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Describe how the yield curve's level and shape affect which Treasury bond is cheapest to deliver.

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

The idea

The conversion factor system assumes a 6% yield, so which bond is cheapest to deliver depends on where yields really are. When yields are above 6%, low-coupon, long-maturity bonds (high duration) tend to be cheapest to deliver. Their prices fall furthest relative to conversion factors fixed at 6%. When yields are below 6%, high-coupon, short-maturity bonds (low duration) tend to be cheapest. The shape of the curve matters too. With an upward-sloping curve the cheapest bond tends to have a long maturity, and with a downward-sloping curve a short maturity. These are rules of thumb, not guarantees, because the ranking can change as yields move. The cost of delivery is the bond price less the futures price times the conversion factor. The usual mistake is reversing the rules. A change in the CTD bond changes the futures contract's duration.

A practice question

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

When yields on deliverable Treasury bonds are well above 6%, which type of bond tends to be cheapest to deliver in the Treasury bond futures contract?

  1. A high-coupon, short-maturity bondWhy not: This is the pattern favoured when yields are below 6%.
  2. A high-coupon, long-maturity bondWhy not: Mixes the two effects; coupon and maturity point in opposite directions in the rule.
  3. A low-coupon, short-maturity bondWhy not: Mixes the two effects; the rule pairs low coupon with long maturity, i.e. high duration.
  4. A low-coupon, long-maturity bond - correct

The conversion factor assumes 6%. When yields are above 6%, high-duration bonds (low coupon, long maturity) have the lowest price relative to their factor and tend to be cheapest to deliver.

A harder one

Treasury yields are flat at 3% across all maturities. The following bonds are deliverable. Which is most likely to be cheapest to deliver?

  1. A 3% coupon bond with 30 years to maturityWhy not: Low coupon and long maturity is favoured when yields are above 6%, not below.
  2. A 6% coupon bond with 20 years to maturityWhy not: A coupon equal to the 6% reference does not benefit from the mismatch between actual yields and 6%.
  3. A 12% coupon bond with 15 years to maturity - correct
  4. A 8% coupon bond with 25 years to maturityWhy not: Long maturity increases duration, which hurts when yields are below 6%.

Below 6%, the conversion factor system favours high-coupon, short-maturity (low-duration) bonds. The 12% bond with the shortest maturity fits that profile.

The hardest tier

Yields are flat at 8% (semiannual compounding). Two bonds with 20 years to maturity are deliverable: Bond A with a 4% coupon (conversion factor 0.7689) and Bond B with a 10% coupon (conversion factor 1.4623). The futures price is 78.00. Ignoring accrued interest, which statement is correct?

  1. Bond B is cheaper to deliver, by about 5.3 per 100 of face value.Why not: Reverses the ranking: the higher-duration low-coupon bond has the lower delivery cost when yields exceed 6%.
  2. Bond A is cheaper to deliver, by about 59.4 per 100 of face value.Why not: Ignores the conversion factors and compares only market prices against the futures price.
  3. The two bonds are equally cheap to deliver because the conversion factors adjust for coupon differences.Why not: The factors adjust for coupon only at a 6% yield; at 8% the bonds are not equivalent.
  4. Bond A is cheaper to deliver, by about 5.3 per 100 of face value. - correct

Price at 8%: A = 60.41, B = 119.79. Cost A = 60.41 - 78 x 0.7689 = 0.44; cost B = 119.79 - 78 x 1.4623 = 5.73. Bond A is cheaper by 5.29.

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