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
Yield Rates & Cash Flow Analysis9
Net Present Value (NPV)
\[NPV(i) = \sum_{t} \frac{CF_t}{(1+i)^t} = \sum_t CF_t\,v^t\]
Accept a project when \(NPV(i)\ge 0\) at the required return \(i\). Compare mutually exclusive projects by higher NPV.
Internal Rate of Return (Yield Rate)
The yield rate \(i\) solves the equation of value
\[\sum_t CF_t\,(1+i)^{-t} = 0\] i.e. PV(inflows) = PV(outflows). Solve numerically (calculator IRR); a sign change in cash flows guarantees a root.
\[\sum_t CF_t\,(1+i)^{-t} = 0\] i.e. PV(inflows) = PV(outflows). Solve numerically (calculator IRR); a sign change in cash flows guarantees a root.
Dollar-Weighted (Money-Weighted) Yield
Exact: solve the equation of value over the period.
Simple-interest approximation:
\[i \approx \dfrac{I}{A + \sum_t C_t\,(1-t)}\] \(A\) = starting balance, \(C_t\) = net deposit at time \(t\) (as a fraction of the year), \(I = B - A - \sum_t C_t\) = interest earned.
Simple-interest approximation:
\[i \approx \dfrac{I}{A + \sum_t C_t\,(1-t)}\] \(A\) = starting balance, \(C_t\) = net deposit at time \(t\) (as a fraction of the year), \(I = B - A - \sum_t C_t\) = interest earned.
Time-Weighted Yield
Link the sub-period growth factors between cash flows:
\[1 + i_{tw} = \prod_{k} \frac{B_k}{B_{k-1}+C_{k-1}}\] \(B_k\) = balance just before the \(k\)-th flow. Independent of the size and timing of deposits (unlike dollar-weighted).
\[1 + i_{tw} = \prod_{k} \frac{B_k}{B_{k-1}+C_{k-1}}\] \(B_k\) = balance just before the \(k\)-th flow. Independent of the size and timing of deposits (unlike dollar-weighted).
Reinvestment at a Different Rate
Single deposit \(P\) earning \(i\), interest reinvested at \(j\):
\[AV = P + P\,i\,s_{\overline{n}|j}\] Level deposits \(X\), interest reinvested at \(j\):
\[AV = nX + X\,i\,(Is)_{\overline{n-1}|j}\]
\[AV = P + P\,i\,s_{\overline{n}|j}\] Level deposits \(X\), interest reinvested at \(j\):
\[AV = nX + X\,i\,(Is)_{\overline{n-1}|j}\]
Portfolio Yield & Average Returns
Overall portfolio yield = market-value-weighted average of the component yields.
Geometric mean return: \(\left(\prod_{k}(1+r_k)\right)^{1/n}-1\)
Arithmetic mean \(\ge\) geometric mean (equal only if all \(r_k\) are equal).
Geometric mean return: \(\left(\prod_{k}(1+r_k)\right)^{1/n}-1\)
Arithmetic mean \(\ge\) geometric mean (equal only if all \(r_k\) are equal).