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.

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

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

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

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

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.

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

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

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

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

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

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.

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

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

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

Continuous Annuities

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

$\lim _{m\to \infty }a_{\overline {n|}}^{(m)}=\lim _{m\to \infty }{\frac {1-v^{n}}{i^{(m)}}}={\frac {1-v^{n}}{\delta }}={\overline {a}}_{\overline {n|}}={\frac {i}{\delta }}a_{\overline {n|}}$ : 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: $A=Pa_{\overline {n|}}+Q{\frac {a_{\overline {n|}}-nv^{n}}{i}}=Pv+(P+Q)v^{2}+(P+2Q)v^{3}+\cdots +(P+(n-1)Q)v^{n}$

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

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

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

$(Ia)_{\overline {n|}}={\frac {{\ddot {a}}_{\overline {n|}}-nv^{n}}{i}}$ : 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.

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

$(I{\ddot {a}})_{\overline {\infty |}}={\frac {1}{d^{2}}}$ : 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 $Q$.

$(Ia)_{\overline {n|}}^{(m)}={\frac {{\ddot {a}}_{\overline {n|}}-nv^{n}}{i^{(m)}}}$ : 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.

$(I^{(m)}a)_{\overline {n|}}^{(m)}={\frac {{\ddot {a}}_{\overline {n|}}^{(m)}-nv^{n}}{i^{(m)}}}$ : 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.

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

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

$\int _{0}^{n}f(t)e^{-\int _{0}^{t}\delta _{r}dr}dt$ : 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 $P_{t}$.

$I_{t}=iB_{t-1}$ : 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

$P=Fra_{{\overline {n|}}i}+Cv^{n}=Cga_{{\overline {n|}}i}+Cv^{n}$ : 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.

$I_{t}=iB_{t-1}$ : 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: $i'={\frac {i-r}{1+r}}$ : 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

${\overline {t}}={\frac {\sum _{t=1}^{n}tR_{t}}{\sum _{t=1}^{n}R_{t}}}$ : method of equated time.

${\overline {d}}={\frac {\sum _{t=1}^{n}tv^{t}R_{t}}{\sum _{t=1}^{n}v^{t}R_{t}}}$ : (Macauley) duration.

Duration

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

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

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

Convexity and (Redington) Immunization

${\overline {c}}={\frac {P''(i)}{P(i)}}$ convexity

To achieve Redington immunization we want: $P'(i)=0$ $P''(i)>0$

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.