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
\(t = \) Term of investment (in years)
\(m = \) Number of conversion periods per year, (a conversion period is the interval of time between successive interest payments)
Annually |
Semiannually |
Quarterly |
Monthly |
Weekly |
Daily |
|---|---|---|---|---|---|
\(m=1\) |
\(m=2\) |
\(m=4\) |
\(m=12\) |
\(m=52\) |
\(m=365\) |
Simple Interest
Interest is always computed based on the original principal.
Interest Earned
\[I = Prt\]
Accumulated Amount
\[A = P(1 + rt)\]
Present Value
The present value of an investment is the principal investment, \(P\), that yields a given accumulated amount, \(A\), in the future.
Discrete Compound Interest
Interest payments are added to the principal at the end of each conversion period and earn interest during future conversion periods.
Accumulated Amount
\[A = P \left(1 + \frac{r}{m}\right)^{mt}\]
Present Value Formula
\[P = A\left(1 + \frac{r}{m}\right)^{-mt}\]
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
\[A = Pe^{rt}\]
Present Value Formula
\[P = Ae^{-rt}\]
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.