Annuities - Exam FM formula sheet

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

9 topics, 32 formula lines. Every line is condensed from the Exam FM formula pages, which stay free to read in full. Print this page to get the sheet on paper; the site chrome drops away.

What is on this sheet

Annuities

Annuity-immediate

Present value: $a_{\overline{n} ∣ i} = \frac{1 - v ^{n}}{i}$
Accumulated value: $s_{\overline{n} ∣ i} = \frac{( 1 + i ) ^{n} - 1}{i}$
Link: $s_{\overline{n} ∣} = a_{\overline{n} ∣} (1 + i)^{n}$
Recursion: $a_{\overline{n} ∣} = v (1 + a_{\overline{n - 1} ∣})$

Annuity-due

Present value: $\overset{a}{¨}_{\overline{n} ∣} = \frac{1 - v ^{n}}{d} = a_{\overline{n} ∣} (1 + i)$
Accumulated value: $\overset{s}{¨}_{\overline{n} ∣} = \frac{( 1 + i ) ^{n} - 1}{d} = s_{\overline{n} ∣} (1 + i)$
Relations: $\overset{a}{¨}_{\overline{n} ∣} = 1 + a_{\overline{n - 1} ∣}, \overset{s}{¨}_{\overline{n} ∣} = s_{\overline{n + 1} ∣} - 1$

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: $_{m ∣} a_{\overline{\infty} ∣} = \frac{v ^{m}}{i}$

Perpetuities

Perpetuity-immediate: $a_{\overline{\infty} ∣} = \frac{1}{i}$
Perpetuity-due: $\overset{a}{¨}_{\overline{\infty} ∣} = \frac{1}{d} = \frac{1 + i}{i}$
Increasing perpetuity: $(I a)_{\overline{\infty} ∣} = \frac{1}{i} + \frac{1}{i ^{2}} = \frac{1 + i}{i ^{2}}$
Geometric (Gordon): $P V = \frac{1}{i - g}$ payments growing at rate g < i

m-thly annuities

Present value: $a_{\overline{n} ∣}^{(m)} = \frac{1 - v ^{n}}{i ^{(m)}}$
Conversion: $a_{\overline{n} ∣}^{(m)} = \frac{i}{i ^{(m)}} a_{\overline{n} ∣}$
Due version: $\overset{a}{¨}_{\overline{n} ∣}^{(m)} = \frac{1 - v ^{n}}{d ^{(m)}}$
Ordering: $a_{\overline{n} ∣} < a_{\overline{n} ∣}^{(m)} < \overset{a}{ˉ}_{\overline{n} ∣} < \overset{a}{¨}_{\overline{n} ∣}^{(m)} < \overset{a}{¨}_{\overline{n} ∣}$

Continuous annuities

Present value: $\overset{a}{ˉ}_{\overline{n} ∣} = \int_{0}^{n} v^{t} d t = \frac{1 - v ^{n}}{δ}$
Accumulated value: $\overset{s}{ˉ}_{\overline{n} ∣} = \frac{( 1 + i ) ^{n} - 1}{δ}$
Conversion: $\overset{a}{ˉ}_{\overline{n} ∣} = \frac{i}{δ} a_{\overline{n} ∣}$

Increasing annuities

Present value: $(I a)_{\overline{n} ∣} = \frac{a ¨ _{\overline{n} ∣} - n v ^{n}}{i}$
Accumulated value: $(I s)_{\overline{n} ∣} = \frac{s ¨ _{\overline{n} ∣} - n}{i}$
Increasing perpetuity: $(I a)_{\overline{\infty} ∣} = \frac{1}{i} + \frac{1}{i ^{2}}$
General arithmetic: $P V = P a_{\overline{n} ∣} + Q \frac{a _{\overline{n} ∣} - n v ^{n}}{i}$ payments P, P+Q, P+2Q, …

Decreasing annuities

Present value: $(D a)_{\overline{n} ∣} = \frac{n - a _{\overline{n} ∣}}{i}$
Accumulated value: $(Ds)_{\overline{n} ∣} = \frac{n ( 1 + i ) ^{n} - s _{\overline{n} ∣}}{i}$
The checking identity: $(I a)_{\overline{n} ∣} + (D a)_{\overline{n} ∣} = (n + 1) a_{\overline{n} ∣}$

Geometric annuities

Present value: $P V = \frac{1 - ( \frac{1 + g}{1 + i} ) ^{n}}{i - g}$ first payment 1 at time 1, growing at rate g
The trick rate: $1 + j = \frac{1 + i}{1 + g}, P V = \frac{a _{\overline{n} ∣ j}}{1 + g}$
When i = g: $P V = n v = \frac{n}{1 + i}$
Growing perpetuity: $P V = \frac{1}{i - g}, g < i$

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.