Financial Math FM/Formulas

Basic Formulas

$a(t)$ : Accumulation function. Measures the amount in a fund with an investment of 1 at time 0 at the end of period t.

$a(t)-a(t-1)$ :amount of growth in period t.

$s_t=\frac{a(t)-a(t-1)}{a(t-1)}$ : rate of growth in period t, also known as the effective rate of interest in period t.

$A(t)=k\cdot a(t)$ : Amount function. Measures the amount in a fund with an investment of k at time 0 at the end of period t. It is simply the constant k times the accumulation function.

Common Accumulation Functions

$a(t)=1+i\cdot t$  : simple interest.

$a(t)=\prod _{j=1}^t(1+i_j)$ : variable interest

$a(t)=(1+i{)}^t$ : compound interest.

$a(t)=e^{t\cdot i}$ : continuous interest.

Present Value and Discounting

$PV=\frac{1}{a(t)}=\frac{1}{(1+i{)}^t}=(1+i{)}^{-t}=v^t$

$d_t=\frac{a(t)-a(t-1)}{a(t)}$ effective rate of discount in year t.

$1-d=v$

$d=\frac{i}{1+i}=i\cdot v$

$i=\frac{d}{1-d}$

Nominal Interest and Discount

$i^{(m)}$ and $d^{(m)}$ are the symbols for nominal rates of interest compounded m-thly.

$1+i={(1+\frac{i^{(m)}}{m})}^m$

$i^{(m)}=m((1+i{)}^{\frac{1}{m}}-1)$

$1-d={(1-\frac{d^{(m)}}{m})}^m$

$d^{(m)}=m(1-(1-d{)}^{\frac{1}{m}})$

Force of Interest

${\delta }_t=\frac{1}{a(t)}\frac{d}{dt}a(t)=\frac{d}{dt}\ln a(t)$ : definition of force of interest.

$a(t)=e^{\underset{0}{\overset{t}{\int }}{\delta }_{r}dr}$

If the Force of Interest is Constant: $a(t)=e^{\delta t}$

$PV=e^{-\delta t}$

$\delta =\ln (1+i)$

Annuities

$a_{\overline{n|}}=\frac{1-v^n}{i}=v+v^2+\cdots +v^n$ : PV of an annuity-immediate.

${\ddot{a}}_{\overline{n|}}=\frac{1-v^n}{d}=1+v+v^2+\cdots +v^{n-1}$ : PV of an annuity-due.

${\ddot{a}}_{\overline{n|}}=(1+i)a_{\overline{n|}}=1+a_{\overline{n-1|}}$

$s_{\overline{n|}}=\frac{(1+i{)}^n-1}{i}=(1+i{)}^{n-1}+(1+i{)}^{n-2}+\cdots +1$ : AV of an annuity-immediate (on the date of the last deposit).

${\ddot{s}}_{\overline{n|}}=\frac{(1+i{)}^n-1}{d}=(1+i{)}^n+(1+i{)}^{n-1}+\cdots +(1+i)$ : AV of an annuity-due (one period after the date of the last deposit).

${\ddot{s}}_{\overline{n|}}=(1+i)s_{\overline{n|}}=s_{\overline{n+1|}}-1$

$a_{\overline{mn|}}=a_{\overline{n|}}+v^n{a}_{\overline{n|}}+v^{2n}{a}_{\overline{n|}}+\cdots +v^{(m-1)n}{a}_{\overline{n|}}$

Perpetuities

$\underset{n\to \mathrm{\infty }}{\lim }{a}_{\overline{n|}}=\underset{n\to \mathrm{\infty }}{\lim }\frac{1-v^n}{i}=\frac{1}{i}=v+v^2+\cdots =a_{\overline{\mathrm{\infty }|}}$ : PV of a perpetuity-immediate.

$\underset{n\to \mathrm{\infty }}{\lim }{\ddot{a}}_{\overline{n|}}=\underset{n\to \mathrm{\infty }}{\lim }\frac{1-v^n}{d}=\frac{1}{d}=1+v+v^2+\cdots ={\ddot{a}}_{\overline{\mathrm{\infty }|}}$ : PV of a perpetuity-due.

${\ddot{a}}_{\overline{\mathrm{\infty }|}}-a_{\overline{\mathrm{\infty }|}}=\frac{1}{d}-\frac{1}{i}=1$

m-thly Annuities & Perpetuities

$a_{\overline{n|}}^{(m)}=\frac{1-v^n}{i^{(m)}}=\frac{i}{i^{(m)}}{a}_{\overline{n|}}=s_{\overline{1|}}^{(m)}{a}_{\overline{n|}}$ : PV of an n-year annuity-immediate of 1 per year payable in m-thly installments.

