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Exam FM formula sheet PDF

Download a four-page financial mathematics formula sheet covering interest conversions, annuities, loans, bonds, duration, convexity, and asset–liability matching. Read the formulas below or save the printable PDF.

4 pages · No email or account · Updated September 14, 2026

Independent ActuaryProof reference, checked against the August 2026 Exam FM syllabus. This is a study aid, not an official SOA handout or a personal reference permitted in the exam.

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Know what is available during the examSOA Exam FM study page and August 2026 Exam FM syllabusThe current syllabus assumes specified calculators are allowed. It also says the listed text references, study notes, notation and terminology note, and other study resources will not be available with the exam; it does not supply a formula sheet.
Retrieve formulas quickly for freeDownload this free four-page Exam FM formula sheetNo email or account. It covers rates, annuities, loans, bonds, duration, convexity, immunization, and cash-flow matching, with notation and conditions beside the formulas.
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Start with the official requirements, free sheet, and free calculator guide. The paid resources do not unlock extra exam-day materials and are not required to use the complete formula reference on this page.

Notation: i is an effective rate per payment period, v=(1+i)⁻¹, and n counts payment periods. An annuity-immediate pays at period ends; an annuity-due pays at period beginnings. Bond coupons and yields must use the same period. The PDF writes annuity-due factors with a ‘due’ label.

Interest rates and accumulation

PDF page 1

Effective interest and discount

v=11+i,d=i1+i,i=d1−dv=\frac1{1+i},\quad d=\frac{i}{1+i},\quad i=\frac d{1-d}

All rates refer to the same period. Interest is measured against starting value; discount against ending value.

Constant force of interest

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

Time t uses the same unit as the force δ.

Nominal interest

1+i=(1+i(m)m)m1+i=\left(1+\frac{i^{(m)}}m\right)^m

i is annual effective; i^(m)/m is the rate for one mth of a year.

Nominal discount

1+i=(1−d(m)m)−m1+i=\left(1-\frac{d^{(m)}}m\right)^{-m}

Convert discount to accumulation before comparing with an interest quotation.

Equivalent payment-period rate

ih=(1+i)h−1i_h=(1+i)^h-1

h is the length of the payment period in years when i is annual effective.

Variable force

a(t)=exp⁡ ⁣(∫0tδ(s) ds),δ(t)=a′(t)a(t)a(t)=\exp\!\left(\int_0^t\delta(s)\,ds\right),\quad\delta(t)=\frac{a\prime(t)}{a(t)}

The accumulation function is positive and differentiable.

Simple interest and discount

a(t)=1+itora(t)=11−dta(t)=1+it\quad\text{or}\quad a(t)=\frac1{1-dt}

These are separate models, not compound accumulation. Simple discount requires 1−dt > 0.

Changing effective rates

a(n)=∏k=1n(1+ik)a(n)=\prod_{k=1}^n(1+i_k)

Multiply accumulation factors; do not use an arithmetic average of rates.

Equations of value and cash-flow returns

PDF page 1

Cash-flow value at time T

VT=∑kCka(T)a(tk)V_T=\sum_k C_k\frac{a(T)}{a(t_k)}

Use one sign convention and one comparison date.

NPV and yield

NPV⁡(i)=∑kCk(1+i)−tk,NPV⁡(i)=0 for IRR\operatorname{NPV}(i)=\sum_kC_k(1+i)^{-t_k},\quad\operatorname{NPV}(i)=0\ \text{for IRR}

An IRR is a root. Cash flows with multiple sign changes can have multiple roots.

Time-weighted return

1+iTW=∏k(1+rk)1+i_{\mathrm{TW}}=\prod_k(1+r_k)

Split at external cash flows; r_k is the investment return over each resulting interval.

Level annuities and payment timing

PDF page 2

Annuity-immediate

an‾∣=1−vni,sn‾∣=(1+i)n−1ia_{\overline n|}=\frac{1-v^n}{i},\quad s_{\overline n|}=\frac{(1+i)^n-1}{i}

Unit payments at 1,…,n. When i = 0, both factors equal n.

Annuity-due

a¨n‾∣=(1+i)an‾∣,s¨n‾∣=(1+i)sn‾∣\ddot a_{\overline n|}=(1+i)a_{\overline n|},\quad\ddot s_{\overline n|}=(1+i)s_{\overline n|}

Payments at 0,…,n−1. The accumulated due factor is also (1+i) times the immediate factor.

