The time interval between the occasions at which
interest is added to
the account is called
the compounding period . The chart below
some of the common compounding periods:
The interest rate, together with the compounding period and the balance in the account, determines how much interest is added in each compounding period. The basic formula is this:
the interest to be added = (interest rate for one period)*(balance at the beginning of the period).Generally, regardless of the compounding period, the interest rate is given as an ANNUAL RATE (sometimes called the nominal rate) labeled with an r. Here is how the interest rate for one period is computed from the nominal rate and the compounding period:
If we put these two formulas together we get
"Nominal" in ordinary English can indicate
something formal, in name only, but not quite reality and
something that needs further description. It fits well here, because
effect of compounding is a real rate of interest slightly higher than
nominal rate of interest. Click here
to the first use of the word "nominal".
What Happens To An Account With Compounded Interest And No Withdrawals?
In words, the balance at the end of a new compounding period is the balance at the end of the preceding period plus the interest that older balance earned during the compounding period. The interest earned is r * (1/K) * Pj,, as described above in the interest calculation for one period. Thus, at the end of the (j+1)th period,
In the last line of the table above, Pj has been factored from the two terms of the previous equality. Here are some examples of the use of this formula, period by period:
For The Saver, There Is An Advantage To Compounding More Frequently. If One Fixes The Nominal Interest Rate And The Total Time The Account Collects Interest, More Frequent Compounding Produces More Interest. In the analysis below,we assume that the total time is a whole number multiple of compounding periods.
With T and r fixed (not changing) for this discussion, view the right-hand side above as a function of real variable K, say f(K). As long as 1+r/K is positive, this function will have a derivative:
This simplifies somewhat:
(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 that r>0 and K>r, we find that
ln(1+r/K) - r/(K+r)) >= (K-r)r2/(2 K2 (K+r)) > 0
["ln" refers to the natural logarithm, the log to the base e.] Note that the derivative exists and is positive when P 0 , r, K, and T are all positive and K > r (which are natural assumptions about a savings account!). Since the derivative is positive, the original function f(K) is increasing. Thus, larger values of K make f(K) larger. If we make K larger and also make KT be an integer, then f(K) happens to coincide with P KT . Thus compounding more frequently produces more interest (subject to the assumption that T is a whole number multiple of the compounding period). If T is not a multiple of the compounding period, the conclusion depends strongly on the account's policies on withdrawals in the middle of a compounding period. For example, in some certificates of deposits the bank may charge a substantial penalty for "early" withdrawal.
What if we are utterly greedy, and insist that the bank compound our interest continuously?
What happens if we make the compounding period a millionth of a second, and ever smaller? Does the amount of interest increase forever without bounds, or do we reach a ceiling (a limit!) as we compound more and more frequently?
As K approaches positive infinity, we have a race between two factors because KT 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 and ln(1+r/K) approaches 0. Thus we seem to have infinity*0 in our limit as K approches positive infinity. Recall that L'Hôspital's rule applies to indeterminate forms 0/0 and infinity/infinity. Rewrite the difficult part of g(K) to take advantage of this rule:
Note that 1/(KT) is approaching 0, so that we have the indeterminate form of 0/0. By L'Hôspital's rule, examine the limit of a new ratio which is the ratio of the separate derivatives of the top and bottom of the indeterminate form:
After simplifying this new ratio, one has
As K approaches positive infinity, this new ratio approaches (rT) * [1/(1+0)] = rT. Thus, g(K) has the limit ln(P0) + rT as K approaches positive infinity. Because ex is a continuous function, we can apply ex to the function g(K) to get f(K) back AND a limit for f(K) which is
Thus, compounding faster and faster does have a finite limit; this finite limit defines what economists (and bankers) mean by continuous compounding. If compounding is continuous at a nominal interest rate of r for a duration T (in years) with an beginning balance of P0, the balance at the end is
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Edited on September 6, 2006.