What Effective Annual Yield Actually Measures
Effective annual yield answers one question: if this instrument's return kept repeating, what would a full year of compounded growth amount to? The formula is EAY = (1 + HPY)^(365/d) − 1, where HPY is the holding period yield and d is the number of days held. A 91-day Treasury bill bought at $9,800 and redeemed at $10,000 earns 2.04% over its life, which annualizes to a true 8.44% once compounding is counted.
The reinvestment assumption is what separates EAY from simple annualization. Simple math multiplies the period yield by the periods per year — 2.04% × (365/91) = 8.19% — and stops there. EAY goes one step further: each matured bill's proceeds roll into the next bill at the same discount, so gains stack on gains. Four consecutive 91-day rollovers turn $9,800 into $10,624.82.
That $824.82 total gain on $9,800 works out to exactly 8.44% for the year — the compounded figure, not the 8.19% simple one. The gap between the two numbers is the compounding premium. It grows as periods get shorter and rates get higher, which is why money market professionals compare instruments on an effective basis before committing funds.
Three Yield Quoting Conventions That Confuse Buyers
Money market instruments get quoted in three different languages. Bank discount yield divides the discount by face value on a 360-day year. Bond equivalent yield divides by the purchase price on a 365-day year. Effective annual yield adds compounding on top. All three can describe the same bill, and they produce three different numbers — ranked lowest to highest in that order.
A concrete example shows the spread. A 91-day, $10,000-face bill quoted at an 8.00% bank discount rate costs $9,797.78, since the discount is $10,000 × 0.08 × 91/360. On a price basis that same bill yields 8.28% bond equivalent and 8.54% effective annual. One instrument, three headline numbers ranging across more than half a point.
Dealers quote the discount rate because it looks cheapest, CDs quote APY because it looks richest, and bonds quote semi-annual yields because of market convention. None of these numbers are wrong — they are just not comparable until they share a basis. The bond equivalent yield calculator handles the discount-to-price-basis conversion, and this tool finishes the job by adding the compounding layer.
From T-Bill Price to True Annual Yield, Step by Step
The calculation runs in three moves. First, holding period yield: HPY = (maturity value − purchase price) / purchase price. Second, simple annualization: multiply HPY by 365/days to get the bond equivalent yield. Third, compounding: raise (1 + HPY) to the power 365/days and subtract one. The tool runs all three and reports the pair side by side with the premium between them.
Maturity length changes the premium noticeably. A 28-day bill bought at $9,950 against a $10,000 face yields 6.55% bond equivalent but 6.75% effective — a 0.20 point gap. Stretch to a 182-day bill bought at $9,700 and the gap narrows to 0.10 points (6.20% versus 6.30%). Shorter instruments compound more times per year, so their effective yield pulls further ahead of the simple figure.
This is why comparing a 28-day bill against a 182-day bill on quoted yields alone misleads. The 28-day paper looks 0.35 points better on the simple comparison (6.55% against 6.20%) and 0.45 points better once compounding counts (6.75% against 6.30%). Rankings can even flip at other rate levels, since the premium scale moves with both maturity and rate. Run both maturities through the tool before deciding which ladder rung to buy.
Semi-Annual Bonds and the (1 + BEY/2) Squared Convention
US corporate and Treasury bonds pay coupons twice a year, so their quoted yield to maturity is a semi-annual rate doubled — the bond equivalent convention. A bond quoted at 4.6% does not pay 4.6% per year in compounded terms; it pays 2.3% every six months, and those two payments compound into a slightly higher annual result.
The conversion is EAY = (1 + BEY/2)^2 − 1. At a 4.6% quote, that is (1.023)^2 − 1 = 4.65%. Per $10,000 invested, the difference between naive and compounded math is $465.29 versus $460 a year. Small at these rate levels, the wedge widens fast: an 8.0% semi-annual quote earns 8.16% effective, and a 6.0% quote earns 6.09%.
Mode 2 of the calculator performs this conversion directly. It matters most when a semi-annual bond competes against an annual-pay instrument or a savings account quoted in APY — comparing raw quotes quietly favors the bond by a few hundredths of a point. For the full discount-cash-flow solve behind a bond's quote, the bond YTM calculator does that heavier work, and the bond price calculator runs the reverse direction from yield to fair value.
