SME
Engineering Economics · Problem type
Effective annual rate
Use this for “A loan charges 9% a year compounded monthly. What is the effective annual rate, and what would it be compounded continuously?”
Chart
iₑ(12) ≈ 9.381
Level continuous = 9.417 never meets the curve
This is an example. Type your own values to start a new problem.
iₑ
%
=
(1
+
%
÷
)
−
1
i∞
Continuous effective rate
calculated · %
Formulas
iₑ
=
100
×
(
(1 +
r
100 ×
m
)
m
−
1)
9.381
=
100
×
((1
+
9
100 × 12
)¹²
−
1)
i∞
=
100
×
(
e
r
÷ 100
−
1)
9.417
=
100
×
(
e
9 ÷ 100
−
1)
Assumptions
•
r is the quoted annual rate (APR); each of the m periods a year earns r ÷ m.
•
The effective rate iₑ is what one year of compounding really adds.
•
Continuous compounding is the limit as m grows without end.
Step-by-step
WE KNOW
r = 9%, m = 12
FIND
effective rate (iₑ), continuous effective rate (i∞)
Step 1 · Find effective rate (iₑ)
Use iₑ = 100 × ((1 + r ÷ (100 × m))ᵐ − 1)
Grow
$1
by
r
m
each
period
for
the
m
periods
of
a
year;
what
it
gained
is
the
effective
rate.
iₑ
=
100
×
((1
+
9
100 × 12
)¹²
−
1)
iₑ
=
100
×
(
(1 +
9
1,200
)
12
−
1)
iₑ
=
100
×
(
(1 + 0.0075)
12
−
1)
iₑ
=
100
×
(
1.0075
12
−
1)
iₑ = 100 × (1.0938 − 1)
iₑ = 100 × 0.0938
iₑ = 9.381%
Step 2 · Find continuous effective rate (i∞)
Use i∞ = 100 × (er ÷ 100 − 1)
As
m
grows,
(1 +
r
m
)
m
closes
in
on
e
r
.
i∞
=
100
×
(
e
9 ÷ 100
−
1)
i∞ = 100 × (1.09417 − 1)
i∞ = 100 × 0.09417
i∞ = 9.417%
Check
Put the numbers back in. Both sides should match.
9.381
=
100
×
((1
+
9
100 × 12
)¹²
−
1)
✓
9.417
=
100
×
(
e
9 ÷ 100
−
1)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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