An interest rate can be a required rate of return (aka, hurdle rate), a discount rate, and/or an opportunity cost. We can deconstruct an interest into its components where the rate, r = real risk-free interest rate + inflation premium + default risk premium + liquidity premium + maturity premium. The nominal rate = real risk-free rate + inflation premium.
The future value, FV(N) = PV*(1 + r_s/m)^(m*N) where r_s is the stated (aka, nominal) rate and m is the number of compound periods per year. If we increase m, beyond monthly and daily, to its limit, m converges on infinity (m = ∞) and we are compounding continuously in which case FV = PV*exp(r*N).
The stated (aka, nominal) interest rate is not the effective rate: for a given stated interest rate, r_s, the future value increases with higher compound frequencies. To compare rates, we should use the effective annual rate (EAR). EAR = (1 + periodic rate)^m - 1, where the periodic rate equals the stated rate divided by m; i.e., periodic rate = r_s/m. Therefore, EAR = (1 + r_s/m)^m - 1. For example, if the stated rate is 8.0% per annum, then if m = 2 (i.e. semiannual compound frequency), the EAR = (1 + 0.080/2)^2 - 1 = 8.160%; and if m = 12 (i.e., monthly compound frequency), the EAR = (1 + 0.080/12)^12 = 8.300%.
The future value, FV(N) = PV*(1 + r_s/m)^(m*N) where r_s is the stated (aka, nominal) rate and m is the number of compound periods per year. If we increase m, beyond monthly and daily, to its limit, m converges on infinity (m = ∞) and we are compounding continuously in which case FV = PV*exp(r*N).
The stated (aka, nominal) interest rate is not the effective rate: for a given stated interest rate, r_s, the future value increases with higher compound frequencies. To compare rates, we should use the effective annual rate (EAR). EAR = (1 + periodic rate)^m - 1, where the periodic rate equals the stated rate divided by m; i.e., periodic rate = r_s/m. Therefore, EAR = (1 + r_s/m)^m - 1. For example, if the stated rate is 8.0% per annum, then if m = 2 (i.e. semiannual compound frequency), the EAR = (1 + 0.080/2)^2 - 1 = 8.160%; and if m = 12 (i.e., monthly compound frequency), the EAR = (1 + 0.080/12)^12 = 8.300%.
