Payment Period

Study guide: Exam FM-2

The payment period is the interval between successive payments of an annuity or loan — monthly, quarterly, annually. Every annuity factor is computed per payment period, so $i$ must be the effective rate for that interval and $n$ the number of such intervals; a rate quoted on any other basis is converted first.

$j = (1+i)^{1/m} - 1$

$j = \left(1 + \frac{i^{(k)}}{k}\right)^{k/m} - 1$

$L = P \cdot a_{\overline{mT}|\,j}$

$m$ is the number of payments per year, $T$ the term in years, $j$ the effective rate per payment period, $i$ the effective annual rate and $i^{(k)}$ a nominal rate convertible $k$ times a year. When $k = m$ the conversion is trivial: $j = i^{(m)}/m$ .
The same conversion works when payments are less frequent than compounding: annual payments at $6\%$ convertible monthly use $j = (1.005)^{12} - 1 = 6.1678\%$ .
An equivalent route keeps an annual rate and uses $m$ -thly annuity symbols: $a^{(m)}_{\overline{T}|} = \dfrac{1 - v^T}{i^{(m)}}$ is the value of $1$ per year paid in $m$ instalments of $1/m$ .
Frequency matters. At a fixed effective annual rate, more frequent payments repay principal sooner, so the total paid per year is smaller: since $i^{(m)} < i$ , $a^{(m)}_{\overline{T}|} > a_{\overline{T}|}$ .
In a loan problem the payment period is one of five linked quantities — term, rate, payment amount, payment period and principal. It is rarely the unknown, but misreading it (a monthly payment against an annual rate) is a classic source of wrong answers.

Monthly Mortgage at a Semiannually Compounded Rate A $300,000 mortgage is repaid by monthly payments over 25 years. The rate is quoted as 5% convertible semiannually. Find the monthly payment.
Answer The rate per half-year is $2.5\%$ ; a month is one-sixth of a half-year.
 $\begin{align*} j &= (1.025)^{1/6} - 1 \\ &= 0.0041239 \\ a_{\overline{300}|\,j} &= \frac{1 - (1.0041239)^{-300}}{0.0041239} \\ &= 171.938 \\ P &= \frac{300{,}000}{171.938} \\ &= 1{,}744.81 \end{align*}$ 
Using $0.05/12 = 0.004167$ per month instead would treat the rate as convertible monthly and overstate the payment.

Annual Versus Quarterly Repayment A $100,000 loan is repaid over 10 years at an effective annual rate of 8%. Compare the annual payment with the total paid per year if payments are made quarterly.
Answer Annual payments:
 $\begin{align*} P_A &= \frac{100{,}000}{a_{\overline{10}|0.08}} \\ &= 14{,}902.95 \end{align*}$ 
Quarterly payments, with $j = (1.08)^{1/4} - 1 = 1.9427\%$ and $40$ periods:
 $\begin{align*} P_Q &= \frac{100{,}000}{a_{\overline{40}|\,j}} \\ &= 3{,}618.91 \\ 4P_Q &= 14{,}475.64 \end{align*}$ 
Check with $i^{(4)} = 4j = 7.7706\%$ : $a^{(4)}_{\overline{10}|} = (1 - 1.08^{-10})/0.077706 = 6.9081566$ and $100{,}000/6.9081566 = 14{,}475.64$ per year. Paying quarterly costs $427.31 less a year, because principal is returned to the lender earlier.