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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)^{t}

i

= annual effective interest rate,

t

= time in years

Present value factor

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

d

= annual effective discount rate

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

d = \frac{i}{1 + i} = 1 - v, v = \frac{1}{1 + i}

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

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

m

= compounding periods per year

Force of interest

δ = ln (1 + i)

e^{δ} = 1 + i, a (t) = e^{δ t}

Simple interest accumulation

a (t) = 1 + i t

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^{\int_{0}^{t} δ (s) d s}

, a(t) = accumulated value of 1,

δ (s)

= force of interest at time s

Effective rate of interest in the nth period

i_{n} = \frac{a ( n ) - a ( n - 1 )}{a ( n - 1 )}

, a(t) = accumulation function, n = period index

Real rate of interest

i^{'} = \frac{i - r}{1 + r}

, i' = real rate, i = nominal (effective) interest rate, r = inflation rate

Future value of a single sum

F V = 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 = \frac{l n ( F V / P V )}{l n ( 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 (t_{1}, t_{2}) = e^{\int_{t_{1}}^{t_{2}} δ (s) d s}

δ (s)

= force of interest at time s,

t_{1}

= start time,

t_{2}

= end time

Compound interest doubling time

n = \frac{l n 2}{l n ( 1 + i )}

, n = years to double, i = annual effective rate

Solving for the unknown effective interest rate

i = (\frac{F V}{P V})^{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 - \frac{d ^{(m)}}{m})^{- 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 - v^{1/ m})

, 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)} < 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 = m ln (1 + \frac{i ^{(m)}}{m})

, δ = force of interest, i^{(m)} = nominal interest rate convertible m-thly, m = compounding frequency

Linear interpolation estimate of the yield rate

i^{*} \approx i_{L} + (i_{H} - i_{L}) \cdot \frac{N P V _{L}}{N P V _{L} - N P V _{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

a_{n ∣} = \frac{1 - v ^{n}}{i}

n

payments of 1 at end of each period;

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

Annuity-due (beginning of period) PV

a ¨_{n ∣} = \frac{1 - v ^{n}}{d} = (1 + i) a_{n ∣}

Annuity-immediate — accumulated value

s_{n ∣} = \frac{( 1 + i ) ^{n} - 1}{i} = (1 + i)^{n} a_{n ∣}

Deferred annuity PV

_{k ∣} a_{n ∣} = v^{k} a_{n ∣}

(annuity starting

k

periods from now)

Increasing annuity-immediate PV

(I a)_{n ∣} = \frac{a ¨ _{n ∣} - n v ^{n}}{i}

Payments:

1, 2, \dots, n

Decreasing annuity-immediate PV

(D a)_{n ∣} = \frac{n - a _{n ∣}}{i}

Payments:

n, n - 1, \dots, 1

Continuously paid annuity PV

a ˉ_{n ∣} = \frac{1 - e ^{- δ n}}{δ}

Mthly annuity-immediate PV

a_{n ∣}^{(m)} = \frac{1 - v ^{n}}{i ^{(m)}}

mn

payments of

1/ m

per period

Geometric annuity present value

P V = \frac{1 - ( \frac{1 + g}{1 + i} ) ^{n}}{i - g}

(i ≠ g), g = payment growth rate, i = effective rate, n = number of payments

Increasing arithmetic perpetuity present value

(I a)_{\infty∣} = \frac{1}{i d}, (I a ¨)_{\infty∣} = \frac{1}{d ^{2}}

, i = effective interest rate, d = effective discount rate

Annuity-due to annuity-immediate relationship

a ¨_{n ∣} = (1 + i) a_{n ∣}

, i = effective interest rate, n = number of payments

Perpetuity present value (immediate and due)

a_{\infty∣} = \frac{1}{i}, a ¨_{\infty∣} = \frac{1}{d}

, i = effective interest rate, d = effective discount rate

Solving for the term of a level annuity

n = - \frac{ln ( 1 - \frac{i \cdot P V}{P} )}{ln ( 1 + i )}

, n = number of payments, i = periodic rate, PV = present value, P = payment; requires

P > i \cdot P V

Arithmetic increasing annuity present value

P V = (P - Q) a_{n ∣} + Q (I a)_{n ∣}

, P = first payment, Q = annual increase, n = number of payments,

a, (I a)

