1
One adjustment, applied twice.
A = P(1 + rm)mt, where m is how many times a year.
annually
m = 1
the plain formula
semi-annually
m = 2
half the rate, twice the periods
quarterly / monthly
m = 4m = 12
same rule, bigger m
Both changes or neither
Dividing the rate without multiplying the periods understates the answer; multiplying
the periods without dividing the rate wildly overstates it. They are two halves of
one adjustment and always travel together.
2
Work out rm and mt before anything else.
m = 2, r = 0.08, t = 3
name them first
rm = 0.04, mt = 6
4% per half-year, six of them
A = 50000(1.04)⁶
now it is the familiar formula
≈ ₱63,265.32
interest ₱13,265.32
Writing "4% for 6 periods" on its own line turns every one of these into the problem
you already solved last lesson.
3
More often pays more, but not much more.
annually
₱12,544.00
10,000 at 12% for 2 years
quarterly
₱12,667.70
₱123.70 more
monthly
₱12,697.35
only ₱29.65 more again
The gains shrink quickly. Going from yearly to quarterly is worth far more than going
from quarterly to monthly, which is why banks rarely bother advertising the difference.
4
Compare offers with the effective annual rate.
8% quarterly: (1 + 0.084)⁴
one year's growth factor
= (1.02)⁴ ≈ 1.08243
effective rate 8.24%
8.2% annually: 1.082
effective rate 8.20%
8% quarterly wins
by about ₱4 per ₱10,000
The advertised, or nominal, rate is not comparable across different compounding
frequencies. The effective annual rate is what puts two offers on the same footing.
5
Variations: money needed now, an effective rate on its own, quarters in months.
The principal is the unknown: deposit how much now, at 8% compounded quarterly, to have ₱50,000 in 3 years?
rm = 0.02, mt = 12
2% a quarter, twelve quarters
50000 = P(1.02)¹²
the familiar formula
P = 50000 ÷ 1.268242 ≈ ₱39,424.66
divide, do not subtract
The same two adjustments come first; only the last line runs backwards.
An effective rate on its own: 12% compounded monthly
(1 + 0.1212)¹² = (1.01)¹²
one year's growth factor
effective rate ≈ 12.68%
subtract 1, write as a percent
The effective rate is always a little above the nominal rate when m is more than 1, and equal to it when m = 1.
Quarters counted in months: ₱20,000 at 10% compounded quarterly for 9 months
rm = 0.104 = 0.025
2.5% a quarter
mt = 9 ÷ 3 = 3 quarters
a quarter is 3 months
A = 20000(1.025)³ ≈ ₱21,537.81
interest ₱1,537.81
Count the periods in the time given: 9 months is 3 quarters, not 9 periods and not 0.75.
6
A period that is not whole years still divides cleanly.
t = 15 months = 1.25 years
convert first
rm = 0.0612 = 0.005
0.5% a month
mt = 12 × 1.25 = 15
fifteen months, fifteen periods
A = 30000(1.005)¹⁵ ≈ ₱32,330.48
interest ₱2,330.48
₱12,667.70 for the opening problem
With monthly compounding the number of periods is simply the number of months, which
is worth noticing: it saves converting to years and back for no reason.
On the exam
Periodical exams ask for the rate per period and the number of periods, then the amount or the interest; entrance tests compare two offers with different compounding, or slip in time given in months. Write rm and mt on their own line before anything else, then use A = P(1 + rm)mt. For a 10-second check, the answer must be a little more than the yearly-compounded amount for the same rate and time, never wildly more.
Remember the flow
Find m · rate per period rm · number of periods mt · A = P(1 + rm)mt
Divide the rate AND multiply the periods. One without the other is always wrong.
