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. 96 formulas across 5 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

96 Formulas
5 Topics
2026 Syllabus
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All Exam FM Formulas

Time Value of Money 19 items
Accumulation factor — compound interest
a(t)=(1+i)ta(t)=(1+i)^t
ii = annual effective interest rate, tt = time in years
Present value factor
v=11+i=1−dv=\dfrac{1}{1+i}=1-d
dd = annual effective discount rate
Relationship: \(d\), \(i\), \(v\)
d=i1+i=1−v,v=11+id=\dfrac{i}{1+i}=1-v,\quad v=\dfrac{1}{1+i}
Nominal rate \(i^{(m)}\) vs effective rate
(1+i(m)m)m=1+i\left(1+\dfrac{i^{(m)}}{m}\right)^m = 1+i
mm = compounding periods per year
Force of interest
δ=ln⁡(1+i)\delta = \ln(1+i)
eδ=1+i,a(t)=eδte^{\delta} = 1+i,\quad a(t)=e^{\delta t}
Simple interest accumulation
a(t)=1+ita(t)=1+it
Used for short periods (< 1 year) and treasury bills
Accumulation function a(t) from time 0 under a variable force of interest
a(t)=e∫0tδ(s) dsa(t) = e^{\int_0^t \delta(s)\,ds}, a(t) = accumulated value of 1, δ(s)\delta(s) = force of interest at time s
Effective rate of interest in the nth period
in=a(n)−a(n−1)a(n−1)i_n = \dfrac{a(n) - a(n-1)}{a(n-1)}, a(t) = accumulation function, n = period index
Real rate of interest
i′=i−r1+ri' = \dfrac{i - r}{1 + r}, i' = real rate, i = nominal (effective) interest rate, r = inflation rate
Future value of a single sum
FV=P(1+i)nFV = P(1 + i)^n, P = principal invested at time 0, i = effective rate per period, n = number of periods
Solving for the unknown number of periods
n=ln⁡(FV/PV)ln⁡(1+i)n = \frac{\ln(FV/PV)}{\ln(1+i)}, n = number of periods, FV = future value, PV = present value, i = effective rate per period
General accumulation factor between times \(t_1\) and \(t_2\) under a variable force of interest
a(t1,t2)=e∫t1t2δ(s) dsa(t_1, t_2) = e^{\int_{t_1}^{t_2} \delta(s)\,ds}, δ(s)\delta(s) = force of interest at time s, t1t_1 = start time, t2t_2 = end time
Compound interest doubling time
n=ln⁡2ln⁡(1+i)n = \frac{\ln 2}{\ln(1+i)}, n = years to double, i = annual effective rate
Solving for the unknown effective interest rate
i=(FVPV)1/n−1i = \left(\frac{FV}{PV}\right)^{1/n} - 1, FV = future value, PV = present value, n = number of periods, i = annual effective rate
Nominal discount rate to effective annual rate
1+i=(1−d(m)m)−m1 + i = \left(1 - \frac{d^{(m)}}{m}\right)^{-m}, i = effective annual interest rate, d^{(m)} = nominal discount rate convertible m-thly, m = compounding frequency
Nominal discount rate from present value factor
d(m)=m(1−v1/m)d^{(m)} = m\left(1 - v^{1/m}\right), d^{(m)} = nominal discount rate convertible m-thly, v = present value factor = 1/(1+i), m = compounding frequency
Ordering of interest and discount rates
d<d(m)<δ<i(m)<id < d^{(m)} < \delta < i^{(m)} < i, d = effective discount, d^{(m)} = nominal discount, δ = force of interest, i^{(m)} = nominal interest, i = effective interest, m ≥ 1
