The time interval between the occasions at whichinterest is added tothe account is called the compounding period . The chart belowdescribessome of the common compounding periods:
| Compounding Period | Descriptive Adverb | Fraction of one year |
| 1 day | daily | 1/365 (ignoring leap years, which have 366 days) |
| 1 month | monthly | 1/12 |
| 3 months | quarterly | 1/4 |
| 6 months | semiannually | 1/2 |
| 1 year | annually | 1 |
The interest rate, together with the compounding periodand the balancein the account, determines how much interest is added in eachcompoundingperiod. The basic formula is this:
the interest to be added = (interestrate for one period)*(balanceat the beginning of the period).Generally, regardless of the compounding period, the interest rate isgivenas an ANNUAL RATE (sometimes called the nominalrate) labeled with an r. Here is how the interestrate for one periodis computed from the nominal rate and the compounding period:
| interest rate for one period | = | (nominal rate)*(compounding period as afraction of a year) |
| = | (nominal rate)/(number of compoundingperiods in one year) |
| the interest to be added | = | (nominal rate)*(compounding period as afraction of a year)*(balanceat the beginning of the compounding period) |
| Compounded | Calculation | Interest Rate For One Period |
| Daily, each day, every 365thof a year | (.06)/365 | 0.000164384 |
| Monthly, each month, every 12thof a year | (.06)/12 | 0.005 |
| Quarterly, every 3 months,every 4th of a year | (.06)/4 | 0.015 |
| Semiannually, every 6 months,every half of a year | (.06)/2 | 0.03 |
| Annually, every year | .06 | .06 |
| 6% means 6 percent (fromMedieval Latin for percentum, meaning "among 100"). 6% means 6 among 100, thus6/100 asa fraction and .06 as a decimal. | ||
Here are some common units forthis calculation:
| ||
| Nominal Interest Rate | Compounded | Interest Rate For One Period | Balance at the beginning of some period | Interest Added at the end of the sameperiod |
| 15%/yr | Daily | 0.000410959=.15/365 | $10,000 | $4.11 |
| 5%/yr | Monthy | 0.004166667=.05/12 | $10,000 | $41.67 |
| 9%/yr | Quarterly | 0.0225=.09/4 | $10,000 | $225 |
| 5.5%/yr | Seminannually | 0.0275=.055/2 | $10,000 | $275 |
| 7.8%/yr86 | Annually | .078=.078/1 | $10,000 | $780 |
1."Nominal" in ordinary English can indicatesomething formal, in name only, but not quite reality andperhapssomething that needs further description. It fits well here, becausetheeffect of compounding is a real rate of interest slightly higher thanthenominal rate of interest. Click hereto returnto the first use of the word "nominal".
What Happens To An Account With Compounded Interest AndNo Withdrawals?
Consider now an account in which P0 is investedat thebeginning of a compounding period, with a nominal interest rate r andcompoundingK times per year (so each compounding period is (1/K)thof oneyear). How much will be in the account after n compoundingperiods?Let P j denote the balance in the account afterj compoundingperiods, including the interest earned in the last of these j periods.NOTE THAT WE HAVE JUST DEFINED A SEQUENCE OF REAL NUMBERS. To reviewwhatthese sequences are, in general, see sequencesof real numbers. Note that we have a recursivedefinition ofthis sequence:
Pj+1=P j + the interest earnedby Pj in one compounding period.
| Pj+1 | = | Pj + theinterest earned by Pj in one compoundingperiod |
| = | Pj + (nominalrate)*(compounding period as a fractionof a year)*Pj | |
| = | Pj + r * (1/K) *Pj | |
| = | Pj + (r/K)* Pj | |
| = | Pj * (1+ r/K) |
| Values of "j" | Pj |
| j=0 | P1 = P0* (1+r/K) |
| j=1 | P2 = P1* (1+r/K) = P0 *(1+r/K) *(1+r/K) = P1 = P0 *(1+r/K)2 |
| j=2 | P3 = P2* (1+r/K) = P0 *(1+r/K)2* (1+r/K) = P0 * (1+r/K)3 |
| j=3 | P4 = P3* (1+r/K) = P0 *(1+r/K)3* (1+r/K) = P0 * (1+r/K)4 |
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For The Saver, There Is An Advantage To CompoundingMore Frequently.If One Fixes The Nominal Interest Rate And The Total Time The AccountCollectsInterest, More Frequent Compounding Produces More Interest.In theanalysis below,we assume that the total time is a whole number multipleof compounding periods.
