# EffectiveInterest

EffectiveInterest[r,q]

gives the effective interest rate corresponding to interest specification r, compounded at time intervals q.

# Details and Options • EffectiveInterest returns an expression suitable for use in TimeValue.
• EffectiveInterest works with numerical or arbitrary symbolic expressions.
• Symbolic expressions returned by EffectiveInterest can be solved for nominal rates, compounding periods, or time parameters.
• In EffectiveInterest[r,q], the interest r can be specified in the following forms:
•  r nominal interest rate {r1,r2,…} schedule of rates applied over unit time intervals {{t1,r1},{t2,r2},…} schedule of forward rates changing at the specified times {p1->r1,p2->r2,…} term structure of interest rates
• EffectiveInterest[r,q] returns an expression in the same form as r.
• EffectiveInterest[r,0] specifies continuous compounding.
• EffectiveInterest[{r1,r2,}] gives the compounded average growth rate (CAGR) corresponding to the rate schedule {r1,r2,}.
• EffectiveInterest[{p1->r1,p2->r2,}] gives the equivalent schedule of future spot rates.

# Examples

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## Basic Examples(7)

Effective rate corresponding to a nominal rate of 5% compounded 4 times per period:

Schedule of nominal rates to effective rates, compounded 12 times per period:

Convert a schedule of nominal rates to effective rates compounded 12 times per period:

Compound annual growth rate (CAGR) corresponding to a schedule of rates:

Convert a term structure of interest rates (yield curve) to a list of implied forward rates and the corresponding intervals over which they are valid:

Solve for the nominal rate corresponding to an effective rate of 5% compounded quarterly:

Use EffectiveInterest with TimeValue:

## Scope(5)

A compounding interval of zero can be used to specify continuous compounding:

An integral compounding frequency may be used to specify compounding of less than once per period. As expected, the effective rate in this case is less than the nominal rate:

Simple interest can be simulated by using an integral compounding interval equal to the growth period:

This is equivalent to the analogous simple interest computation:

EffectiveInterest works with symbolic parameters:

Solutions to equations involving EffectiveInterest can be found in terms of symbolic parameters:

EffectiveInterest from a TimeSeries:

## Generalizations & Extensions(1)

Study the convergence of the future value of an amount as interest compounding approaches infinity:

## Applications(1)

Lender A quotes the nominal interest rate on a loan at 8% per year with continuous compounding. Lender B quotes their rate using quarterly compounding. Convert lender A's rate to an equivalent rate with quarterly compounding so that the two rates may be compared: