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Work out the price of a US Treasury bill given its discount rate.

FRM® Part I · Financial Markets and Products · Interest Rate Futures · FRM1.FM.19.b

The idea

A Treasury bill pays no coupon and is quoted on a discount rate, which measures the discount from face value as an annualised percentage of face value on an Actual/360 basis. Let d be the discount rate in percent and n the actual number of days to maturity. The price per 100 of face value then follows: Price = 100 − d × n ÷ 360. For example, 91 days at 4% gives a price of 100 − 4 × 91 ÷ 360 = 98.99. Reversing it gives the discount rate implied by a quoted price: d = (100 − Price) × 360 ÷ n. The discount rate understates the investor's actual return because it is measured on face value, not on the price paid, and uses a 360-day year. The bond-equivalent yield, BEY, is a simple yield on the price paid over a 365-day year, so it corrects both: BEY = (100 − Price) ÷ Price × 365 ÷ n. The common errors are subtracting d without the n ÷ 360 scaling, using 365 in the price formula, or treating d as a yield on price.

Formulas

Treasury bill price from its discount rate

P = 100 − d × n ÷ 360

d
the quoted discount rate, in percent
n
the actual days to maturity

Per 100 of face, on a 360-day year. Using 365, or dropping the n ÷ 360 scaling, are the usual errors.

Bond-equivalent yield of a Treasury bill

BEY = (100 − P) ÷ P × 365 ÷ n

P
the bill's price per 100 of face
n
the actual days to maturity

A simple yield on the price paid over a 365-day year, so it is above the discount rate, which uses face value and 360 days.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which description of the discount rate quoted on a US Treasury bill is correct?

  1. The annualised discount from face value, expressed as a percentage of face value on an Actual/360 basis - correct
  2. The annualised return on the price actually paid, on an Actual/365 basisWhy not: This describes the bond-equivalent yield; the discount rate is measured on face value and uses 360 days.
  3. The continuously compounded yield to maturityWhy not: Bills are quoted on a simple discount basis; no compounding is assumed.
  4. The semiannually compounded yield comparable to a Treasury coupon bondWhy not: Semiannual compounding is a bond yield convention and not the way the bill discount rate is calculated.

The discount rate is (face - price)/face x 360/n. Because it is based on face value rather than price and uses a 360-day year, it understates the actual yield.

A harder one

A Treasury bill with face value USD 1,000,000 matures in 182 days and is quoted at a discount rate of 6.00%. The price paid is closest to:

  1. USD 970,082Why not: Uses 365 days in the day count fraction instead of 360.
  2. USD 970,560Why not: Treats the discount rate as a simple yield and discounts the face value, which overstates the price.
  3. USD 969,667 - correct
  4. USD 940,000Why not: Applies the full annual discount without the day count fraction 182/360.

Price = 1,000,000 x (1 - 0.06 x 182/360) = 969,667.

The hardest tier

A 120-day Treasury bill is quoted at a discount rate of 5.00%. Its bond-equivalent yield (simple yield on price, 365-day year) is closest to:

  1. 5.00%Why not: Equates the yield with the discount rate; the yield is measured on the lower price and so is higher.
  2. 5.16% - correct
  3. 5.07%Why not: Adjusts only for the 365/360 day count and forgets that the return is measured on price, not face value.
  4. 5.08%Why not: Measures the return on price but keeps a 360-day year; the bond-equivalent yield uses 365.

Price = 100 - 5 x 120/360 = 98.3333. BEY = (100 - 98.3333)/98.3333 x 365/120 = 5.16%.

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