Financial Math FM/Formulas

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

General Definitions edit

  : 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

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

  : interest paid at the end of year  .

  : principal 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  .

  : principal 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. To scale the interest, principal, and balance owed at time  , multiply the above formulas for  ,  , 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 (Macaulay) Duration edit

  : method of equated time.

  : (Macauley) duration.

Duration edit

  : PV of a cash flow at interest rate  .

  : volatility/modified duration.

  : alternate definition of (Macaulay) 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.