Force of interest from nominal interest rate
δ=mln⁡(1+i(m)m)\delta = m\ln\left(1 + \frac{i^{(m)}}{m}\right), δ = force of interest, i^{(m)} = nominal interest rate convertible m-thly, m = compounding frequency
Linear interpolation estimate of the yield rate
i∗≈iL+(iH−iL)⋅NPVLNPVL−NPVH i^* \approx i_L + (i_H - i_L) \cdot \frac{NPV_L}{NPV_L - NPV_H} , i* = yield rate, i_L/i_H = low/high test rates, NPV_L/NPV_H = NPV at each rate
Annuities and Non-Contingent Cash Flows 23 items
Annuity-immediate (end of period) PV
an‾∣=1−vnia_{\overline{n}|}=\dfrac{1-v^n}{i}
nn payments of 1 at end of each period; v=(1+i)−1v=(1+i)^{-1}
Annuity-due (beginning of period) PV
a¨n‾∣=1−vnd=(1+i) an‾∣\ddot{a}_{\overline{n}|}=\dfrac{1-v^n}{d}=(1+i)\,a_{\overline{n}|}
Annuity-immediate — accumulated value
sn‾∣=(1+i)n−1i=(1+i)n an‾∣s_{\overline{n}|}=\dfrac{(1+i)^n-1}{i}=(1+i)^n\,a_{\overline{n}|}
Deferred annuity PV
k∣an‾∣=vk an‾∣_{k|}a_{\overline{n}|}=v^k\,a_{\overline{n}|}
(annuity starting kk periods from now)
Increasing annuity-immediate PV
(Ia)n‾∣=a¨n‾∣−nvni(I a)_{\overline{n}|}=\dfrac{\ddot{a}_{\overline{n}|}-n v^n}{i}
Payments: 1,2,…,n1,2,\ldots,n
Decreasing annuity-immediate PV
(Da)n‾∣=n−an‾∣i(D a)_{\overline{n}|}=\dfrac{n - a_{\overline{n}|}}{i}
Payments: n,n−1,…,1n,n-1,\ldots,1
Continuously paid annuity PV
aˉn‾∣=1−e−δnδ\bar{a}_{\overline{n}|}=\dfrac{1-e^{-\delta n}}{\delta}
Mthly annuity-immediate PV
an‾∣(m)=1−vni(m)a_{\overline{n}|}^{(m)}=\dfrac{1-v^n}{i^{(m)}}
mnmn payments of 1/m1/m per period
Geometric annuity present value
PV=1−(1+g1+i)ni−gPV = \frac{1 - \left(\frac{1+g}{1+i}\right)^n}{i - g} (i ≠ g), g = payment growth rate, i = effective rate, n = number of payments
Increasing arithmetic perpetuity present value
(Ia)∞∣‾=1id,(Ia¨)∞∣‾=1d2(Ia)_{\overline{\infty\vert}} = \frac{1}{id}, \quad (I\ddot{a})_{\overline{\infty\vert}} = \frac{1}{d^2}, i = effective interest rate, d = effective discount rate
Annuity-due to annuity-immediate relationship
a¨n∣‾=(1+i) an∣‾\ddot{a}_{\overline{n\vert}} = (1+i)\, a_{\overline{n\vert}}, i = effective interest rate, n = number of payments
Perpetuity present value (immediate and due)
a∞∣‾=1i,a¨∞∣‾=1da_{\overline{\infty\vert}} = \frac{1}{i}, \quad \ddot{a}_{\overline{\infty\vert}} = \frac{1}{d}, i = effective interest rate, d = effective discount rate
Solving for the term of a level annuity
n=−ln⁡(1−i⋅PVP)ln⁡(1+i)n = -\dfrac{\ln\left(1 - \frac{i \cdot PV}{P}\right)}{\ln(1+i)}, n = number of payments, i = periodic rate, PV = present value, P = payment; requires P>i⋅PVP > i \cdot PV
Arithmetic increasing annuity present value
PV=(P−Q) an∣‾+Q (Ia)n∣‾PV = (P - Q)\,a_{\overline{n\vert}} + Q\,(Ia)_{\overline{n\vert}}, P = first payment, Q = annual increase, n = number of payments, a,(Ia)a,(Ia) = level and increasing annuity factors
Annuity payment from present value
P=PVan∣‾P = \dfrac{PV}{a_{\overline{n\vert}}}, P = level payment, PV = present value, an∣‾a_{\overline{n\vert}} = annuity-immediate factor (use a¨n∣‾\ddot{a}_{\overline{n\vert}} for due)
