Financial Math FM/Formulas

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Basic Formulas[edit]

\ 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[edit]

\ 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[edit]

\ PV = \frac{1}{a(t)} = \frac{1}{(1+i)^t}=(1+t)^{-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[edit]

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[edit]

\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 and Perpetuities[edit]

Annuities[edit]

a_{\overline{n|}}=\frac{1-v^n}{i}=v+v^2+..+v^n : PV of an annuity-immediate.
\ddot{a}_{\overline{n|}}=\frac{1-v^n}{d}=1+v+v^2+..+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}+..+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}+..+(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|}}+..+v^{(m-1)n} a_{\overline{n|}}

Perpetuities[edit]

\lim_{n\to\infty} a_{\overline{n|}} = \lim_{n\to\infty} \frac{1-v^n}{i}=\frac{1}{i}=v+v^2+...= 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+...= \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[edit]

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[edit]

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+..+(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[edit]

\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

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

General Definitions[edit]

R_t=I_t+P_t : payment made at the end of year t, split into the interest I_t and the principle 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}) : principle 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} : principle 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. In this case, multiply 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[edit]

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 (\frac{B_{t_k}}{B_{t_k}+C_{t_{k-1}}})  : Time Weighted

Sinking Funds[edit]

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[edit]

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[edit]

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 (Macauley) Duration[edit]

\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.

Volatility and Modified Duration[edit]

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 (Macauley) duration.

Convexity and (Redington) Immunization[edit]

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

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

Options[edit]

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.