Notation and Terminology#
Basic Notation and Terminology
\(P = \) Principal (i.e., value of initial deposit)
\(A = \) Accumulated amount (i.e., sum of the principal and interest)
\(r = \) Nominal interest rate
\(m = \) Number of conversion periods per year, (a conversion period is the interval of time between successive interest payments)
Annually |
Semiannually |
Quarterly |
Monthly |
Weekly |
Daily |
|---|---|---|---|---|---|
\(m=1\) |
\(m=2\) |
\(m=4\) |
\(m=12\) |
\(m=52\) |
\(m=365\) |
\(t = \) Term of investment (in years)
Simple Interest
Interest is always computed based on the original principal.
Interest Earned |
Accumulated Amount |
|---|---|
\(I = Prt\) |
\(A = P(1 + rt)\) |
Discrete Compound Interest
Interest payments are added to the principal at the end of each conversion period and therefore earn interest during future conversion periods.
Accumulated Amount |
Present Value Formula |
|---|---|
\(A = P \left(1 + \frac{r}{m}\right)^{mt}\) |
\(P = A\left(1 + \frac{r}{m}\right)^{-mt}\) |
Continuous Compound Interest
Continuous compounding of interest is equivalent to a discrete compounding of interest where \(m\), the number of conversion periods per year, goes to infinity.
Accumulated Amount |
Present Value Formula |
|---|---|
\(A = Pe^{rt}\) |
\(P = Ae^{-rt}\) |
Effective Rate of Interest
The effective interest rate, \(r_{\text{eff}}\), is the simple interest rate that produces the same accumulated amount in 1 year as the nominal rate, \(r\), compounded \(m\) times a year.