${\ddot{a}}_{\overline{n|}}^{(m)}=\frac{1-v^n}{d^{(m)}}=\frac{i}{d^{(m)}}{a}_{\overline{n|}}={\ddot{s}}_{\overline{1|}}^{(m)}{a}_{\overline{n|}}$ : PV of an n-year annuity-due of 1 per year payable in m-thly installments.

$s_{\overline{n|}}^{(m)}=\frac{(1+i{)}^n-1}{i^{(m)}}$ : AV of an n-year annuity-immediate of 1 per year payable in m-thly installments.

${\ddot{s}}_{\overline{n|}}^{(m)}=\frac{(1+i{)}^n-1}{d^{(m)}}$ : AV of an n-year annuity-due of 1 per year payable in m-thly installments.

$\underset{n\to \mathrm{\infty }}{\lim }{a}_{\overline{n|}}^{(m)}=\underset{n\to \mathrm{\infty }}{\lim }\frac{1-v^n}{i^{(m)}}=\frac{1}{i^{(m)}}=a_{\overline{\mathrm{\infty }|}}^{(m)}$ : PV of a perpetuity-immediate of 1 per year payable in m-thly installments.

$\underset{n\to \mathrm{\infty }}{\lim }{\ddot{a}}_{\overline{n|}}^{(m)}=\underset{n\to \mathrm{\infty }}{\lim }\frac{1-v^n}{d^{(m)}}=\frac{1}{d^{(m)}}={\ddot{a}}_{\overline{\mathrm{\infty }|}}^{(m)}$ : PV of a perpetuity-due of 1 per year payable in m-thly installments.

${\ddot{a}}_{\overline{\mathrm{\infty }|}}^{(m)}-a_{\overline{\mathrm{\infty }|}}^{(m)}=\frac{1}{d^{(m)}}-\frac{1}{i^{(m)}}=\frac{1}{m}$

Continuous Annuities

Since $\underset{m\to \mathrm{\infty }}{\lim }{i}^{(m)}=\underset{m\to \mathrm{\infty }}{\lim }{d}^{(m)}=\delta$,

$\underset{m\to \mathrm{\infty }}{\lim }{a}_{\overline{n|}}^{(m)}=\underset{m\to \mathrm{\infty }}{\lim }\frac{1-v^n}{i^{(m)}}=\frac{1-v^n}{\delta }={\bar{a}}_{\overline{n|}}=\frac{i}{\delta }{a}_{\overline{n|}}$ : PV of an annuity (immediate or due) of 1 per year paid continuously.

Payments in Arithmetic Progression: In general, the PV of a series of $n$ payments, where the first payment is $P$ and each additional payment increases by $Q$ can be represented by: $A=P{a}_{\overline{n|}}+Q\frac{a_{\overline{n|}}-n{v}^n}{i}=Pv+(P+Q)v^2+(P+2Q)v^3+\cdots +(P+(n-1)Q)v^n$

Similarly: $\ddot{A}=P{\ddot{a}}_{\overline{n|}}+Q\frac{a_{\overline{n|}}-n{v}^n}{d}$

$S=P{s}_{\overline{n|}}+Q\frac{s_{\overline{n|}}-n}{i}$ : AV of a series of $n$ payments, where the first payment is $P$ and each additional payment increases by $Q$.

$\ddot{S}=P{\ddot{s}}_{\overline{n|}}+Q\frac{s_{\overline{n|}}-n}{d}$

$(Ia{)}_{\overline{n|}}=\frac{{\ddot{a}}_{\overline{n|}}-n{v}^n}{i}$ : PV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute $d$ for $i$ in denominator to get due form.

$(Is{)}_{\overline{n|}}=\frac{{\ddot{s}}_{\overline{n|}}-n}{i}$ : AV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute $d$ for $i$ in denominator to get due form.

$(Da{)}_{\overline{n|}}=\frac{n-a_{\overline{n|}}}{i}$ : PV of an annuity-immediate with first payment $n$ and each additional payment decreasing by 1; substitute $d$ for $i$ in denominator to get due form.

$(Ds{)}_{\overline{n|}}=\frac{n(1+i{)}^n-s_{\overline{n|}}}{i}$ : AV of an annuity-immediate with first payment $n$ and each additional payment decreasing by 1; substitute $d$ for $i$ in denominator to get due form.

$(Ia{)}_{\overline{\mathrm{\infty }|}}=\frac{1}{id}=\frac{1}{i}+\frac{1}{i^2}$ : PV of a perpetuity-immediate with first payment 1 and each additional payment increasing by 1.

