1
Simple or general depends on whether two clocks agree.
monthly pay, monthly compounding
simple
use the formula as it stands
annual pay, monthly compounding
general
convert the rate first
monthly pay, quarterly compounding
general
also needs converting
Nothing new is being learned here. The whole job is to restate the interest in the
rhythm of the payments, and then last lesson's formula applies unchanged.
2
Ask what one payment period actually does to the money.
12% compounded monthly
1% a month
one year = 12 months
so twelve of those
(1.01)¹² = 1.126825
the growth factor
i = 0.126825, so 12.68%
subtract the 1 back off
Do not just divide or multiply the nominal rate
12% a year compounded monthly is not 12% a year in effect. Interest joins the
balance eleven times before the year ends and earns on itself, and that is the whole
0.68%. Rates compound; they do not add.
3
Convert, then use the ordinary formula.
i = 0.126825, n = 5
the converted rate
F = 10000[(1.126825)⁵ − 10.126825]
last lesson's formula
(1.126825)⁵ ≈ 1.816697
the growth over 5 years
F ≈ ₱64,395
against ₱50,000 paid in
Once the rate has been converted there is no such thing as a general annuity any more.
That is the point of the conversion.
4
A deferred annuity starts paying later.
ordinary
first payment after 1 period
the standard case
deferred 3 periods
first payment after 4
three idle periods first
the effect
discount the whole annuity back
divide by (1 + i) raised to the delay
A pension bought at 25 that first pays out at 65 is a deferred annuity. So is a
loan with a grace period before the first instalment.
5
Variations: a rate converted down, a half-year rate, a deferral that changes nothing.
Converting down: payments monthly, interest 8% compounded quarterly
2% a quarter, and a quarter is 3 months
the compounding clock
(1 + i)³ = 1.02
three months must grow like one quarter
i = (1.02)1/3 − 1 ≈ 0.0066227
a cube root
about 0.6623% a month
not 2% ÷ 3 = 0.6667%
When the payment is shorter than the compounding period, take a root instead of raising to a power.
A half-year rate: payments every 6 months, interest 10% compounded annually
(1 + i)² = 1.10
two half-years must grow like one year
i = √1.10 − 1 ≈ 0.048809
a square root
about 4.88% a half-year
not 5%
Halving the rate to 5% would compound to 10.25% over the year, more than the bank pays.
A deferral that changes nothing: future value of ₱2,000 a year for 3 years at 5%
F = 2000[1.05³ − 10.05]
the ordinary formula
= 2000 × 3.1525 = ₱6,305
at the last payment
Whether the first payment is next year or in ten years, the fund at the moment of the last payment is the same ₱6,305. Deferral only changes a PRESENT value.
6
Value the annuity, then move it back.
P = 6000[1 − 1.05⁻³0.05]
value at the start of the payments
= 6000 × 2.723248 = ₱16,339.49
but that is a value 3 years from now
÷ (1.05)³
discount back three periods
≈ ₱14,114.66 today
the present value
12.68% equivalent annual rate
The ordinary formula always gives a value one period before the first payment.
With the first payment four years out, that puts the figure at year 3, so it needs
three more periods of discounting — not four. Counting that delay wrongly is the one
real trap in deferred annuities.
On the exam
Periodical exams ask whether an annuity is simple or general, for an equivalent rate, or for the present value of a deferred annuity; entrance tests mostly ask the definitions and the "how many periods of discounting" count. Convert the rate to the payment period first, then use last lesson's formulas, and count deferral from one period before the first payment. For a 10-second check, an equivalent annual rate must be a little ABOVE the nominal rate, never below it.
Remember the flow
Do the clocks agree? · If not, convert the rate to the payment period · Use the simple formula · Discount back for any deferral
The ordinary formula lands one period before the first payment. Count from there.
