Payment Amount

Study guide: Exam FM-2

The payment amount $P$ is the level amount paid each period under an annuity or an amortized loan; it is found by dividing the value of the payment stream by the matching annuity factor, and each loan payment splits into interest on the outstanding balance plus repayment of principal.

$P = \frac{L}{a_{\overline{n}|i}}$

$P = \frac{PV}{\ddot{a}_{\overline{n}|i}}$

$P = \frac{FV}{s_{\overline{n}|i}}$

$L$ is the loan principal, $n$ the number of payments and $i$ the effective rate per payment period. Use $a_{\overline{n}|}$ for payments at the end of each period (Annuity Immediate), $\ddot{a}_{\overline{n}|} = (1+i)\,a_{\overline{n}|}$ for payments at the start (Annuity Due), and $s_{\overline{n}|}$ to accumulate to a target. If the rate is quoted on another period, convert it first — see Payment Period.
Interest and principal in the $k$ -th payment of a level-payment loan:

$I_k = i \cdot OB_{k-1} = P\left(1 - v^{\,n-k+1}\right)$

$PR_k = P\,v^{\,n-k+1}$

The principal portions grow by the factor $(1+i)$ each period while the interest portions shrink — the pattern laid out row by row in an Amortization Schedule. The Outstanding Balance after $k$ payments is $OB_k = P\,a_{\overline{n-k}|}$ .
Given any four of principal, rate, payment amount, term and payment period, the fifth follows. When the term is not a whole number of level payments, the last one is a Drop Payment or Balloon Payment.
Namesake. This is the FM annuity/loan payment. The page Payment is Exam P's insurance payment $Y$ — what an insurer pays on a loss after a deductible and limit.

Monthly Payment and Its Interest/Principal Split A $30,000 loan is repaid with level monthly payments over 5 years at 6% convertible monthly. Find the payment, and the interest and principal in the 25th payment.
Answer The rate per month is $0.06/12 = 0.005$ and $n = 60$ .
 $\begin{align*} P &= \frac{30{,}000}{a_{\overline{60}|0.005}} \\ &= \frac{30{,}000}{51.7256} \\ &= 579.98 \end{align*}$ 
For $k = 25$ there are $n - k + 1 = 36$ payments left including this one:
 $\begin{align*} PR_{25} &= 579.98\,(1.005)^{-36} \\ &= 484.66 \\ I_{25} &= 579.98 - 484.66 \\ &= 95.32 \end{align*}$ 
Check: $OB_{24} = P\,a_{\overline{36}|0.005} = 19{,}064.66$ and $0.005 \times 19{,}064.66 = 95.32$ . Two years in, about 84% of each payment is already repaying principal.

Structured Settlement Paid in Advance An insurer settles a bodily injury claim by buying an annuity for $500,000. It pays the claimant a level amount at the start of each year for 20 years, at an effective rate of 4%. Find the annual payment.
Answer
 $\begin{align*} \ddot{a}_{\overline{20}|0.04} &= 1.04 \times 13.5903 \\ &= 14.1339 \\ P &= \frac{500{,}000}{14.1339} \\ &= 35{,}375.84 \end{align*}$ 
The claimant receives $35,375.84 a year. Paid at year-end instead, the payment would be $1.04$ times larger, $36,790.88, because each payment waits a year longer.