$3\mathrm{.}4\mathrm{.}5($Alternative derivation of Makeham's Formula$)$ A loan of amount $L$ is repaid by a series of principal payments along with interest on the outstanding balance at the end of each period. If the rate of interest is i per period and the loan is valued on the issue date at a rate of $j$ per period, denote the present value on the issue date by $A_j(i).$

Let $K_j$ denote the present value at rate $j$ of the principal payments.

LOAN REPAYMENT $<mn\; id="EditorElement\_212400">2</mn><mn\; id="EditorElement\_212401">4</mn><mn\; id="EditorElement\_212402">5</mn>$

Then $A_j(i)-K_j$ is the present value $($at rate $j)$ of the interest payments. Show that $($i$)\frac{A_j(i)-K_j}{A_j(i^{\text{'}})-K_j}=\frac{i}{i^7},$ and $($ii$)A_j(j)=L.$ Use $($i$)$ and $($ii$)$ to show that $A_j(i)=K_j+\frac{i}{j}(L-K_j).$