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Compute the conversion factor of a US Treasury bond futures contract and explain what it represents.

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

The idea

The conversion factor (CF) puts all deliverable Treasury bonds on a common footing. It is the price per 1 of face value at which the bond would trade if it yielded 6% per year with semiannual compounding. The bond's maturity is first rounded down to the nearest three months. If that leaves three months over a whole number of half-years, the first coupon is three months away, and the accrued interest for those three months, half a semiannual coupon, is subtracted from the value. A coupon above 6% gives a CF above 1, and a coupon below 6% gives a CF below 1. Exactly 6% with whole half-years gives 1. The CF is fixed in advance and does not change with market yields, so it equalises bonds only when yields are 6%. Typical errors are discounting annually at 6% and forgetting to double the number of periods.

Formulas

Conversion factor with whole half-years to maturity

CF = [(c ÷ 2) × (1 − 1.03⁻ⁿ) ÷ 0.03 + 100 × 1.03⁻ⁿ] ÷ 100

c
the annual coupon per 100 of face
n
the number of half-years left, after rounding maturity down to three months

Discount at 3% per half-year, which is 6% a year compounded semiannually. With three months left over, the first coupon is three months away and half a coupon of accrued interest is subtracted.

A practice question

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

In the Treasury bond futures market, the conversion factor of a deliverable bond is best described as:

  1. the bond's market price divided by the futures priceWhy not: The conversion factor is fixed by formula at a 6% yield; it is not a market ratio.
  2. the price per 1 of face value at which the bond would trade if it yielded 6% (semiannual compounding) - correct
  3. the bond's current yield divided by 6%Why not: It is a price per unit of face value built from the bond's cash flows, not a ratio of yields.
  4. the ratio of the bond's duration to that of the cheapest-to-deliver bondWhy not: Duration plays no part in the conversion factor; it depends on coupon, maturity and a 6% yield.

The CF is the price of 1 face value of the bond at a 6% yield with semiannual compounding, with maturity rounded down to the nearest three months. It scales the futures price into an invoice price.

A harder one

A deliverable Treasury bond has an 8% coupon (paid semiannually) and exactly 20 years to maturity. Its conversion factor (6% yield, semiannual compounding) is closest to:

  1. 1.2294Why not: Compounds annually: 20 annual periods at 6% with an 8 coupon, rather than 40 semiannual periods at 3%.
  2. 1.2311 - correct
  3. 1.1488Why not: Uses 20 semiannual periods (10 years) instead of 40, forgetting to double the years.
  4. 1.3333Why not: Divides the coupon by 6 as a shortcut; the factor is a present value and not a simple ratio.

CF = [4 x annuity(40, 3%) + 100/1.03^40]/100 = 1.2311.

The hardest tier

A deliverable Treasury bond has an 11% coupon and 17 years 5 months to maturity. Maturity is rounded down to the nearest three months. The conversion factor (6% yield, semiannual compounding, subtracting accrued interest where the first coupon is three months away) is closest to:

  1. 1.5601Why not: Forgets to subtract the 3 months of accrued interest (half the semiannual coupon of 5.5).
  2. 1.5283Why not: Rounds maturity down to the nearest whole half-year (17 years) instead of the nearest three months.
  3. 1.5372Why not: Rounds the maturity up to 17 years 6 months rather than down to 17 years 3 months.
  4. 1.5326 - correct

Rounded maturity 17y3m, so the first coupon is 3 months away. Value at 6% semiannual with that stub, less 2.75 accrued (half of 5.5), divided by 100 = 1.5326.

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