Florence, 1402
A merchant borrows 100 florins at 2% per month. What annual rate is the merchant paying? The moneylender says: "2% times 12 months -- that's 24% per year. Very reasonable." But the moneylender compounds monthly. After one year, the merchant owes 100 × 1.0212 = 126.82 florins -- an effective rate of 26.82%, not 24%. The moneylender quoted the AIR. The truth is the EAR. The difference -- 2.82 percentage points -- is money in the moneylender's pocket, and the merchant never sees it in the quoted rate. That convention persists today. (You'll also see this called "APR" in consumer lending contexts -- the idea is the same, though the legal definition of APR varies by country and can include fees. Our textbook uses AIR to keep things clean.)
1.00% per month
The Right Thing
EAR (compound up)
(1 + 0.01)12 − 1
12.68%
True annual rate
The Wrong Thing
AIR (just multiply)
1.00% × 12
12.00%
Stated rate (understates truth)
The Understatement
68
basis points hidden
5.4%
understated by
$68
extra per $10k/yr
Frequency m (per yr) Periodic Rate AIR (the wrong thing) EAR (the right thing) Gap (bp)
Where You'll Encounter AIR in the Wild

Corporate Bonds

Bonds pay semiannual coupons. A "6% bond" pays 3% every six months. The stated 6% is an AIR -- the EAR is 6.09%.

r_per = 3%, m = 2

Credit Cards

A card with 1.5% monthly rate is advertised as "18% AIR." The actual annual cost is 19.56% -- 156 bp higher. (Consumer lenders call this the "APR," same idea.)

r_per = 1.5%, m = 12

Money Market / Savings

Banks compound daily. A "5% AIR" savings account actually earns 5.13% EAR -- in your favor this time, but the bank still quotes AIR.

r_per = 0.0137%, m = 365
Rate Converter

When you encounter an AIR in the wild, extract the periodic rate, then compound it properly to get the EAR.

→ ÷ m →
→ compound →
Using m = 12 periods per year
The takeaway: The AIR is not a rate. It's a convention -- a lazy multiplication that ignores compounding. The periodic rate is the primitive. The EAR is the periodic rate compounded honestly up to a year. The AIR is the periodic rate multiplied dishonestly up to a year. They coincide only when m = 1 (annual compounding), because there's nothing to compound within the year. Everywhere else, the AIR understates the truth. Know what you're looking at, extract the periodic rate, and compound it yourself. (In consumer lending, this same concept is called "APR." The term varies by industry and jurisdiction, but the math is identical.)