# Financial Math FM/Formulas

## Basic Formulas

${\displaystyle \ a(t)}$ : Accumulation function. Measures the amount in a fund with an investment of 1 at time 0 at the end of period t.
${\displaystyle \ a(t)-a(t-1)}$ :amount of growth in period t.
${\displaystyle \ s_{t}={\frac {a(t)-a(t-1)}{a(t-1)}}}$ : rate of growth in period t, also known as the effective rate of interest in period t.
${\displaystyle \ A(t)=k\cdot a(t)}$ : 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

${\displaystyle \ a(t)=1+i\cdot t}$ : simple interest.
${\displaystyle \ a(t)=\prod _{j=1}^{t}(1+i_{j})}$ : variable interest
${\displaystyle \ a(t)=(1+i)^{t}}$ : compound interest.
${\displaystyle \ a(t)=e^{t\cdot i}}$ : continuous interest.

### Present Value and Discounting

${\displaystyle \ PV={\frac {1}{a(t)}}={\frac {1}{(1+i)^{t}}}=(1+i)^{-t}=v^{t}}$
${\displaystyle \ d_{t}={\frac {a(t)-a(t-1)}{a(t)}}}$ effective rate of discount in year t.
${\displaystyle \ 1-d=v}$
${\displaystyle \ d={\frac {i}{1+i}}=i\cdot v}$
${\displaystyle \ i={\frac {d}{1-d}}}$

### Nominal Interest and Discount

${\displaystyle i^{(m)}}$ and ${\displaystyle d^{(m)}}$ are the symbols for nominal rates of interest compounded m-thly.
${\displaystyle 1+i=(1+{\frac {i^{(m)}}{m}})^{m}}$
${\displaystyle i^{(m)}=m((1+i)^{\frac {1}{m}}-1)}$
${\displaystyle 1-d=(1-{\frac {d^{(m)}}{m}})^{m}}$
${\displaystyle d^{(m)}=m(1-(1-d)^{\frac {1}{m}})}$

### Force of Interest

${\displaystyle \delta _{t}={\frac {1}{a(t)}}{\frac {d}{dt}}a(t)={\frac {d}{dt}}\ln a(t)}$ : definition of force of interest.
${\displaystyle a(t)=e^{\int _{0}^{t}\delta _{r}\,dr}}$

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

${\displaystyle PV=e^{-\delta t}}$
${\displaystyle \delta =\ln(1+i)}$

### Annuities

${\displaystyle a_{\overline {n|}}={\frac {1-v^{n}}{i}}=v+v^{2}+\cdots +v^{n}}$ : PV of an annuity-immediate.
${\displaystyle {\ddot {a}}_{\overline {n|}}={\frac {1-v^{n}}{d}}=1+v+v^{2}+\cdots +v^{n-1}}$ : PV of an annuity-due.
${\displaystyle {\ddot {a}}_{\overline {n|}}=(1+i)a_{\overline {n|}}=1+a_{\overline {n-1|}}}$
${\displaystyle s_{\overline {n|}}={\frac {(1+i)^{n}-1}{i}}=(1+i)^{n-1}+(1+i)^{n-2}+\cdots +1}$ : AV of an annuity-immediate (on the date of the last deposit).
${\displaystyle {\ddot {s}}_{\overline {n|}}={\frac {(1+i)^{n}-1}{d}}=(1+i)^{n}+(1+i)^{n-1}+\cdots +(1+i)}$ : AV of an annuity-due (one period after the date of the last deposit).
${\displaystyle {\ddot {s}}_{\overline {n|}}=(1+i)s_{\overline {n|}}=s_{\overline {n+1|}}-1}$
${\displaystyle a_{\overline {mn|}}=a_{\overline {n|}}+v^{n}a_{\overline {n|}}+v^{2n}a_{\overline {n|}}+\cdots +v^{(m-1)n}a_{\overline {n|}}}$

