howard-lynn-ye Yield measures and bill quotes
"5.00%" on a Treasury bill screen is a discount rate. It is not a yield, and it is not comparable to anything else on the page. Four numbers describe the same bill and they are all different.
Every figure below is printed by scripts/yield_measures.py (runs in 0.94 s; QuantLib
optional). ✅ Measured means this file produced it on 2026-09-09 with QuantLib 1.43, scipy 1.13.0,
Python 3.11.3.
The rule: a discount rate divides the gain by FACE on a 360-day year; a yield divides it by PRICE on a 365-day year. Convert with 31 CFR 356 Appendix B — the simple formula up to one half-year, the QUADRATIC beyond it. On a callable, quote yield to worst, not yield to maturity.
1. 🚨 The same 5.00% bill, four ways
✅ Measured:
| days | price | discount rate | money-market (add-on) | bond-equivalent | Treasury investment rate | d → BEY | BEY → IR |
|---|---|---|---|---|---|---|---|
| 28 | 99.611111 | 5.0000% | 5.0195% | 5.0892% | 5.0892% | +8.9 bp | +0.0 |
| 91 | 98.736111 | 5.0000% | 5.0640% | 5.1343% | 5.1343% | +13.4 bp | +0.0 |
| 182 | 97.472222 | 5.0000% | 5.1297% | 5.2009% | 5.2009% | +20.1 bp | +0.0 |
| 364 | 94.944444 | 5.0000% | 5.2662% | 5.3394% | 🚨 5.2701% | +33.9 bp | 🚨 −6.9 bp |
Three different denominators are in play:
- discount rate =
(100 − P)/100 × 360/days— the gain over face, 360-day year. - money-market / add-on / CD-equivalent =
(100 − P)/P × 360/days— over price, still 360. - bond-equivalent yield =
(100 − P)/P × 365/days— over price, 365-day year.
🚨 The discount rate is the only one of the three that is not a return on anything you can invest. You pay P and receive 100; the return is measured on P.
The gap grows with the level, not just the tenor — ✅ measured at a fixed 182 days
| discount rate | 1.00% | 3.00% | 5.00% | 8.00% | 12.00% |
|---|---|---|---|---|---|
| bond-equivalent yield | 1.0190% | 3.0885% | 5.2009% | 8.4530% | 12.9524% |
| gap | +1.9 bp | +8.9 bp | +20.1 bp | +45.3 bp | +95.2 bp |
At 12% the two quotes are 95 bp apart on the same instrument. A conversion tested on a 2020 bill and reused in 2023 is wrong by four times as much.
2. ✅ Treasury's own conversion is TWO formulas, and the regulation prints both examples
✅ Source-verified at eCFR 31 CFR 356 Appendix B, section VI (fetched 2026-09-09), Computation of Purchase Price, Discount Rate, and Investment Rate (Coupon-Equivalent Yield) for Treasury Bills:
B.1 — not more than one half-year to maturity
i = (100 - P)/P * y/r
B.2 — more than one half-year to maturity
P [1 + (r - y/2)(i/y)] (1 + i/2) = 100
solved as a quadratic: b = r/y, a = (r/2y) - 0.25, c = (P - 100)/P
i = (-b + sqrt(b^2 - 4ac)) / (2a)
with r = days to maturity and y = 365, or 366 if the year following the ISSUE date
contains 29 February — not the year the bill matures in.
🔑 The quadratic exists because a bill longer than six months has to be made comparable to a
security that pays a coupon halfway through and reinvests it. That is the (1 + i/2) factor.
✅ The regulation's own two worked examples, reproduced by this file:
| P | r | regulation | treasury_investment_rate |
|
|---|---|---|---|---|
| cash-management bill 1990-06-01 → 1990-06-21 | 99.559444 | 20 | 8.076% | 8.075725% → 8.076% |
| 52-week bill 1990-06-07 → 1991-06-06 | 92.265000 | 364 | 8.237% | 8.237324% → 8.237% |
🚨 Applying the short formula to that 52-week bill gives 8.406492% — 16.9 bp too high. The
formulas do not blend at the boundary; there is a branch at r > y/2 and you have to take it.
⚠️ FRED's DTB4WK/DTB3/DTB6/DTB1YR are discount rates; DTB3 is not DGS3MO. The
*_CBOND-style constant-maturity series are yields. Mixing them inside one time series is the
same 20 bp error moving in and out of the data.