Deferred annuity

PV⁡=vman‾∣\operatorname{PV}=v^m a_{\overline n|}

Unit payments at m+1,…,m+n; m is the deferral to the annuity valuation date.

Perpetuities

a∞‾∣=1i,a¨∞‾∣=1da_{\overline\infty|}=\frac1i,\quad\ddot a_{\overline\infty|}=\frac1d

Requires i > 0. Immediate starts at time 1; due starts now.

mthly annuity

an‾∣(m)=1−vni(m)a^{(m)}_{\overline n|}=\frac{1-v^n}{i^{(m)}}

Payments of 1/m at times 1/m,…,n; i is annual effective and mn is an integer.

Continuous annuity

aˉn‾∣=1−e−δnδ\bar a_{\overline n|}=\frac{1-e^{-\delta n}}\delta

Unit payment rate over 0 to n with constant force δ. At δ = 0 the value is n.

Increasing, decreasing, and geometric payments

PDF page 2

Increasing annuity-immediate

(Ia)n‾∣=a¨n‾∣−nvni(Ia)_{\overline n|}=\frac{\ddot a_{\overline n|}-nv^n}{i}

Payments 1,2,…,n at times 1,2,…,n. For i = 0 use n(n+1)/2.

Decreasing annuity-immediate

(Da)n‾∣=(n+1)an‾∣−(Ia)n‾∣(Da)_{\overline n|}=(n+1)a_{\overline n|}-(Ia)_{\overline n|}

Payments n,n−1,…,1.

Arithmetic-gradient payments

PV⁡=Pan‾∣+Q((Ia)n‾∣−an‾∣)\operatorname{PV}=P a_{\overline n|}+Q\bigl((Ia)_{\overline n|}-a_{\overline n|}\bigr)

Payment at time t is P+(t−1)Q.

Geometric payments

PV⁡=P11−((1+g)/(1+i))ni−g\operatorname{PV}=P_1\frac{1-((1+g)/(1+i))^n}{i-g}

First payment P₁ at time 1, growth rate g. If i = g, PV = nP₁/(1+i).

Growing perpetuity

PV⁡=P1i−g\operatorname{PV}=\frac{P_1}{i-g}

First payment P₁ at time 1; requires i > g and i,g > −1.

Loans and sinking funds

PDF page 3

Level loan payment

P=Lan‾∣P=\frac{L}{a_{\overline n|}}

Loan L, n end-of-period payments, effective loan rate i per period.

Prospective outstanding balance

Bk=Pan−k‾∣B_k=P a_{\overline{n-k}|}

Immediately after payment k. There are n−k remaining payments.

Retrospective outstanding balance

Bk=L(1+i)k−Psk‾∣B_k=L(1+i)^k-Ps_{\overline k|}

Should equal the prospective balance under the same payment assumptions.

Interest and principal portions

Ik=iBk−1,PR⁡k=P−IkI_k=iB_{k-1},\quad\operatorname{PR}_k=P-I_k

Interest uses the balance before payment k.

Principal progression

PR⁡k=Pvn−k+1\operatorname{PR}_k=Pv^{n-k+1}

For a level-payment loan; each next principal portion is (1+i) times the previous one.

Sinking-fund deposit

D=Lsn‾∣j,periodic outlay=Li+DD=\frac{L}{s_{\overline n|j}},\quad\text{periodic outlay}=Li+D

Interest-only loan L at rate i, accumulated fund repays L after n deposits earning rate j.

Bond prices and book values

PDF page 3

Coupon bond price

P=Can‾∣j+RvjnP=C a_{\overline n|j}+Rv_j^n

C is coupon per period, R redemption value, j yield per coupon period, v_j=(1+j)⁻¹.

Premium or discount

P−R=(C−Rj)an‾∣jP-R=(C-Rj)a_{\overline n|j}

P = R when C = Rj. Compare coupon amount with yield times redemption, not unlike rate conventions.

Book-value recursion

Bk+1=Bk(1+j)−CB_{k+1}=B_k(1+j)-C

B_k is immediately after coupon k; this recurrence applies before final redemption.

Price between coupon dates

Pdirty=Bk(1+j)hP_{\mathrm{dirty}}=B_k(1+j)^h

h is the elapsed fraction of a coupon period. Under straight-line accrued interest, clean price = dirty price − hC.

