Financial Math FM/Formulas

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Basic Formulas[edit]

 : Accumulation function. Measures the amount in a fund with an investment of 1 at time 0 at the end of period t.
 :amount of growth in period t.
 : rate of growth in period t, also known as the effective rate of interest in period 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]

 : simple interest.
 : variable interest
 : compound interest.
 : continuous interest.

Present Value and Discounting[edit]

effective rate of discount in year t.

Nominal Interest and Discount[edit]

and are the symbols for nominal rates of interest compounded m-thly.

Force of Interest[edit]

 : definition of force of interest.

If the Force of Interest is Constant:

Annuities and Perpetuities[edit]

Annuities[edit]

 : PV of an annuity-immediate.
 : PV of an annuity-due.
 : AV of an annuity-immediate (on the date of the last deposit).
 : AV of an annuity-due (one period after the date of the last deposit).

Perpetuities[edit]

 : PV of a perpetuity-immediate.
 : PV of a perpetuity-due.

m-thly Annuities & Perpetuities[edit]

 : PV of an n-year annuity-immediate of 1 per year payable in m-thly installments.

 : PV of an n-year annuity-due of 1 per year payable in m-thly installments.

 : AV of an n-year annuity-immediate of 1 per year payable in m-thly installments.

 : AV of an n-year annuity-due of 1 per year payable in m-thly installments.

 : PV of a perpetuity-immediate of 1 per year payable in m-thly installments.

 : PV of a perpetuity-due of 1 per year payable in m-thly installments.

Continuous Annuities[edit]

Since ,

 : 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 payments, where the first payment is and each additional payment increases by can be represented by:

Similarly:

 : AV of a series of payments, where the first payment is and each additional payment increases by .

 : PV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute for in denominator to get due form.

 : AV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute for in denominator to get due form.

 : PV of an annuity-immediate with first payment and each additional payment decreasing by 1; substitute for in denominator to get due form.

 : AV of an annuity-immediate with first payment and each additional payment decreasing by 1; substitute for in denominator to get due form.

 : PV of a perpetuity-immediate with first payment 1 and each additional payment increasing by 1.

 : PV of a perpetuity-due with first payment 1 and each additional payment increasing by 1.

Additional Useful Results:  : PV of a perpetuity-immediate with first payment and each additional payment increasing by .

 : PV of an annuity-immediate with m-thly payments of in the first year and each additional year increasing until there are m-thly payments of in the nth year.

 : PV of an annuity-immediate with payments of at the end of the first mth of the first year, at the end of the second mth of the first year, and each additional payment increasing until there is a payment of at the end of the last mth of the nth year.

 : PV of an annuity with continuous payments that are continuously increasing. Annual rate of payment is at time .

 : PV of an annuity with a continuously variable rate of payments and a constant interest rate.

 : PV of an annuity with a continuously variable rate of payment and a continuously variable rate of interest.

Payments in Geometric Progression[edit]

 : PV of an annuity-immediate with an initial payment of 1 and each additional payment increasing by a factor of . Chapter 5

Definitions:  : payment at time . A negative value is an investment and a positive value is a return.

 : PV of a cash flow at interest rate . Chapter 6

General Definitions[edit]

 : payment made at the end of year , split into the interest and the principle repaid .

 : interest paid at the end of year .

 : principle repaid at the end of year .

 : balance remaining at the end of year , just after payment is made.

On a Loan Being Paid with Level Payments:  : interest paid at the end of year on a loan of .

 : principle repaid at the end of year on a loan of .

 : balance remaining at the end of year on a loan of , just after payment is made.

For a loan of , level payments of will pay off the loan in years. In this case, multiply , , and by , ie etc.

Yield Rates[edit]

 : Dollar weighted
 : Time Weighted

Sinking Funds[edit]

 : total yearly payment with the sinking fund method, where is the interest paid to the lender and is the deposit into the sinking fund that will accumulate to in years. is the interest rate for the loan and is the interest rate that the sinking fund earns.

Bonds[edit]

Definitions:  : Price paid for a bond.

 : Par/face value of a bond.

 : Redemption value of a bond.

 : coupon rate for a bond.

 : modified coupon rate.

 : yield rate on a bond.

 : PV of .

 : number of coupon payments.

 : base amount of a bond.

Determination of Bond Prices[edit]

 : price paid for a bond to yield .

 : Premium/Discount formula for the price of a bond.

 : premium paid for a bond if .

 : discount paid for a bond if .

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:  : coupon payment.

 : interest earned from the coupon payment.

 : adjustment amount for amortization of premium ("write down") or

 : adjustment amount for accumulation of discount ("write up").

 : book value of bond after adjustment from the most recent coupon paid.

Price Between Coupon Dates: For a bond sold at time after the coupon payment at time and before the coupon payment at time :  : "flat price" of the bond, ie the money that actually exchanges hands on the sale of the bond.

 : "market price" of the bond, ie the price quoted in a financial newspaper.

Approximations of Yield Rates on a Bond:  : Bond Salesman's Method.

Price of Other Securities:  : price of a perpetual bond or preferred stock.

 : theoretical price of a stock that is expected to return a dividend of with each subsequent dividend increasing by , . Chapter 9

Recognition of Inflation:  : real rate of interest, where is the effective rate of interest and is the rate of inflation.

Method of Equated Time and (Macauley) Duration[edit]

 : method of equated time.

 : (Macauley) duration.

Volatility and Modified Duration[edit]

 : PV of a cash flow at interest rate .

 : volatility/modified duration.

 : alternate definition of (Macauley) duration.

Convexity and (Redington) Immunization[edit]

convexity

To achieve Redington immunization we want:

Options[edit]

Put–Call parity

where

is the value of the call at time ,
is the value of the put,
is the value of the share,
is the strike price, and
value of a bond that matures at time . If a stock pays dividends, they should be included in , because option prices are typically not adjusted for ordinary dividends.