

COMPOUND INTEREST
Bank deposits, over time, usually have compound interest .
That
is, interest is computed on an account such as a savings account or a
checking
account and the interest is added to the account. Because the interest
is added to the account (the alternative would be to mail the interest
to the customer), the interest itself earns interest during the next
time
period for computing interest. This is what is meant when it is said
that
the interest compounds . See Salas and Hille, page
448449.
The time interval between the occasions at which
interest is added to
the account is called
the compounding period . The chart below
describes
some 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 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:
interest rate for one period 
= 
(nominal rate)*(compounding period as a
fraction of a year) 

= 
(nominal rate)/(number of compounding
periods in one year) 
If we put these two formulas together we get
the interest to be added 
= 
(nominal rate)*(compounding period as a
fraction of a year)*(balance
at the beginning of the compounding period) 
Interest Rate For One Period
with various periods and a nominal annual rate of 6%
per year
Compounded 
Calculation 
Interest Rate For One Period 
Daily, each day, every 365^{th}
of a year 
(.06)/365 
0.000164384 
Monthly, each month, every 12^{th}
of a year 
(.06)/12 
0.005 
Quarterly, every 3 months,
every 4^{th} 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 (from
Medieval Latin for per
centum, meaning "among 100"). 6% means 6 among 100, thus
6/100 as
a fraction and .06 as a decimal. 
Here are some common units for
this calculation:

nominal annual rate has units of
reciprocal year: for example,
0.06/year

the compounding period is converted
to years: for example,
3 months is converted to (1/4) year.

the interest rate for one period is
a pure number because the unit
of years cancel in the calculation:
(.06/year)*[(1/4)year]=.06/4.

Some Examples With Various Interest
Rates And Compounding Periods
Nominal Interest Rate 
Compounded 
Interest Rate For One Period 
Balance at the beginning of some period 
Interest Added at the end of the same
period 
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 indicate
something formal, in name only, but not quite reality and
perhaps
something that needs further description. It fits well here, because
the
effect of compounding is a real rate of interest slightly higher than
the
nominal rate of interest. Click here
to return
to the first use of the word "nominal".
What Happens To An Account With Compounded Interest And
No Withdrawals?
Consider now an account in which P_{0 }is invested
at the
beginning of a compounding period, with a nominal interest rate r and
compounding
K times per year (so each compounding period is (1/K)^{th}
of one
year). How much will be in the account after n compounding
periods?
Let P_{ j }denote the balance in the account after
j compounding
periods, including the interest earned in the last of these j periods.
NOTE THAT WE HAVE JUST DEFINED A SEQUENCE OF REAL NUMBERS. To review
what
these sequences are, in general, see sequences
of real numbers. Note that we have a recursive
definition of
this sequence:
P_{j+1}=P
_{j }+ the interest earned
by P_{j }in one compounding period. 
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)
* P_{j,}, as described above in the interest
calculation for one
period. Thus, at the end of the (j+1)^{th}
period,
P_{j+1} 
= 
P_{j }+ the
interest earned by P_{j} in one compounding
period 

= 
P_{j }+ (nominal
rate)*(compounding period as a fraction
of a year)*P_{j} 

= 
P_{j }+ r * (1/K) *
P_{j} 

= 
P_{j} + (r/K)
* P_{j} 

= 
P_{j} * (1
+ r/K) 
In the last line of the table above, P_{j} has been
factored from
the two terms of the previous equality. Here are some
examples of
the use of this formula, period by period:
Values of "j" 
P_{j} 
j=0 
P_{1} = P_{0}
* (1+r/K) 
j=1 
P_{2} = P_{1}
* (1+r/K) = P_{0} *(1+r/K) *
(1+r/K) = P_{1} = P_{0} *
(1+r/K)^{2} 
j=2 
P_{3} = P_{2}
* (1+r/K) = P_{0} *(1+r/K)^{2}
* (1+r/K) = P_{0} * (1+r/K)^{3} 
j=3 
P_{4} = P_{3}
* (1+r/K) = P_{0} *(1+r/K)^{3}
* (1+r/K) = P_{0} * (1+r/K)^{4} 
In general
P_{j} = P_{0}
* (1+r/K)^{j}
for nonnegative whole numbers j. The rare person may wonder
how
we can leap to this conclusion about an infinite number of possible
j's,
given only four examples! This formula can be proved for all
of the
infinite number of possible j's by using the principle of
mathematical
induction.

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.
If one fixes the initial balance (P _{0}), the
nominal interest
rate (r) and the duration of the deposit (T, in years) , you earn more
interest with more compounding periods per year (K). The
number of
compounding periods that make up T will be KT. To avoid
fractions
of compounding periods, which were not analyzed above, assume that K is
such that KT is a whole number. Then, by the formula above,
P _{KT }= P _{0
}* (1+r/K)^{KT}.
With T and r fixed (not changing) for this discussion, view the
righthand
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:
(d/dK)[f(K)] = P _{0 }*
(1+r/K)^{KT }* [ T * ln(1+r/K) + K * T*
(1/(1+r/K))*(r/(K^{2 })) ].
^{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)) >= (Kr)r^{2}/(2 K^{2}
(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?}
^{
To answer these questions, consider g(K) = ln(f(K)):
}
^{g(K) = ln(P0)
+
(KT) * ln(1+r/K).}
^{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:
}
^{g(K) = ln(P0)
+
ln(1+r/K) / [1/(KT)].}
^{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:
}
^{{[1/(1+r/K)](r/(K2)}
/ {(KT)2*T}}
^{After simplifying this new ratio, one has
}
^{[1/(1+r/K)] * (r/T) * [(KT)2]
/ (K2) =
(rT) * [1/(1+r/K)].}
^{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
}
^{e[ln(P0)+rT]=P
0*erT.}
^{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
}
^{P 0*erT.}
^{Your comments and questions are welcome.
Please use the email address
at www.math.hawaii.edu
Edited on September 6, 2006. } 