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.

$$P(i)=\sum{v^tR_t}$$ : 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.

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

Duration
$$P(i)=\sum{v^tR_t}$$ : 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.