Exam FM -- Financial Mathematics
Formula Reference
All formulas tested on SOA Exam FM, organized by topic.
No formula sheet is provided during Exam FM. A BAII Plus or BAII Plus Professional calculator is strongly recommended.
Interest Measurement1
All-in-One Relationship
\[(1+i)^t = \left(1+\frac{i^{(m)}}{m}\right)^{mt} = (1-d)^{-t} = e^{\delta t}\]
Rate Definitions
Effective rate of interest: \(i_t = \dfrac{A(t)-A(t-1)}{A(t-1)}\)
Effective rate of discount: \(d_t = \dfrac{A(t)-A(t-1)}{A(t)}\)
Discount factor: \(v = \dfrac{1}{1+i} = 1-d\)
\(d = \dfrac{i}{1+i} = iv\)
Effective rate of discount: \(d_t = \dfrac{A(t)-A(t-1)}{A(t)}\)
Discount factor: \(v = \dfrac{1}{1+i} = 1-d\)
\(d = \dfrac{i}{1+i} = iv\)
Force of Interest
\(\delta_t = \dfrac{a'(t)}{a(t)}\)
Accumulate 1 from \(t_1\) to \(t_2\):
\[AV = \exp\!\left(\int_{t_1}^{t_2}\delta_u\,du\right)\]
Accumulate 1 from \(t_1\) to \(t_2\):
\[AV = \exp\!\left(\int_{t_1}^{t_2}\delta_u\,du\right)\]
Inflation & Real Rate
\(i_{\text{real}} = \dfrac{1+i}{1+\pi}-1\)
\(i = (1+i_{\text{real}})(1+\pi)-1\)
\(i = (1+i_{\text{real}})(1+\pi)-1\)
Annuities2
Annuity-Immediate (end of period)
\(PV = a_{\overline{n}|} = \dfrac{1-v^n}{i}\)
\(AV = s_{\overline{n}|} = \dfrac{(1+i)^n-1}{i}\)
\(AV = s_{\overline{n}|} = \dfrac{(1+i)^n-1}{i}\)
Annuity-Due (beginning of period)
\(PV = \ddot{a}_{\overline{n}|} = \dfrac{1-v^n}{d}\)
\(AV = \ddot{s}_{\overline{n}|} = \dfrac{(1+i)^n-1}{d}\)
\(AV = \ddot{s}_{\overline{n}|} = \dfrac{(1+i)^n-1}{d}\)
Immediate vs. Due
\(\ddot{a}_{\overline{n}|} = a_{\overline{n}|}(1+i) = 1+a_{\overline{n-1}|}\)
\(\ddot{s}_{\overline{n}|} = s_{\overline{n}|}(1+i) = s_{\overline{n+1}|}-1\)
\(\ddot{s}_{\overline{n}|} = s_{\overline{n}|}(1+i) = s_{\overline{n+1}|}-1\)
Deferred Annuity & Perpetuity
Deferred: \({}_{m|}a_{\overline{n}|} = v^m\cdot a_{\overline{n}|} = a_{\overline{m+n}|}-a_{\overline{m}|}\)
Perpetuity-immediate: \(a_{\overline{\infty}|} = \dfrac{1}{i}\)
Perpetuity-due: \(\ddot{a}_{\overline{\infty}|} = \dfrac{1}{d}\)
Perpetuity-immediate: \(a_{\overline{\infty}|} = \dfrac{1}{i}\)
Perpetuity-due: \(\ddot{a}_{\overline{\infty}|} = \dfrac{1}{d}\)
Continuous Annuity
\(\bar{a}_{\overline{n}|} = \displaystyle\int_0^n v^t\,dt = \dfrac{1-v^n}{\delta} = \dfrac{i}{\delta}a_{\overline{n}|}\)
More General Annuities3
Arithmetic Progression
Payments \(P, P+Q, P+2Q, \ldots\):
\[PV = Pa_{\overline{n}|} + Q\frac{\ddot{a}_{\overline{n}|}-nv^n}{i}\] Calculator: \(PMT = P+Q/i,\quad FV = -Qn/i\)
\[PV = Pa_{\overline{n}|} + Q\frac{\ddot{a}_{\overline{n}|}-nv^n}{i}\] Calculator: \(PMT = P+Q/i,\quad FV = -Qn/i\)
Increasing / Decreasing
\((Ia)_{\overline{n}|} = \dfrac{\ddot{a}_{\overline{n}|}-nv^n}{i}\)
\((Da)_{\overline{n}|} = \dfrac{n-a_{\overline{n}|}}{i}\)
Increasing perpetuity: \((Ia)_{\overline{\infty}|} = \dfrac{1}{id}\)
\((Da)_{\overline{n}|} = \dfrac{n-a_{\overline{n}|}}{i}\)
Increasing perpetuity: \((Ia)_{\overline{\infty}|} = \dfrac{1}{id}\)
Geometric Progression
Payments \(1,(1+k),(1+k)^2,\ldots\):
\[PV = \frac{1-\left(\frac{1+k}{1+i}\right)^n}{i-k},\quad i\neq k\]
\[PV = \frac{1-\left(\frac{1+k}{1+i}\right)^n}{i-k},\quad i\neq k\]
j-Effective Method
When payments differ in frequency from interest period, convert to effective rate per payment interval.
