After completing this reading, you should be able to:

- Identify the most commonly used day count conventions, describe the markets that each one is typically used in, and apply each to an interest calculation.
- Calculate the conversion of a discount rate to a price for a US Treasury bill.
- Differentiate between the clean and dirty price for a US Treasury bond; calculate the accrued interest and dirty price on a US Treasury bond.
- Explain and calculate a US Treasury bond futures contract conversion factor.
- Calculate the cost of delivering a bond into a Treasury bond futures contract.
- Describe the impact of the level and shape of the yield curve on the cheapest-to-deliver Treasury bond decision.
- Calculate the theoretical futures price for a Treasury bond futures contract.
- Calculate the final contract price on a Eurodollar futures contract.
- Describe and compute the Eurodollar futures contract convexity adjustment.
- Explain how Eurodollar futures can be used to extend the LIBOR zero curve.
- Calculate the duration-based hedge ratio and create a duration-based hedging strategy using interest rate futures.
- Explain the limitations of using a duration-based hedging strategy.

## Day Count Conventions

A day count convention dictates how interest accrues over time in a variety of financial instruments, including bonds, swaps, and loans. It determines how interest is calculated at the end of each period. It’s usually expressed as a fraction \({ A }/{ B }\). \(A\) defines the way in which the number of days between the two days is calculated, usually a notional or 30. \(B\) defines the way in which the total number of days in the reference period is measured, usually 360 or 365.

$$ Accrued\quad interest=coupon\times \frac { number\quad of\quad days\quad between\quad dates }{ number\quad of\quad days\quad in\quad reference\quad period } $$

### Common day count conventions:

- Actual/actual
- 30/360
- Actual/360

The actual/actual convention means that the accrued interest is based on the ratio of the actual days elapsed to the actual number of days in the period between coupon payments. For example, suppose we have a \(bond\) paying a coupon rate of \(10\%\) per annum on a principal of \($100\) on March 1st and September 1st. This implies a coupon of \($5\) on each of these dates. If we want to compute the accrued interest as of May 31st, we will have to determine the actual number of days between May 31st and the last coupon date, i.e., March 1st. That’s \(91\quad days\quad \left( = 30 + 30 + 31 \right) \). The reference period, March 1st to Sept 1st, has \(184\) actual days. Thus,

$$ Accrued\quad interest=\frac { 91 }{ 184 } \times $5 $$

The actual/actual is used for **Treasury bonds**.

The 30/360 convention follows the same logic but assumes **all** months have exactly 30 days. It’s used for **corporate and municipal bonds**.

The actual/360 convention is used for money market instruments, e.g., T-bills, commercial paper, and certificates of deposit.

## Clean Price vs. Dirty Price

The **dirty price** of a bond is the price that includes the present value of all of the bond’s cash flows, including the interest accruing on the next coupon payment date. It’s the price the issuer of the bond must be paid by the investor in order to dispense with the bond. The dirty price is comprised of the quoted price and accrued interest.

Example:

A \(bond\) quoted at \(102-20\) has accrued \($2.54\) in interest over the last six months. Determine the cash price of the bond.

Answer

$$ Cash\quad price=dirty\quad price=quoted\quad price+accrued\quad interest=102+\frac { 20 }{ 32 } +2.54=$105.165 $$

The **clean price** of a bond is the price that excludes the interest that has accrued since issue or the most recent coupon payment. It’s also known as the quoted price.

$$ Clean\quad price=dirty\quad price–accrued\quad interest. $$

## T-bill Prices

Like other money market instruments, T-bills are issued at a discount to par value, on an actual/360 day count basis. The quoted price is, in fact, the discount rate. The quoted price, \(P\), can be expressed as:

$$ P=\frac { 360 }{ n } \left( 100-Y \right) $$

Where:

\(n\) = number of days to maturity

\(Y\) = the bill’s **cash** price

Alternatively, you might be asked to compute a T-bill’s cash price, in which case you should just make \(Y\) the subject of the formula:

$$ Y=100-\frac { Pn }{ 360 } $$

### Example

A 120-day treasury bill with a cash price of 99 would have a quoted price of:

$$ P=\frac { 360 }{ 120 } \left( 100-99 \right) = 3 $$

## U.S. Treasury Bonds Futures Contracts

Deliverable securities for T-Bond futures contracts are bonds with remaining terms to maturity of 15 years or more. The cash received by the short position in a T-bond futures contract is given by:

$$ Cash \quad received=\left( QFP\times CF \right) +AI $$

Where:

\(QFP\) = quoted futures price/settlement price

\(CF\) = conversion factor

\(AI\) = accrued interest since the last coupon date on the bond delivered

The conversion factor is given by:

$$ CF=\frac { discounted\quad bond\quad price-accrued\quad interest }{ face\quad value } $$

For instance, if a bond has a present value of \($125\), accrued interest of \($5\), and \($100\) face value, then:

$$ CF=\frac { 125-5 }{ 100 } =1.25 $$

Conversion factors for different bonds are issued on a daily basis by the Chicago Board of Trade. A conversion factor is actually the approximate decimal price at which $1 par of a bond would trade if it had a 6% yield to maturity (YTM).

