{
  "deckName": "Exam FM — Formula Sheet",
  "examCode": "Exam FM",
  "cards": [
    {
      "front": "Discount factor and accumulation: $v$, $a(t)$ under compound interest.",
      "back": "$v=\\dfrac{1}{1+i}=(1+i)^{-1}$, so $v^{n}=(1+i)^{-n}$. Compound accumulation $a(t)=(1+i)^{t}$; simple interest $a(t)=1+it$.",
      "tag": "interest-measurement"
    },
    {
      "front": "Rate of discount $d$: relate to $i$ and $v$.",
      "back": "$d=\\dfrac{i}{1+i}=iv=1-v$. Inversely $i=\\dfrac{d}{1-d}$. Key identity: $i-d=id$.",
      "tag": "interest-measurement"
    },
    {
      "front": "The $1+i$ chain of equivalences (interest, discount, force).",
      "back": "$1+i=\\dfrac{1}{v}=(1-d)^{-1}=e^{\\delta}=\\left(1+\\dfrac{i^{(m)}}{m}\\right)^{m}=\\left(1-\\dfrac{d^{(m)}}{m}\\right)^{-m}$.",
      "tag": "interest-measurement"
    },
    {
      "front": "Force of interest $\\delta$ and effective rate $i$ (constant force).",
      "back": "$1+i=e^{\\delta}$, so $\\delta=\\ln(1+i)$ and $i=e^{\\delta}-1$. Also $\\delta=-\\ln(1-d)$ and $v^{n}=e^{-\\delta n}$.",
      "tag": "force-of-interest"
    },
    {
      "front": "Nominal interest $i^{(m)}$ convertible $m$-thly: convert to effective $i$.",
      "back": "$1+i=\\left(1+\\dfrac{i^{(m)}}{m}\\right)^{m}$, so $i^{(m)}=m\\left[(1+i)^{1/m}-1\\right]$.",
      "tag": "interest-measurement"
    },
    {
      "front": "Nominal discount $d^{(m)}$ convertible $m$-thly: convert to effective.",
      "back": "$1-d=\\left(1-\\dfrac{d^{(m)}}{m}\\right)^{m}$, so $d^{(m)}=m\\left[1-(1-d)^{1/m}\\right]=m\\left[1-v^{1/m}\\right]$.",
      "tag": "interest-measurement"
    },
    {
      "front": "Limits of nominal rates as $m\\to\\infty$.",
      "back": "$\\lim_{m\\to\\infty} i^{(m)}=\\lim_{m\\to\\infty} d^{(m)}=\\delta$. The force of interest is the continuously-compounded rate.",
      "tag": "force-of-interest"
    },
    {
      "front": "Simple interest vs simple discount accumulation/PV.",
      "back": "Simple interest: $a(t)=1+it$. Simple (bank) discount: present value $=A(1-dt)$ for $0\\leq t<1/d$; accumulation $a(t)=(1-dt)^{-1}$.",
      "tag": "interest-measurement"
    },
    {
      "front": "Rule of 72: approximate doubling time.",
      "back": "$n\\approx\\dfrac{72}{100\\,i}=\\dfrac{0.72}{i}$ (with $i$ as a decimal). Exact: $n=\\dfrac{\\ln 2}{\\ln(1+i)}$.",
      "tag": "interest-measurement"
    },
    {
      "front": "Time-varying force of interest: accumulation and present value.",
      "back": "$a(t)=\\exp\\!\\left(\\displaystyle\\int_{0}^{t}\\delta_{s}\\,ds\\right)$. Present value at $0$ of $1$ at $t$: $v(t)=\\exp\\!\\left(-\\displaystyle\\int_{0}^{t}\\delta_{s}\\,ds\\right)$.",
      "tag": "force-of-interest"
    },
    {
      "front": "Recover force of interest from the accumulation function.",
      "back": "$\\delta_{t}=\\dfrac{a'(t)}{a(t)}=\\dfrac{d}{dt}\\ln a(t)=\\dfrac{A'(t)}{A(t)}$.",
      "tag": "force-of-interest"
    },
    {
      "front": "Annuity-immediate: present and accumulated value of $1$ per period.",
      "back": "$a_{\\overline{n|}}=\\dfrac{1-v^{n}}{i}$ and $s_{\\overline{n|}}=\\dfrac{(1+i)^{n}-1}{i}$. Note $s_{\\overline{n|}}=a_{\\overline{n|}}(1+i)^{n}$.",
      "tag": "annuities"
    },
    {
      "front": "Annuity-due: present and accumulated value of $1$ per period.",
