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Spot where a fixed-income security with certain cash flows offers an arbitrage opportunity.

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

The idea

A fixed-income security with certain cash flows offers an arbitrage when its quoted price differs from the value of the same cash flows priced off other securities, or when a set of prices breaks a basic bound. The method is to value the cash flows with discount factors taken from other instruments, such as STRIPS prices, and compare the result with the quote. If the bond is cheap, buy it and sell the replicating cash flows. If it is rich, sell it and buy them. The profit today is the price gap scaled by the face amount, and the later cash flows cancel. Discount factors must also fall as maturity lengthens and cannot exceed one, because cash held in a safe earns at least zero. A longer zero priced above a shorter one is an arbitrage. Mixing up rich and cheap is the common slip.

Formulas

Riskless profit from a mispriced bond

Profit = |V − P| × Face ÷ 100

V
the value of the bond's cash flows priced off other securities, per 100 face
P
the bond's quoted price per 100 face
Face
the face amount traded

Buy the bond and sell the replication if P is below V; sell the bond and buy the replication if P is above V. Mixing up rich and cheap is the common slip.

A practice question

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

Which description best fits an arbitrage opportunity in fixed-income markets?

  1. A trade with a positive expected profit but a real chance of making a lossWhy not: Describes a speculative bet; an arbitrage has no chance of loss.
  2. Buying the bond with the highest yield among all those available in the marketWhy not: Confuses a high yield with riskless profit; yield compensates for risk.
  3. Any trade that earns a return above the risk-free rate on the capital committedWhy not: Confuses excess return with arbitrage, which needs no risk and no investment.
  4. A trade that costs nothing, never loses money and gains in at least one case - correct

An arbitrage needs no net investment and carries no risk of loss, with a gain in at least one scenario. Positive expected profit alone does not qualify.

A harder one

A one-year Treasury note pays a 5% coupon semiannually, so its cash flows per 100 face value are 2.50 at six months and 102.50 at one year, and it trades at 101.50. Six-month and one-year strips trade at 98.80 and 97.40 per 100 face value. The riskless profit per 100 face value from the best arbitrage trade is closest to:

  1. 5.71Why not: Uses 5.00 and 105.00 as the cash flows, treating the annual coupon as paid at each date.
  2. 0.81 - correct
  3. 2.20Why not: Swaps the dates, valuing the 2.50 at the one-year price and the 102.50 at the six-month price.
  4. 1.63Why not: Values the last flow as 100 instead of 102.50, which understates the strip value.

Strip value = 2.50 x 0.9880 + 102.50 x 0.9740 = 2.470 + 99.835 = 102.305. The note at 101.50 is cheap by 0.805: buy it and sell the strips.

The hardest tier

Treasury strips trade at 0.5, 1.0, 1.5 and 2.0 years at 99.20, 98.30, 97.30 and 96.20 per 100 face value. A 2-year Treasury note paying a 4% coupon semiannually (2, 2, 2 and 102 per 100 face value) trades at 104.60. A dealer executes the best arbitrage on USD 50 million face value of the note. The profit is closest to:

  1. USD 290,000 - correct
  2. USD 352,000Why not: Values every cash flow, 108 in total, at the two-year strip price.
  3. USD 1,252,000Why not: Treats the last cash flow as 100 and omits the final coupon, overstating the gap.
  4. USD 3,620,000Why not: Pays 4.00 at each half-year date, so the replicating value is too high and the gap too large.

Strip value = 2 x (0.9920 + 0.9830 + 0.9730) + 102 x 0.9620 = 5.896 + 98.124 = 104.02. The note is rich by 0.58 per 100: sell it and buy strips. On 50 million face, profit = 0.58 x 500,000 = 290,000.

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