Outstanding Balance

Study guide: Exam FM-2

Outstanding balance (or outstanding loan balance) is the amount still owed on a loan at a given time, calculated by either the prospective or retrospective method.

Prospective method (present value of future payments):

$OB_k = P \cdot a_{\overline{n-k}|i}$

Retrospective method (accumulated value of loan minus accumulated payments):

$OB_k = L(1+i)^k - P \cdot s_{\overline{k}|i}$

The outstanding balance $OB_k$ is the remaining principal owed immediately after the $k$ -th payment on a loan of original amount $L$ , level payment $P$ , interest rate $i$ per period, and $n$ total payments.
Both formulas give the same result. The prospective method needs only the remaining payments; the retrospective method needs only the original loan and the payments already made, which makes it the one to use when the future payments are not fully known — for example a loan settled by a final balloon payment of unknown size.
Note that the formula $OB_k = L - P\,a_{\overline{k}|i}$ is incorrect in general; it would only apply if interest were zero.

Outstanding Balance After 3 Payments A $ $20{,}000$ loan is repaid with level annual payments over 6 years at $i = 8\%$ . Find the outstanding balance immediately after the 3rd payment using both methods.
Answer Payment: $P = 20000 / a_{\overline{6}|8\%} = 20000 / 4.6228797 = \$4{,}326.3077$ , about $ $4{,}326.31$ a year.

Prospective: $OB_3 = P \cdot a_{\overline{3}|8\%} = 4326.3077 \times 2.577097 = \$11{,}149.31$

Retrospective:
 $\begin{align*} OB_3 &= 20000(1.08)^3 - 4326.3077\,s_{\overline{3}|8\%} \\ &= 20000(1.259712) - 4326.3077(3.2464) \\ &= 25194.24 - 14044.93 \\ &= \$11{,}149.31 \end{align*}$ 
Both methods give the same balance. (Rounding $P$ to $4{,}326.31$ before using it moves the results by at most a cent — $11{,}149.32$ prospectively, $11{,}149.31$ retrospectively — which is rounding, not a difference between the methods.)