I am having trouble solving the following problem which seems simple, but I cannot quite get it right.
Smith can repay a loan $L=250,000$ in one of two ways
1), 30 annual payments based on amortization at $i=.12$
2), 30 annual interest payments to the lender at rate $i=.10$, along with 30 level annual deposits to a sinking fund earning rate $j$
find the value of $j$ that makes the schemes equivalent
I am not 100% clear what they mean by equivalent. Smith is borrowing $L$ initially so it cannot be the present value of the loan. The two guesses that I have is the "total outlay being equivalent" and the "future value" being equivalent. I was not sure if I would get the same result, so I am doing the following calculations assuming the former case.
1), 30 level payments $K$ for an amortized loan is simply
$$K=\frac{L}{a_{\overline{30}\rceil .12}} \approx 31035.91$$
Thus the total outlay is again, simply $30K$.
2), Each outlay of the other scheme $M$ is
$$M=Li+\frac{L}{s_{\overline{30}\rceil j}}$$
so the total outlay through 30 yrs is equal to $30M$
So I am thinking that since these values are equal to each other I am solving for the equation for $j$ where
$$\frac{1}{a_{\overline{30}\rceil .12}}=.12+\frac{1}{s_{\overline{30}\rceil j}}$$
I used a graphing calculator to solve for $j$, but it gave me $j=i$ which is not the answer (It's supposed to be .021322)
Thanks.
*Edited
Aha! I think I see where your error is. $i=0.10$ in the equation involving the annuity symbols.