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Exam FM — Formula Sheet Flashcards

A condensed, must-know-formula recall deck for SOA Exam FM — one formula per card covering interest measurement, force of interest, annuities, loans, bonds, cash flows, duration, immunization, and swaps for fast cram-style review.

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Annuities — Level Annuities (Level) Practice Annuities — Varying & Continuous Annuities (Varying & Continuous) Practice Bonds Bonds Practice

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  1. interest-measurement
    Discount factor and accumulation: $v$, $a(t)$ under compound interest.
    $v=\dfrac{1}{1+i}=(1+i)^{-1}$, so $v^{n}=(1+i)^{-n}$. Compound accumulation $a(t)=(1+i)^{t}$; simple interest $a(t)=1+it$.
  2. interest-measurement
    Rate of discount $d$: relate to $i$ and $v$.
    $d=\dfrac{i}{1+i}=iv=1-v$. Inversely $i=\dfrac{d}{1-d}$. Key identity: $i-d=id$.
  3. interest-measurement
    The $1+i$ chain of equivalences (interest, discount, force).
    $1+i=\dfrac{1}{v}=(1-d)^{-1}=e^{\delta}=\left(1+\dfrac{i^{(m)}}{m}\right)^{m}=\left(1-\dfrac{d^{(m)}}{m}\right)^{-m}$.
  4. force-of-interest
    Force of interest $\delta$ and effective rate $i$ (constant force).
    $1+i=e^{\delta}$, so $\delta=\ln(1+i)$ and $i=e^{\delta}-1$. Also $\delta=-\ln(1-d)$ and $v^{n}=e^{-\delta n}$.
  5. interest-measurement
    Nominal interest $i^{(m)}$ convertible $m$-thly: convert to effective $i$.
    $1+i=\left(1+\dfrac{i^{(m)}}{m}\right)^{m}$, so $i^{(m)}=m\left[(1+i)^{1/m}-1\right]$.
  6. interest-measurement
    Nominal discount $d^{(m)}$ convertible $m$-thly: convert to effective.
    $1-d=\left(1-\dfrac{d^{(m)}}{m}\right)^{m}$, so $d^{(m)}=m\left[1-(1-d)^{1/m}\right]=m\left[1-v^{1/m}\right]$.
  7. force-of-interest
    Limits of nominal rates as $m\to\infty$.
    $\lim_{m\to\infty} i^{(m)}=\lim_{m\to\infty} d^{(m)}=\delta$. The force of interest is the continuously-compounded rate.
  8. interest-measurement
    Simple interest vs simple discount accumulation/PV.
    Simple interest: $a(t)=1+it$. Simple (bank) discount: present value $=A(1-dt)$ for $0\leq t<1/d$; accumulation $a(t)=(1-dt)^{-1}$.
  9. interest-measurement
    Rule of 72: approximate doubling time.
    $n\approx\dfrac{72}{100\,i}=\dfrac{0.72}{i}$ (with $i$ as a decimal). Exact: $n=\dfrac{\ln 2}{\ln(1+i)}$.
  10. force-of-interest
    Time-varying force of interest: accumulation and present value.
    $a(t)=\exp\!\left(\displaystyle\int_{0}^{t}\delta_{s}\,ds\right)$. Present value at $0$ of $1$ at $t$: $v(t)=\exp\!\left(-\displaystyle\int_{0}^{t}\delta_{s}\,ds\right)$.
  11. force-of-interest
    Recover force of interest from the accumulation function.
    $\delta_{t}=\dfrac{a'(t)}{a(t)}=\dfrac{d}{dt}\ln a(t)=\dfrac{A'(t)}{A(t)}$.
  12. annuities
    Annuity-immediate: present and accumulated value of $1$ per period.
    $a_{\overline{n|}}=\dfrac{1-v^{n}}{i}$ and $s_{\overline{n|}}=\dfrac{(1+i)^{n}-1}{i}$. Note $s_{\overline{n|}}=a_{\overline{n|}}(1+i)^{n}$.
  13. annuities
    Annuity-due: present and accumulated value of $1$ per period.
    $\ddot{a}_{\overline{n|}}=\dfrac{1-v^{n}}{d}$ and $\ddot{s}_{\overline{n|}}=\dfrac{(1+i)^{n}-1}{d}$.
