Math 451 Unit 2 Sinking Fund Interactive Examples
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Feb 20, 2024
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Math 451
Unit 2 Sinking Fund
Interactive Examples
Joe is opening an educational fund for his son who needs to go to graduate school for 2 years from now. Joe has the money to invest in two different funds, A and B, now. The first fund (A) matures every year with a rate of return of 4%. The second fund (B) matures every 2 years with a
rate of return of 9%. His son requires $19,000 for each of the 3 years that he is in graduate school. Joe wants to invest in the funds and pay the required education expenses from the amount maturing from the prior investments. Let A1 be the amount of money invested in fund A for the first year, B1 be the amount of money invested in fund B for the first year, and so on.
Question 1
What is year 3 cash flow?
1.04A2 + 1.09B1 − A3 − B3 = 19,000
1.04A2 + 1.09B1 + A3 + B3 = 19,000
1.04A2 + 1.09B1 + A3 − B3 = 19,000
1.04A2 + 1.09B1 − A3 + B3 = 19,000
Solution
The correct answer is:
1.04A2 + 1.09B1 − A3 − B3 = 19,000
Correct
Hide solution
Question 2
When investing in fixed income securities, to provide for future expenses, a higher yield is better. Setting aside considerations of risk and only considering maturities, the problem is that a longer maturity may not provide the funds when they are needed.
Suppose you need to provide funds for 4 years, beginning in year 3. In this case, you need $3,000 at the beginning of each of those years.
Assume that you can hold cash (equivalently 1-year bonds) and 2-year bonds. Suppose that there are no transaction costs and that the yields on cash and 2-year bonds are 5% and 9%, respectively.
Once the last payment of $3,000 in 6 years is made, there will be no further obligations. The questions to be answered are as follows:
What is the minimum investment to accomplish this?
For periods 1–5, how do you re-invest the proceeds of maturing bonds?
Due to new costs at the bank, there is a rule that bonds may only be bought in multiples of $100. With this restriction and using the solver ability to include an integer constraint, what is the minimum amount that must be invested at time 0 to achieve the required payments of $3,000 in years 3–6?
Hints:
Set up decision variable for the number of bonds of each kind to buy.
Keep a running cash balance. It can never fall below 0.
Make an initial deposit before time 0. Use that to fund the purchases.
The initial deposit is also a decision variable.
The initial deposit is what should be minimized.
How much is required? Round up to the nearest dollar. Be sure to use the Generalized Reduced Gradient (GRG) nonlinear method. Try it several times with different starting values for the decision variables.
$7,597
$8,374
$8,850
$8,889
Solution
The correct answer is:
$8,374
Correct
Hide solution
Joe is opening an educational fund for his son who needs to go to graduate school for 2 years from now. Joe has the money to invest in two different funds, A and B, now. The first fund (A) matures every year with a rate of return of 4%. The second fund (B) matures every 2 years with a
rate of return of 9%. His son requires $19,000 for each of the 3 years that he is in graduate school. Joe wants to invest in the funds and pay the required education expenses from the amount maturing from the prior investments. Let A1 be the amount of money invested in fund A for the first year, B1 be the amount of money invested in fund B for the first year, and so on.
Question 3
What is year 4 cash flow?
1.04A3 + 1.09B2 + A4 = 19,000
1.04A3 − 1.09B2 − A4 = 19,000
1.04A3 + 1.09B2 − A4 = 19,000
1.04A3 − 1.09B2 + A4 = 19,000
Solution
The correct answer is:
1.04A3 + 1.09B2 − A4 = 19,000
Correct
Hide solution
Question 4
When investing in fixed income securities, to provide for future expenses, a higher yield is better. Setting aside considerations of risk and only considering maturities, the problem is that a longer maturity may not provide the funds when they are needed.
Suppose you need to provide funds in 3–5 years from now. The problem scales, so you need to figure out how to provide $1,000 in each of those years. Multiplying by 10 will give the solution to providing $10,000 at those same times.
Assume that you can hold cash (equivalently 1-year bonds) and 2-year bonds. Suppose that there are no transaction costs and that the yields on cash and 2-year bonds are 5% and 9%, respectively.
The cash flows will look like the following:
Once the last payment of $1,000 in 5 years is made, there will be nothing left. The questions to be answered are as follows:
What is the minimum investment to accomplish this?
For periods 1–4, how do you re-invest the proceeds of maturing bonds?
The decision variables are
and
, where
is the amount invested in the short-term bond in year n, and D
n
is the amount invested in the long-term bond in year n. What is the minimum required investment? Round to the nearest dollar.
$2,185
$2,775
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$3,000
None of the options are correct
Solution
The correct answer is:
$2,185
Wrong
Hide solution
Joe is opening an educational fund for his son who needs to go to graduate school for 2 years from now. Joe has the money to invest in two different funds, A and B, now. The first fund (A) matures every year with a rate of return of 4%. The second fund (B) matures every 2 years with a
rate of return of 9%. His son requires $19,000 for each of the 3 years that he is in graduate school. Joe wants to invest in the funds and pay the required education expenses from the amount maturing from the prior investments. Let A1 be the amount of money invested in fund A for the first year, B1 be the amount of money invested in fund B for the first year, and so on.
Question 5
What is year 5 cash flow?
−1.04A4 − 1.09B3 = 19,000
−1.04A4 + 1.09B3 = 19,000
1.04A4 + 1.09B3 = 19,000
1.04A4 − 1.09B3 = 19,000
Solution
The correct answer is:
1.04A4 + 1.09B3 = 19,000
Correct
Hide solution