Level annuities

2 free worked examples

Solved in full, free to everyone, and never rotated. Read these before deciding whether the rest of the drill is worth paying for.

1. Exam FMLevel annuitiesCore

   Find the present value of 12 annual payments of 2,500, the first one year from now, at 6% effective.

   1. A$20,000.00$
   2. B$20,959.61$
   3. C$21,500.00$
   4. D$22,217.19$
   5. E$30,000.00$

   Solution

   1. Payments at the END of each year make this an annuity-immediate: $PV=2,500a_{\overline{12}|0.06}$.
   2. $a_{\overline{12}|0.06}=\frac{1-1.06^{-12}}{0.06}=8.383844$.
   3. $PV=2,500\times 8.383844$.
   4. $=20,959.61$. The annuity-due version would be 6% larger, at 22,217.19.

   Trap. Using the due factor when the first payment is a year away.

2. Exam FMLevel annuitiesCore

   Which relationship is always true?

   1. A${\ddot{a}}_{\overline{n}|}=a_{\overline{n}|}(1+i{)}^n$
   2. B${\ddot{a}}_{\overline{n}|}=a_{\overline{n}|}(1+i)$
   3. C${\ddot{a}}_{\overline{n}|}=a_{\overline{n}|}+n$
   4. D${\ddot{a}}_{\overline{n}|}=a_{\overline{n}|}/(1+i)$
   5. E${\ddot{a}}_{\overline{n}|}=a_{\overline{n+1}|}$

   Solution

   1. An annuity-due pays each instalment exactly ONE period earlier than the corresponding annuity-immediate.
   2. Shifting every payment one period earlier multiplies its present value by $(1+i)$ - once, not $n$ times.
   3. So ${\ddot{a}}_{\overline{n}|}=a_{\overline{n}|}(1+i)$, equivalently $\frac{1-v^n}{d}$.
   4. The related identity ${\ddot{a}}_{\overline{n}|}=1+a_{\overline{n-1}|}$ says the same thing from the other direction: a payment now plus a shorter immediate annuity.

   Trap. Raising (1+i) to the power n, which shifts the whole annuity n periods rather than one.