Monthly from annual: \(j = (1+i)^{1/12}-1\)
Monthly from annual: \(j = (1+i)^{1/12}-1\)
Loans & Amortization4
Outstanding Balance
Prospective: \(B_t = R\,a_{\overline{n-t}|}\)
Retrospective: \(B_t = L(1+i)^t - R\,s_{\overline{t}|}\)
Retrospective: \(B_t = L(1+i)^t - R\,s_{\overline{t}|}\)
Amortization Components
\(I_t = i\cdot B_{t-1}\)
\(P_t = R_t - I_t\)
\(B_t = B_{t-1}(1+i)-R_t\)
Level payments: \(P_{t+k} = P_t(1+i)^k\)
\(P_t = R_t - I_t\)
\(B_t = B_{t-1}(1+i)-R_t\)
Level payments: \(P_{t+k} = P_t(1+i)^k\)
Unit Loan of \(a_{\overline{n}|}\)
Period \(t\) interest: \(1-v^{n-t+1}\)
Period \(t\) principal: \(v^{n-t+1}\)
Total payment: 1
Period \(t\) principal: \(v^{n-t+1}\)
Total payment: 1
Bonds5
Bond Pricing
Basic: \(P = Fr\,a_{\overline{n}|i} + Cv^n\)
Premium/Discount: \(P = C + (Fr-Ci)\,a_{\overline{n}|i}\)
\(P>C\) (premium): \(Fr>Ci\)
\(P
Premium/Discount: \(P = C + (Fr-Ci)\,a_{\overline{n}|i}\)
\(P>C\) (premium): \(Fr>Ci\)
\(P
Notation
\(F\) = face/par value, \(r\) = coupon rate/period
\(Fr\) = coupon, \(C\) = redemption value
\(i\) = yield/period, \(n\) = coupons
\(F=C\) unless otherwise stated
\(Fr\) = coupon, \(C\) = redemption value
\(i\) = yield/period, \(n\) = coupons
\(F=C\) unless otherwise stated
Bond Amortization
Book value: \(B_t = Fr\,a_{\overline{n-t}|i} + Cv^{n-t}\)
Interest: \(i\cdot B_{t-1}\)
Write-down/up: \(|(Fr-Ci)v^{n-t+1}|\)
Interest: \(i\cdot B_{t-1}\)
Write-down/up: \(|(Fr-Ci)v^{n-t+1}|\)
Callable Bonds
Price for all possible redemption dates at given yield.
Premium bond: call on FIRST date
Discount bond: call on LAST date
Lowest price = max guaranteed yield
Premium bond: call on FIRST date
Discount bond: call on LAST date
Lowest price = max guaranteed yield
Spot Rates & Forward Rates6
Spot-Forward Relationship
\[(1+s_n)^n\cdot(1+f_{[n,n+m]})^m = (1+s_{n+m})^{n+m}\]
\[(1+s_n)^n = (1+f_{[0,1]})(1+f_{[1,2]})\cdots(1+f_{[n-1,n]})\]
Definitions
\(s_t\) = t-year spot rate
\(f_{[t_1,t_2]}\) = forward rate from \(t_1\) to \(t_2\), annual
Price bonds by discounting each cash flow at its maturity spot rate.
\(f_{[t_1,t_2]}\) = forward rate from \(t_1\) to \(t_2\), annual
Price bonds by discounting each cash flow at its maturity spot rate.