3. Current yield ignores the pull to par
✅ Measured — annual coupon over clean price, against the true yield to maturity:
| coupon | clean | years | current yield | YTM | gap |
|---|---|---|---|---|---|
| 3.00% | 85.00 | 5 | 3.5294% | 6.5681% | 🚨 +303.9 bp |
| 3.00% | 85.00 | 20 | 3.5294% | 4.1070% | +57.8 bp |
| 8.00% | 118.00 | 5 | 6.7797% | 3.9930% | 🚨 −278.7 bp |
| 8.00% | 118.00 | 20 | 6.7797% | 6.3927% | −38.7 bp |
| 5.00% | 100.00 | 10 | 5.0000% | 5.0000% | +0.0 |
Current yield is exactly right at par and nowhere else, and it is worst on SHORT bonds away from par, because that is where the pull to par is spread over the fewest years. It is a carry number, not a return.
4. 🚨 Yield to maturity on a callable above par is a fiction
✅ Measured: a 5% bond, 10 years to maturity, clean 108.00, callable at 102 in 2 years, 101 in 3 years, 100 in 5 years:
| redemption | at | yield |
|---|---|---|
| 102.00 | 2.0y | 🚨 1.8909% ← the worst |
| 101.00 | 3.0y | 2.5366% |
| 100.00 | 5.0y | 3.2534% |
| 100.00 | 10.0y (maturity) | 4.0205% |
🚨 Quoting the yield to maturity overstates by 213.0 bp on a bond the issuer will call. Yield to worst is the minimum over every redemption date at that date's own call price — a call at 102 is not a call at 100, and repricing to par understates the yield.
Yield to worst is still not the answer, only the honest floor: it assumes a deterministic exercise. Once you want the option's value, you need an OAS on a lattice, which lives with option-adjusted spread work, not here.
5. ✅ Checked against QuantLib
| check | mine | QuantLib 1.43 | |diff| |
|---|---|---|---|
| 91d bill BEY as a simple ACT/365 rate → compound factor | 1.0128006752 | 1.0128006752 | 0.0e+00 |
| 364d bill, same | 1.0532475132 | 1.0532475132 | 0.0e+00 |
| 10y 5% semiannual clean price at y = 4.50% | 103.9909280925 | 103.9909280925 | 1.6e-13 |
| and back to a yield | 0.0450000000 | 0.0450000000 | 2.1e-17 |
🚨 The price check only agrees with paymentConvention = ql.Unadjusted. ✅ QuantLib's default
Following bumps coupons off weekends and prices the same bond at 103.9870971915, −0.003831
lower — which reads as a broken yield formula and is not one. Set the convention deliberately.
6. What the script gives you
scripts/yield_measures.py — scipy (brentq) only at import; QuantLib inside one function.
| Function | Does |
|---|---|
bill_price_from_discount(d, days) / bill_discount_from_price |
§1, the 360-on-face quote |
money_market_yield(P, days) / bond_equivalent_yield(P, days, year) |
§1, the two add-on rates |
treasury_investment_rate(P, r, y=365) |
§2, both branches of 31 CFR 356 App. B VI |
bill_table(discount, day_counts) |
the §1 table with the bp gaps |
bond_price_from_yield(coupon, y, n, w, freq) |
street-convention dirty/clean/accrued, w = fraction of the period left |
ytm(clean, coupon, n, w, freq) |
§3 and §5, by brentq |
current_yield(coupon, clean) |
§3 |
yield_to_worst(clean, coupon, calls, n_maturity) |
§4; each call priced at its own redemption |
quantlib_cross_checks(...) |
§5, or None |
Where this sits
../bond-conventions-and-accrued/SKILL.md— the accrued and thewinbond_price_from_yieldcome from there; ACT/ACT ICMA without its schedule is 🚨 $2,002.76 per $1mm.../ex-dividend-and-rebate-interest/SKILL.md— the UK DMO price/yield formula, and what a yield means when the next cash flow is zero.../duration-convexity-and-dv01/SKILL.md— what to do with the yield once you have it, and which duration a library actually returned.../sofr-and-rfr-compounding/SKILL.md— the other family of quotes that are not yields: compounded-in-arrears averages and the SOFR Index.../../../fin-models/skills/term-structure-models/SKILL.md— a YIELD is one number for one bond; a zero curve is a different object, and the same discount factor is six different zero rates there.../../../fin-core/skills/fundamental-and-macro-data/SKILL.md— whereDTB3,DGS3MOand the Treasury par curve come from, and their revision behaviour.../../../fin-libraries/skills/lib-quantlib/SKILL.md—BondFunctions.bondYieldneeds aql.BondPrice, andevaluationDateis a global.