      "back": "$\\ddot{a}_{\\overline{n|}}=\\dfrac{1-v^{n}}{d}$ and $\\ddot{s}_{\\overline{n|}}=\\dfrac{(1+i)^{n}-1}{d}$.",
      "tag": "annuities"
    },
    {
      "front": "Immediate-to-due conversion (PV and AV).",
      "back": "$\\ddot{a}_{\\overline{n|}}=(1+i)\\,a_{\\overline{n|}}$ and $\\ddot{s}_{\\overline{n|}}=(1+i)\\,s_{\\overline{n|}}$. Multiply by $(1+i)$ to go immediate $\\to$ due.",
      "tag": "annuities"
    },
    {
      "front": "Useful annuity identities (due $=$ 1 + shorter immediate).",
      "back": "$\\ddot{a}_{\\overline{n|}}=1+a_{\\overline{n-1|}}$ and $s_{\\overline{n|}}=\\ddot{s}_{\\overline{n-1|}}+1$. Also $\\dfrac{1}{a_{\\overline{n|}}}=\\dfrac{1}{s_{\\overline{n|}}}+i$.",
      "tag": "annuities"
    },
    {
      "front": "Perpetuity-immediate and perpetuity-due present values.",
      "back": "$a_{\\overline{\\infty|}}=\\dfrac{1}{i}$ and $\\ddot{a}_{\\overline{\\infty|}}=\\dfrac{1}{d}$. (Due $=$ immediate $\\times(1+i)=\\dfrac{1}{i}+1$.)",
      "tag": "annuities"
    },
    {
      "front": "Deferred annuity-immediate: $m$-period deferred PV.",
      "back": "$_{m|}a_{\\overline{n|}}=v^{m}\\,a_{\\overline{n|}}=a_{\\overline{m+n|}}-a_{\\overline{m|}}$.",
      "tag": "annuities"
    },
    {
      "front": "Increasing annuity-immediate $(Ia)_{\\overline{n|}}$ (payments $1,2,\\ldots,n$).",
      "back": "$(Ia)_{\\overline{n|}}=\\dfrac{\\ddot{a}_{\\overline{n|}}-n\\,v^{n}}{i}$. Increasing perpetuity-immediate: $(Ia)_{\\overline{\\infty|}}=\\dfrac{1}{i}+\\dfrac{1}{i^{2}}=\\dfrac{1}{id}$.",
      "tag": "varying-annuities"
    },
    {
      "front": "Decreasing annuity-immediate $(Da)_{\\overline{n|}}$ (payments $n,n-1,\\ldots,1$).",
      "back": "$(Da)_{\\overline{n|}}=\\dfrac{n-a_{\\overline{n|}}}{i}$. Identity: $(Ia)_{\\overline{n|}}+(Da)_{\\overline{n|}}=(n+1)\\,a_{\\overline{n|}}$.",
      "tag": "varying-annuities"
    },
    {
      "front": "Geometric (compound-growth) annuity-immediate: payments grow at rate $k$.",
      "back": "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 $=\\dfrac{n}{1+i}$.",
      "tag": "varying-annuities"
    },
    {
      "front": "Geometric growing perpetuity-immediate (Gordon growth).",
      "back": "PV $=\\dfrac{1}{i-k}$ for $i>k$ (first payment $1$ at time $1$, growing at rate $k$). Diverges if $k\\geq i$.",
      "tag": "varying-annuities"
    },
    {
      "front": "Continuous annuity $\\bar{a}_{\\overline{n|}}$ (continuous payment at rate $1$).",
      "back": "$\\bar{a}_{\\overline{n|}}=\\dfrac{1-v^{n}}{\\delta}$ and $\\bar{s}_{\\overline{n|}}=\\dfrac{(1+i)^{n}-1}{\\delta}$. Also $\\bar{a}_{\\overline{n|}}=\\dfrac{i}{\\delta}\\,a_{\\overline{n|}}$.",
      "tag": "varying-annuities"
    },
    {
      "front": "Continuously increasing continuous annuity $(\\bar{I}\\bar{a})_{\\overline{n|}}$.",
      "back": "$(\\bar{I}\\bar{a})_{\\overline{n|}}=\\dfrac{\\bar{a}_{\\overline{n|}}-n\\,v^{n}}{\\delta}$. Continuously increasing perpetuity: $(\\bar{I}\\bar{a})_{\\overline{\\infty|}}=\\dfrac{1}{\\delta^{2}}$.",
      "tag": "varying-annuities"
    },
    {
      "front": "Annuity payable $m$-thly (annuity-immediate), $1$ per year total.",
      "back": "$a_{\\overline{n|}}^{(m)}=\\dfrac{1-v^{n}}{i^{(m)}}=\\dfrac{i}{i^{(m)}}\\,a_{\\overline{n|}}$ and $\\ddot{a}_{\\overline{n|}}^{(m)}=\\dfrac{1-v^{n}}{d^{(m)}}=\\dfrac{i}{d^{(m)}}\\,a_{\\overline{n|}}$.",
      "tag": "varying-annuities"