$(I\ddot{a}{)}_{\overline{\mathrm{\infty }|}}=\frac{1}{d^2}$ : PV of a perpetuity-due with first payment 1 and each additional payment increasing by 1.

$(Ia{)}_{\overline{n|}}+(Da{)}_{\overline{n|}}=(n+1)a_{\overline{n|}}$

Additional Useful Results: $\frac{P}{i}+\frac{Q}{i^2}$ : PV of a perpetuity-immediate with first payment $P$ and each additional payment increasing by $Q$.

$(Ia{)}_{\overline{n|}}^{(m)}=\frac{{\ddot{a}}_{\overline{n|}}-n{v}^n}{i^{(m)}}$ : PV of an annuity-immediate with m-thly payments of $\frac{1}{m}$ in the first year and each additional year increasing until there are m-thly payments of $\frac{n}{m}$ in the nth year.

$(I^{(m)}a{)}_{\overline{n|}}^{(m)}=\frac{{\ddot{a}}_{\overline{n|}}^{(m)}-n{v}^n}{i^{(m)}}$ : PV of an annuity-immediate with payments of $\frac{1}{m^2}$ at the end of the first mth of the first year, $\frac{2}{m^2}$ at the end of the second mth of the first year, and each additional payment increasing until there is a payment of $\frac{mn}{m^2}$ at the end of the last mth of the nth year.

$(\bar{I}\bar{a}{)}_{\overline{n|}}=\frac{{\bar{a}}_{\overline{n|}}-n{v}^n}{\delta }$ : PV of an annuity with continuous payments that are continuously increasing. Annual rate of payment is $t$ at time $t$.

$\underset{0}{\overset{n}{\int }}f(t)v^{t}dt$ : PV of an annuity with a continuously variable rate of payments and a constant interest rate.

$\underset{0}{\overset{n}{\int }}f(t)e^{-\underset{0}{\overset{t}{\int }}{\delta }_{r}dr}dt$ : PV of an annuity with a continuously variable rate of payment and a continuously variable rate of interest.

Payments in Geometric Progression

$\frac{1-(\frac{1+k}{1+i}{)}^n}{i-k}$ : PV of an annuity-immediate with an initial payment of 1 and each additional payment increasing by a factor of $(1+k)$. Chapter 5

General Definitions

$R_t$ : payment at time $t$. A negative value is an investment and a positive value is a return.

$P(i)=\sum v^t{R}_t$ : PV of a cash flow at interest rate $i$. Chapter 6

$R_t=I_t+P_t$ : payment made at the end of year $t$, split into the interest $I_t$ and the principal repaid $P_t$.

$I_t=i{B}_{t-1}$ : interest paid at the end of year $t$.

$P_t=R_t-I_t=(1+i)P_{t-1}+(R_t-R_{t-1})$ : principal repaid at the end of year $t$.

$B_t=B_{t-1}-P_t$ : balance remaining at the end of year $t$, just after payment is made.

On a Loan Being Paid with Level Payments:

$I_t=1-v^{n-t+1}$ : interest paid at the end of year $t$ on a loan of $a_{\overline{n|}}$.

$P_t=v^{n-t+1}$ : principal repaid at the end of year $t$ on a loan of $a_{\overline{n|}}$.

$B_t=a_{\overline{n-t|}}$ : balance remaining at the end of year $t$ on a loan of $a_{\overline{n|}}$, just after payment is made.

For a loan of $L$, level payments of $\frac{L}{a_{\overline{n|}}}$ will pay off the loan in $n$ years. To scale the interest, principal, and balance owed at time $t$, multiply the above formulas for $I_t$, $P_t$, and $B_t$ by $\frac{L}{a_{\overline{n|}}}$, ie $B_t=\frac{L}{a_{\overline{n|}}}{a}_{\overline{n-t|}}$ etc.

Yield Rates

$I=B-A-C$

$i=\frac{I}{A+\sum _{t_k}{C}_{t_k}(1-t_k)}$ : dollar-weighted

$(1+i)=\prod _{t_k}^t\left(\frac{B_{t_k}}{B_{t_{k-1}}+C_{t_{k-1}}}\right)$ : time-weighted

Sinking Funds

$PMT=Li+\frac{L}{s_{\overline{n|}j}}$ : total yearly payment with the sinking fund method, where $Li$ is the interest paid to the lender and $\frac{L}{s_{\overline{n|}j}}$ is the deposit into the sinking fund that will accumulate to $L$ in $n$ years. $i$ is the interest rate for the loan and $j$ is the interest rate that the sinking fund earns.