Geometric growing perpetuity present value
PV=Pi−gPV = \frac{P}{i - g} (valid only when g < i), P = first payment, g = growth rate per period, i = effective interest rate
Decreasing annuity-immediate accumulated value
(Ds)n∣‾=n(1+i)n−sn∣‾i(Ds)_{\overline{n\vert}} = \frac{n(1+i)^n - s_{\overline{n\vert}}}{i}, n = number of payments, i = effective rate, s = annuity-immediate accumulated value factor
Modified rate for valuing a geometric annuity as level
j=i−g1+gj = \frac{i - g}{1 + g}, j = equivalent level rate, i = effective interest rate, g = geometric growth rate per period
Increasing annuity-immediate accumulated value
(Is)n∣‾=s¨n∣‾−ni(Is)_{\overline{n\vert}} = \frac{\ddot{s}_{\overline{n\vert}} - n}{i}, n = number of payments, i = effective rate, ä = annuity-due accumulated value factor
Accumulated value of an annuity payable m-thly
sn∣‾(m)=(1+i)n−1i(m)=ii(m) sn∣‾s_{\overline{n\vert}}^{(m)} = \frac{(1+i)^n - 1}{i^{(m)}} = \frac{i}{i^{(m)}}\,s_{\overline{n\vert}}, i = annual effective rate, i^{(m)} = nominal rate compounded m-thly, n = years
Present value of a continuously varying payment stream
PV=∫0nf(t) exp⁡ ⁣(−∫0tδ(s) ds)dtPV = \int_0^n f(t)\,\exp\!\left(-\int_0^t \delta(s)\,ds\right)dt, f(t) = payment rate at time t, δ(s) = force of interest, n = term
Continuously increasing annuity present value
(Iˉaˉ)n∣‾=aˉn∣‾−nvnδ(\bar{I}\bar{a})_{\overline{n\vert}} = \frac{\bar{a}_{\overline{n\vert}} - nv^n}{\delta}, aˉn∣‾\bar{a}_{\overline{n\vert}} = level continuous annuity PV, v = present value factor, n = term, δ = force of interest
Continuous perpetuity present value
aˉ∞∣‾=1δ\bar{a}_{\overline{\infty\vert}} = \frac{1}{\delta}, δ = force of interest =ln⁡(1+i)= \ln(1+i)
Loans 15 items
Prospective loan balance
Bt=P an−t‾∣B_t = P\,a_{\overline{n-t}|}
PV of remaining n−tn-t payments; PP=level payment
Retrospective loan balance
Bt=L(1+i)t−P st‾∣B_t = L(1+i)^t - P\,s_{\overline{t}|}
LL=original loan, PP=level payment
Interest and principal portions of payment \(k\)
Interest: Ik=i Bk−1I_k = i\,B_{k-1}
Principal: PRk=P−IkPR_k = P - I_k
Sinking fund method — annual cost
Annual cost =iL+Lsn‾∣j= iL + \dfrac{L}{s_{\overline{n}|j}}
ii=loan rate, jj=sinking fund rate, LL=loan
Principal repaid over a block of payments
∑t=j+1kPt=Bj−Bk\sum_{t=j+1}^k P_t = B_j - B_k, PtP_t = principal in payment t, BjB_j = balance after payment j, BkB_k = balance after payment k
Final drop or balloon payment
Rfinal=Bn−1(1+i)R_{\text{final}} = B_{n-1}(1+i), RfinalR_{\text{final}} = exact last payment, Bn−1B_{n-1} = balance after the last full payment, i = periodic rate; drop if < R, balloon if > R
Total interest paid over the life of a loan
∑t=1nIt=nR−L\sum_{t=1}^n I_t = nR - L, ItI_t = interest in payment t, n = number of payments, R = level payment, L = loan principal
Amortized loan level payment
