Actuarial Science - Amortization

Question

Kevin takes out a $10$-year loan of $L$, which he pays by the amortization method at an annual effective interest rate of $i$. Kevin makes payments of $1000$ at the end of each year. The total amount of interest repaid during the life of the loan is also equal to $L$. Calculate the amount of interest repaid during the first year of the loan.

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Here's the screenshot of the official solution:

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My question is simple, why couldn't I use this equation for total interest instead of the correct one above: $L(1+i)^{10}-L=L$ ? I think $L(1+i)^{10}$ is the accumulated value after $10$ years and if we subtract the original loan amount $L$ from it, we should get the total amount of interest, right? However, the answer indicates that I am wrong. And I am so confused right now. Someone please help!

Answer 1

Your solution is not correct because the loan is repaid in annual installments at the end of each year for 10 years, rather than as a single balloon payment at the end of the loan term, which is the situation that your equation describes.

I would have written the solution in a slightly different order, first solving for the present value of the loan $L$. Since the total amount of interest paid on the loan equals the total payments minus the total borrowed, and we are also told this equals the value of the loan itself, we simply have $$1000(10) - L = L,$$ or $L = 5000$. Then we note that the sum of the present value of all payments on the loan is $$L = 1000 a_{\overline{10\,|}\,i} = 1000(v + v^2 + \cdots + v^{10}),$$ where $v = (1+i)^{-1}$, and it is straightforward to solve for $i$.