## Cheapest to Deliver Bond

The cheapest to deliver (CTD) the bond refers to the cheapest bond that could be delivered to the long position in line with contractual specifications. Determining the CTB bond is necessitated by a discrepancy between the market price of a security and the conversion factor used to determine the value of the security being delivered. Thus, picking one bond for delivery over another can be advantageous to the short position.

$$ CTD=Current\quad Bond\quad Price–Settlement\quad Price\quad \times \quad Conversion\quad Factor $$

(settlement price = quoted futures price)

CTB calculations are relevant in all cases where multiple financial instruments can satisfy the contract.

The theoretical Futures Price for a Treasury bond futures, \({ F }_{ 0 }\), is given by:

$$ { F }_{ 0 }=\left( { S }_{ 0 }-I \right) { e }^{ rT } $$

Where:

\({ S }_{ 0 }\)=spot price of the bond

\(I\)=present value of cash flows, i.e., coupons

\(r\)=risk-free rate of interest

\(T\)=time to maturity

## The final Price of Eurodollar Futures Contracts

Eurodollars are U.S. dollars deposited in banks outside the United States. Eurodollar futures provide a valuable tool for hedging fluctuations in short-term U.S. dollar interest rates. These type of futures have a maturity term of 3 months and largely reflect market expectations for that period.

The final price of a Eurodollar futures contract is determined by LIBOR on the last trading day. Eurodollar futures contract settle in cash and are based on a Eurodollar deposit of $1million.

The minimum price change is one “tick,” which is equivalent to one interest rate basis point = 0.01 price points = $25 per contract.

$$ Eurodollar\quad futures\quad price=$10,000\left[ 100-\left( 0.25 \right) \left( 100-Z \right) \right] $$

Where \(Z\) = quoted price for a Eurodollar futures contract

For example, if the quoted price \(Z\) is 98.5, then:

$$ Eurodollar\quad futures\quad price=$10,000\left[ 100-\left( 0.25 \right) \left( 100-98.50 \right) \right] =$996,250 $$

The three-month forward LIBOR for each contract is \(100 – Z\). In practice, however, daily marking-to-market can result in differences between actual forward rates and those implied by fixtures contracts. To reduce this difference, we use a convexity adjustment:

$$ Actual\quad forward\quad rate=forward\quad rate\quad implied\quad by\quad futures-\left( \frac { 1 }{ 2 } \times { \sigma }^{ 2 }\times { T }_{ 1 }\times { T }_{ 2 } \right) $$

Where:

\({ T }_{ 1 }\) = maturity on the futures contract

\({ T }_{ 2 }\) = time to the maturity of the rate underlying the contract (90 days)

\(\sigma\)= annual standard deviation of the change in the rate underlying the futures contract, or 90-day LIBOR

## Duration-based Hedge Ratio

A duration-based hedge ratio is a hedge ratio constructed when interest rate futures contracts are used to hedge positions in an interest-dependent asset, usually bonds money market securities.

The number of futures contracts (\(N\)) required to hedge against a given change in yield, \(\left( \Delta y \right) \) is:

$$ N=-\frac { P\times DP }{ FC\times DF } $$

Where:

\(P\) = forward value of the fixed-income portfolio being hedged

\(DP\) = duration of the portfolio at the maturity date of the hedge

\(FC\) = futures contract price

\(DF\) = duration of the asset underlying the futures

The negative sign implies that the number of contracts taken up must be the opposite of the original position. If the investor is **short** the portfolio, for example, they must **long** \(N\) contracts to produce a position with zero duration.

### Example

A pension fund has a $25 million portfolio of Treasury bonds with a portfolio duration of 6.1. The cheapest to deliver bond has a duration of 4.7. The six-month treasury bond futures price is 127. The number of futures contracts to fully hedge the portfolio is:

$$ N=-\frac { P\times DP }{ FC\times DF } $$

$$ N=-\frac { $25,000,000\times 6.1 }{ $127,000\times 4.7} = -255 $$

If yields rise over the next 6 months, it’s bad news for the portfolio as it will lose value. Suppose the new value is $23.5 million, then it’s good news in the futures market as the gains can be used to offset the spot market losses.

In fact, if the hedge is executed properly and the yield curve changes are parallel, then it is possible to gain $1.5 million in the derivatives market. The total portfolio value after the hedge is $25 million.

**Limitations of a Duration-based Hedging Strategy**

The major limitation of employing a duration-based hedging strategy has much to do with the fact that duration measures are only accurate for small changes in yield. For large changes in yield, the price/yield relationship is not linear but is actually convex. Thus, using the strategy in the face of large moves in yield will result in “underhedging.”