R=Lan∣i‾R = \dfrac{L}{a_{\overline{n\vert i}}}, R = level end-of-period payment, L = loan principal, an∣i‾a_{\overline{n\vert i}} = annuity-immediate factor, n = number of payments, i = periodic rate
Recursive loan balance update
Bt=Bt−1(1+it)−RtB_t = B_{t-1}(1+i_t) - R_t, B_t = balance after payment t, i_t = rate charged in period t, R_t = payment made at time t
Total interest over a block of payments
∑t=j+1kIt=(k−j)R−(Bj−Bk)\sum_{t=j+1}^{k} I_t = (k-j)R - (B_j - B_k), R = level payment, B_j = balance after payment j, B_k = balance after payment k
Refinanced loan new payment
R′=Btan′∣i′‾R' = \dfrac{B_t}{a_{\overline{n'\vert i'}}}, B_t = outstanding balance at refinance date, n' = new term in periods, i' = new periodic rate, R' = new level payment
Lender's realized yield with reinvestment
i=(R⋅sn∣j‾L)1/n−1i = \left(\dfrac{R \cdot s_{\overline{n\vert j}}}{L}\right)^{1/n} - 1, L = loan amount, R = payment, j = reinvestment rate, n = number of periods, i = realized annual yield
Interest-only period payment
RIO=iLR_{IO} = iL, i = periodic rate, L = loan principal (balance stays at L since no principal is repaid)
Total interest across a loan restructure
Total interest=tR+n′R′−L\text{Total interest} = tR + n'R' - L, t = phase-1 payments, R = original payment, n' = phase-2 payments, R' = new payment, L = original principal
Amortizing payment after an interest-only period
R=Lan−k∣i‾R = \dfrac{L}{a_{\overline{n-k\vert i}}}, L = principal, n = total term, k = interest-only periods, i = periodic rate, R = level payment
Bonds 18 items
Bond price formula
P=Fr an‾∣+CvnP = Fr\,a_{\overline{n}|} + Cv^n
FF=face, rr=coupon rate, CC=redemption, nn=periods, yield ii
Premium/discount bond formula
P−C=(Fr−Ci) an‾∣P - C = (Fr - Ci)\,a_{\overline{n}|}
Premium if Fr>CiFr>Ci; discount if Fr<CiFr<Ci
Makeham bond formula
P=K+gi(C−K)P = K + \dfrac{g}{i}(C-K)
K=CvnK=Cv^n, g=Fr/Cg=Fr/C (modified coupon rate)
Bond book value at time t
BVt=Fr⋅an−t∣i‾+C⋅vn−t=C+(Fr−Ci)⋅an−t∣i‾BV_t = Fr \cdot a_{\overline{n-t\vert i}} + C \cdot v^{n-t} = C + (Fr - Ci) \cdot a_{\overline{n-t\vert i}}, F = face value, r = coupon rate, C = redemption, i = original yield, n = total periods, t = elapsed periods
Discount accumulation write-up in period t
Write-upt=i⋅BVt−1−Fr=(Ci−Fr)⋅vn−t+1\text{Write-up}_t = i \cdot BV_{t-1} - Fr = (Ci - Fr) \cdot v^{n-t+1}, Fr = coupon, C = redemption, i = yield, BV = book value, n = total periods, t = period
Premium write-down in period t
Write-downt=Fr−i⋅BVt−1=(Fr−Ci)⋅vn−t+1\text{Write-down}_t = Fr - i \cdot BV_{t-1} = (Fr - Ci) \cdot v^{n-t+1}, Fr = coupon, C = redemption, i = yield, BV = book value, n = total periods, t = period
Accumulated value of a bond with coupon reinvestment
AV=Fr⋅sn∣j‾+CAV = Fr \cdot s_{\overline{n\vert j}} + C, Fr = coupon per period, j = reinvestment rate per period, n = number of coupons, C = redemption value
Solving a bond for its coupon rate
r=P−CvnF an∣i‾r = \dfrac{P - C v^n}{F\, a_{\overline{n\vert i}}}, P = price, C = redemption value, v = 1/(1+i), n = coupons, F = face, i = yield, r = per-period coupon rate
