Periodic to Continuous Interest Rate Formula

Formula

r = m x LN(1 + i / m)
Variables used in the annuity formula
r = Continuous interest rate
i = Periodic interest rate
m = Number of times compounding takes place in a period
LN = Natural logarithm

Use

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

If an amount is invested at an annual rate of 5% compounded monthly, then 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%

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, then its 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

If an amount is invested at an annual rate of 6% compounded quarterly, then 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%

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

If an amount is invested at an annual rate of 8% compounded annually, then 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%

This means that annual compounding at a rate of 8% is the same as continuous compounding at a rate of 7.6961%.

The periodic to continuous interest rate formula is one example of an annuity formula used in time value of money calculations, discover another at the link below.

Periodic to Continuous Interest Rate Formula November 6th, 2016Team

You May Also Like


Related pages


payment voucher format wordexamples of suspencerent accrual journal entrypv formula calculatorexample of contra entrysum of years digits methodfixed factory overheadthe formula for computing annual straight line depreciation iscost of capital calculation excelhow to maintain petty cash bookdefine debtorcredit noediscount factor excelaccruals and deferred income balance sheetreplenishment of petty cash fundhow to calculate stock splitprepayment accounting entryfob shipping point fob destinationhow to calculate the quick ratiobad debt provision vathow to journalize payrolljournal entry for credit sales with discountissuing stock journal entryhow to calculate remaining mortgage balancehow to calculate weighted average contribution marginamortised cost accountingjournal entry in tallyrecording petty cashreceivables turnover ratiosbookkeeping sampleschart of accounts for non profit organizationcalculate mark upwhat is the final step in the accounting cyclelink between balance sheet and income statementcontinuously compounding interest calculatorhow to calculate variable expensesmargin versus mark upsales return and allowances normal balancedouble entry system advantages and disadvantagesdouble declining depreciation method examplejournal entry for accrued salariesaccounting journal entries examples pdfwhat is lifo reserveaccounting transaction analysisdefine consignee and consignorledger sampledepreciation half year conventionfuture value of perpetuity formulathe journal entry to adjust supplies issample money receipt formatvouchers templatesjournal entries debits and creditswhat is journal voucher with exampletabel pvaccounting excel templateskenexa prove it test tutorialprovision of doubtful accountsaccrued expenses double entrydays sales in receivables definitionexample of inventory turnoveraged creditorstemplate for income statement and balance sheetwhat is operating cycle in accountingincome tax expense journal entryaccounting journal and ledgerpresent value of a perpetuity calculatorprepaid insurance accountingpetty cash count formamortize bond discountdefine contra accountformula for growing annuityrental ledger samplewhat is a debtors control account