EAY vs EAR vs APY: Untangling the Alphabet Soup
Effective annual rate starts from a stated nominal rate and a compounding schedule: EAR = (1 + r/n)^n − 1. Effective annual yield applies identical mathematics but starts from an instrument's actual cash flows — what you paid and what you received — then annualizes the result. Same formula family, different inputs, and the inputs decide which tool fits the job.
Annual percentage yield is the bank-disclosure name for EAR on deposit products, required under US Truth in Savings rules so competing accounts can be compared on one number. A savings account advertising a 4.9% rate compounded monthly carries a 5.01% APY, for instance. The disclosure number already includes the compounding effect, which is exactly the adjustment this tool makes for market instruments.
The practical split: use the EAR calculator when converting a stated rate between compounding frequencies, the APY calculator when sizing up deposit accounts, and this tool when the starting point is a price, a payout, or a semi-annual market quote. Feeding a bank's nominal rate into a yield annualizer — or a T-bill price into a rate converter — produces numbers that answer the wrong question.
Money Market Quirks: 360-Day Years and Discount Bases
The bank discount convention uses a 360-day year and divides by face value, and both choices shrink the quoted number. The same physical bill moves from an 8.00% discount quote to 8.28% bond equivalent just by switching to a 365-day year and a purchase-price basis, before any compounding enters. Nothing about the investment changed — only the arithmetic convention.
Certificates of deposit carry their own wrinkle: some quote rates on a 360-day basis while compounding monthly or daily, which is why two CDs with identical 5.0% rate headlines can post slightly different APYs. Reading the fine print about the day-count basis matters whenever yields sit within a few hundredths of each other, as they routinely do in money funds.
Compounding frequency is the last dial. Daily compounding at a 5.0% nominal rate lifts the effective figure to 5.13%, and the compound interest calculator projects those balances across the years. The theoretical ceiling is continuous compounding, e^r − 1, which sits only a hair above daily compounding at any ordinary rate — the continuous compound calculator works through that limit in detail.
Comparing Instruments on One Number
The point of reducing everything to EAY is honest comparison. A 91-day bill at 8.44% effective, a savings account at 5.01% APY, and a bond at 4.65% effective from a 4.6% quote all sit on the same yardstick, so the ranking needs no mental adjustment. Without the conversion, each quote's different basis quietly tilts the comparison.
Municipal bonds add a tax layer. A muni yielding 3.4% tax-free competes against taxable paper only after adjustment — an investor in the 32% federal bracket needs roughly 5.0% taxable to match it. The taxable equivalent yield calculator runs that comparison, and pairing it with an effective-yield basis keeps both the tax adjustment and the compounding adjustment honest.
EAY is strictly a one-year lens. For holdings spanning multiple years — dividend stocks, rental property, a fund held since 2019 — the annualized figure that respects the full timeline is the compound annual growth rate, and the CAGR calculator is built for that job. EAY annualizes periods up to a year cleanly; beyond that, compounding across unequal periods makes the geometric mean the right tool.
Real Yield: What EAY Leaves Out
An 8.44% nominal EAY alongside 3.0% inflation delivers a real return of about 5.3% — (1.0844 / 1.03) − 1. The compounding premium that EAY captures on the nominal side applies to prices too, which is why inflation-adjusted comparisons deserve the same care. The buying power calculator translates that erosion into concrete dollar terms.
Taxes claim their share next. US Treasury interest skips state income tax, a genuine edge for investors in high-tax states like California or New York, while corporate CD interest gets taxed at every level. Two instruments with identical EAYs can land nearly half a point apart after tax depending on the investor's brackets — compare after-tax yields, not headline ones.
The last caveat is reinvestment risk. EAY's annualization assumes every maturing bill rolls at the same rate, but a 91-day ladder reprices four times a year and rates move. Falling rates shrink realized returns below the projected EAY; rising rates improve them. Treat the calculated figure as the break-even yardstick for comparison, not a promise of realized wealth.