Term of Annuity

Study guide: Exam FM-2

The term of an annuity is the fixed period of time over which its payments are made, measured in payment periods: an annuity of n payments, one per period, has a term of n periods. For an annuity-immediate the term runs from one period before the first payment to the last payment; for an annuity-due, from the first payment to one period after the last. A 30-year monthly mortgage has a term of 360 months.

• In the annuity factor a_{\overline{n}|i}, n is the term — the number of payment periods.
• Given the present value, the level payment P and the rate i, the term of an annuity-immediate is found with logarithms:

n = \frac{\ln(1 - i \cdot \text{PV}/P)}{\ln(1/(1+i))}

= \frac{-\ln(1 - i \cdot \text{PV}/P)}{\ln(1+i)}

• For an annuity-due, \text{PV} = P\,\ddot{a}_{\overline{n}|} = P(1-v^n)/d, so replace i by d inside the logarithm: n = -\ln(1 - d \cdot \text{PV}/P)/\ln(1+i).
• When n is not a whole number, \lfloor n \rfloor regular payments are made plus one smaller payment — added to the last regular payment (a balloon payment) or made one period after it (a drop payment).
• A Perpetuity is an annuity with an infinite term.

Solving for Unknown Term A $10,000 loan at 6\% is repaid with level annual payments of $1,500, the first one year from now, for as long as needed, followed by a smaller final payment one year after the last regular one. How many regular payments are made, and how large is the final payment?

Answer

\begin{align*}
a_{\overline{n}|} &= \frac{10{,}000}{1{,}500} \\
&= 6.666667 \\
n &= \frac{-\ln(1 - 0.06 \times 6.666667)}{\ln(1.06)} \\
&= \frac{-\ln(0.60)}{0.0582689} \\
&= \frac{0.5108256}{0.0582689} \\
&= 8.77
\end{align*}

So 8 regular payments are made, and a smaller ninth at time 9 — a drop payment. It is the balance just after the 8th payment, carried one more year:

\begin{align*}
OB_8 &= 10{,}000(1.06)^8 - 1{,}500\,s_{\overline{8}|} \\
&= 15{,}938.48 - 1{,}500(9.897468) \\
&= 15{,}938.48 - 14{,}846.20 \\
&= 1{,}092.28 \\
\text{Drop} &= 1{,}092.28 \times 1.06 \\
&= 1{,}157.82
\end{align*}

Check: 1{,}500\,a_{\overline{8}|} + 1{,}157.82\,v^9 = 9{,}314.69 + 685.31 = 10{,}000.