Price from the discount rate, then the bond-equivalent yield, the money market yield and the effective annual yield, with the quadratic the Treasury uses for bills longer than six months.
A Treasury bill is quoted at a discount rate, but its yield is the discount divided by the price actually paid, annualised on 365 days. A 13-week bill quoted at 3.92 per cent costs 99.009111 per 100 and yields 4.014 per cent on a bond-equivalent basis, 9.4 basis points more than the quote. That bond-equivalent figure, called the investment rate on Treasury auction results, is the one to set beside a bond's yield or a fund's distribution rate.
Worked in full in The Credit Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The discount rate is a pricing convention, not a return. It divides the discount by the face value instead of the price, and it uses a 360-day year. Both choices make it smaller than the return the buyer earns, and the gap grows with the bill's maturity.
| Bill | Days to maturity | Discount rate quoted |
|---|---|---|
| 13-week | 91 | 3.92% |
| 26-week | 182 | 3.88% |
| 52-week | 364 | 3.80% |
Step 1: price from the discount rate.
Price = 100 × (1 − d × t / 360)
13-week: 100 × (1 − 3.92% × 91 / 360) = 100 × 0.990091 = 99.009111
In Excel: =TBILLPRICE(settlement, maturity, 3.92%).
Step 2: bond-equivalent yield, for bills of 182 days or less. Divide the discount by the price, not by 100, and annualise on 365 days.
BEY = (100 − P) / P × 365 / t
13-week: 0.990889 / 99.009111 = 0.010008; × 365 / 91 = 4.014%
In Excel: =TBILLEQ(settlement, maturity, 3.92%). The money market yield, used to compare with deposits and commercial paper, keeps the price in the denominator but annualises on 360 days: 3.959 per cent.
Step 3: bills longer than half a year. A 52-week bill has no coupon date, but a coupon bond of the same maturity would pay interest at six months and compound it. The Treasury's investment rate for these bills solves a quadratic so the figure is comparable with a semiannual bond yield.
BEY = [−2a + 2 × √(a2 − (2a − 1)(1 − 100/P))] / (2a − 1), with a = t / 365
52-week: price 96.157778, a = 0.997260, BEY = 3.967%
The simple formula of step 2 would give 4.007 per cent, overstating the comparable yield by 4 basis points because it ignores the semiannual compounding a bond enjoys.
| Bill | Discount rate | Price | Money market yield | Bond-equivalent yield | Effective annual | BEY less discount, bp |
|---|---|---|---|---|---|---|
| 13-week | 3.92% | 99.009111 | 3.959% | 4.014% | 4.075% | 9.4 |
| 26-week | 3.88% | 98.038444 | 3.958% | 4.013% | 4.053% | 13.3 |
| 52-week | 3.80% | 96.157778 | 3.952% | 3.967% | 4.007% | 16.7 |
Read on the quote, the 13-week bill pays 12 basis points more than the 52-week one. Read on the bond-equivalent basis, the gap is 4.7 basis points, because the discount convention understates long bills more than short ones. The effective annual yield, (100 / P)365/t − 1, assumes the bill is rolled at the same rate for a full year and is the right figure to compare with an annual percentage yield on a savings account.
The gap also grows with the level of rates, because the discount convention understates more when there is more discount to understate.
| Discount rate | Bond-equivalent yield | Gap, bp |
|---|---|---|
| 3.00% | 3.065% | 6.5 |
| 3.92% | 4.014% | 9.4 |
| 4.50% | 4.615% | 11.5 |
| 5.00% | 5.134% | 13.4 |
A reserve is held for liquidity first and yield second, so the comparison that matters is with the other places the cash could sit at the same risk: a deposit, a government money market fund, or the bill itself bought at auction or in the secondary market.
An investor holding $52,000 as a reserve and buying 13-week bills at 99.009111 gets $52,520.42 of face, and $520.42 of interest at maturity. Rolled four times at the same rate, the reserve earns $2,113.14 over 364 days, 4.0637 per cent: more than the 4.014 per cent bond-equivalent yield, because each roll reinvests the interest from the last.
The bill is the hurdle every other line in a credit portfolio has to clear after its losses and fees. A high-yield bond whose expected yield falls below the bill's is paying the investor to take a risk the bill does not, and not enough for it. That comparison, at a bill rate of 4.00 per cent, is worked in yield after expected default losses.
Price = 100 × (1 − d × t / 360); bond-equivalent yield = (100 − P) / P × 365 / t up to 182 days, and the quadratic beyond. A 13-week bill quoted at 3.92 per cent yields 4.014 per cent, and that is the number to compare. How a portfolio's displayed yield is weighted across lines, bills included, is in how to calculate the weighted average yield of a bond portfolio, and the whole seventeen-line portfolio, with its bill reserve, is in the free workbooks for this book.
The discount rate divides the discount by face value and annualises on 360 days. The investment rate, or bond-equivalent yield, divides it by the price paid and annualises on 365 days. For an illustrative 13-week bill the discount rate is 3.92 per cent and the investment rate 4.014 per cent.
Price = 100 x (1 - discount rate x days / 360). A 26-week bill of 182 days quoted at 3.88 per cent costs 100 x (1 - 0.0388 x 182 / 360), or 98.038444 per 100 of face. Its bond-equivalent yield is 4.013 per cent.
Because a coupon bond of the same maturity would compound interest at six months, so the Treasury solves a quadratic to make the bill's yield comparable. For an illustrative 52-week bill priced at 96.157778 the result is 3.967 per cent, against 4.007 per cent from the simple formula.
This article is one calculation from The Credit Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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