Solving a bond for its term
an∣i‾=P−CFr−Cia_{\overline{n\vert i}} = \dfrac{P-C}{Fr-Ci}, then vn=1−i an∣i‾v^n = 1 - i\,a_{\overline{n\vert i}}; P = price, C = redemption, Fr = coupon, Ci = yield income, i = yield, n = coupons
Total premium amortized through time t
P−BVt=(Fr−Ci)(an∣‾−an−t∣‾)P - BV_t = (Fr - Ci)(a_{\overline{n\vert}} - a_{\overline{n-t\vert}}), P = price, BV_t = book value at t, Fr = coupon, Ci = yield income, n = coupons, t = periods elapsed
Bond salesman's yield approximation
i≈Fr+(C−P)/n(P+C)/2i \approx \dfrac{Fr + (C-P)/n}{(P+C)/2}, F = face, r = per-period coupon rate, C = redemption value, P = price, n = number of coupons, i = per-period yield
Call premium
Call premium=Ck−F \text{Call premium} = C_k - F , C_k = call price at date k, F = par value
Callable bond worst-case price
P=min⁡k[Fr⋅ak∣i‾+Ck⋅vk] P = \min_{k} \left[Fr \cdot a_{\overline{k\vert i}} + C_k \cdot v^k\right] , F = par, r = coupon rate, i = min yield, a = annuity factor, C_k = call price at date k, v = present value factor, k = call date
Call price premium-discount threshold
C∗=Fr/i C^* = Fr/i , C* = threshold call price, F = par, r = coupon rate, i = minimum yield per period; call prices below C* price at earliest call, above C* price at maturity
Bond book value adjustment in a coupon period
At=Fr−i⋅BVt−1=(Fr−Ci) vn−t+1A_t = Fr - i\cdot BV_{t-1} = (Fr - Ci)\,v^{n-t+1}, Fr = coupon, C = redemption value, i = yield, v = 1/(1+i), n = periods, t = period
Coupon minus yield interest from a known adjustment
Fr−Ci=Atvn−t+1Fr - Ci = \dfrac{A_t}{v^{n-t+1}}, AtA_t = adjustment in period t, Fr = coupon, Ci = yield interest on par, v = 1/(1+i), n = periods, t = period
Interest earned in a coupon period of a bond
It=i⋅BVt−1I_t = i \cdot BV_{t-1}, ItI_t = interest earned in period t, i = original purchase yield, BVt−1BV_{t-1} = book value at start of period
Sum of bond amortization adjustments
∑t=1nAt=∣P−C∣\sum_{t=1}^{n} A_t = |P - C|, AtA_t = period adjustment, P = purchase price, C = redemption value; the total premium or discount
General Cash Flows, Portfolios, and Asset-Liability Management 21 items
Macaulay duration
DMac=∑tt vt CFt∑tvt CFtD_{\text{Mac}} = \dfrac{\sum_t t\,v^t\,CF_t}{\sum_t v^t\,CF_t}
Weighted average time of cash flows
Modified duration
Dmod=DMac1+iD_{\text{mod}} = \dfrac{D_{\text{Mac}}}{1+i}
ΔP≈−Dmod P Δi\Delta P \approx -D_{\text{mod}}\,P\,\Delta i
Convexity
C=1Pd2Pdi2C = \dfrac{1}{P}\dfrac{d^2P}{di^2}
Price approximation: ΔP≈−Dmod P Δi+12C P (Δi)2\Delta P \approx -D_{\text{mod}}\,P\,\Delta i + \tfrac{1}{2}C\,P\,(\Delta i)^2
Immunization (Redington) conditions
1. PVassets=PVliabilitiesPV_{\text{assets}}=PV_{\text{liabilities}}
2. Dassets=DliabilitiesD_{\text{assets}}=D_{\text{liabilities}} (durations match)
3. Cassets>CliabilitiesC_{\text{assets}}>C_{\text{liabilities}} strictly (convexity)
First two zero surplus and its slope; strict convexity makes the current rate a local minimum.