= level and increasing annuity factors

Annuity payment from present value

P = \frac{P V}{a _{n ∣}}

, P = level payment, PV = present value,

a_{n ∣}

= annuity-immediate factor (use

a ¨_{n ∣}

for due)

Geometric growing perpetuity present value

P V = \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

(D s)_{n ∣} = \frac{n ( 1 + i ) ^{n} - s _{n ∣}}{i}

, n = number of payments, i = effective rate, s = annuity-immediate accumulated value factor

Modified rate for valuing a geometric annuity as level

j = \frac{i - g}{1 + g}

, j = equivalent level rate, i = effective interest rate, g = geometric growth rate per period

Increasing annuity-immediate accumulated value

(I s)_{n ∣} = \frac{s ¨ _{n ∣} - n}{i}

, n = number of payments, i = effective rate, ä = annuity-due accumulated value factor

Accumulated value of an annuity payable m-thly

s_{n ∣}^{(m)} = \frac{( 1 + i ) ^{n} - 1}{i ^{(m)}} = \frac{i}{i ^{(m)}} s_{n ∣}

, i = annual effective rate, i^{(m)} = nominal rate compounded m-thly, n = years

Present value of a continuously varying payment stream

P V = \int_{0}^{n} f (t) exp (- \int_{0}^{t} δ (s) d s) d t

, f(t) = payment rate at time t, δ(s) = force of interest, n = term

Continuously increasing annuity present value

(I ˉ a ˉ)_{n ∣} = \frac{a ˉ _{n ∣} - n v ^{n}}{δ}

a ˉ_{n ∣}

= level continuous annuity PV, v = present value factor, n = term, δ = force of interest

Continuous perpetuity present value

a ˉ_{\infty∣} = \frac{1}{δ}

, δ = force of interest

= ln (1 + i)

Loans 15 items

Prospective loan balance

B_{t} = P a_{n - t ∣}

PV of remaining

n - t

payments;

P

=level payment

Retrospective loan balance

B_{t} = L (1 + i)^{t} - P s_{t ∣}

L

=original loan,

P

=level payment

Interest and principal portions of payment $k$

Interest:

I_{k} = i B_{k - 1}

Principal:

P R_{k} = P - I_{k}

Sinking fund method — annual cost

Annual cost

= i L + \frac{L}{s _{n ∣ j}}

i

=loan rate,

j

=sinking fund rate,

L

=loan

Principal repaid over a block of payments

\sum_{t = j + 1}^{k} P_{t} = B_{j} - B_{k}

P_{t}

= principal in payment t,

B_{j}

= balance after payment j,

B_{k}

= balance after payment k

Final drop or balloon payment

R_{final} = B_{n - 1} (1 + i)

R_{final}

= exact last payment,

B_{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

\sum_{t = 1}^{n} I_{t} = n R - L

I_{t}

= interest in payment t, n = number of payments, R = level payment, L = loan principal

Amortized loan level payment

R = \frac{L}{a _{n ∣ i}}

, R = level end-of-period payment, L = loan principal,

a_{n ∣ i}

= annuity-immediate factor, n = number of payments, i = periodic rate

Recursive loan balance update

B_{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

\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^{'} = \frac{B _{t}}{a _{n^{'} ∣ 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 = (\frac{R \cdot s _{n ∣ j}}{L})^{1/ n} - 1

, L = loan amount, R = payment, j = reinvestment rate, n = number of periods, i = realized annual yield

Interest-only period payment

R_{I O} = i L

, 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

, 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 = \frac{L}{a _{n - k ∣ i}}

, L = principal, n = total term, k = interest-only periods, i = periodic rate, R = level payment

Bonds 18 items

Bond price formula

P = F r a_{n ∣} + C v^{n}

F

=face,

r

=coupon rate,

C

=redemption,

n

=periods, yield

i

Premium/discount bond formula

P - C = (F r - C i) a_{n ∣}

Premium if

F r > C i

; discount if

F r < C i

Makeham bond formula

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

K = C v^{n}

g = F r / C

(modified coupon rate)

