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I have the following problem I'd like to get some confirmation on. I've seen one answer on chegg that don't seem to line up with my solution at all, but I believe I have the right idea, just not sure if I'm solving it right. Here is the problem:

Henry is repaying a loan at an effective rate of 5% a year. The payments at the end of each year for 10 years are 1000 each. In addition to the loan payments. Henry pays premiums for loan insurance at the beginning of each year. The first premium is 0.5% of the original loan balance, the second premium is 0.5% of the loan balance immediately after the first loan payment, etc., and the tenth premium is 0.5% of the loan balance immediately after the 9th loan payment. The present value of the premiums at 5% is X. Determine X.

Here is a walkthrough to my solution:

If the payments are 1000 each for 10 years, then the present value of them would be

$1000* \frac{1-(1.05^{-10})}{.05} = 7721.73$

Using an amortization method or calculating

$7721.73*1.05-1000=7107.82, $

$7107.82*1.05-1000=6463.21, etc.$

we get the loan balances as

$7721,73, 7107,82, ... 952.37, 0$

at $t=0, 1, ... 9, 10$ respectively.

Multiplying each of those numbers by 0.5% would give $38.61, 33.85, ... 3.07$ and the sum of all their present values would give

$38.61 + \frac{33.86}{1.05} + ... + \frac{3.07}{1.05^{9}} = 196.87$

So I end up getting 196.87 as my present vaule of the premiums paid at 5%. Can anyone confirm this was the right method to use? Not completely necessary if you verify the answer but that would be much appreciated. Thank you!

PS. I can't find a tag that suits this problem, so feel free to add one!

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I do not think that you are interpreting the phrase "loan balance" the way that a bank will, with respect to paying premiums for loan insurance.

Let $k \in \{0, 1, 2, \cdots, 9\}.$

After the $k$-th payment, which will occur $k$ years in the future, the loan balance at that time will be interpreted as

$$(10 - k) \times 1000.$$

The idea behind this is that if you ignore any consideration of the present value of the money that you still owe, after the $k$-th payment, you still have $(10-k)$ payments remaining.

Based on this interpretation of loan balance, the insurance premium payment immediately following the $k$-th loan payment will be

$$P(k) = 0.005 \times (10-k) \times 1000.$$

The expected value (today) of the payment $P(k)$ is

$$E(k) = \frac{P(k)}{(1.05)^k}.$$

Thus,

$$X = \sum_{k=0}^9 E(k).$$

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