Annuities - Exam FM formula sheet

Every Exam FM formula in annuities on one printable sheet, condensed from the 9 reference pages that teach them.

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Annuities

Annuity-immediate

Present value: $a_{\overline{n}|i} = \dfrac{1 - v^{n}}{i}$
Accumulated value: $s_{\overline{n}|i} = \dfrac{(1+i)^{n} - 1}{i}$
Link: $s_{\overline{n}|} = a_{\overline{n}|}(1+i)^{n}$
Recursion: $a_{\overline{n}|} = v\left(1 + a_{\overline{n-1}|}\right)$

Annuity-due

Present value: $\ddot{a}_{\overline{n}|} = \dfrac{1 - v^{n}}{d} = a_{\overline{n}|}(1+i)$
Accumulated value: $\ddot{s}_{\overline{n}|} = \dfrac{(1+i)^{n} - 1}{d} = s_{\overline{n}|}(1+i)$
Relations: $\ddot{a}_{\overline{n}|} = 1 + a_{\overline{n-1}|}, \qquad \ddot{s}_{\overline{n}|} = s_{\overline{n+1}|} - 1$

Deferred annuities

Deferred immediate: ${}_{m|}a_{\overline{n}|} = v^{m} a_{\overline{n}|} = a_{\overline{m+n}|} - a_{\overline{m}|}$
First payment date: $t = m + 1$ for a deferred annuity-immediate
Deferred perpetuity: ${}_{m|}a_{\overline{\infty}|} = \dfrac{v^{m}}{i}$

Perpetuities

Perpetuity-immediate: $a_{\overline{\infty}|} = \dfrac{1}{i}$
Perpetuity-due: $\ddot{a}_{\overline{\infty}|} = \dfrac{1}{d} = \dfrac{1+i}{i}$
Increasing perpetuity: $(Ia)_{\overline{\infty}|} = \dfrac{1}{i} + \dfrac{1}{i^{2}} = \dfrac{1+i}{i^{2}}$
Geometric (Gordon): $PV = \dfrac{1}{i-g}$ payments growing at rate g < i

m-thly annuities

Present value: $a^{(m)}_{\overline{n}|} = \dfrac{1 - v^{n}}{i^{(m)}}$
Conversion: $a^{(m)}_{\overline{n}|} = \dfrac{i}{i^{(m)}}\,a_{\overline{n}|}$
Due version: $\ddot{a}^{(m)}_{\overline{n}|} = \dfrac{1-v^{n}}{d^{(m)}}$
Ordering: $a_{\overline{n}|} < a^{(m)}_{\overline{n}|} < \bar{a}_{\overline{n}|} < \ddot{a}^{(m)}_{\overline{n}|} < \ddot{a}_{\overline{n}|}$

Continuous annuities

Present value: $\bar{a}_{\overline{n}|} = \int_0^{n} v^{t}\,dt = \dfrac{1-v^{n}}{\delta}$
Accumulated value: $\bar{s}_{\overline{n}|} = \dfrac{(1+i)^{n}-1}{\delta}$
Conversion: $\bar{a}_{\overline{n}|} = \dfrac{i}{\delta}\,a_{\overline{n}|}$

Increasing annuities

Present value: $(Ia)_{\overline{n}|} = \dfrac{\ddot{a}_{\overline{n}|} - n v^{n}}{i}$
Accumulated value: $(Is)_{\overline{n}|} = \dfrac{\ddot{s}_{\overline{n}|} - n}{i}$
Increasing perpetuity: $(Ia)_{\overline{\infty}|} = \dfrac{1}{i} + \dfrac{1}{i^{2}}$
General arithmetic: $PV = P\,a_{\overline{n}|} + Q\,\dfrac{a_{\overline{n}|} - n v^{n}}{i}$ payments P, P+Q, P+2Q, …

Decreasing annuities

Present value: $(Da)_{\overline{n}|} = \dfrac{n - a_{\overline{n}|}}{i}$
Accumulated value: $(Ds)_{\overline{n}|} = \dfrac{n(1+i)^{n} - s_{\overline{n}|}}{i}$
The checking identity: $(Ia)_{\overline{n}|} + (Da)_{\overline{n}|} = (n+1)\,a_{\overline{n}|}$

Geometric annuities

Present value: $PV = \dfrac{1 - \left(\dfrac{1+g}{1+i}\right)^{n}}{i - g}$ first payment 1 at time 1, growing at rate g
The trick rate: $1 + j = \dfrac{1+i}{1+g}, \qquad PV = \dfrac{a_{\overline{n}|j}}{1+g}$
When i = g: $PV = n\,v = \dfrac{n}{1+i}$
Growing perpetuity: $PV = \dfrac{1}{i-g}, \quad g < i$

Annuities - Exam FM formula sheet. Formulas condensed from the free reference pages on this site. Not affiliated with or endorsed by the Society of Actuaries.