Duration & Convexity7
Macaulay Duration
\[MacD = \frac{\sum_{t} t\cdot v^t\cdot CF_t}{\sum_{t} v^t\cdot CF_t}\]
Modified Duration
\[ModD = \frac{\sum_{t} t\cdot v^{t+1}\cdot CF_t}{\sum_{t} v^t\cdot CF_t} = MacD\cdot v\]
Quick Reference
Zero-coupon n-year: \(MacD = n\)
n-year par bond: \(MacD = \ddot{a}_{\overline{n}|}\)
Geom. increasing perp.: \(MacD = \dfrac{1+i}{i-k}\)
n-year par bond: \(MacD = \ddot{a}_{\overline{n}|}\)
Geom. increasing perp.: \(MacD = \dfrac{1+i}{i-k}\)
Price Approximations
Modified: \(P(i_1)\approx P(i_0)[1-(i_1-i_0)(ModD)]\)
Macaulay: \(P(i_1)\approx P(i_0)\left(\dfrac{1+i_0}{1+i_1}\right)^{MacD}\)
Macaulay: \(P(i_1)\approx P(i_0)\left(\dfrac{1+i_0}{1+i_1}\right)^{MacD}\)
Portfolio Duration
\[MacD_P = \frac{P_1}{P}MacD_1 + \cdots + \frac{P_m}{P}MacD_m\]
Passage of time: \(MacD_{t_2} = MacD_{t_1} - (t_2-t_1)\)
Convexity
\[ModC = \frac{\sum t(t+1)v^{t+2}\cdot CF_t}{\sum v^t\cdot CF_t}\]
\(ModC = v^2(MacC + MacD)\)
Zero-coupon: \(MacC = n^2\)
Zero-coupon: \(MacC = n^2\)
Immunization8
Redington vs Full
| Condition | Redington | Full |
|---|---|---|
| \(PV_A = PV_L\) | Required | Required |
| \(MacD_A = MacD_L\) | Required | Required |
| \(C_A > C_L\) | Required | -- |
| Asset CFs bracket liability | -- | Required |
| Protects against | Small \(\Delta i\) | Any \(\Delta i\) |
Immunization Shortcut
Weight in shorter bond:
\[w = \frac{t_2-t_L}{t_2-t_1}\] \(t_1\) = shorter, \(t_2\) = longer, \(t_L\) = liability duration
\(1-w\) = weight in longer bond
\[w = \frac{t_2-t_L}{t_2-t_1}\] \(t_1\) = shorter, \(t_2\) = longer, \(t_L\) = liability duration
\(1-w\) = weight in longer bond
Interest Rate Swaps9
Plain Vanilla Swap
Fixed-rate payer pays \(R\) (fixed), receives floating (LIBOR/SOFR).
Swap rate R makes PV of fixed payments = PV of floating payments:
\[R = \frac{1-v^n}{\displaystyle\sum_{t=1}^n v^t} = \frac{1-v^n}{a_{\overline{n}|}}\]
Swap rate R makes PV of fixed payments = PV of floating payments:
\[R = \frac{1-v^n}{\displaystyle\sum_{t=1}^n v^t} = \frac{1-v^n}{a_{\overline{n}|}}\]
Swap Value & Settlements
Net payment at each period \(t\):
Fixed payer: (floating rate - R) x notional
Value of existing swap = PV of net future payments
Swap = portfolio of FRAs (forward rate agreements)
Fixed payer: (floating rate - R) x notional
Value of existing swap = PV of net future payments
Swap = portfolio of FRAs (forward rate agreements)
Derivatives & Options10
Put-Call Parity
\[C - P = S_0 - Ke^{-rT} = S_0 - Kv^T\]
\(C\) = call price, \(P\) = put price
\(S_0\) = current stock price, \(K\) = strike
\(r\) = risk-free rate, \(T\) = time to expiry
\(S_0\) = current stock price, \(K\) = strike
\(r\) = risk-free rate, \(T\) = time to expiry
Option Payoffs at Expiry
Long call: \(\max(S_T - K, 0)\)
Short call: \(-\max(S_T - K, 0)\)
Long put: \(\max(K - S_T, 0)\)
Short put: \(-\max(K - S_T, 0)\)
Short call: \(-\max(S_T - K, 0)\)
Long put: \(\max(K - S_T, 0)\)
Short put: \(-\max(K - S_T, 0)\)
Forwards & Futures
Forward price: \(F_0 = S_0(1+i)^T\)
Forward payoff (long): \(S_T - F_0\)
Prepaid forward: \(F_0^P = S_0 - PV(\text{dividends})\)
Forward payoff (long): \(S_T - F_0\)
Prepaid forward: \(F_0^P = S_0 - PV(\text{dividends})\)
Common Strategies
Bull spread: long call K1, short call K2 (\(K1 < K2\))
Bear spread: long put K2, short put K1
Straddle: long call + long put same K
Collar: long put K1, short call K2 (\(K1 < K2\))
Bear spread: long put K2, short put K1
Straddle: long call + long put same K
Collar: long put K1, short call K2 (\(K1 < K2\))