Question

Treasury spot rates (expressed as semiannually pay yields to maturity) are as follows: 6 months 1%, 1 year 1.5%, 1.5 year 2% 2 year: 2.5% and 2.5 year: 2.25%. A 2.5 year, 3.5% Treasury bond is trading at $1,043. What is the arbitrage trade and how much would profit would you earn from doing the trade?

Answer 1

Face Value = $1000

Term = 2.5 years

Coupon rate = 3.5% (semi-annual)

6-month spot rate: 1%.

1 year spot rate: 1.5%.

1.5 year spot rate: 2%.

2 year spot rate: 2.5%

2.5 year spot rate: 2.25%

Solution

Coupon per period = (Coupon rate / No of coupon payments per year) * Face value

Coupon per period = (3.5% / 2) * $1000

Coupon per period = $17.5

Bond price = Coupon / (1 + 6-month spot rate / 2)¹ + Coupon / (1 + 1 year spot rate / 2)² + Coupon / (1 + 1.5 year spot rate / 2)³ + Coupon / (1 + 2 year spot rate / 2)⁴ + Coupon / (1 + 2.5 year spot rate / 2)⁵ + Face value / (1 + 2.5 year spot rate / 2)⁵

Bond price = $17.5 / (1 + 1% / 2)¹ + $17.5 / (1 + 1.5% / 2)² + $17.5 / (1 + 2% / 2)³ + $17.5 / (1 + 2.5% / 2)⁴ + $17.5 / (1 + 2.25% / 2)⁵ + $1000 / (1 + 2.25% / 2)⁵

Bond price = $17.41 + $17.24 + $16.98 + $16.65 + $16.55 + $945.60

Bond price = $1,030.44

The price calculated based on the spot curve gives us a price = $1030.44 implying a yield to maturity = 2.2412%

So the T-bond is currently overpriced at $1043 implying a yield to maturity = 1.7350%

I would short(sell) the T-bond till it comes back to it true intrinsic value = $1030.44

The profit from this trade = Market Price of the T-bond Short - Intrinsic value of T-bond

The profit from this trade = $1,043 - $1,030.44

The profit from this trade = $12.56