$L=(PMT-Li)s_{\overline{n|}j}$

Bonds

Definitions: $P$ : Price paid for a bond.

$F$ : Par/face value of a bond.

$C$ : Redemption value of a bond.

$r$ : coupon rate for a bond.

$g=\frac{Fr}{C}$ : modified coupon rate.

$i$ : yield rate on a bond.

$K$ : PV of $C$.

$n$ : number of coupon payments.

$G=\frac{Fr}{i}$ : base amount of a bond.

$Fr=Cg$

Determination of Bond Prices

$P=Fr{a}_{\overline{n|}i}+C{v}^n=Cg{a}_{\overline{n|}i}+C{v}^n$ : price paid for a bond to yield $i$.

$P=C+(Fr-Ci)a_{\overline{n|}i}=C+(Cg-Ci)a_{\overline{n|}i}$ : Premium/Discount formula for the price of a bond.

$P-C=(Fr-Ci)a_{\overline{n|}i}=(Cg-Ci)a_{\overline{n|}i}$ : premium paid for a bond if $g>i$.

$C-P=(Ci-Fr)a_{\overline{n|}i}=(Ci-Cg)a_{\overline{n|}i}$ : discount paid for a bond if $g<i$.

Bond Amortization: When a bond is purchased at a premium or discount the difference between the price paid and the redemption value can be amortized over the remaining term of the bond. Using the terms from chapter 6: $R_t$ : coupon payment.

$I_t=i{B}_{t-1}$ : interest earned from the coupon payment.

$P_t=R_t-I_t=(Fr-Ci)v^{n-t+1}=(Cg-Ci)v^{n-t+1}$ : adjustment amount for amortization of premium ("write down") or

$P_t=I_t-R_t=(Ci-Fr)v^{n-t+1}=(Ci-Cg)v^{n-t+1}$ : adjustment amount for accumulation of discount ("write up").

$B_t=B_{t-1}-P_t$ : book value of bond after adjustment from the most recent coupon paid.

Price Between Coupon Dates: For a bond sold at time $k$ after the coupon payment at time $t$ and before the coupon payment at time $t+1$: $B_{t+k}^f=B_t(1+i{)}^k=(B_{t+1}+Fr)v^{1-k}$ : "flat price" of the bond, ie the money that actually exchanges hands on the sale of the bond.

$B_{t+k}^m=B_{t+k}^f-kFr=B_t(1+i{)}^k-kFr$ : "market price" of the bond, ie the price quoted in a financial newspaper.

Approximations of Yield Rates on a Bond: $i\approx \frac{nFr+C-P}{\frac{n}{2}(P+C)}$ : Bond Salesman's Method.

Price of Other Securities: $P=\frac{Fr}{i}$ : price of a perpetual bond or preferred stock.

$P=\frac{D}{i-k}$ : theoretical price of a stock that is expected to return a dividend of $D$ with each subsequent dividend increasing by $(1+k)$, $k<i$. Chapter 9

Recognition of Inflation: $i^{\prime }=\frac{i-r}{1+r}$ : real rate of interest, where $i$ is the effective rate of interest and $r$ is the rate of inflation.

Method of Equated Time and (Macaulay) Duration

$\bar{t}=\frac{\sum _{t=1}^{n}t{R}_t}{\sum _{t=1}^n{R}_t}$ : method of equated time.

$\bar{d}=\frac{\sum _{t=1}^{n}t{v}^t{R}_t}{\sum _{t=1}^n{v}^t{R}_t}$ : (Macauley) duration.

Duration

$P(i)=\sum v^t{R}_t$ : PV of a cash flow at interest rate $i$.

$\bar{v}=-\frac{P^{\prime }(i)}{P(i)}=v\bar{d}=\frac{\bar{d}}{1+i}$ : volatility/modified duration.

$\bar{d}=-(1+i)\frac{P^{\prime }(i)}{P(i)}$ : alternate definition of (Macaulay) duration.

Convexity and (Redington) Immunization

$\bar{c}=\frac{P^{″}(i)}{P(i)}$ convexity

To achieve Redington immunization we want: $P^{\prime }(i)=0$ $P^{″}(i)>0$

Options

Put–Call parity

$C(t)-P(t)=S(t)-K\cdot B(t,T)$

where

$C(t)$ is the value of the call at time $t$,

$P(t)$ is the value of the put,

$S(t)$ is the value of the share,

$K$ is the strike price, and

$B(t,T)$ value of a bond that matures at time $T$. If a stock pays dividends, they should be included in $B(t,T)$, because option prices are typically not adjusted for ordinary dividends.