Title: Effective Annual Interest Rate (EAR): What It Is, How to Calculate It, and How to Use It
Source: Investopedia — “Effective Interest Rate” (Theresa Chiechi) — https://www.investopedia.com/terms/e/effectiveinterest.asp
Overview
The effective annual interest rate (EAR), also called the effective rate or annual equivalent rate (AER), is the actual annual return on an investment or the real annual cost of a loan after accounting for the effect of compounding. Because interest can be compounded more than once per year (monthly, daily, continuously, etc.), the EAR is usually higher than the nominal (stated) rate unless compounding is annual.
Why EAR matters
– It reveals the true annual return on savings or the true annual cost of borrowing.
– It allows apples-to-apples comparisons of offers that have different compounding frequencies or different nominal rates.
– Borrowers who ignore EAR may underestimate the cost of credit; investors who ignore it may over- or under-estimate returns.
Key formula(s)
– Discrete compounding:
EAR = (1 + i/n)^n − 1
where i = nominal annual interest rate (in decimal), n = number of compounding periods per year.
– Continuous compounding:
EAR_continuous = e^i − 1
where e ≈ 2.71828 and i is the nominal annual rate.
Simple numeric examples
1) Monthly compounding, 10% nominal:
EAR = (1 + 0.10/12)^12 − 1 ≈ 0.104713 = 10.4713%
2) Semiannual compounding, 10.1% nominal:
EAR = (1 + 0.101/2)^2 − 1 ≈ 0.103550 = 10.3550%
If you invest $5,000,000:
– At 10% nominal, compounded monthly (EAR ≈ 10.4713%), the interest earned ≈ $5,000,000 × 0.104713 = $523,565.
– At 10.1% nominal, compounded semiannually (EAR ≈ 10.3550%), the interest earned ≈ $5,000,000 × 0.103550 = $517,750.
Choosing the higher nominal rate (10.1%) would be worse in this case because of less frequent compounding.
Effect of compounding frequency
– All else equal, as n (compounding frequency) increases, EAR increases:
yearly < semiannual < quarterly < monthly < daily < continuous (limit).
– However, there is an upper bound: the continuous compounding case (EAR = e^i − 1).
Conversions and related formulas
– Convert EAR back to a nominal rate with n compounding periods per year:
nominal i = n × ((1 + EAR)^(1/n) − 1)
– To compute EAR from APR (nominal annual percentage rate) when you know compounding frequency, use the EAR formula with APR as i. Note: APR and EAR measure different things—APR is often a quoted rate used for disclosure that may include certain fees but typically does not show compounding’s full effect.
Practical steps — how to calculate and compare offers
1. Gather data:
– Nominal (quoted) annual rate (i) or APR.
– Compounding frequency (n): yearly (1), semiannual (2), quarterly (4), monthly (12), daily (usually 360 or 365), etc.
– Any one-time or recurring fees, taxes, or charges.
2. Compute EAR:
– Use EAR formula: (1 + i/n)^n − 1 (or e^i − 1 for continuous).
– Tools:
– Calculator: compute directly.
– Excel: =EFFECT(nominal_rate, npery) or compute (1+nominal/n) ^ n – 1.
– Python: import math; EAR = (1 + i/n)**n – 1 or math.exp(i) – 1 for continuous.
3. Compare EARs:
– Choose the investment with the highest EAR.
– For loans, choose the loan with the lowest EAR, after adjusting for fees and payment structure.
4. Factor in fees, taxes, and payment timing:
– Add transaction/service/account fees, or compute the effective yield after fees.
– Consider taxes and after-tax rate of return; EAR ignores taxes.
– For loans, build an amortization schedule to compute total interest paid over the loan term rather than relying only on EAR.
5. Check special disclosures:
– Consumer loans often show APR (which may incorporate some fees); use APR and the lender’s disclosed compounding method to compute EAR. For true cost comparison, include fees and frequency of compounding.
Excel / calculator quick reference
– Discrete EAR: =(1 + nominal/n) ^ n – 1
– Excel built-in: =EFFECT(nominal_rate, npery)
– Convert EAR to nominal with m compounding periods per year:
nominal = m * ((1 + EAR)^(1/m) − 1)
Python snippet
– Discrete compounding:
import math
def ear_from_nominal(i, n):
return (1 + i / n) ** n – 1
– Continuous:
def ear_continuous(i):
return math.exp(i) – 1
Common real-world uses
– Comparing savings accounts, CDs, or money market yields with different compounding.
– Comparing loan offers (mortgages, auto loans, personal loans) that have different compounding periods.
– Understanding credit card costs: many cards quote a periodic APR; convert to EAR to see true cost given daily or monthly compounding.
– Comparing investment returns when rates are quoted on different bases.
Limits and caveats
– EAR assumes a constant interest rate for the period being measured; real interest rates can change.
– EAR ignores fees, account maintenance costs, and taxes, which can materially change after-fee and after-tax returns.
– For consumer disclosures, APR and EAR serve different regulatory and informational purposes; both may be relevant.
– For bonds, use yield to maturity (YTM) to capture coupons, price paid, and time to maturity — YTM is the internal rate of return, not simply an EAR from a quoted coupon rate. Discount yield is a different convention used for certain short-term government securities (e.g., Treasury bills) and isn’t directly comparable to EAR without appropriate conversion.
Is a higher EAR better?
– For investors: yes — higher EAR means higher effective return, all else equal.
– For borrowers: no — a higher EAR means a higher effective borrowing cost.
– Always compare after accounting for fees, payment structure, and risk.
How to use this in real life — a checklist
– When offered a nominal rate, ask and confirm compounding frequency.
– Compute the EAR to compare offers on the same footing.
– Include fees and taxes by computing net (after-fee, after-tax) EAR if needed.
– For long-term loans, use amortization calculators to evaluate total interest and monthly payment impact.
– Ask lenders for both APR and EAR, and verify which is being advertised.
Bottom line
EAR reveals the true annual rate that results from compounding. It’s the right metric when you need to compare financial products that compound at different frequencies. However, it’s only one piece of the decision: fees, taxes, payment timing, and changing interest rates also matter. For loans, combine EAR comparisons with amortization schedules and fee disclosures; for investments, compute after-fee and after-tax effective returns.
For further reading (source)
Investopedia, “Effective Interest Rate” (Theresa Chiechi): https://www.investopedia.com/terms/e/effectiveinterest.asp
If you’d like, I can:
– Calculate EARs for specific offers you have (give nominal rates and compounding frequency).
– Provide an Excel template or amortization schedule to compare total loan costs.
– Convert APRs from your credit-card or loan documents into EARs.