Understanding Bond Duration in Excel

Bond duration is a measure of how sensitive a bond’s price is to changes in interest rates. It also tells you how long it will take for a bond to pay back its initial cost through its cash flows, such as interest payments and principal repayment.

There are different types of duration, but the most common ones are Macaulay duration and modified duration. Macaulay duration is the weighted average time until you receive all the cash flows of a bond. For example, if a bond pays interest every year and matures in 10 years, its Macaulay duration will be somewhere between 0 and 10 years, depending on the coupon rate and the yield. The higher the coupon rate and the lower the yield, the shorter the Macaulay duration. This is because you will get more money sooner, and thus recover your initial cost faster.

Modified duration is a measure of how much a bond’s price will change if the interest rate changes by 1%. For example, if a bond has a modified duration of 5, it means that if the interest rate goes up by 1%, the bond’s price will go down by 5%. Conversely, if the interest rate goes down by 1%, the bond’s price will go up by 5%. The higher the modified duration, the more volatile the bond’s price is.

Basic Theory:

Bond duration represents the weighted average time it takes for the bond’s cash flows to be repaid. It helps investors assess interest rate risk and make informed investment decisions. Duration is expressed in years and is a crucial metric for understanding how bond prices react to interest rate changes.

The formula for Macaulay duration (D) is given by:

$D = \frac{\sum_{t=1}^{n} \frac{t \cdot CF_t}{(1+y)^t}}{\sum_{t=1}^{n} \frac{CF_t}{(1+y)^t}}$

Where:

$CF_t$ is the cash flow at time $t$ ,
$y$ is the yield to maturity,
$n$ is the number of periods.

Procedures:

Gather Bond Information:
- Bond Face Value
- Coupon Rate
- Yield to Maturity (YTM)
- Number of Periods ( $n$ )
- Frequency of Payments (usually semi-annual)
Calculate Cash Flows:
- For a bond with a coupon, the cash flows include both coupon payments and the final principal repayment.
Set Up Excel Spreadsheet:
- Create a table with columns for period ( $t$ ), cash flows ( $CF_t$ ), and discount factors ( $1/(1+y)^t$ ).
Apply the Duration Formula in Excel:
- Use Excel functions (e.g., SUMPRODUCT) to calculate the weighted average of time and cash flows.

Explanation:

Scenario:

Consider a bond with a face value of $1,000, a 5% coupon rate, maturing in 5 years, and a yield to maturity of 6% with semi-annual payments.

Excel Table:

Period	Coupon Payment	Principal Payment	Total Cash Flow	Discount Factor	Weighted Cash Flow
1	$25	$0	$25	0.9715	$24.2875
2	$25	$0	$25	0.9434	$23.585
3	$25	$0	$25	0.9174	$22.935
4	$25	$0	$25	0.893	$22.325
5	$25	$1,000	$1,025	0.8706	$899.0425

Excel Formula for Duration:

$D = \frac{\text{SUMPRODUCT}(t \cdot CF_t, 1/(1+y)^t)}{\text{SUMPRODUCT}(CF_t, 1/(1+y)^t)}$

Calculation:

$D = \frac{(1\cdot 24.2875) + (2\cdot 23.585) + (3\cdot 22.935) + (4\cdot 22.325) + (5\cdot 899.0425)}{25.2325}$

Result:

The Macaulay duration of the bond is approximately 4.91 years.

Other Approaches:

Modified Duration:
- Adjusts Macaulay duration for changes in yield. It is calculated as $MD = \frac{D}{1 + \frac{y}{m}}$ , where $m$ is the number of compounding periods per year.
Effective Duration:
- Measures the bond’s sensitivity to changes in interest rates by incorporating the bond’s embedded options.
Excel Functions:
- Excel offers built-in functions like DURATION to calculate Macaulay duration and MDURATION for modified duration, simplifying the process.

Understanding Bond Duration in Excel

Basic Theory:

Procedures:

Gather Bond Information:

Calculate Cash Flows:

Set Up Excel Spreadsheet:

Apply the Duration Formula in Excel:

Explanation:

Scenario:

Calculation:

Result:

Other Approaches:

Modified Duration:

Effective Duration:

Excel Functions:

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