Understanding Bond Futures Duration and Coupon Payments in Excel

Coupons are the periodic interest payments that a bond issuer makes to the bondholder until the bond matures. The coupon rate is the annual interest rate expressed as a percentage of the bond’s face value. For example, a bond with a face value of $1,000 and a coupon rate of 5% pays $50 every year to the bondholder.

Bond futures are contracts that allow investors to buy or sell a specific bond at a predetermined price and date in the future. Bond futures are standardized by specifying the characteristics of the bond that can be delivered, such as the maturity range, the coupon range, and the conversion factor.

The conversion factor is a number that adjusts the price of the bond to reflect its coupon rate relative to the coupon rate of the bond futures contract. The conversion factor is used to calculate the invoice price of the bond, which is the amount that the buyer of the bond futures contract pays to the seller when the bond is delivered. The invoice price is equal to the bond futures price multiplied by the conversion factor, plus the accrued interest.

The accrued interest is the amount of interest that has accumulated on the bond since the last coupon payment date. The accrued interest is paid by the buyer of the bond futures contract to the seller, because the buyer will receive the next coupon payment from the bond issuer.

The seller of the bond futures contract has the option to choose which bond to deliver from a basket of eligible bonds. The seller will typically choose the bond that has the lowest cost of delivery, which is also known as the cheapest-to-deliver (CTD) bond. The cost of delivery depends on factors such as the bond’s price, coupon rate, conversion factor, and implied repo rate.

The implied repo rate is the rate of return that the seller of the bond futures contract can earn by buying the bond in the cash market, delivering it to the buyer of the bond futures contract, and investing the proceeds in a repurchase agreement (repo). A repo is a transaction in which one party sells a security to another party and agrees to buy it back at a later date for a higher price. The implied repo rate is inversely related to the cost of delivery, meaning that a higher implied repo rate implies a lower cost of delivery.

The CTD bond may change over time as the bond prices, coupon rates, conversion factors, and implied repo rates fluctuate. The CTD bond also affects the basis, which is the difference between the bond futures price and the bond cash price adjusted by the conversion factor. The basis reflects the expectations of the future bond prices and interest rates, as well as the optionality of the bond futures contract.

Basic Theory:

Bond Coupons: Bonds typically pay periodic interest payments called coupons. These coupons represent the bond’s interest rate multiplied by its face value.
Bond Futures Duration: Duration is a measure of a bond’s sensitivity to interest rate changes. It helps investors understand the potential impact of interest rate fluctuations on the bond’s price.

Procedures:

Calculate Macaulay Duration:Macaulay Duration is the weighted average time until a bond’s cash flows are received. It is calculated as the sum of present values of all future cash flows, divided by the bond’s current price.
$D_{\text{Mac}} = \frac{\sum_{t=1}^{n} \frac{t \times C + FV}{(1 + r)^t}}{P}$
Calculate Modified Duration:Modified Duration adjusts Macaulay Duration for interest rate changes. It is calculated as Macaulay Duration divided by $1 + \frac{r}{n}$ , where $n$ is the number of compounding periods per year.
$D_{\text{Mod}} = \frac{D_{\text{Mac}}}{1 + \frac{r}{n}}$
Calculate Bond Futures Duration:For bond futures, the duration is the sensitivity of the bond’s price to changes in interest rates. It is calculated by multiplying the modified duration by the bond’s conversion factor.
$D_{\text{Fut}} = D_{\text{Mod}} \times \text{Conversion Factor}$

Comprehensive Explanation:

Let’s consider a scenario where you have a 10-year bond with a face value of $1,000, an annual coupon rate of 5%, and a yield to maturity of 4%. The bond pays semi-annual coupons.

Scenario:

Face Value ( $FV$ ): $1,000
Annual Coupon Rate ( $C$ ): 5%
Yield to Maturity ( $r$ ): 4%
Bond Maturity ( $n$ ): 10 years
Coupon Payments per Year ( $m$ ): 2 (semi-annual)

Calculation in Excel:

Period	Cash Flow	Present Value Factor	Present Value
1	$25	$( 1 + 0.02 ) ^{1} 1$	$1.02 25$
2	$25	$( 1 + 0.02 ) ^{2} 1$	$1.0404 25$
…	…	…	…
20	$25 + $1,000	$( 1 + 0.02 ) ^{20} 1$	$1.485944 1 , 025$

Results:

Macaulay Duration ( $D_{\text{Mac}}$ ): Approximately 8.65 years
Modified Duration ( $D_{\text{Mod}}$ ): Approximately 8.47 years
Conversion Factor: Assume 0.75 (for illustration purposes)
Bond Futures Duration ( $D_{\text{Fut}}$ ): Approximately $8.47 \times 0.75 = 6.35$ years

Other Approaches:

Effective Duration: An alternative measure that considers the bond’s cash flows and the impact of changes in yield.
Excel Functions: Utilize Excel functions such as PV, FV, RATE, and others to streamline the calculation process.