#### Perpetuities

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

### m-thly Annuities & Perpetuities

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

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

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

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

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

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

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

#### Continuous Annuities

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

${\displaystyle \lim _{m\to \infty }a_{\overline {n|}}^{(m)}=\lim _{m\to \infty }{\frac {1-v^{n}}{i^{(m)}}}={\frac {1-v^{n}}{\delta }}={\overline {a}}_{\overline {n|}}={\frac {i}{\delta }}a_{\overline {n|}}}$ : 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 ${\displaystyle n}$ payments, where the first payment is ${\displaystyle P}$ and each additional payment increases by ${\displaystyle Q}$ can be represented by: ${\displaystyle A=Pa_{\overline {n|}}+Q{\frac {a_{\overline {n|}}-nv^{n}}{i}}=Pv+(P+Q)v^{2}+(P+2Q)v^{3}+\cdots +(P+(n-1)Q)v^{n}}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Payments in Geometric Progression

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

### General Definitions

${\displaystyle R_{t}}$ : payment at time ${\displaystyle t}$. A negative value is an investment and a positive value is a return.

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

${\displaystyle R_{t}=I_{t}+P_{t}}$ : payment made at the end of year ${\displaystyle t}$, split into the interest ${\displaystyle I_{t}}$ and the principal repaid ${\displaystyle P_{t}}$.

${\displaystyle I_{t}=iB_{t-1}}$ : interest paid at the end of year ${\displaystyle t}$.

${\displaystyle P_{t}=R_{t}-I_{t}=(1+i)P_{t-1}+(R_{t}-R_{t-1})}$ : principal repaid at the end of year ${\displaystyle t}$.

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

On a Loan Being Paid with Level Payments:

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

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

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

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

## Yield Rates

${\displaystyle I=B-A-C}$
${\displaystyle i={\frac {I}{A+\sum _{t_{k}}C_{t_{k}}(1-t_{k})}}}$ : dollar-weighted
${\displaystyle (1+i)=\prod _{t_{k}}^{t}\left({\frac {B_{t_{k}}}{B_{t_{k-1}}+C_{t_{k-1}}}}\right)}$ : time-weighted

## Sinking Funds

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

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

## Bonds

Definitions: ${\displaystyle P}$ : Price paid for a bond.

${\displaystyle F}$ : Par/face value of a bond.

${\displaystyle C}$ : Redemption value of a bond.

${\displaystyle r}$ : coupon rate for a bond.

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

${\displaystyle i}$ : yield rate on a bond.

${\displaystyle K}$ : PV of ${\displaystyle C}$.

${\displaystyle n}$ : number of coupon payments.

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

${\displaystyle Fr=Cg}$

### Determination of Bond Prices

${\displaystyle P=Fra_{{\overline {n|}}i}+Cv^{n}=Cga_{{\overline {n|}}i}+Cv^{n}}$ : price paid for a bond to yield ${\displaystyle i}$.

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

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

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

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: ${\displaystyle R_{t}}$ : coupon payment.

${\displaystyle I_{t}=iB_{t-1}}$ : interest earned from the coupon payment.

${\displaystyle P_{t}=R_{t}-I_{t}=(Fr-Ci)v^{n-t+1}=(Cg-Ci)v^{n-t+1}}$ : adjustment amount for amortization of premium ("write down") or

${\displaystyle P_{t}=I_{t}-R_{t}=(Ci-Fr)v^{n-t+1}=(Ci-Cg)v^{n-t+1}}$ : adjustment amount for accumulation of discount ("write up").

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

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

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

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

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

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

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

## Method of Equated Time and (Macaulay) Duration

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

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

#### Volatility and Modified Duration

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

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

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

## Convexity and (Redington) Immunization

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

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

## Options

Put–Call parity

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

where

${\displaystyle C(t)}$ is the value of the call at time ${\displaystyle t}$,
${\displaystyle P(t)}$ is the value of the put,
${\displaystyle S(t)}$ is the value of the share,
${\displaystyle K}$ is the strike price, and
${\displaystyle B(t,T)}$ value of a bond that matures at time ${\displaystyle T}$. If a stock pays dividends, they should be included in ${\displaystyle B(t,T)}$, because option prices are typically not adjusted for ordinary dividends.