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Having trouble understanding the solution for this question.

A borrower is repaying a loan at 5% effective with payments at the end of each year for 12 years, such that the payment at the end of the first year is $200, at the end of the second year is 190 and so forth until the payment at the end of the 10th year is 110.

(i) Find the amount of the loan.

The solution is as follows:

$L = 100*a_{\overline10|} + 10(Da)_{\overline10|}$

Since the payments start at 200, why is it not

$L = 200*a_{\overline10|} + 10(Da)_{\overline10|}$?

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user270494
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  • Could you write out the expressions-without using actuarial notation ? And what are the results in both cases ? – callculus42 Apr 13 '16 at 02:43
  • Simple answer: $200 a_{\overline{10}|}$ is greater than the payment stream (since after the first payment, all are less than $200$); and $10 (Da)_{\overline{10}|}$ is positive, so adding it gets you even further from the value of the loan that was described. – David K Apr 13 '16 at 04:53

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To get the result in the first statement, each payment is broken into two pieces: $$ \begin{align*} \$200 &= \$100 + \$100\\ \$190 &= \$100 + \$90\\ &\,\,\vdots\\ \$110 &= \$100 + \$10\\ \end{align*} $$ so the present value is the present value of the constant $\$100$ stream of payments plus the present value of the decreasing stream of payments.

If you break the payments down as $$ \begin{align*} \$200 &= \$200 - \$0\\ \$190 &= \$200 - \$10\\ &\,\,\vdots\\ \$110 &= \$200 - \$90\\ \end{align*} $$ then you could write the present value as $$200\cdot a_{\overline{10}\vert}-10\cdot d \cdot(I\"{a})_{\overline{9}\vert},$$ which is close to your second statement, but not quite the same.

Laars Helenius
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  • Don't you want to subtract an increasing annuity? – David K Apr 13 '16 at 04:56
  • Of course! It is fixed now. – Laars Helenius Apr 13 '16 at 11:16
  • @LaarsHelenius For the decreasing annuity and increasing annuity, how do we know it starts at 100 and 0 respectively? Like how does putting a 10 in front of the decreasing annuity make it start at 100? – user270494 Apr 13 '16 at 18:12
  • So I have revised the second expression again and I think it is fixed now. The thing about all the actuarial symbols is that they represent finite geometric series, so they always start with 1, in general. So the the deductions occur in the 2nd through 10th years so the increasing annuity's PV needs an additional year of discount. – Laars Helenius Apr 13 '16 at 20:45