Question

Fixed Income Arbitrage is to find mispricing of coupon bonds through the system of linear equations where # of equations > # of unknowns. Suppose there are only three treasury coupon bonds on the market today:

Bond A: 2-year 2% treasury coupon bond, trading today at price $937.

Bond B: 2-year 3% treasury coupon bond, trading today at price $970.

Bond C: 2-year 4% treasury coupon bond, trading today at price $974.

Let ??0,1 and ??0,2 be the two unknowns, where ??0,1 denotes the price today of a 1-year STRIPS, and ??0,2 the price today of a 2-year STRIPS. Face value of STRIPS is $1. Questions:

1) Since each coupon bond is essentially a portfolio of STRIPS, write the 3-equation 2-unknown system.

2) If market price of STRIPS are: ??0,1 = 0.95 and ??0,2 = 0.9, plug into the 3- equation 2- unknown system, is there arbitrage opportunities in any of the coupon bonds, in which one(s)?

Answer 1

A coupon paying bond can be expressed as a combination of STRIPS of different maturity. Say for example a 2 year 2% coupon bond will pay:

• 2% x Par value = 2% x 1,000 = $ 20 as first coupon at t = 1; this is equivalent to 20 nos. of 1 year STRIPS
• $ 20 as second coupon + Par value of 1,000 at t= 2; this is equivalent to 1,020 nos. of 2 year STRIPS

Hence, Price of this bond should be = Price of 20 nos. of 1 year STRIP + Price of 1020 nos. of 2 year STRIPS

937 = 20PP_0,1 + 1,020PP_0,2

Part (1)

Three equations in two unknows will be:

Hence, out first equation will be: Eqn (1) ------------- 20PP_0,1 + 1,020PP_0,2 = 937

second equation will be: Eqn (2) ---------------------30PP_0,1 + 1,030PP_0,2 = 970

and third equation will be: Eqn (2) -------------------40PP_0,1 + 1,040PP_0,2 = 974

Part (2)

PP_0,1 = 0.95 and PP_0,2 = 0.9

LHS of Eqn (1) = 20PP_0,1 + 1,020PP_0,2 = 20 x 9.95 + 1,020 x 0.90 = 937 = RHS of Eqn (1)

LHS of Eqn (2) = 30PP_0,1 + 1,030PP_0,2 = 30 x 9.95 + 1,030 x 0.90 = 955.50 < RHS of Eqn (2)

LHS of Eqn (3) = 40PP_0,1 + 1,040PP_0,2 = 40 x 9.95 + 1,040 x 0.90 = 974 = RHS of Eqn (3)

Eqn (2) is not satisfied that means Bond B is not priced appropriately and hence there exists an arbitrage opportunity in Bond B