MAT.116
4.1

Simple Interest

Linear Function

r

Typically a function of time, though it can be thought of as a function of principal or interest rate

Simple Interest Formula

r

I = PrtI = simple interest (dollars)P = principal (dollars)r = interest rate (convert to decimal) (annual)t = time (years)Simple interest is only ever accrued based on the original principal.

Accumulated Amount Formula

r

A = P(1 + rt)A = accumulated amount (principal + interest) (dollars)P = principal (dollars)r = interest rate (convert to decimal) (annual)t = time (years)

Compound Interest

Exponential Function

Nominal Interest Rates

r

NominalStatedAPR

Conversion Periods

r

We assume the conversion period is a constant length of time from one interest calculation to the next.We represent the number of conversion periods in one year by the letter m.Typical conversion periods are the following:Annual, m=1Semiannual, m=2Quarterly, m=4Monthly, m=12Daily, m=365

Compound Interest Formula (Accumulated Amount)

r

A = P(1 + r/m)^(mt)A = accumulated amount (principal + interest) (dollars)P = principal (dollars)r = nominal interest rate (convert to decimal) (annual)m = conversion periods per yeart = time (years)

Continuous Compounding of Interest

Infinite Interest?

Continuous Compound Interest Formula (Accumulated Amount)

r

A = Pe^(rt)A = accumulated amount (principal + interest) (dollars)P = principal (dollars)r = nominal interest rate (convert to decimal) (annual)t = time (years)

Uses

r

It is not practical for everyday interest calculations.It is useful for certain theoretical work in finance.It is also useful for quantities that don't increase at set intervals, like inflation and population growth.

Using Logarithms to Solve Problems in Finance

Guidelines

r

When the variable is an exponent, logarithms must be used to bring the variable out of the exponent.The relevant rule islog_b m^n = n log_b m.However, if the variable is in the base of an expression with a fixed exponent, we can use roots to help in isolating the variable.The relevant rule, assuming m represents nonnegative values, is (m^n)^(1/n) = m.

Present Value

Terminology

r

Future value = accumulated amountPresent value = principalThese terms are tricky because they are really only relative to each other; that is, future value comes after present value in time.

Present Value Formula for Compound Interest

r

P = A(1 + r/m)^(-mt)A = future value (principal + interest) (dollars)P = present value (dollars)r = nominal interest rate (convert to decimal) (annual)m = conversion periods per yeart = time (years)

Present Value Formula for Continuous Compound Interest

r

P = Ae^(-rt)A = future value (principal + interest) (dollars)P = present value (dollars)r = nominal interest rate (convert to decimal) (annual)t = time (years)

Effective Rate of Interest

Synonyms

r

Effective rateAPYEquivalent simple interest rate (for one year)

Effective Rate of Interest Formula

r

r_eff = (1 + r/m)^m - 1r_eff = APY (convert back to percent)r = APR (convert to decimal)m = conversion periods per year