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Use several fixed-income securities to build a replicating portfolio that reproduces the cash flows of a given fixed-income security.

FRM® Part I · Valuation and Risk Models · Pricing Conventions, Discounting, and Arbitrage · FRM1.VR.9.e

The idea

A replicating portfolio is a mix of long and short positions in other securities whose cash flows match a target security date by date. Build it by working backward from the longest maturity. Choose the position in the instrument that matures last so its final cash flow equals the target's, see what that position leaves at each earlier date, and correct the excess or shortfall with instruments maturing then, using short positions where needed. Zero-coupon bonds replicate any bond directly, holding a face amount equal to each cash flow. By the law of one price, the target's price equals the cost of the replicating portfolio, and any gap is an arbitrage, assuming no trading costs, shorting allowed and equal credit quality. The usual errors are matching only the final payment, and forgetting that the longer bond's coupons scale with the size of the position.

A practice question

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

What is a replicating portfolio for a fixed-income security?

  1. A portfolio with the same duration as the target, whatever its cash flows may beWhy not: Confuses replication with duration matching, which reproduces only first-order rate sensitivity.
  2. A portfolio with the same yield to maturity as the targetWhy not: Matches a single number rather than the cash flows themselves.
  3. Long and short positions in other securities that reproduce the target's cash flows on every date - correct
  4. A portfolio of bonds from the same issuer and with a similar maturity to the targetWhy not: Same issuer does not make the cash flows identical.

Replication means matching cash flows date by date, so that by the law of one price the portfolio and the target must cost the same.

A harder one

A risk manager replicates 100 face value of a two-year zero-coupon bond using a one-year 3% annual coupon bond, which pays 103 per 100 face value, and a two-year 5% annual coupon bond, which pays 5 and 105 per 100 face value. Long positions in the two-year bond are sized to match the 100 payment. The face value of the one-year bond to be sold short is closest to:

  1. 4.76Why not: Ignores the one-year bond's own coupon and assumes it repays only its face value.
  2. 4.62 - correct
  3. 5.00Why not: Does not scale the 5 coupon by the 100 / 105 position size.
  4. 4.54Why not: Treats the one-year bond as paying 105 at maturity, the two-year bond's coupon rate.

Holding 100 / 105 = 0.9524 of the two-year bond pays a year-one coupon of 4.762. A one-year bond pays 1.03 per unit of face, so short 4.762 / 1.03 = 4.62 face.

The hardest tier

A 1.5-year Treasury note pays a 4% coupon semiannually, so its cash flows per 100 face value are 2, 2 and 102. It is replicated using a 1.5-year 6% coupon note paying 3, 3 and 103, priced at 101.05, together with six-month and one-year zero-coupon bonds priced at 0.975 and 0.950 per unit of face value. Using the replication, the price per 100 face value of the 4% note is closest to:

  1. 99.13Why not: Holds a full unit of the 6% note, ignoring the final cash flow mismatch of 103 against 102.
  2. 101.94Why not: Takes the zero-coupon positions long instead of short.
  3. 98.20 - correct
  4. 100.07Why not: Scales the 6% note by 102 / 103 but ignores the extra coupons it pays.

Match the last flow: hold 102 / 103 = 0.9903 of the 6% note, costing 100.063. Its coupons of 2.971 exceed the 2 needed by 0.971 at each date, so sell zeros with face 0.971: 0.971 x (0.975 + 0.950) = 1.869. Price = 100.063 - 1.869 = 98.20.

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

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