Calculating the yearly payment in a loan. Amortization with unequal payments.

Question

The following is the problem that I am working on.

A 30-yr loan of $1,000$ is repaid with payments at the end of each year. Each of the first ten payments equals the amount of interest due. Each of the nest ten payments equals $150\%$ of the amount of interest due. Each of the last ten payments is $X$. The lender charges interest at an effective annual rate of $10\%$. Calculate $X$.

The following is my attempt.

The present value of the loan $L=1,000$ can be found as

$$L = (Liv+ \cdots +Liv^{10})+(1.5Liv^{11}+ \cdots +1.5Liv^{20})+(Xv^{21} + \cdots +Xv^{30})$$

where $i=10\%$ and $v=(1+i)^{-1}$.

This simplifies to

$$L = a_{\overline{10}\rceil .10}( Li(1+1.5v^{10})+Xv^{20})$$

So to solve of $X$,

$$X = (1.1)^{20}(\frac{L}{a_{\overline{10}\rceil .10}}-Li(1+1.5v^{10})) \approx 33.05$$

However, the solution is supposed to be $97.44$.

What is the part that I am not doing correctly?

Answer 1

Perhaps you misunderstood the question.

In particular, note that the payments are flat in the first decade and last decade but vary in the middle decade. Apart from tiny rounding issues, $97.44$ appears to be correct for payments in the third decase.

Year Start  Interest Payment End
 1  1000.00 100.00  -100.00 1000.00
 2  1000.00 100.00  -100.00 1000.00
 3  1000.00 100.00  -100.00 1000.00
 4  1000.00 100.00  -100.00 1000.00
 5  1000.00 100.00  -100.00 1000.00
 6  1000.00 100.00  -100.00 1000.00
 7  1000.00 100.00  -100.00 1000.00
 8  1000.00 100.00  -100.00 1000.00
 9  1000.00 100.00  -100.00 1000.00
10  1000.00 100.00  -100.00 1000.00
11  1000.00 100.00  -150.00  950.00
12   950.00  95.00  -142.50  902.50
13   902.50  90.25  -135.38  857.37
14   857.37  85.74  -128.61  814.50
15   814.50  81.45  -122.18  773.77
16   773.77  77.38  -116.07  735.08
17   735.08  73.51  -110.27  698.32
18   698.32  69.83  -104.75  663.40
19   663.40  66.34  - 99.51  630.23
20   630.23  63.02  - 94.53  598.72
21   598.72  59.87  - 97.44  561.15
22   561.15  56.12  - 97.44  519.83
23   519.83  51.98  - 97.44  474.37
24   474.37  47.44  - 97.44  424.37
25   424.37  42.44  - 97.44  369.37
26   369.37  36.94  - 97.44  308.87
27   308.87  30.89  - 97.44  242.32
28   242.32  24.23  - 97.44  169.11
29   169.11  16.91  - 97.44   88.58
30    88.58   8.86  - 97.44    0.00

If you want to calculate $X$ directly, the amount outstanding after ten years is $1000$. The amount outstanding after twenty years is $1000 \times 0.95^{10}$. Since you then pay off what remains at a flat rate $X$ for the final decade, you have $X+1.1X+1.1^2X+\cdots+X\times1.1^9 = 1000 \times 0.95^{10} \times 1.1 ^ {10}$ which means $$X=1000 \times 0.95^{10} \times 1.1 ^ {10} \times \frac{1.1-1}{1.1^{10}-1} \approx 97.44.$$