Formula and Use
Periodic rates relate to interest compounded at regular intervals, such as monthly, quarterly, or annually. In contrast continuous rates are rates that are compounded infinitely frequently. This means that interest is added to the balance continuously over time. The periodic to continuous interest rate formula is used to convert a periodic interest rate (i) with compounding taking place (m) times in a period, into a continuous interest rate (r).
Example 1: Using the Periodic to Continuous Interest Rate Formula
To illustrate suppose an amount is invested at an annual rate of 5% compounded monthly. The equivalent continuous interest rate is given as follows:
Continuous interest rate = r = m x LN(1 + i / m) i = 5% annual m = 12 (monthly compounding) Continuous interest rate = r = 12 x LN(1 + 5% / 12) Continuous interest rate = r = 4.9896%
Consequently this means that monthly compounding at a rate of 5% is the same as continuous compounding at a rate of 4.9896%.
To show that the two rates do in fact give the same result, suppose for example an amount of 5,000 is invested at a rate of 5% and compounded monthly for 3 years. The future value is given as follows:
FV = PV x (1 + i)n PV = 5,000 i = 5%/12 (monthly rate) n = 3 x 12 = 36 months FV = 5000 x (1 + 5%/12)36 FV = 5,807.36
If the same amount is invested with continuous compounding, then using our calculated rate above, the future value is given by
FV = PV x ern PV = 5,000 r = 4.9896% n = 3 FV = 5000 x e(4.9896% x 3) FV = 5,807.36
The future value is each case is the same.
Example 2: Using the Periodic to Continuous Interest Rate Formula
To further illustrate suppose an amount is invested at an annual rate of 6% compounded quarterly. The equivalent continuous interest rate is given as follows:
Continuous interest rate = r = m x LN(1 + i / m) i = 6% annual m = 4 (quarterly compounding) Continuous interest rate = r = 4 x LN(1 + 6% / 4) Continuous interest rate = r = 5.9554%
Consequently this means that quarterly compounding at a rate of 6% is the same as continuous compounding at a rate of 5.9554%.
Example 3: Using the Periodic to Continuous Interest Rate Formula
Likewise suppose an amount is invested at an annual rate of 8% compounded annually. The equivalent continuous interest rate is given as follows:
Continuous interest rate = r = m x LN(1 + i / m) i = 8% annual m = 1 (annual compounding) Continuous interest rate = r = 1 x LN(1 + 8% / 1) Continuous interest rate = r = 7.6961%
Consequently the annual compounding at a rate of 8% is the same as continuous compounding at a rate of 7.6961%.
The periodic to continuous rate formula is one example of an annuity formula used in time value of money calculations. Additionally discover another at the link below.
About the Author
Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a degree from Loughborough University.