Question

Suppose an investor would like to buy 200 Treasury notes. The investor wants notes with an annual coupon rate of 7%, a 3-year maturity, and semi-annual coupon payments.

a. (5 pts) If there were no such Treasury note available, propose a portfolio for this investor (using only Zeroes with maturities up to 3 years) that replicates the cash flows from investing in the Treasury notes above.

b. (5 pts) Assuming the yield curve is flat at 4.0% for bonds with maturities of up to 3 years, calculate the prices of the Zeroes in your portfolio from part (a). Using these prices, compute the no-arbitrage price of a Treasury note.

c. (5 pts) Now suppose there is a 3-year, 7% coupon rate Treasury note available that has a YTM of 4.5%. Would the investor above prefer to buy 200 Treasury notes or the portfolio of Zeroes identified in part (a)?

d. (5 pts) Find a costless and riskless trading strategy that makes an instantaneous profit by buying or selling the Treasury note and the portfolio of zeroes.

e. (5 pts) If this costless strategy required that you put up 50% collateral for the short sales, would you be willing to use all of your available capital for collateral in this strategy?

Answer 1

a]

The cash flows from each Treasury note are :

• Semiannual coupon payment = face value * coupon rate / 2 = $1,000 * 7% / 2 = $35
• Face value repaid at maturity = $1,000
• Cash flow in 1st to 5th semiannual period = $35
• Cash flow in 6th semiannual period = $35 + $1,000 = $1,035
• These cash flows are multiplied by 200 to get the cash flow on 200 Treasury notes

These cash flows can be replicated by zeroes (with $1,000 face value) as below :

• Buy 7 Zero bonds ($35 * 200 / $1,000), with 6 months to 2.5 years maturity
• Buy 207 Zero bonds ($1,035 * 200 / $1,000), with 3 years maturity

b]

Price of zero coupon bond = face value / (1 + YTM)^{years to maturity}

no-arbitrage price of a Treasury note = sum of prices of zero coupon bonds used to replicate Treasury note

no-arbitrage price of a Treasury note = $1,085.18

c]

The investor would prefer to buy 200 Treasury notes as the YTM of the Treasury notes is higher than the YTM on zeroes

d]

First, we calculate the price of the Treasury note.

Price of a bond is the present value of its cash flows. The cash flows are the coupon payments and the face value receivable on maturity

Price of bond is calculated using PV function in Excel :

rate = 4.5%/2 (Semiannual YTM of bonds = annual YTM / 2)

nper = 3 * 2 (3 years remaining until maturity with 2 semiannual coupon payments each year)

pmt = 1000 * 7% / 2 (semiannual coupon payment = face value * coupon rate / 2)

fv = 1000 (face value receivable on maturity)

PV is calculated to be $1,069.43

d]

No-arbitrage price of a Treasury note = $1,085.18

Price of the Treasury note is $1,069.43

The Treasury note is underpriced relative to the portfolio of zeroes.

An arbitrage profit can be earned by buying the Treasury note and selling the portfolio of zeroes.

Arbitrage profit = $1,085.18 - $1,069.43 = $15.75