    },
    {
      "front": "Outstanding loan balance: prospective and retrospective.",
      "back": "Level payment $X$, original term $n$, after $t$ payments: prospective $B_{t}=X\\,a_{\\overline{n-t|}}$; retrospective $B_{t}=L(1+i)^{t}-X\\,s_{\\overline{t|}}$.",
      "tag": "loans"
    },
    {
      "front": "Amortization: interest and principal portions of payment $t$.",
      "back": "Interest paid $I_{t}=i\\,B_{t-1}=X\\left(1-v^{n-t+1}\\right)$. Principal repaid $P_{t}=X\\,v^{n-t+1}=X-I_{t}$.",
      "tag": "loans"
    },
    {
      "front": "Principal repayments form a geometric sequence — relate $P_{t+1}$ to $P_{t}$.",
      "back": "$P_{t+1}=P_{t}(1+i)$. Total principal $\\sum_{t=1}^{n}P_{t}=L$ (original loan).",
      "tag": "loans"
    },
    {
      "front": "Sinking fund: total periodic outlay to service a loan.",
      "back": "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 $=L\\,i+\\dfrac{L}{s_{\\overline{n|}j}}$.",
      "tag": "loans"
    },
    {
      "front": "Bond price — basic (cash-flow) formula.",
      "back": "$P=Fr\\,a_{\\overline{n|}}+C\\,v^{n}$, where $Fr$ is the periodic coupon, $C$ the redemption value, $i$ the periodic yield, $n$ the number of coupons.",
      "tag": "bonds"
    },
    {
      "front": "Bond price — premium/discount (alternative) formula.",
      "back": "$P=C+(Fr-Ci)\\,a_{\\overline{n|}}$. Premium if $Fr>Ci$ ($P>C$); discount if $Fr<Ci$ ($P<C$).",
      "tag": "bonds"
    },
    {
      "front": "Makeham's bond price formula.",
      "back": "$P=K+\\dfrac{g}{i}(C-K)$, where $K=C\\,v^{n}$ (PV of redemption) and $g=\\dfrac{Fr}{C}$ is the modified coupon rate.",
      "tag": "bonds"
    },
    {
      "front": "Bond book value (prospective) just after coupon $t$.",
      "back": "$B_{t}=Fr\\,a_{\\overline{n-t|}}+C\\,v^{n-t}$. Then $B_{0}=P$ and $B_{n}=C$.",
      "tag": "bonds"
    },
    {
      "front": "Bond amortization of premium/discount in period $t$.",
      "back": "Coupon $Fr$; interest earned $=i\\,B_{t-1}$. Write-down (premium) $=Fr-i\\,B_{t-1}=(Fr-Ci)v^{n-t+1}$. Book change $B_{t}-B_{t-1}=i\\,B_{t-1}-Fr$.",
      "tag": "bonds"
    },
    {
      "front": "Callable bond pricing rule (to be conservative).",
      "back": "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.",
      "tag": "bonds"
    },
    {
      "front": "Net present value (NPV) of a cash-flow stream.",
      "back": "$\\text{NPV}=\\displaystyle\\sum_{t=0}^{n} C_{t}\\,v^{t}=\\sum_{t=0}^{n}\\dfrac{C_{t}}{(1+i)^{t}}$, where $C_{t}$ is the net cash flow at time $t$ (sign-aware).",
      "tag": "cash-flows"
    },
    {
      "front": "Internal rate of return (IRR / yield rate).",
      "back": "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).",
      "tag": "cash-flows"
    },
    {
      "front": "Dollar-weighted (money-weighted) rate of return — simple-interest approx.",
      "back": "$i\\approx\\dfrac{I}{A+\\sum_{t} C_{t}(1-t)}$, where $I$ is interest earned, $A$ the starting balance, and $C_{t}$ the net deposit at time $t$ (fraction of year). Denominator is exposure.",
      "tag": "cash-flows"
    },
    {
      "front": "Time-weighted rate of return.",
      "back": "$1+i_{tw}=\\displaystyle\\prod_{k}\\left(1+j_{k}\\right)$ 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).",
      "tag": "cash-flows"
    },
    {
      "front": "Spot rates and present value off the spot curve.",
