Charta is on Google Play: practise on the go. Get the app
Charta

Create a duration-based hedging strategy in interest rate futures, starting with working out the hedge ratio.

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

The idea

A duration-based hedge matches the price sensitivity of the portfolio to that of the futures. The number of contracts is N = (P × DP) ÷ (F × DF). Here P is the portfolio value, DP its duration, F the value of one futures contract, and DF the duration of the cheapest-to-deliver bond at futures delivery. A quote of 110 on a 100,000 face contract means 110,000 per contract. Sell futures if the portfolio loses when yields rise. Buy futures if you will purchase bonds later and fear lower yields. To move duration from DP to a target DT, the number of contracts is N = (DT − DP) × P ÷ (F × DF). A negative result means sell. Common mistakes are using the 100,000 face value in place of the futures price, inverting the ratio of durations, and getting the sign wrong. On the exam, decide the direction first, then do the arithmetic.

Formulas

Futures contracts for a duration hedge

N = (P × DP) ÷ (F × DF)

P
the value of the portfolio
DP
the portfolio's duration at the end of the hedge
F
the value of one contract, futures price × contract size
DF
the cheapest-to-deliver bond's duration at futures delivery

Assumes small parallel shifts. F is the futures price times size, not the face value; sell to protect a held portfolio, buy to hedge a future purchase.

Futures contracts to reach a target duration

N = (DT − DP) × P ÷ (F × DF)

DT
the target duration
DP
the portfolio's current duration
P
the value of the portfolio
F
the value of one contract
DF
the cheapest-to-deliver bond's duration at delivery

Negative N means sell futures to cut duration; positive means buy to extend it.

A practice question

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

The standard formula for the number of futures contracts in a duration-based hedge is N = (P x D_P)/(F x D_F). What does F represent?

  1. The face value of the portfolio being hedgedWhy not: Portfolio value is already P; F is the value of a single futures contract.
  2. The value of one futures contract (futures price x contract size) - correct
  3. The forward yield on the cheapest-to-deliver bondWhy not: F is a price-based contract value, not a yield.
  4. The conversion factor of the cheapest-to-deliver bondWhy not: The conversion factor does not appear in the basic hedge ratio; F is the contract's value.

N = P x D_P /(F x D_F) matches the dollar duration of the portfolio to that of the futures, where F is the futures price times the contract size and D_F is the duration of the underlying bond at delivery.

A harder one

A manager holds a USD 50 million bond portfolio with a duration of 6.2 and plans to hedge fully with T-bond futures at a price of 112.50 (100,000 face per contract). The cheapest-to-deliver bond has a duration of 8.9. The number of contracts to sell is closest to:

  1. 444Why not: Ignores the duration ratio, matching only the value of the portfolio to the futures.
  2. 638Why not: Inverts the duration ratio (D_F/D_P instead of D_P/D_F).
  3. 310 - correct
  4. 348Why not: Uses the face value of 100,000 per contract instead of the futures price 112,500.

N = 50,000,000 x 6.2 /(112,500 x 8.9) = 309.6, so sell about 310 contracts.

The hardest tier

A USD 75 million portfolio will have a duration of 7.4 at the end of a three-month hedge. The T-bond futures price is 105-24 (that is, 105.75; 105,750 per contract). The cheapest-to-deliver bond has a duration of 7.2 today but 6.8 at futures delivery. The number of contracts to sell is closest to:

  1. 729Why not: Uses the CTD bond's duration today instead of its duration at futures delivery.
  2. 816Why not: Uses a face value of 100,000 per contract instead of the futures contract value of 105,750.
  3. 652Why not: Inverts the duration ratio, putting the futures duration in the numerator.
  4. 772 - correct

N = P x D_P /(F x D_F) with D_F measured at futures delivery = 75,000,000 x 7.4 /(105,750 x 6.8) = 771.8.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Interest Rate Futures