With T and r fixed (not changing) for this discussion, view theright-handside above as a function of real variable K, say f(K). As long as 1+r/Kis positive, this function will have a derivative:P KT = P 0 * (1+r/K)KT.
This simplifies somewhat:(d/dK)[f(K)] = P 0 *(1+r/K)KT * [ T * ln(1+r/K) + K * T*(1/(1+r/K))*(-r/(K2 )) ].
(d/dK)[f(K)] = P 0 *(1+r/K)KT * T* [ln(1+r/K) - r/(K+r))]
It well known that for x in the interval [0,1), we have ln(1+x) >= x - x2/2.If we substitute r/K for x and assume thatr>0 and K>r, we find thatln(1+r/K)- r/(K+r)) >= (K-r)r2/(2 K2(K+r)) > 0
["ln" refers to the natural logarithm, the log to thebasee.] Notethat the derivative exists and is positive when P 0 ,r, K,and T are all positive and K > r (which are natural assumptionsabout a savingsaccount!).Since thederivative is positive, the original function f(K) is increasing. Thus,larger values of K make f(K) larger. If we make K larger andalsomake KT be an integer, then f(K) happens to coincide with P KT. Thus compounding more frequently produces more interest(subjectto the assumption that T is a whole number multiple of the compoundingperiod). If T is not a multiple of the compounding period,the conclusiondepends strongly on the account's policies on withdrawals in the middleof a compounding period. For example, in some certificates ofdepositsthe bank may charge a substantial penalty for "early" withdrawal.
What if we are utterly greedy, andinsist thatthe bank compound ourinterest continuously?
What happens if we make the compoundingperiod amillionth of a second,and ever smaller? Does the amount of interest increase forever withoutbounds, or do we reach a ceiling (a limit!) as we compound more andmorefrequently?
To answer these questions, consider g(K) = ln(f(K)):
As K approaches positive infinity, we have a racebetween two factorsbecauseKT is also approaching positive infinity (we assume that T is positive)while r/K approaches 0. As r/K approaches 0, 1+r/K approaches 1 andln(1+r/K)approaches 0. Thus we seem to have infinity*0 in our limit as Kapprochespositive infinity. Recall that L'Hôspital's rule applies toindeterminateforms 0/0 and infinity/infinity. Rewrite the difficult part of g(K) totake advantage of this rule:g(K) = ln(P0)+(KT) * ln(1+r/K).
Note that 1/(KT) is approaching 0, so that we have theindeterminateformof 0/0. By L'Hôspital's rule, examine the limit of a newratio whichis the ratio of the separate derivatives of the top and bottom of theindeterminateform:g(K) = ln(P0)+ln(1+r/K) / [1/(KT)].
After simplifying this new ratio, one has{[1/(1+r/K)](-r/(K2)}/ {-(KT)-2*T}
As K approaches positive infinity, this new ratioapproaches (rT) *[1/(1+0)]= rT. Thus, g(K) has the limit ln(P0) + rT as Kapproaches positiveinfinity. Because ex is a continuous function,we can applyex to the function g(K) to get f(K) back AND alimit for f(K)which is[1/(1+r/K)] * (r/T) * [(KT)2]/ (K2) =(rT) * [1/(1+r/K)].
Thus, compounding faster and faster does have a finitelimit; thisfinitelimit defines what economists (and bankers) mean by continuouscompounding.If compounding is continuous at a nominal interest rate of r for adurationT (in years) with an beginning balance of P0,the balance atthe end ise[ln(P0)+rT]=P 0*erT.
Your comments and questions are welcome. Please use the email addressat www.math.hawaii.eduP 0*erT.
Edited on September 6, 2006.
