📚 Cheat Sheet - CF1

Cheat Sheet

⏱️ Time Value of Money

Accumulation & Amount Functions

• $A(t)=Ka(t)$

• $A(0)=K$, $a(0)=1$

Effective Interest Rate

• $i_t=\frac{a(t)-a(t-1)}{a(t-1)}=\frac{A(t)-A(t-1)}{A(t-1)}$

Simple Interest

• $a(t)=1+it$

• $i_t=\frac{i}{1+i(t-1)}$

Compound Interest

• $a(t)=(1+i{)}^t$

• $i_t=i$

Discount Rate

• $d_t=\frac{a(t)-a(t-1)}{a(t)}$

• $d=\frac{i}{1+i}$

• $v=\frac{1}{1+i}$

• $1-d=v$

Nominal Rates

• $(1+\frac{i^{(m)}}{m}{)}^m=1+i$

• $(1-\frac{d^{(m)}}{m}{)}^m=1-d$

• $i^{(m)}=m\left((1+i{)}^{1/m}-1\right)$

• $d^{(m)}=m\left(1-(1-d{)}^{1/m}\right)$

Force of Interest

• ${\delta }_t=\frac{a^{\prime }(t)}{a(t)}=\frac{d}{dt}\ln a(t)$

• $a(t)=e^{{\int }_0^t{\delta }_{r}dr}$

• Constant: $\delta =\ln (1+i)$

Present & Accumulated Value

• $PV$ of $1$ due in $t$:

  • $(1+i{)}^{-t}=v^t=e^{-\delta t}=(1-d{)}^t$

• $AV$ of $1$ at time $t$:

  • $(1+i{)}^t=e^{\delta t}=(1-d{)}^{-t}$

• $AV$ from $t_1$ to $t_2$:

  • $\frac{a(t_2)}{a(t_1)}=e^{{\int }_{t_1}^{t_2}{\delta }_{t}dt}$

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📐 Annuities

Annuity-Immediate

• $a_n=\frac{1-v^n}{i}$

• $s_n=\frac{(1+i{)}^n-1}{i}$

• $s_n=(1+i{)}^n{a}_n$

Annuity-Due

• ${\ddot{a}}_n=\frac{1-v^n}{d}=(1+i)a_n$

• ${\ddot{s}}_n=\frac{(1+i{)}^n-1}{d}=(1+i)s_n$

Continuous Annuity

• ${\bar{a}}_n=\frac{1-v^n}{\delta }={\int }_0^n{e}^{-\delta t}dt$

• ${\bar{s}}_n=\frac{(1+i{)}^n-1}{\delta }$

Deferred Annuity

• ${}_m|a_n=v^m{a}_n=a_{m+n}-a_m$

Perpetuity

• $a_{\infty }=\frac{1}{i}$

• ${\ddot{a}}_{\infty }=\frac{1}{d}$

• ${\bar{a}}_{\infty }=\frac{1}{\delta }$

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📈 Increasing & Decreasing Annuities

Increasing (Arithmetic)

• $(Ia{)}_n=\frac{{\ddot{a}}_n-n{v}^n}{i}$

• $(Is{)}_n=\frac{s_{n+1}-(n+1)}{i}$

• $(I\ddot{a}{)}_n=\frac{{\ddot{a}}_n-n{v}^n}{d}$

• $(I\ddot{s}{)}_n=\frac{s_{n+1}-(n+1)}{d}$

Decreasing

• $(Da{)}_n=\frac{n-a_n}{i}$

• $(Ds{)}_n=\frac{n(1+i{)}^n-s_n}{i}$

• $(D\ddot{a}{)}_n=\frac{n-a_n}{d}$

• $(D\ddot{s}{)}_n=\frac{n(1+i{)}^n-s_n}{d}$

Increasing Perpetuity

• $(Ia{)}_{\infty }=\frac{1}{i^2}=\frac{1}{id}$

• $(I\ddot{a}{)}_{\infty }=\frac{1}{d^2}$

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🔁 m-thly Annuities

• $a_n^{(m)}=\frac{1-v^n}{i^{(m)}}$

• ${\ddot{a}}_n^{(m)}=\frac{1-v^n}{d^{(m)}}$

• $(Ia{)}_n^{(m)}=\frac{{\ddot{a}}_n-n{v}^n}{i^{(m)}}$

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📉 Continuously Increasing Annuity

• $(\bar{I}\bar{a}{)}_n=\frac{{\bar{a}}_n-n{v}^n}{\delta }={\int }_0^{n}t{e}^{-\delta t}dt$

• $(\bar{I}\bar{s}{)}_n=\frac{{\bar{s}}_n-n}{\delta }$

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🔄 Varying Payments

• $PV={\int }_0^{n}f(t)e^{-\delta t}dt$

• If $\delta$ varies:

  • $PV={\int }_0^{n}f(t)e^{-{\int }_0^t{\delta }_{r}dr}dt$

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📐 Geometric Annuities

Immediate

• $PV=\frac{1-{\left(\frac{1+k}{1+i}\right)}^n}{i-k}$

• $FV=\frac{(1+i{)}^n-(1+k{)}^n}{i-k}$

Due

• $PV=\frac{1-{\left(\frac{1+k}{1+i}\right)}^n}{d-kv}$

• $FV=\frac{(1+i{)}^n-(1+k{)}^n}{d-kv}$

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🏦 Loans

Level Payment Loan

• $L=P{a}_n$

• $P=P{r}_t+I_t$

• $I_t=i{B}_{t-1}$

• $P{r}_t=P-I_t$

• $B_t=B_{t-1}-P{r}_t$

• Prospective: $B_t=P{a}_{n-t}$

• Retrospective: $B_t=L(1+i{)}^t-P{s}_t$

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📜 Bonds

Price

• $P=Fr{a}_n+C{v}^n$

• $P=C+(Fr-Ci)a_n$

• $K=C{v}^n$, $g=\frac{Fr}{C}$

• $P=K+\frac{g}{i}(C-K)$

Book Value

• $B_t=Fr{a}_{n-t}+C{v}^{n-t}$

Premium / Discount

• If $g>i$: Premium $=(Fr-Ci)a_n$

• If $g<i$: Discount $=(Ci-Fr)a_n$

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📊 Spot & Forward Rates

Spot Rate

• $P_t=(1+r_t{)}^{-t}$

Forward Rate

• $f_{[t,t+1]}=\frac{(1+r_{t+1}{)}^{t+1}}{(1+r_t{)}^t}-1$

• $f_{[t,t+m]}={\left(\frac{(1+r_{t+m}{)}^{t+m}}{(1+r_t{)}^t}\right)}^{1/m}-1$

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⏳ Duration & Convexity

Duration

• $D_{mac}=\frac{\sum t{A}_t{v}^t}{\sum A_t{v}^t}$

• $D_{mod}=\frac{D_{mac}}{1+i}$

Special cases:

• Perpetuity: $D_{mac}=\frac{1+i}{i}=\frac{1}{d}$

• Level annuity: $D_{mac}=\frac{(Ia{)}_n}{a_n}$

• Par bond: $D_{mac}={\ddot{a}}_n$

Convexity

• $C_{mac}=\frac{d^{2}P/d{\delta }^2}{P}$

• $C_{mod}=C_{mac}+\frac{D_{mac}}{(1+i{)}^2}$

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🛡️ Immunization

Redington Conditions

• $PV(A)=PV(L)$

• $D_{mod}(A)=D_{mod}(L)$

• $C_{mod}(A)>C_{mod}(L)$

Exact Matching

• Match amount and timing of all cash flows exactly

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