Bond book value at time t

B V_{t} = F r \cdot a_{n - t ∣ i} + C \cdot v^{n - t} = C + (F r - C i) \cdot a_{n - t ∣ 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-up_{t} = i \cdot B V_{t - 1} - F r = (C i - F r) \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-down_{t} = F r - i \cdot B V_{t - 1} = (F r - C i) \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

A V = F r \cdot s_{n ∣ 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 = \frac{P - C v ^{n}}{F a _{n ∣ 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

a_{n ∣ i} = \frac{P - C}{F r - C i}

, then

v^{n} = 1 - i a_{n ∣ i}

; P = price, C = redemption, Fr = coupon, Ci = yield income, i = yield, n = coupons

Total premium amortized through time t

P - B V_{t} = (F r - C i) (a_{n ∣} - a_{n - t ∣})

, P = price, BV_t = book value at t, Fr = coupon, Ci = yield income, n = coupons, t = periods elapsed

Bond salesman's yield approximation

i \approx \frac{F r + ( 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 = C_{k} - F

, C_k = call price at date k, F = par value

Callable bond worst-case price

P = min_{k} [F r \cdot a_{k ∣ i} + C_{k} \cdot v^{k}]

, 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^{*} = F r / 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

A_{t} = F r - i \cdot B V_{t - 1} = (F r - C i) 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

F r - C i = \frac{A _{t}}{v ^{n - t + 1}}

A_{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

I_{t} = i \cdot B V_{t - 1}

I_{t}

= interest earned in period t, i = original purchase yield,

B V_{t - 1}

= book value at start of period

Sum of bond amortization adjustments

\sum_{t = 1}^{n} A_{t} = ∣ P - C ∣

A_{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

D_{Mac} = \frac{\sum _{t} t v ^{t} C F _{t}}{\sum _{t} v ^{t} C F _{t}}

Weighted average time of cash flows

Modified duration

D_{mod} = \frac{D _{Mac}}{1 + i}

Δ P \approx - D_{mod} P Δ i

Convexity

C = \frac{1}{P} \frac{d ^{2} P}{d i ^{2}}

Price approximation:

Δ P \approx - D_{mod} P Δ i + \frac{1}{2} C P (Δ i)^{2}

Immunization (Redington) conditions

PVassets​=PVliabilities​
Dassets​=Dliabilities​ (durations match)
Cassets​>Cliabilities​ strictly (convexity)
First two zero surplus and its slope; strict convexity makes the current rate a local minimum.

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

N P V = \sum_{t = 0}^{n} v^{t} C F_{t}

Accept project if

N P V > 0

Second-order price approximation with convexity

\frac{Δ P}{P} \approx - D_{mod} Δ i + \frac{1}{2} C_{mod} (Δ i)^{2}

, D_mod = modified duration, C_mod = modified convexity, Δi = change in yield rate

First-order price approximation using modified duration

\frac{Δ P}{P} \approx - D_{mod} \cdot Δ i

, ΔP/P = relative price change, D_mod = modified duration, Δi = change in yield rate (decimal)

Modified convexity from Macaulay measures

C_{mod} = \frac{C _{mac} + D _{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

P V = \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) \approx P (i_{0}) (\frac{1 + i _{0}}{1 + i})^{D_{mac}}

, P = price, i = new yield, i₀ = base yield, D_mac = Macaulay duration

Portfolio modified duration

D_{mod}^{port} = \sum_{j} w_{j} D_{mod, j}

, w_j = PV_j/ΣPV = PV weight of asset j, D_mod,j = modified duration of asset j

Modified convexity

C_{mod} = \frac{\sum _{t} t ( t + 1 ) C F _{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

P V = \sum_{t} \frac{C F _{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

P_{2} = \frac{D - t _{1}}{t _{2} - t _{1}} \cdot P V_{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

P_{1} + P_{2} = P V_{L}, t_{1} P_{1} + t_{2} P_{2} = D \cdot P V_{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

C_{mod} = \frac{t ( t + 1 )}{( 1 + i ) ^{2}}

, t = time to maturity, i = effective periodic yield

Macaulay convexity of a two-zero asset portfolio

C_{mac} (A) = \frac{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

f_{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

A F = \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 + s_{n})^{n} = (1 + f_{0, 1}) (1 + f_{1, 2}) \dots (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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