## Discussion on Treasury Bonds

- 14th May, 2019
- 13:20 PM

In this blog, we will try to understand the basics of bonds and debt instruments by answering the below 2 questions

**Question 1 - **By what amount will the market value of a Treasury bond futures contract change if interest rates rise from 5% to 5.25%? The underlying Treasury bond has duration of 10.48 years, and the Treasury bond futures contract is currently being quoted at 113-06. (Remember that Treasury bonds are quoted in 32nds.)

**Question 2- **Morning View National Bank reports that its assets have duration of 7 years, and its liabilities average 1.75 years in duration. To hedge this duration gap, management plans to employ Treasury bond futures, which are currently quoted at 112-170 and have duration of 10.36 years. Morning View’s latest financial report shows total assets of $100 million and liabilities of $88 million. Approximately how many futures contracts will the bank need to cover its overall exposure?

**Pertinent Literature**

Among various types of bonds, the paper covers treasury bonds only. Treasury bond or T-bond is a type of marketable, fixed coupon paying United States government debt security. It’s time to maturity is generally more than 10 years. Treasury bond generally makes semi-annual payment to bond holder or bond subscriber. Treasury bonds are considered to be risk free as the same may be issued by United States government. Hence, the default risk is minimal. It should be noted that treasury bonds are quoted in 32nd therefore the quote of x- y is $ x. y/32 (Johnson ,2010).

Bond prices are estimated as present value of future cash flows (Fabozzi,1998). Therefore price of the bond may be represented as

Where

F = Face value of the bond

ia = Appropriate Yield to maturity or market interest rate.

C = F * Annual coupon Rate/Number of coupon payment in a year = Coupon payment after every period (periodic interest payment)

N = Number of payments

M = Maturity Value of the bond which is in general face value of the bond.

P = Market price of bond.

It may be observed that coupon rate, coupon payment frequency and maturity value of the bond is fixed and will not be changed during the tenure of the bond provided there is no embedded option in the bond. Therefore market interest rate will remain the only sensitive factor. Since market interest rate is in denominator therefore increase in market interest rate or yield to maturity reduces the market price of the bond and decrease in interest rate or yield to maturity increases the market price of the bond.

For capturing the price sensitivity of bond with respect to interest rate, there is parameter named duration of the bond. It is defined as the change in bonds price for every unit change in interest rate. In general higher the duration, price of bond will drop for increase in interest rate. Therefore it may be considered as representative of interest rate risk (Fabozzi,1998). There are set of bond fundamentals which affect duration. These are time to maturity and coupon rate. Time to maturity is positively correlated with duration which means higher the time to maturity, higher the duration. However, coupon rate is negatively correlated with duration i.e. higher the coupon rate lower is the duration.

Mathematically, it may be represented as follows:

Where Dis duration of bond,Pis price of the bond, ΔP is change in price while ΔY is change in interest rate.

Since duration of the bond is the reflection of interest rate risk, therefore for hedging interest rate risk, derivatives are purchased in a way that total duration of the portfolio become zero. It should be noted that when hedges are constructed using interest rates, the interest rates and future prices should move in opposite direction. Therefore, if interest rates falls and investor is expected to lose money, investor should long future contract as any loss in portfolio would be offset by gain in future prices (Veronesi, 2010).

The concepts discussed above is used for solving the two example problem in subsequent sections.

Example problem

The first problem requires to estimate the change in market value of a Treasury bond futures contract provided there is an increase in interest rate. As per question there is an increase in interest rate to 0.25%. The question states that interest rates rise from 5% to 5.25%. The underlying Treasury bond has a duration of 10.48 years. Treasury bond futures contract is currently being quoted at 113-06 i.e. $ 113.1875. Applying the formula as mentioned above, change in market value of a Treasury bond futures contract is equal to negative of duration multiplied by change in interest rate multiplied with current Treasury bond futures contract

Change in Treasury bond futures contract = - (10.48) x (0.25/100) x (113.1875)

Change in Treasury bond futures contract = - $ 2.9655

Therefore, it may be concluded that if interest rate increases from 5.0% to 5.25%, market value of a Treasury bond futures contract is reduced by $ 2.9655. Alternatively, the market value of Treasury bond futures contract would reduce to $ 110.222 from $ 113.1875

The second example requires to hedge the duration gap. Duration gap is a term which is generally used by banks, pension funds and other financial institutions to measure the risk associated with changes in interest rate. This is primarily occurs owing to asset liability mismatches. Duration gap may be defined as follows (Skinner, 2004)

In this case duration of earning assets is 7 years while that of liabilities is 1.75. Also balance sheet of National Bank indicates total assets of $100 million and liabilities of $88 million. Therefore duration gap is equal to 7 – 1.75 x (88/100) = 5.46 . Further, management wants to hedge the duration gap using Treasury bond futures which is quoted at 112-170 i.e. $ (112 + 170/320) x 1000 = $ 112,531.25 with a duration 10.36 years. Also Morning View’s latest financial report shows total assets of $100 million and liabilities of $88 million. So number of contracts may be calculated as follows:

Number of contracts = (7 – 1.75 x (88/100)) x 100/( 10.36 x 112,531.25/10^6) = 5.46 x 85.776 = 468.338 contracts i.e. 469 contracts.

Morning View National Bank is exposed with a +5.46 years Duration Gap which indicates that in order to protect the bank against an interest rate rise, Morning View will want to adopt a short- hedge. They would need 469 of the futures contracts which are named above in order to cover their exposure. In simpler terms, banks need to sell 469 future contracts.

**Conclusion**

It may be concluded that duration is a metric of measuring interest rate risk. Bond prices are inversely related to interest rate i.e. increase in interest rate decreases the market price of bond and decrease in interest rate increases the market price of bond. Portfolio is prone to interest rate risk if there exist any dIn any portfolio, interest rate risk gets mitigated if there is zero duration gap.

**References**

Batten, J. A.; Peter G. S. (2006). Developing Foreign Bond Markets: The Arirang Bond Experience in Korea. IIS Discussion Papers (138). Retrieved 2007-07-06.

**Fabozzi F (1998).** Valuation of fixed income securities and derivatives (3rd ed.). John Wiley. ISBN 978-1-883249-25-0

**Johnson R. S. (2010).** Bond Evaluation, Selection, and Management (2nd ed.). John Wiley. ISBN 0470478357

**Skinner F. (2004). **Pricing and Hedging Interest and Credit Risk Sensitive Instruments. Butterworth-Heinemann. pp. 218–. ISBN 9780080473956.

**Veronesi P.(2010). **Fixed Income Securities: Valuation, Risk, and Risk Management. John Wiley. ISBN 978-0470109106