Forward interest rate
(1+in)n(1+fn,m)m=(1+in+m)n+m(1+i_n)^n(1+f_{n,m})^m=(1+i_{n+m})^{n+m}
fn,mf_{n,m} = mm-year forward rate starting in nn years
Net present value
NPV=∑t=0nvt CFtNPV = \sum_{t=0}^{n} v^t\,CF_t
Accept project if NPV>0NPV>0
Second-order price approximation with convexity
ΔPP≈−Dmod Δi+12Cmod(Δi)2\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta i + \tfrac{1}{2}C_{\text{mod}}(\Delta i)^2, D_mod = modified duration, C_mod = modified convexity, Δi = change in yield rate
First-order price approximation using modified duration
ΔPP≈−Dmod⋅Δi\frac{\Delta P}{P} \approx -D_{\text{mod}} \cdot \Delta i, ΔP/P = relative price change, D_mod = modified duration, Δi = change in yield rate (decimal)
Modified convexity from Macaulay measures
Cmod=Cmac+Dmac(1+i)2C_{\text{mod}} = \frac{C_{\text{mac}} + D_{\text{mac}}}{(1+i)^2}, C_mod = modified convexity, C_mac = Macaulay convexity, D_mac = Macaulay duration, i = effective yield
Spot rate price of a zero-coupon bond
PV=1(1+st)tPV = \frac{1}{(1 + s_t)^t}, PV = present value of $1 at time t, s_t = spot rate for maturity t, t = time to maturity in years
First-order Macaulay approximation (multiplicative)
P(i)≈P(i0)(1+i01+i)Dmac P(i) \approx P(i_0)\left(\frac{1+i_0}{1+i}\right)^{D_{\text{mac}}} , P = price, i = new yield, i₀ = base yield, D_mac = Macaulay duration
Portfolio modified duration
Dmodport=∑jwjDmod,j D_{\text{mod}}^{\text{port}} = \sum_j w_j D_{\text{mod},j} , w_j = PV_j/ΣPV = PV weight of asset j, D_mod,j = modified duration of asset j
Modified convexity
Cmod=∑tt(t+1)CFtvt+2P C_{\text{mod}} = \frac{\sum_t t(t+1) CF_t v^{t+2}}{P} , CF_t = cash flow at t, v = 1/(1+i), P = price, i = yield
Present value under a non-flat yield curve
PV=∑tCFt(1+st)t PV = \sum_t \frac{CF_t}{(1+s_t)^t} , CF_t = cash flow at time t, s_t = spot rate for term t
Immunization asset amount shortcut for two zeros
P2=D−t1t2−t1⋅PVLP_2 = \dfrac{D - t_1}{t_2 - t_1} \cdot PV_L, P_2 = PV in longer zero, D = liability Macaulay duration, t_1,t_2 = zero maturities, PV_L = liability present value
Duration-matching two-equation system for two zeros
P1+P2=PVL,  t1P1+t2P2=D⋅PVLP_1 + P_2 = PV_L,\; t_1 P_1 + t_2 P_2 = D \cdot PV_L, P_1,P_2 = PVs invested, t_1,t_2 = maturities, D = liability duration, PV_L = liability PV
Modified convexity of a zero-coupon bond
Cmod=t(t+1)(1+i)2C_{\text{mod}} = \dfrac{t(t+1)}{(1+i)^2}, t = time to maturity, i = effective periodic yield
Macaulay convexity of a two-zero asset portfolio
Cmac(A)=t12P1+t22P2P1+P2C_{\text{mac}}(A) = \dfrac{t_1^2 P_1 + t_2^2 P_2}{P_1 + P_2}, t_1,t_2 = zero maturities, P_1,P_2 = PVs invested at each
One-year forward rate from adjacent spot rates
ft,t+1=(1+st+1)t+1(1+st)t−1f_{t,t+1} = \frac{(1+s_{t+1})^{t+1}}{(1+s_t)^t} - 1, f = forward rate from t to t+1, s = spot rate, t = time in years (larger accumulation factor on top)
Accumulation factor from a zero priced as a fraction of redemption
AF=1pAF = \frac{1}{p}, AF = accumulation factor (redemption per dollar invested), p = price as a fraction of redemption value (p is the discount factor)
Spot rate as geometric mean of one-year forwards
(1+sn)n=(1+f0,1)(1+f1,2)⋯(1+fn−1,n)(1+s_n)^n = (1+f_{0,1})(1+f_{1,2})\cdots(1+f_{n-1,n}), s_n = n-year spot rate, f = one-year forward rate, f_{0,1} = s_1, n = number of years

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