  14. annuities
    Immediate-to-due conversion (PV and AV).
    $\ddot{a}_{\overline{n|}}=(1+i)\,a_{\overline{n|}}$ and $\ddot{s}_{\overline{n|}}=(1+i)\,s_{\overline{n|}}$. Multiply by $(1+i)$ to go immediate $\to$ due.
  15. annuities
    Useful annuity identities (due $=$ 1 + shorter immediate).
    $\ddot{a}_{\overline{n|}}=1+a_{\overline{n-1|}}$ and $s_{\overline{n|}}=\ddot{s}_{\overline{n-1|}}+1$. Also $\dfrac{1}{a_{\overline{n|}}}=\dfrac{1}{s_{\overline{n|}}}+i$.
  16. annuities
    Perpetuity-immediate and perpetuity-due present values.
    $a_{\overline{\infty|}}=\dfrac{1}{i}$ and $\ddot{a}_{\overline{\infty|}}=\dfrac{1}{d}$. (Due $=$ immediate $\times(1+i)=\dfrac{1}{i}+1$.)
  17. annuities
    Deferred annuity-immediate: $m$-period deferred PV.
    $_{m|}a_{\overline{n|}}=v^{m}\,a_{\overline{n|}}=a_{\overline{m+n|}}-a_{\overline{m|}}$.
  18. varying-annuities
    Increasing annuity-immediate $(Ia)_{\overline{n|}}$ (payments $1,2,\ldots,n$).
    $(Ia)_{\overline{n|}}=\dfrac{\ddot{a}_{\overline{n|}}-n\,v^{n}}{i}$. Increasing perpetuity-immediate: $(Ia)_{\overline{\infty|}}=\dfrac{1}{i}+\dfrac{1}{i^{2}}=\dfrac{1}{id}$.
  19. varying-annuities
    Decreasing annuity-immediate $(Da)_{\overline{n|}}$ (payments $n,n-1,\ldots,1$).
    $(Da)_{\overline{n|}}=\dfrac{n-a_{\overline{n|}}}{i}$. Identity: $(Ia)_{\overline{n|}}+(Da)_{\overline{n|}}=(n+1)\,a_{\overline{n|}}$.
  20. varying-annuities
    Geometric (compound-growth) annuity-immediate: payments grow at rate $k$.
    First payment $1$, growing by factor $(1+k)$: PV $=\dfrac{1-\left(\frac{1+k}{1+i}\right)^{n}}{i-k}$ (for $i\neq k$). If $i=k$: PV $=\dfrac{n}{1+i}$.
  21. varying-annuities
    Geometric growing perpetuity-immediate (Gordon growth).
    PV $=\dfrac{1}{i-k}$ for $i>k$ (first payment $1$ at time $1$, growing at rate $k$). Diverges if $k\geq i$.
  22. varying-annuities
    Continuous annuity $\bar{a}_{\overline{n|}}$ (continuous payment at rate $1$).
    $\bar{a}_{\overline{n|}}=\dfrac{1-v^{n}}{\delta}$ and $\bar{s}_{\overline{n|}}=\dfrac{(1+i)^{n}-1}{\delta}$. Also $\bar{a}_{\overline{n|}}=\dfrac{i}{\delta}\,a_{\overline{n|}}$.
  23. varying-annuities
    Continuously increasing continuous annuity $(\bar{I}\bar{a})_{\overline{n|}}$.
    $(\bar{I}\bar{a})_{\overline{n|}}=\dfrac{\bar{a}_{\overline{n|}}-n\,v^{n}}{\delta}$. Continuously increasing perpetuity: $(\bar{I}\bar{a})_{\overline{\infty|}}=\dfrac{1}{\delta^{2}}$.
  24. varying-annuities
    Annuity payable $m$-thly (annuity-immediate), $1$ per year total.
    $a_{\overline{n|}}^{(m)}=\dfrac{1-v^{n}}{i^{(m)}}=\dfrac{i}{i^{(m)}}\,a_{\overline{n|}}$ and $\ddot{a}_{\overline{n|}}^{(m)}=\dfrac{1-v^{n}}{d^{(m)}}=\dfrac{i}{d^{(m)}}\,a_{\overline{n|}}$.