Callable bond and guaranteed yield

Pmax⁡=min⁡k∈C(Cak‾∣j+Rkvjk)P_{\max}=\min_{k\in\mathcal C}\left(Ca_{\overline k|j}+R_kv_j^k\right)

Test permitted call dates and maturity with their respective redemption amounts R_k.

Duration, convexity, and yield sensitivity

PDF page 4

PV weights and Macaulay duration

wt=CtvtP,DMac=∑ttwtw_t=\frac{C_tv^t}{P},\quad D_{\mathrm{Mac}}=\sum_ttw_t

P = Σ C_t v^t and P ≠ 0. Time is measured in yield periods.

Modified duration

Dmod=−P′(i)P(i)=DMac1+iD_{\mathrm{mod}}=-\frac{P\prime(i)}{P(i)}=\frac{D_{\mathrm{Mac}}}{1+i}

Cash flows are fixed and discounted with one effective yield i per period.

Convexity

C=P′′(i)P(i)=∑tt(t+1)Ct(1+i)−t−2P(i)\mathcal C=\frac{P\prime\prime(i)}{P(i)}=\frac{\sum_t t(t+1)C_t(1+i)^{-t-2}}{P(i)}

This is modified convexity for changes in effective yield i.

Price-change approximation

ΔPP≈−DmodΔi+12C(Δi)2\frac{\Delta P}{P}\approx-D_{\mathrm{mod}}\Delta i+\frac12\mathcal C(\Delta i)^2

A local approximation with fixed cash flows. Use Δi as a decimal, in the same yield convention.

Portfolio duration

Dportfolio=∑jPj∑kPkDjD_{\mathrm{portfolio}}=\sum_j\frac{P_j}{\sum_kP_k}D_j

Use present-value weights and matching definitions of duration.

Asset–liability matching and immunization

PDF page 4

Redington conditions

A(i0)=L(i0),A′(i0)=L′(i0),A′′(i0)>L′′(i0)A(i_0)=L(i_0),\quad A\prime(i_0)=L\prime(i_0),\quad A\prime\prime(i_0)>L\prime\prime(i_0)

These give a local surplus minimum against small parallel changes in the common yield.

Two-asset duration weights

w=DL−D2D1−D2,1−w=D1−DLD1−D2w=\frac{D_L-D_2}{D_1-D_2},\quad 1-w=\frac{D_1-D_L}{D_1-D_2}

Weights are PV fractions. Match PV as well; matching duration alone does not establish immunization.

Full immunization for a single liability

∑jAjvtj=LvT,∑jtjAjvtj=TLvT\sum_j A_jv^{t_j}=Lv^T,\quad\sum_j t_jA_jv^{t_j}=TLv^T

For a single liability at T, positive asset cash flows at dates on both sides of T, and a common flat yield. Match PV and duration.

Cash-flow matching

At=Ltat each payment dateA_t=L_t\quad\text{at each payment date}

Exact matching by date removes the need to reinvest an unmatched interim cash flow.

Real return

1+ireal=1+inominal1+π1+i_{\mathrm{real}}=\frac{1+i_{\mathrm{nominal}}}{1+\pi}

π is inflation for the same period. Nominal here means before inflation, not convertible mthly.

Which formula should you use?

These four original examples show the decisions that come before substitution. Each includes a check and a common mistake.

The examples are on this page; the four-page download remains the compact formula reference.

Convert a nominal rate to the payment period

When to use it: The annual quotation and the payment interval differ.

A nominal annual interest rate is 12%, compounded monthly. Find the monthly effective rate and the annual effective rate.

Divide the nominal rate by its 12 compounding periods, then compound the monthly accumulation factor for one year.

imonth=0.12/12=0.01i_{\text{month}}=0.12/12=0.01
iyear=(1.01)12−1≈0.1268250301i_{\text{year}}=(1.01)^{12}-1\approx0.1268250301

Check: A balance of 1,000 grows to 1,126.825030 after 12 monthly multiplications by 1.01.

Common mistake: An annual effective rate of 12% would instead imply a monthly rate of (1.12)^(1/12)−1. The words nominal and effective change the calculation.

Return to the formulas · Interest-rate problems

Change the nominal quotation in the free rate converter.

Locate the first payment before valuing an annuity

When to use it: The first payment is today: use an annuity-due or sum from time zero.

Twelve monthly payments of 100 occur at months 0 through 11. At an effective monthly interest rate of 1%, find their value at month 0.

