04%] b) a 6 1/8% account compounded daily [6.32%] c) a
9% account compounded semi-annually [9.20%] [This is a
problem that we should have done in class where we are calculating
a yield after being given a rate an a compounding interval.
The YIELD is the effect on the RATE after compounding. Use
the future value of a single sum equation and leave off the "P."
We are not using principal. At the end of the calculation
subtract 1.
( 1 + i )n
-1.
16. An investor saves $500 per quarter in an IRA account for 30
years at 7 3/4% interest compounded quarterly. At the end of 30
years, she wants to pay herself equal (annual) payments for the
next 25 years. If money continues to be worth 7.75% (now
compounded annually), how much will the annual payments be?
[This is the combination problem that we did in class where you
are faced with the future value of an annuity to save the money,
then to withdraw it (live off of it) you use the present value of
an annuity and solve for the cash flow ("P"). $21,299.13]
17. A city’s population is expected to increase at a rate of
4.95% per year for the next ten years. If the current population
is 322,000, what is the expected population at the end of the next
ten years? [Future value of a single sum. 522,012]
18. Find the amount of $6000 for 8 years at 8% compounded a)
annually, b) semiannually, c) quarterly, d) monthly.
[a=$11,105.58, b=$11,237.89, c=$11,307.24, d=$11,357.74]
Future values of a single sum.
19. $2000 is deposited into an account earning 6.75%. What is the
balance of this account in 6 years, 8 months if interest is
compounded monthly? [Future value of a single sum. (time is
6 8/12 yrs.) $3,132.67]
20. An investment of $4000 is made for 12 years. During the first
5 years the interest rate is 7% compounded semiannually; the rate
then drops to 6% for the remainder of the time. What is the final
amount? [Use the future value of a single sum, twice. $8,534.64]
21. The University is given a gift of $400,000 for construction of
a business building. The University receives 8% on the money for 9
years then the rate drops to 7%. If the building is constructed 25
years after the gift was received, how much is in the fund at that
time? [Use the future value of a single sum, twice. Use the first
future value at 8%, then take the result of that calculation and
use it in the second future value problem using the 7% rate. $2,360,555.60]
22. What principal is needed to accumulate $3000 in 8 years at
4.5% interest compounded semiannually? [Either use the future
value of a single sum and solve for "P" OR use the present value
of a single sum. $2,101.40]
23. What is the present value at 4% compounded quarterly of
$12,000 due in 18 months? [Simple present value of a single sum. $11,304.54]
24. A person owns a note for $2500 due in 5 years. What should a
buyer pay for the note if money is worth 5% compounded quarterly?
[Present value of a single sum. $1,950.02]
25. Find the present value of $2000 due in 15 months if money is
worth 7% compounded semiannually? [Present value of a single sum. $1,835.46]
26. A University alumni wanted to set aside an amount such that
$250,000 will accumulate in 50 years to give to the University.
What should she set aside now to provide for the gift if money is
worth 6% compounded monthly? [Present value of a single sum. $12,540.16]
27. An investor deposits $1000 per year into an account earning
8.125% compounded annually for 14 years. At that time, she stops
contributing, the balance in the account continues to grow for
another 11 years (at the same interest rate). What is the balance
at the end of the term? [The first part of the problem is an
annuity (future value); when she stopped contributing, the
remainder of the problem is the future value of a single sum. $57,697.45]
28. An investor’s IRA grows with $2,000 annual deposits and
earns 9% for 30 years. The sum is used to live off of for the next
30 years. What is the value of the annuity before the withdrawals
and what is the amount of the annual withdrawals? [Take the future
value of an annuity to accumulate the money. $272,615.08,
Then take the present value of an annuity and solve for "P" to get
the annual payment. $26,535.36]
29. $500 rent payments are paid to a landlord. What is the future value
of 5 years of monthly payments if money is worth 6.5% compounded
monthly? [Future value of an annuity. $35,336.98]
30. $200 monthly deposits are made into a mutual fund that earns
12.25% compounded monthly. What is the value of the account at the
end of 7.25 years? [Future value of an annuity $27,813.53]
31. What is the monthly payment of a $28,000 car loan at 9.62%
interest for 5 years? [Remember that to find the payment on a
loan, use the present value of an annuity and solve for "P". $589.70] How much is paid to the principal?
[Principal is the amount financed. $28,000] How much is paid to interest? [$7,382]
32. The landlord holding a $400 per month, four year lease wants
to sell the obligation to a bank. With money worth 7.15%,
(compounded monthly), how much will the bank pay for the lease?
[The bank will pay the present value of an annuity = $16,655.63] What does the bank earn from this transaction?
[$2,544.37]
33. To retire with $1,000,000 in 40 years, how much would have to
be deposited monthly to meet the goal if money is worth 7.375%?
[Use the future value of an annuity (the money is accumulating
into the future) and solve for the payment "P" = $342.69] How much principal is paid into the account?
[$164,491.20] How much interest was earned? [$835,508.80]
34. What is the current cash value of $2500 annual payments for 8
years with money worth 6.175% compounded annually? [Current means
present value (annuity) $15,417.53]
35. In 5 years, your annual salary will be $47,500 if you receive
8.2% annual raises. What is your current salary? [Present value of
a single sum. $32,030]
36. How long would it take to turn $2500 into $10,000 if a 15%
return (compounded monthly) was available? [Use the future value
of a single sum and solve for time ("n") 111.595 months ]
37. How long would it take for $1800 to grow to $5000 with money
worth 9 1/2%, compounded daily? [Use the future value of a single
sum, solve for "n" = 3925.80 days ]
38. How long would it take for money to double at 11% interest
compounded quarterly. [ 25.55 quarters ]
39. In 1998, a mutual fund investing in computer stocks turned a
$1000 investment into $1993.30. What was the rate of return if
money was compounded monthly? [ 71% ]
40. $11,500 invested in America On Line (AOL) in 1992 would now
have $5,000,000.00 (that’s right, five million). What is the
rate of return assuming a 7 year term and annual compounding? [
138.175% ]
41. The DOW (Stock Market) rose from 7539 to 9181 in 1998. What is
the rate of return assuming annual compounding? [Future value of a
single sum. 21.78%]
42. A $2000 IRA deposit is made annually into an account earning
16%. How long would it take to raise $500,000? [Future value of an
annuity. Solve for "n" = 25.02 years]
43. How long would it take to save $100,000 if $200 is deposited
in an account earning 8 1/2% compounded monthly? [Future value of
an annuity, solve for "n" = 214.397 months]
44. How long would it take to save $10,000 with $100 monthly
deposits earning 9 ¼% compounded monthly? [Future value of an
annuity, solve for "n". 74.4 months]
45. What is the future value of $5000 invested for 10 years at 8%
compounded continuously? [Future value of a single sum $11,127.70]
46. What is the future value of $1500 invested for 6 ½ years at 7
¼% compounded continuously? [$2402.99]
47. What is the present value of $10,000 at 8 1/8% compounded
continuously for 7 ¾ years? [Present value of a single sum $5,327.58]
48. What is the present value of $1000 at 11 3/8% interest
compounded continuously for 5 years 3 months? [Present value of a
single sum $550.36]
49. How long would it take money to double at 15% compounded
continuously? [Use the "Pert" formula and solve for 't' 4.62098 years]
50. $1000 is invested for 6 years producing $2500. What is the
rate of return if money was compounded continuously? [Use the
"Pert" formula, solve for "r" 15.2715%]
51. $10,000 turns into $20,000 in 4 ½ years. What was the rate of
return under continuous compounding? [Use the "Pert" formula,
solve for "r". 15.4032%]