  25. loans
    Outstanding loan balance: prospective and retrospective.
    Level payment $X$, original term $n$, after $t$ payments: prospective $B_{t}=X\,a_{\overline{n-t|}}$; retrospective $B_{t}=L(1+i)^{t}-X\,s_{\overline{t|}}$.
  26. loans
    Amortization: interest and principal portions of payment $t$.
    Interest paid $I_{t}=i\,B_{t-1}=X\left(1-v^{n-t+1}\right)$. Principal repaid $P_{t}=X\,v^{n-t+1}=X-I_{t}$.
  27. loans
    Principal repayments form a geometric sequence — relate $P_{t+1}$ to $P_{t}$.
    $P_{t+1}=P_{t}(1+i)$. Total principal $\sum_{t=1}^{n}P_{t}=L$ (original loan).
  28. loans
    Sinking fund: total periodic outlay to service a loan.
    Pay lender interest $L\cdot i$ each period and deposit $D$ into a fund earning $j$ so $D\,s_{\overline{n|}j}=L$. Total cost per period $=L\,i+\dfrac{L}{s_{\overline{n|}j}}$.
  29. bonds
    Bond price — basic (cash-flow) formula.
    $P=Fr\,a_{\overline{n|}}+C\,v^{n}$, where $Fr$ is the periodic coupon, $C$ the redemption value, $i$ the periodic yield, $n$ the number of coupons.
  30. bonds
    Bond price — premium/discount (alternative) formula.
    $P=C+(Fr-Ci)\,a_{\overline{n|}}$. Premium if $Fr>Ci$ ($P>C$); discount if $Fr<Ci$ ($P<C$).
  31. bonds
    Makeham's bond price formula.
    $P=K+\dfrac{g}{i}(C-K)$, where $K=C\,v^{n}$ (PV of redemption) and $g=\dfrac{Fr}{C}$ is the modified coupon rate.
  32. bonds
    Bond book value (prospective) just after coupon $t$.
    $B_{t}=Fr\,a_{\overline{n-t|}}+C\,v^{n-t}$. Then $B_{0}=P$ and $B_{n}=C$.
  33. bonds
    Bond amortization of premium/discount in period $t$.
    Coupon $Fr$; interest earned $=i\,B_{t-1}$. Write-down (premium) $=Fr-i\,B_{t-1}=(Fr-Ci)v^{n-t+1}$. Book change $B_{t}-B_{t-1}=i\,B_{t-1}-Fr$.
  34. bonds
    Callable bond pricing rule (to be conservative).
    Price at the worst-case call date. Premium bond ($Fr>Ci$): assume the earliest call. Discount bond ($Fr<Ci$): assume the latest call (maturity). Use the lowest resulting price.
  35. cash-flows
    Net present value (NPV) of a cash-flow stream.
    $\text{NPV}=\displaystyle\sum_{t=0}^{n} C_{t}\,v^{t}=\sum_{t=0}^{n}\dfrac{C_{t}}{(1+i)^{t}}$, where $C_{t}$ is the net cash flow at time $t$ (sign-aware).
  36. cash-flows
    Internal rate of return (IRR / yield rate).
    The rate $i$ solving $\displaystyle\sum_{t=0}^{n} C_{t}\,(1+i)^{-t}=0$, i.e. $\text{NPV}(i)=0$. Equivalently PV(inflows) $=$ PV(outflows).
  37. cash-flows
    Dollar-weighted (money-weighted) rate of return — simple-interest approx.
    $i\approx\dfrac{I}{A+\sum_{t} C_{t}(1-t)}$, where $I$ is interest earned, $A$ the starting balance, and $C_{t}$ the net deposit at time $t$ (fraction of year). Denominator is exposure.
  38. cash-flows
    Time-weighted rate of return.
    $1+i_{tw}=\displaystyle\prod_{k}\left(1+j_{k}\right)$ where each subperiod return $j_{k}=\dfrac{B_{k}^{\text{end}}}{B_{k-1}^{\text{end}}+\text{deposit}_{k}}-1$ (value just before each cash flow over value just after the previous).
  39. term-structure
    Spot rates and present value off the spot curve.
    $s_{t}$ = annual effective spot rate for term $t$. PV of $1$ at time $t$: $P_{t}=(1+s_{t})^{-t}$. PV of a stream: $\sum_t C_t (1+s_t)^{-t}$.
