Semiannual
compounding
What is the effect if interest is paid twice
a year, onehalf of the stated interest rate after
six months and another half of the state rate
after 12 months. In one year, $100 at 8% interest
compounded semiannually will be:
Month 
P x (1 + %) 
Value at end
of period 
6 
$100 x (1 + .04) 
$104 
12 
$104 x (1 + .04) 
$108.16 
Although the nominal rate was 8%, the effective
rate was $8.16. No wonder banks that offer compound
interest advertise effective rates to borrowers!
Quarterly compounding
The more the compounding, the greater the effective
rate. Consider $100 at 8% interest compounded
quarterly:
Month 
P x (1 + %)

Value at end
of period 
3 
$100 x (1 + .02) 
$102 
6 
$102 x (1 + .02) 
$104.04 
9 
$104.04 x (1 + .02)

$106.12 
12 
$106.12 x (1 + .02)

$108.24 
Now the effective rate is 8.24%.
We could have gotten the same result using a
modified version of our future value
formula:
FV
= PV (1 + i/m)^{nm} 
FV 
future value
at the end of period 
PV 
the present value (or
initial principal) 
i 
the interest rate paid
each period 
n 
the number of periods

m 
the number of coumpouding
periods 
Continuous compounding
You can even compute future values assuming continuous
compounding. Using the formula
FV = PV*(1 + i/m)^{n*m}
(where m is the frequency of compounding)
it is possible to use some calculus
to compute future values when interest is compounded
continuously. The formula is:
FV
= PV*e^{i}^{*}^{n}
(where e is the exponential function, 2.7183)

FV 
future value at the end
of period 
PV 
the present value (or
initial principal) 
i 
the interest rate paid
each period 
n 
the number of periods

This means that if you invest $100
at 8% compounded continually, your effective rate
is approximately 8.33% (after one year, $100 becomes
$108.3288).

A note on APR
"Truth in lending"
laws require that lenders disclose the
annual percentage rate (APR)
when making certain consumer loans, such
as for credit cards. This is the nominal
rate computed by multiplying the periodic
rate by the number of periods in one year.
This means a bank credit card that advertises
an 18% rate is charging nominally 1.5% per
month, but this is then compounded continuously.
This, of course,is misleading! The effective
rate is more than 18%  it's actually
about 19.56%. That is if you took out a
loan for $10,000 you would have to pay $11,9560
to repay it after a year, not the $11,800
that the advertised rate suggested. See
Regulation
Z.
How did the banking lobby convince lawmakers
that disclosure of the nominal
APR is appropriate for consumer loans,
but that disclosure of the higher effective
rate is appropriate for deposit
savings accounts? 
