a man owes the following obligations

Question

a man owes the following obligations a. 2000 due today, b 4000 due in 9 months with accumulated simple interest from today at 12% c. 10000 due in 2 years with accumulated interest from today at 10% (m=2) if the man wishes to pay these debts by a payment of 3000 in 1 year and 5000 in 1 1/2 years and a last payment in 3 years, how much should the last payment be if money is worth 14% compounded quarterly. the answer is 12,815.88. i just want to know how to arrive at that final answer.

Answer 1

The easiest way is to choose year 3 as the focal date and move every debt and payment to that date.

Money is worth 14% compounded quarterly, so the quarterly rate is:

i=.144=.035i=\frac{.14}{4}=.035

First find the actual amounts of the original debts when they are due.

The $2,000 is due today:

20002000

The $4,000 debt earns 12% simple interest for 9 months:

4000(1+.12(912))=43604000\left(1+.12\left(\frac{9}{12}\right)\right) =4360

The $10,000 debt earns 10% compounded semiannually for 2 years:

10000(1+.102)4=12155.0610000\left(1+\frac{.10}{2}\right)^4 =12155.06

Now move all three debts forward to year 3.

Debt due today grows for 12 quarters:

2000(1.035)12=3022.142000(1.035)^{12}=3022.14

The $4,360 debt is due at 9 months, so it grows from month 9 to year 3, which is 9 quarters:

4360(1.035)9=5943.544360(1.035)^9=5943.54

The $12,155.06 debt is due at year 2, so it grows for 4 more quarters:

12155.06(1.035)4=13946.9012155.06(1.035)^4=13946.90

Total value of the debts at year 3:

3022.14+5943.54+13946.90=22912.583022.14+5943.54+13946.90 =22912.58

Now move the replacement payments to year 3.

The $3,000 payment at year 1 grows for 8 quarters:

3000(1.035)8=3950.433000(1.035)^8=3950.43

The $5,000 payment at year 1.5 grows for 6 quarters:

5000(1.035)6=6146.275000(1.035)^6=6146.27

Let XX be the final payment at year 3.

Set the values equal:

3950.43+6146.27+X=22912.583950.43+6146.27+X=22912.58 X=22912.58−10096.70X=22912.58-10096.70 X=12815.88\boxed{X=12815.88}

So the final payment should be $12,815.88.