How to calculate EAR and EPR
I have a supplementary note on APR, EAR, and EPR for Classes 3 and 4, which explains
the concept and derivation of EPR, and how to use it. It seems these rates are still
confusing to many of you. This additional note hopefully helps clear that confusion.
First, I address some frequently asked questions. Then, I discuss an example with detailed
calculations with formulas as well as TVM calculator work.
1. FAQs
What is APR?
APR (Annual Percentage Rate) is the quoted (or stated) annual interest rate. It is “what
you see.”
What is EAR?
EAR (Effective Annual Rate) is the yearly rate that a borrower (saver) effectively pays
(earns) given an APR with certain compounding intervals. It is “what you get.”
Can EAR (and APR) be calculated with TVM functions in a financial calculator?
Yes, you can use the interest rate conversion menu.
For Casio, press Menu / TVM / F5: Conversion. Enter inputs for n (C/Y), and I%. If APR
is given, press F1 to get EFF, i.e. the effective rate or EAR. If EAR is given, press F2 to
get APR.
For TI BA II+, press 2ND ICONV. Press the arrows up or down to enter inputs and
compute the output. NOM is the nominal rate, i.e. APR, and EFF is the effective rate, i.e.
EAR.
What is EPR?
EPR (Effective Periodic Rate) is the applicable rate used in the formulas for PV and FV
(of a single lump sum, or of annuities), given an APR with certain compounding intervals
(C/Y) and certain payment intervals (P/Y).
EPR can be calculated with either of the following equations.
1
EPR = ( EAR + 1) PY − 1
(1)
CY
APR ⎞ PY
⎛
EPR = ⎜ 1 +
⎟ −1
CY ⎠
⎝
or
(2)
where
CY = number of compounding periods in a year (C/Y)
PY = number of payments in a year (P/Y)
•
If an APR is quoted with annual compounding and annual payments (C/Y = P/Y = 1),
then EPR is the APR itself. Plug CY = PY = 1 into Equation (2) above, we have
1
APR ⎞1
⎛
EPR = ⎜ 1 +
⎟ − 1 = (1 + APR ) − 1 = APR
1 ⎠
⎝
Managerial Finance I (CFIN300)
Instructor: Henry Pham
1
How to calculate EAR and EPR
•
If an APR is quoted with the same compounding and payment intervals, but not
annually (C/Y = P/Y ≠ 1), then EPR is simply equal to APR divided by C/Y (or by
P/Y). Plug PY = CY into Equation (2) above, we have
CY
APR ⎞ CY
APR ⎞
APR APR
⎛
⎛
EPR = ⎜ 1 +
=
⎟ − 1 = ⎜1 +
⎟ −1 =
CY ⎠
CY ⎠
CY
PY
⎝
⎝
For example, given an APR of 9% with quarterly compounding and payments, EPR =
0.09 / 4 = 0.0225 = 2.5% (per quarter).
•
If an APR is quoted with different compounding and payment intervals (C/Y ≠ P/Y),
then EPR is calculated with the full formula.
Why is EPR needed if APR is already given?
EPR is needed to calculate periodic amounts, e.g. regular loan payments, interest or
principal paid per period, or compute PV and FV of annuities. EPR specifically address
those periodic intervals.
Can EPR be calculated with TVM functions in a financial calculator?
Yes, you can, but you don’t need to. Once the given inputs of APR, C/Y and P/Y are
correctly entered into the calculator, you can compute your output right away. The
calculator will take care of EPR. But if you insist, I’ll show how to compute EPR with
TVM functions (see below).
What is the relationship between EAR and EPR?
When the periodic amount is annual (P/Y = 1), EPR is the same as EAR. Plug PY = 1
into Equation (2) above, we have
APR ⎞
⎛
EPR = ⎜ 1 +
⎟
CY ⎠
⎝
Managerial Finance I (CFIN300)
Instructor: Henry Pham
CY
1
APR ⎞
⎛
− 1 = ⎜1 +
⎟
CY ⎠
⎝
CY
− 1 = EAR
2
How to calculate EAR and EPR
2. Example 6.13 in the textbook (page 158, 7 ed.; page 160, 6 ed.; page 154, 5ed)
2.1. The calculator solution for this question is fairly short with the following inputs:
N = 300; I/Y (or I%) = 6 (P/Y = 12, and C/Y = 2); PV = 100,000; FV = 0
Compute PMT = -639.81 (The answer of 639.83 in the textbook is due to
rounding error.)
There’s no need to calculate EPR.
2.2. However, the formula approach would involve EPR. Here, I only show how to
calculate EPR. After getting EPR, you can apply it to the formula for present value of
annuities to compute the monthly payment in the same way as shown in the textbook.
Alternative 1
Step 1: Calculate EAR
APR ⎞
⎛
EAR = ⎜ 1 +
⎟
CY ⎠
⎝
CY
2
⎛ 0.06 ⎞
− 1 = ⎜1 +
⎟ − 1 = 0.0609 = 6.09%
2 ⎠
⎝
Calculator:
Casio: Menu / TVM / F5. n = 2; I% = 6. Press F1 to get EFF = 6.09
TI BA II+: 2ND ICONV . Press the arrows up and down to enter inputs and compute
output. C/Y = 2; NOM = 6. CPT EFF = 6.09
Step 2: Calculate EPR
1
1
EPR = ( EAR + 1) PY − 1 = ( 0.0609 + 1)12 − 1 = 0.004939 = 0.4939%
Calculator:
N = 12; PV = -1; PMT = 0; FV = 1.0609; P/Y = 1; C/Y = 1.
Compute I/Y (or I%) = 0.4939
Rationale for calculator work: Given an EAR of 6.09% (above), $1 today (PV = 1) will grow to $1×(1 + 0.0609) = $1.0609 in one year (FV = 1.0609). As we set
N = 12 (months), and P/Y = C/Y = 1, the solution for I/Y (or I%) will be a
monthly quoted rate, which is EPR.
Alternative 2 (I prefer this one as it is shorter.)
CY
2
APR ⎞ PY
⎛
⎛ 0.06 ⎞12
EPR = ⎜ 1 +
⎟ − 1 = ⎜1 +
⎟ − 1 = 0.004939 = 0.4939%
CY ⎠
2 ⎠
⎝
⎝
Calculator:
N = 6; PV = -1; PMT = 0; FV = 1.03; P/Y = 1; C/Y = 1.
Compute I/Y (or I%) = 0.4939
Rationale for calculator work: Given an APR of 6% compounded semiannually,
half of the interest (3%) is accrued for six months. So, $1 today (PV = -1) will
grow to $1×(1 + 0.03) = $1.03 in six months (FV = 1.03). As we set N = 6
(months), and P/Y = C/Y = 1, the solution for I/Y (or I%) will be a monthly
quoted rate, which is EPR.
Managerial Finance I (CFIN300)
Instructor: Henry Pham
3