Recursion

Recursion expresses the value of a loan balance or annuity at one time step in terms of its value at the immediately preceding step. For a loan with outstanding balance $B_{k-1}$ at the start of period $k$ , level payment $P$ , and periodic effective rate $i$ :

$B_k = B_{k-1}(1+i) - P$

$a_{\overline{n}|} = v\bigl(1 + a_{\overline{n-1}|}\bigr)$

The loan recursion rolls the balance forward by one period's interest, then subtracts the payment; it starts from $B_0$ equal to the original loan amount.
The annuity recursion treats an $n$ -payment annuity-immediate as one payment plus a deferred $(n-1)$ -payment annuity.
Useful for building amortization schedules, proving annuity formulas by induction, and finding outstanding balances without evaluating long sums.

Outstanding Loan Balance After 2 Payments A loan of $ $5{,}000$ is repaid with level annual payments over 4 years at $i = 8\%$ . Find the outstanding balance after the 2nd payment using recursion.
Answer First find the level payment $P$ : $P = \frac{5{,}000}{a_{\overline{4}|0.08}} = \frac{5{,}000}{3.31213} \approx \$1{,}509.60$ Then apply the recursion from $B_0 = 5{,}000$ :
 $\begin{align*} B_1 &= 5{,}000(1.08) - 1{,}509.60 = \$3{,}890.40 \\ B_2 &= 3{,}890.40(1.08) - 1{,}509.60 = \$2{,}692.03 \end{align*}$ 
The outstanding balance after 2 payments is approximately $ $2{,}692.03$ .