  40. term-structure
    Forward rate $f_{[t,t+1]}$ from spot rates.
    $(1+s_{t+1})^{t+1}=(1+s_{t})^{t}\,(1+f_{[t,t+1]})$, so $1+f_{[t,t+1]}=\dfrac{(1+s_{t+1})^{t+1}}{(1+s_{t})^{t}}=\dfrac{P_{t}}{P_{t+1}}$.
  41. duration
    Macaulay duration $D_{mac}$ of a cash-flow stream.
    $D_{mac}=\dfrac{\sum_{t} t\,v^{t} C_{t}}{\sum_{t} v^{t} C_{t}}=\dfrac{\sum_{t} t\,v^{t} C_{t}}{P}$ — the PV-weighted average time of the cash flows.
  42. duration
    Modified duration $D_{mod}$ and link to Macaulay duration.
    $D_{mod}=-\dfrac{1}{P}\dfrac{dP}{di}=\dfrac{D_{mac}}{1+i}$. Under continuous compounding (force $\delta$), $D_{mod}=D_{mac}$.
  43. duration
    First-order (duration) price-change estimate.
    $\dfrac{\Delta P}{P}\approx -D_{mod}\,\Delta i$, i.e. $\Delta P\approx -P\,D_{mod}\,\Delta i$. (Dollar duration $=P\,D_{mod}$.)
  44. duration
    Macaulay duration of special instruments.
    Zero-coupon bond maturing at $n$: $D_{mac}=n$. Level perpetuity-immediate: $D_{mac}=\dfrac{1+i}{i}$.
  45. duration
    Portfolio (aggregate) duration.
    $D_{port}=\displaystyle\sum_{k} w_{k}\,D_{k}$, where $w_{k}=\dfrac{P_{k}}{\sum_j P_j}$ is the present-value weight of asset $k$. Same weighting for modified or Macaulay.
  46. duration
    Convexity $C$ and the second-order price-change estimate.
    $C=\dfrac{1}{P}\dfrac{d^{2}P}{di^{2}}=\dfrac{\sum_{t} t(t+1)v^{t+2}C_{t}}{P}$. Estimate: $\dfrac{\Delta P}{P}\approx -D_{mod}\,\Delta i+\tfrac{1}{2}C\,(\Delta i)^{2}$.
  47. immunization
    Redington immunization — the three conditions.
    (1) $PV_{A}=PV_{L}$; (2) $D_{A}=D_{L}$ (equal duration, equivalently $PV_A'=PV_L'$); (3) $C_{A}>C_{L}$ (asset convexity exceeds liability convexity). Protects against small rate moves.
  48. immunization
    Full immunization conditions.
    (1) $PV_{A}=PV_{L}$; (2) $D_{A}=D_{L}$; (3) one asset cash flow on each side of the liability date (an asset before and an asset after the liability). Protects against any size rate move.
  49. immunization
    Exact (dedication / cash-flow) matching.
    Choose assets so asset cash flows equal liability cash flows at every date: $A_{t}=L_{t}$ for all $t$. No interest-rate risk at all; no convexity condition needed.
  50. swaps
    Level interest-rate swap rate $R$ (from discount factors).
    $R=\dfrac{1-P_{n}}{\displaystyle\sum_{t=1}^{n}P_{t}}$, where $P_{t}=(1+s_{t})^{-t}$. Numerator $1-P_n$ = PV of floating leg; denominator = PV of an annuity of $1$.
  51. swaps
    Swap as forward-rate weighted average; market value after inception.
    $R=\dfrac{\sum_{t=1}^{n} P_{t}\,f_{[t-1,t]}}{\sum_{t=1}^{n} P_{t}}$ (PV-weighted average of forward rates). Market value to fixed payer at time $0$ for new rate $R$ vs contract $R_0$: $(R-R_{0})\displaystyle\sum_{t} P_{t}$ per unit notional.
  52. interest-measurement
    Equation of value: the master accumulation/discounting balance.
    At any comparison date, PV(inflows) $=$ PV(outflows). Accumulate or discount each cash flow with $(1+i)^{\pm t}$ (or $\exp\!\int\delta$) to that date, then equate.