      "back": "$s_{t}$ = annual effective spot rate for term $t$. PV of $1$ at time $t$: $P_{t}=(1+s_{t})^{-t}$. PV of a stream: $\\sum_t C_t (1+s_t)^{-t}$.",
      "tag": "term-structure"
    },
    {
      "front": "Forward rate $f_{[t,t+1]}$ from spot rates.",
      "back": "$(1+s_{t+1})^{t+1}=(1+s_{t})^{t}\\,(1+f_{[t,t+1]})$, so $1+f_{[t,t+1]}=\\dfrac{(1+s_{t+1})^{t+1}}{(1+s_{t})^{t}}=\\dfrac{P_{t}}{P_{t+1}}$.",
      "tag": "term-structure"
    },
    {
      "front": "Macaulay duration $D_{mac}$ of a cash-flow stream.",
      "back": "$D_{mac}=\\dfrac{\\sum_{t} t\\,v^{t} C_{t}}{\\sum_{t} v^{t} C_{t}}=\\dfrac{\\sum_{t} t\\,v^{t} C_{t}}{P}$ — the PV-weighted average time of the cash flows.",
      "tag": "duration"
    },
    {
      "front": "Modified duration $D_{mod}$ and link to Macaulay duration.",
      "back": "$D_{mod}=-\\dfrac{1}{P}\\dfrac{dP}{di}=\\dfrac{D_{mac}}{1+i}$. Under continuous compounding (force $\\delta$), $D_{mod}=D_{mac}$.",
      "tag": "duration"
    },
    {
      "front": "First-order (duration) price-change estimate.",
      "back": "$\\dfrac{\\Delta P}{P}\\approx -D_{mod}\\,\\Delta i$, i.e. $\\Delta P\\approx -P\\,D_{mod}\\,\\Delta i$. (Dollar duration $=P\\,D_{mod}$.)",
      "tag": "duration"
    },
    {
      "front": "Macaulay duration of special instruments.",
      "back": "Zero-coupon bond maturing at $n$: $D_{mac}=n$. Level perpetuity-immediate: $D_{mac}=\\dfrac{1+i}{i}$.",
      "tag": "duration"
    },
    {
      "front": "Portfolio (aggregate) duration.",
      "back": "$D_{port}=\\displaystyle\\sum_{k} w_{k}\\,D_{k}$, where $w_{k}=\\dfrac{P_{k}}{\\sum_j P_j}$ is the present-value weight of asset $k$. Same weighting for modified or Macaulay.",
      "tag": "duration"
    },
    {
      "front": "Convexity $C$ and the second-order price-change estimate.",
      "back": "$C=\\dfrac{1}{P}\\dfrac{d^{2}P}{di^{2}}=\\dfrac{\\sum_{t} t(t+1)v^{t+2}C_{t}}{P}$. Estimate: $\\dfrac{\\Delta P}{P}\\approx -D_{mod}\\,\\Delta i+\\tfrac{1}{2}C\\,(\\Delta i)^{2}$.",
      "tag": "duration"
    },
    {
      "front": "Redington immunization — the three conditions.",
      "back": "(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.",
      "tag": "immunization"
    },
    {
      "front": "Full immunization conditions.",
      "back": "(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.",
      "tag": "immunization"
    },
    {
      "front": "Exact (dedication / cash-flow) matching.",
      "back": "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.",
      "tag": "immunization"
    },
    {
      "front": "Level interest-rate swap rate $R$ (from discount factors).",
      "back": "$R=\\dfrac{1-P_{n}}{\\displaystyle\\sum_{t=1}^{n}P_{t}}$, where $P_{t}=(1+s_{t})^{-t}$. Numerator $1-P_n$ = PV of floating leg; denominator = PV of an annuity of $1$.",
      "tag": "swaps"
    },
    {
      "front": "Swap as forward-rate weighted average; market value after inception.",
      "back": "$R=\\dfrac{\\sum_{t=1}^{n} P_{t}\\,f_{[t-1,t]}}{\\sum_{t=1}^{n} P_{t}}$ (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.",
      "tag": "swaps"
    },
    {
      "front": "Equation of value: the master accumulation/discounting balance.",
      "back": "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.",
      "tag": "interest-measurement"
    }
  ]
}