An annuity-immediate with 12 payments starts at month 1. Moving all its payments one month earlier multiplies the value by 1.01.

PV=100(1.01)1−(1.01)−120.01≈1136.762825PV=100(1.01)\frac{1-(1.01)^{-12}}{0.01}\approx1136.762825
PV=100∑k=011(1.01)−kPV=100\sum_{k=0}^{11}(1.01)^{-k}

Check: The direct sum begins with an undiscounted 100 and agrees with the annuity-due value. Its value is below the undiscounted total of 1,200.

Common mistake: Using 12 payments at months 1–12 values a different stream. Neither changing N to 11 nor adding a thirteenth payment repairs that timing error.

Return to the formulas · Annuity and perpetuity problems

Count only the payments remaining after a loan payment

When to use it: A balance is requested immediately after payment k; value n−k remaining payments.

A loan of 10,000 is repaid by five equal year-end payments at 5% effective annual interest. Find the balance immediately after the second payment. Use unrounded payments in the calculation.

First value all five payments at origination. After payment two, three payments remain, with the next one a year away.

R=10000(1−1.05−5)/0.05≈2309.747981R=\frac{10000}{(1-1.05^{-5})/0.05}\approx2309.747981
B2=R1−1.05−30.05≈6290.016638B_2=R\frac{1-1.05^{-3}}{0.05}\approx6290.016638

Check: Accumulating the original loan and subtracting the first two payments gives B₂=10,000(1.05)²−R(1.05+1), also 6,290.016638.

Common mistake: Including the second payment again produces a balance before that payment. Rounding the payment early also changes the final cents.

Return to the formulas · Loan balance problems

Compare the loan balance immediately before and after payment two.

Separate coupon rate from yield

When to use it: The coupon determines the cash payment; the yield discounts that payment.

A bond pays annual coupons of 40 and redeems for 1,000 with the third coupon. At an effective annual yield of 5%, find the price immediately after the previous coupon date.

There are coupons at times 1, 2 and 3, plus redemption at time 3. Discount all four amounts at the yield.

P=401−1.05−30.05+1000(1.05)−3≈972.767520P=40\frac{1-1.05^{-3}}{0.05}+1000(1.05)^{-3}\approx972.767520
P=401.05+401.052+10401.053P=\frac{40}{1.05}+\frac{40}{1.05^2}+\frac{1040}{1.05^3}

Check: The coupon rate is 4%, below the 5% yield, so a price below the 1,000 redemption value is consistent. The direct cash-flow sum agrees.

Common mistake: Discounting at the 4% coupon rate substitutes the payment-setting convention for the required investment yield.

Return to the formulas · Bond valuation problems

What is included in this Exam FM cheat sheet?

The reference covers the five current FM syllabus areas: time value of money, annuities, loans, bonds, and cash-flow portfolios with asset–liability management. Rate conversions and formula conditions appear alongside the expressions. The sheet distinguishes a bond coupon amount from its yield and a loan interest rate from a sinking-fund earning rate.

Does SOA provide this formula sheet on exam day?

No. This downloadable sheet is an independent preparation resource. Use the official syllabus and exam instructions to check which tables and materials are supplied. Practice recalling the cash-flow relationships with your permitted calculator; do not plan on taking this sheet into the examination room.

Free reference or a paid study PDF?

The free four-page PDF gives formulas, notation, and conditions. It contains no full sample-solution manual or timed practice exam. Use it to find an identity quickly; use a worked solution when you need the reasoning that leads to that identity.

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Financial Mathematics Proof Manual, USD 49. Check its included content and actual page previews before purchasing.

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Formula and calculator cards

Financial Mathematics Formula & Calculator Workflow Cards, USD 19. The product page shows the card format and included material.

See the paid card set →

For BA II Plus settings and worksheet operations, use the separate Exam FM calculator guide.

Choosing your first exam? Read Exam P vs FM: which should you take first? Or open the Exam P formula sheet.

Source and format comparison reviewed September 26, 2026; worked examples added September 24, 2026. Published by ActuaryProof; report a correction.

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ActuaryProof publishes the independent explanations on this page. Official exam facts on this page come from the sources above; schedules, study ledgers, and checking protocols are ActuaryProof editorial tools unless stated otherwise. Read about the publisher or report a suspected error.

Primary sources last checked: September 26, 2026.