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MATH 400 (Theory of Interest) Final Exam
Review Questions
The actual exam will consist of the questions here except that the portions in
red will be different.
1.
The following investment project is being considered:
Year
Outlay
Return
0
$10,000
$0
1
2,000
2,000
2
1,000
8,000
3
3,000
3,000
4
0
12,000
(a) Find the value of the project one year before it goes into
operation, if interest is 10% compounded annually.
(b) Find the yield rate of the project.
(c) Find how the return of $12,000 for year 4 would have to
change in order for the yield rate to be 15%.
2. A man invests $10,000. At the end of year 1 the
investment is worth $11,000, and he adds another $5,000.
At the end of year 2 the investments are worth $18,000, and
he adds another $5,000. At the end of year 3 the
investments are worth $20,000 and he removes $10,000.
At the end of year 4 the investments are worth $11,000.
(a) Estimate the effective annual dollar-weighted yield rate
based on approximating (1 + i)n with 1 + in .
(b) Estimate the effective annual dollar-weighted yield rate
based on approximating the time when all activity took
place with the mid-way point of the project.
(c) Find the time-weighted yield rate.
3.
A man borrows $10,000 at 6% compounded monthly.
(a) Find the loan balance after 4 years, if he repays the loan
in equal monthly payments over 10 years.
(b) Find the loan balance after 4 years, if he repays the loan
at the rate of $100 at the end of each month.
(c) Find the loan balance after 4 years, if he repays the loan
at the rate of $50 on the 15th and 30th of each month.
(d) Find the loan balance after 4 years, if he repays the loan
at the rate of $300 at the end of each quarter.
4. A man borrows $4,000 at 6% compounded
semi-annually and wishes to repay the loan in semi-annual
payments.
(a) Suppose he wishes to amortize the loan in four equal
payments. Complete the amortization table, adjusting the
final payment as necessary.
interest principal
outstanding
period payment paid
repaid
balance
1
2
3
4
4.-continued
(b) Suppose he wishes to make payments of $1000 for as
long as necessary, with one larger balloon payment at the
end. Complete the amortization table, adjusting the final
payment as necessary.
interest principal
outstanding
period payment paid
repaid
balance
1
2
3
4
4.-continued
(c) Suppose he wishes to make payments of $1000 for as
long as necessary, with one smaller payment at the end.
Complete the amortization table, adjusting the final
payment as necessary.
interest principal
outstanding
period payment paid
repaid
balance
1
2
3
4
5. Five-year certificates of deposit are offered as follows:
deposits earn interest at 9% effective, but interest can be
reinvested at only 6% effective.
(a) For a single deposit with all interest reinvested for the
life of the certificate, find the overall effective rate of
interest.
(b) Find the single deposit necessary to accumulate $6000
at the end of the 5 years.
(c) Suppose you are allowed to add to the certificate of
deposit at each annual anniversary date. Find the annual
deposit at the beginning of each year necessary to
accumulate $6000 at the end of the 5 years.
6. A man borrows $10,000 at 12% interest compounded
monthly.
(a) Find the effective rate of interest.
(b) Suppose he intends to pay the entire amount (principal
and all interest) in one lump sum at the end of 8 years. He
intends do this by making monthly deposits at the end of
each month into a sinking fund that pays 9% compounded
monthly. Find the necessary monthly payment into the
sinking fund.
6.-continued
(c) Suppose he makes monthly payments of $200 at the end
of each month for the first year, $300 at the end of each
month for the second year, and $400 at the end of each
month for the third year. Find the outstanding balance at
the end of 3 years.
(d) Suppose he sets up regular monthly payments to
amortize the loan in 8 years. With the regular payment at
exactly 4 years, he pays an extra $100. If he then continues
with the regular monthly payments, find the outstanding
balance at exactly 5 years.
7. A woman placed $1000 in an account. After 4 years
the amount has grown to $1500.
(a) Suppose the account earned interest compounded
quarterly. Find the nominal yearly rate of interest.
(b) Suppose the account earned interest compounded
continuously. Find the nominal yearly rate of interest.
(c) Suppose the account earned simple interest. Find the
nominal yearly rate of interest.
(d) Suppose the account earned simple discount. Find the
nominal yearly rate of discount.
8. A woman borrows $48,000 at 12% compounded
monthly.
(a) Suppose she decides to pay back the loan in equal
monthly payments over 30 years. Find the amount of the
180th payment, and find how much of that payment is
interest.
(b) Suppose she decides to pay back the loan by paying
$500 per month for as long as it takes, with the final
payment being a smaller irregular amount. Find the
amount of the 180th payment, and find how much of that
payment is interest.
(c) Suppose she decides to pay back the loan by paying
$300 the first month and increasing each future monthly
payment by $3 for as long as it takes, with the final
payment being a smaller irregular amount. Find the
amount of the 180th payment, and find how much of that
payment is interest.
9. A man has $25,000 available to set up annuities. He
wants charity A to receive $X a year at the end of every
year for 20 years and charity B to receive $Y at the end of
every year in perpetuity. All interest is 8% compounded
semi-annually.
(a) Suppose X = $2000 and Y = $1000. How much of the
$25,000 is needed to set up two annuities, one for each
charity, or is $25,000 not enough to do this?
(b) Suppose the man decides to divide the $25,000 between
the two charities so that X = Y. Find the value for X and Y.
9.-continued
(c) Suppose the man decides to divide the $25,000 equally
between the two charities. Find the values of X and Y.
(d) Suppose the man decides to set up Y = $900 in
perpetuity for charity B. Find the number of years charity
A will be able to receive X = $2000 per year.
10. Consider the amount function A(t) = (t + 30)/10. Find
the corresponding accumulation function a(t), and find a
formula for each of In and in.
11. If the force of interest is t = 0.1/(1+t)2 , find the
accumulation function.
Answers to MATH 400 (Theory of Interest) Final Exam
Review Questions
1.
(a) $3,619.35
(b) 0.2247
(c) $8,232.56
2.
(a) 0.7273
(b) 0.10
(c) 0.1837
3.
(a) $6,698.92
(b) $7,295.11
(c) $7,288.35
(d) $7,322.02
4.
(a)
(b)
(c)
5.
(a) 0.0855
(b) $3,980.53
(c) $928.36
6.
(a) 0.1268
(b) $185.85
(c) $1,726.72
(b) 0.1017
(c) 0.125
(d) $5,573.66
7.
(a) 0.1027
(d) nominal yearly rate of discount = 1/12
8.
(a) $412.20
9.
(a) $21,957.24
Y = $4,897.04
10. a(t) = (t + 30)/30
11. a(t) = e0.1t / (1 + t)
(b) $381.27
(b) $2040.00
(c) $424.63
(c) X = $1,288.35 and
(d) any number of years
In = 1/10
in = 1/(n + 29)