A Treasury bill quoted at a 4.50% bank discount rate actually pays you closer to 4.61% on the money you invest. That’s the whole story of the bank discount rate vs the coupon equivalent yield: two ways of expressing the same return, one built for quoting prices and one built for measuring what you earned. The discount rate is always the lower number, and it’s the one that misleads you. The coupon equivalent yield is the one to trust when you’re deciding whether a bill is worth buying.
What the Bank Discount Rate Actually Measures
The bank discount rate is the standard quoting convention for short-term securities sold below face value, including Treasury bills, commercial paper, and banker’s acceptances.1sites.uni.edu. Bank Discount, Coupon Equivalent, and Compound Yields These instruments pay no periodic interest. You buy them below face value and collect the full face amount at maturity. The gap is your return.
The formula annualizes that gap using face value and a 360-day year:
BD = ((Face Value − Price) / Face Value) × (360 / Days to Maturity)
Both choices in that formula work against you as an investor. Dividing by face value overstates your capital at risk, because you paid less than face value. And a 360-day year, a convention from an era when bankers preferred round numbers, crams a full year of return into a shorter frame, further understating the annualized rate.1sites.uni.edu. Bank Discount, Coupon Equivalent, and Compound Yields
The bank discount rate is really a pricing tool. Given a discount rate, face value, and days to maturity, a dealer can translate directly into a dollar price using the TreasuryDirect formula:
Price = Face Value × (1 − (Discount Rate × Days to Maturity) / 360)2TreasuryDirect. Understanding Pricing and Interest Rates
So a $10,000 26-week bill quoted at a 4.50% discount rate costs $9,772.50. You earn $227.50, but calling that “4.50%” against a $10,000 face value hides the fact that the $227.50 came from a smaller investment over fewer than 365 days.
What the Coupon Equivalent Yield Measures
The coupon equivalent yield, also called the bond equivalent yield or investment yield, fixes both distortions. It divides return by the price you actually paid and annualizes over 365 days.1sites.uni.edu. Bank Discount, Coupon Equivalent, and Compound Yields The Treasury labels this figure the “investment rate” when it publishes auction results.3TreasuryDirect. Price, Yield and Rate Calculations for a Treasury Bill
For bills with 182 days or less to maturity:
CEY = ((Face Value − Price) / Price) × (365 / Days to Maturity)3TreasuryDirect. Price, Yield and Rate Calculations for a Treasury Bill
Purchase price replaces face value in the denominator. 365 replaces 360 in the annualization. Both changes push the number up, and the higher number is the one that describes your actual return on capital. It’s also the number you should compare against yields on Treasury notes, corporate bonds, CDs, or anything else quoted on an investment-yield basis.1sites.uni.edu. Bank Discount, Coupon Equivalent, and Compound Yields
Why the Coupon Equivalent Yield Is Always Higher
The gap between the two rates comes from two independent sources that stack on top of each other.
- The denominators differ. The bank discount rate divides by face value; the coupon equivalent yield divides by purchase price, which is always lower for a discount security. The same dollar return, divided by a smaller number, becomes a larger percentage. This effect grows as the discount deepens, which happens when rates rise or maturities lengthen.
- The day counts differ. Scaling from 360 to 365 multiplies the annualized rate by 365/360, or about 1.0139. That factor alone accounts for roughly 1.4% of the difference, no matter what the price is.
Neither rate accounts for compounding. Both treat your return as simple interest, sitting idle rather than reinvested. On short maturities that assumption barely matters. On a 52-week bill it can.
Converting a Discount Rate to a Coupon Equivalent Yield
You don’t need the dollar price to move between the two. For bills of 182 days or fewer, the algebra collapses to one conversion:
CEY = (365 × BD) / (360 − (BD × Days to Maturity))
The numerator rescales days. The denominator shifts the base from face value to purchase price.
Worked Example: 91-Day T-Bill at 4.50%
A 91-day T-bill quoted at a 4.50% bank discount rate. What are you really earning?
The price on a $10,000 bill: $10,000 × (1 − (0.045 × 91) / 360) = $9,886.25.2TreasuryDirect. Understanding Pricing and Interest Rates Your discount income is $113.75.
The coupon equivalent yield, using the shortcut:
CEY = (365 × 0.045) / (360 − (0.045 × 91)) = 16.425 / 355.905 = 0.04614, or about 4.614%.
Verify it the long way: ($113.75 / $9,886.25) × (365 / 91) = 4.614%. Same answer, because the shortcut is just algebra on the same relationship.
The gap here is roughly 11 basis points. On an individual buy that’s a rounding difference. Rolled across a $1 million short-term portfolio, it’s about $1,140 a year you’d overlook if you took the discount rate at face value.
The Wrinkle for Bills Longer Than 182 Days
The simple CEY formula only works for bills maturing in half a year or less. For 52-week bills the Treasury uses a more complex calculation that folds in semi-annual compounding, matching the way coupon-bearing notes and bonds pay every six months.1sites.uni.edu. Bank Discount, Coupon Equivalent, and Compound Yields
The point is comparability. A 10-year Treasury note’s quoted yield assumes you reinvest each semi-annual coupon at the same rate. To make a 52-week bill’s return directly comparable, the calculation has to assume the same reinvestment. The Treasury’s official formula for bills over 182 days solves a quadratic that bakes in this compounding.3TreasuryDirect. Price, Yield and Rate Calculations for a Treasury Bill
You almost never need to solve it yourself. Auction results report the investment rate. Most brokerage platforms show it. If you’re computing your own numbers on a 26- or 52-week bill and you use the short formula anyway, know that you’ll slightly understate the true yield because you’ve ignored compounding. The longer the maturity and the higher the rate, the larger the miss.
Where You’ll See Each Rate
Treasury auction announcements are the clearest place to see both side by side. Every bill auction reports a “high rate” (the bank discount rate used to set the price) and an “investment rate” (the coupon equivalent yield). The investment rate is always the higher of the two.2TreasuryDirect. Understanding Pricing and Interest Rates
Brokerage displays are less consistent. Some platforms show only the discount rate, because that’s the market convention. Others lead with the investment yield, because it’s more useful for comparison. When you’re weighing a T-bill against a money market fund, a CD, or a short bond fund, work from the coupon equivalent yield. Putting a discount rate next to a bond yield is a category error; the two aren’t measuring the same thing.
A Tax Note the Yields Don’t Capture
Neither rate reflects taxes, and Treasury bill interest gets a break other short-term instruments don’t. The discount is taxed as interest, not as a capital gain, even though you buy low and receive full face at maturity. That interest is subject to federal tax but exempt from state and local income tax.4Internal Revenue Service. Topic no. 403, Interest Received In a high-tax state, a T-bill’s coupon equivalent yield of 4.6% can beat a fully taxable CD at 4.8% on an after-tax basis. The comparison isn’t complete until you factor that in.
The Practical Rule
Use the bank discount rate only to understand price. Use the coupon equivalent yield to understand return. For bills of six months or less, the conversion is a single line of arithmetic. For longer bills, take the investment rate straight from the auction results rather than applying the short-term formula and hoping compounding won’t bite. And when you compare a T-bill to anything taxable at the state level, remember that neither published rate captures the exemption that often tips the decision.