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What's covered on the Exam FM formula sheet?

Every formula is grouped by official syllabus topic, with the formula in math notation plus a one-line note on when to use it (or a watch-out from CAIA, CFA, or other prep-provider commentary). Coverage is calibrated to the 2026 syllabus and refreshed when the corpus changes.

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The full question bank with detailed solutions, mixed practice, readiness tracking, lessons (where available), and the formula sheet are all free forever. Fellow ($59/quarter or $149/year per track) unlocks timed mock exams, spaced-repetition flashcards, performance analytics, AI essay grading, and a personalized study plan.

Free SOA Exam FM (Financial Mathematics) Formula Sheet (2026)

Every Exam FM formula you need on the test, grouped by topic, rendered with full math notation. 30 formulas across 5 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

30 Formulas

5 Topics

2026 Syllabus

Free Forever

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All Exam FM Formulas

Time Value of Money 6 items

Accumulation factor — compound interest

a(t)=(1+i)^t

i

= annual effective interest rate,

t

= time in years

Present value factor

v=\dfrac{1}{1+i}=1-d

d

= annual effective discount rate

Relationship: $d$, $i$, $v$

d=\dfrac{i}{1+i}=1-v,\quad v=\dfrac{1}{1+i}

Nominal rate $i^{(m)}$ vs effective rate

\left(1+\dfrac{i^{(m)}}{m}\right)^m = 1+i

m

= compounding periods per year

Force of interest

\delta = \ln(1+i)

e^{\delta} = 1+i,\quad a(t)=e^{\delta t}

Simple interest accumulation

a(t)=1+it

Used for short periods (< 1 year) and treasury bills

Annuities and Non-Contingent Cash Flows 11 items

Annuity-immediate (end of period) PV

a_{\overline{n}|}=\dfrac{1-v^n}{i}

n

payments of 1 at end of each period;

v=(1+i)^{-1}

Annuity-due (beginning of period) PV

\ddot{a}_{\overline{n}|}=\dfrac{1-v^n}{d}=(1+i)\,a_{\overline{n}|}

Annuity-immediate — accumulated value

s_{\overline{n}|}=\dfrac{(1+i)^n-1}{i}=(1+i)^n\,a_{\overline{n}|}

Annuity-due — accumulated value

\ddot{s}_{\overline{n}|}=\dfrac{(1+i)^n-1}{d}

Perpetuity-immediate PV

a_{\overline{\infty}|}=\dfrac{1}{i}

Perpetuity-due PV

\ddot{a}_{\overline{\infty}|}=\dfrac{1}{d}

Deferred annuity PV

_{k|}a_{\overline{n}|}=v^k\,a_{\overline{n}|}

(annuity starting

k

periods from now)

Increasing annuity-immediate PV

(I a)_{\overline{n}|}=\dfrac{\ddot{a}_{\overline{n}|}-n v^n}{i}

Payments:

1,2,\ldots,n

Decreasing annuity-immediate PV

(D a)_{\overline{n}|}=\dfrac{n - a_{\overline{n}|}}{i}

Payments:

n,n-1,\ldots,1

Continuously paid annuity PV

\bar{a}_{\overline{n}|}=\dfrac{1-e^{-\delta n}}{\delta}

Mthly annuity-immediate PV

a_{\overline{n}|}^{(m)}=\dfrac{1-v^n}{i^{(m)}}

mn

payments of

1/m

per period

Loans 4 items

Prospective loan balance

B_t = P\,a_{\overline{n-t}|}

PV of remaining

n-t

payments;

P

=level payment

Retrospective loan balance

B_t = L(1+i)^t - P\,s_{\overline{t}|}

L

=original loan,

P

=level payment

Interest and principal portions of payment $k$

Interest:

I_k = i\,B_{k-1}

Principal:

PR_k = P - I_k

Sinking fund method — annual cost

Annual cost

= iL + \dfrac{L}{s_{\overline{n}|j}}

i

=loan rate,

j

=sinking fund rate,

L

=loan

Bonds 3 items

Bond price formula

P = Fr\,a_{\overline{n}|} + Cv^n

F

=face,

r

=coupon rate,

C

=redemption,

n

=periods, yield

i

Premium/discount bond formula

P - C = (Fr - Ci)\,a_{\overline{n}|}

Premium if

Fr>Ci

; discount if

Fr<Ci

Makeham bond formula

P = K + \dfrac{g}{i}(C-K)

K=Cv^n

g=Fr/C

(modified coupon rate)

General Cash Flows, Portfolios, and Asset-Liability Management 6 items

Macaulay duration

D_{\text{Mac}} = \dfrac{\sum_t t\,v^t\,CF_t}{\sum_t v^t\,CF_t}

Weighted average time of cash flows

Modified duration

D_{\text{mod}} = \dfrac{D_{\text{Mac}}}{1+i}

\Delta P \approx -D_{\text{mod}}\,P\,\Delta i

Convexity

C = \dfrac{1}{P}\dfrac{d^2P}{di^2}

Price approximation:

\Delta P \approx -D_{\text{mod}}\,P\,\Delta i + \tfrac{1}{2}C\,P\,(\Delta i)^2

Immunization (Redington) conditions

PVassets=PVliabilitiesPV_{\text{assets}}=PV_{\text{liabilities}}PVassets​=PVliabilities​
Dassets=DliabilitiesD_{\text{assets}}=D_{\text{liabilities}}Dassets​=Dliabilities​ (durations match)
Cassets>CliabilitiesC_{\text{assets}}>C_{\text{liabilities}}Cassets​>Cliabilities​ (convexity condition)

Forward interest rate

(1+i_n)^n(1+f_{n,m})^m=(1+i_{n+m})^{n+m}

f_{n,m}

m

-year forward rate starting in

n

years

Net present value

NPV = \sum_{t=0}^{n} v^t\,CF_t

Accept project if

NPV>0

Free SOA Exam FM (Financial Mathematics) Formula Sheet (2026)

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