# Macaulay Duration Calculation for a 2-Year Bond Given: - **Par value (\(F\))**: \$100 - **Coupon rate**: 8% (paid semi-annually) - **Coupon payment (\(C\))**: \(.08 \times 100 / 2 = \$40\) per period - **Number of periods (\(n\))**: 2 years × 2 = 4 periods - **Yield to maturity (YTM)**: 6% per year = 3% per period (compounded semi-annually) ## Step 1: Calculate Present Value of Each Cash Flow | Period (\(t\)) | Cash Flow (\(CF_t\)) | PV Factor \((1 + .03)^{-t}\) | PV of CF (\(CF_t \times PV\)) | |:--------------:|:-------------------:|:-----------------------------:|:-----------------------------:| | 1 | \$40 | \(1.03^{-1} = .9709\) | \$38.84 | | 2 | \$40 | \(1.03^{-2} = .9426\) | \$37.70 | | 3 | \$40 | \(1.03^{-3} = .9151\) | \$36.60 | | 4 | \$104 | \(1.03^{-4} = .8885\) | \$924.03 | **Total Present Value (Bond Price):** \[ \text{Price} = 38.84 + 37.70 + 36.60 + 924.03 = \$1037.17 \] ## Step 2: Calculate Macaulay Duration \[ D = \frac{\sum_{t=1}^{n} t \times PV(CF_t)}{\text{Bond Price}} \] \[ \begin{align*} \text{Numerator} &= 1 \times 38.84 + 2 \times 37.70 + 3 \times 36.60 + 4 \times 924.03 \\ &= 38.84 + 75.40 + 109.80 + 3696.12 \\ &= 392.16 \end{align*} \] \[ D = \frac{392.16}{1037.17} = 3.78 \text{ semi-annual periods} \] **Convert to years:** \[ \text{Macaulay Duration (years)} = \frac{3.78}{2} = 1.89 \text{ years} \] ## Step 3: Select the Correct Answer - **1.89 years** is the Macaulay duration. --- ## **Final Answer** > **Macaulay duration of the bond is: 1.89 years.** --- **Correct choice:** `1.89 years`

# Macaulay Duration Calculation for a 2-Year Bond ## Objective Determine the Macaulay duration of a 2-year bond with a par value of $1000, an 8% coupon rate paid semi-annually, and a YTM of 6%. ## Given Data - **Par value (\(F\))**: \$1000 - **Coupon rate**: 8% - **Coupon payment (\(C\))**: \[ C = \frac{0.08 \times 1000}{2} = \$40 \text{ (per period)} \] - **Number of periods (\(n\))**: \[ n = 2 \text{ years} \times 2 = 4 \text{ periods} \] - **Yield to maturity (YTM)**: \[ YTM = 0.06 \text{ per year} = 0.03 \text{ per period (semi-annually)} \] ## Step 1: Calculate Present Value of Each Cash Flow | Period (\(t\)) | Cash Flow (\(CF_t\)) | PV Factor \((1 + 0.03)^{-t}\) | PV of Cash Flow (\(CF_t \times PV\)) | |:--------------:|:-------------------:|:------------------------------:|:------------------------------------:| | 1 | \$40 | \(1.03^{-1} = 0.9709\) | \$38.84 | | 2 | \$40 | \(1.03^{-2} = 0.9426\) | \$37.70 | | 3 | \$40 | \(1.03^{-3} = 0.9151\) | \$36.60 | | 4 | \$104 | \(1.03^{-4} = 0.8885\) | \$924.03 | **Total Present Value (Bond Price):** \[ \text{Price} = 38.84 + 37.70 + 36.60 + 924.03 = \$1037.17 \] ## Step 2: Calculate Macaulay Duration The formula for Macaulay duration (\(D\)) is: \[ D = \frac{\sum_{t=1}^{n} t \times PV(CF_t)}{\text{Bond Price}} \] ### Numerator Calculation \[ \begin{align*} \text{Numerator} &= 1 \times 38.84 + 2 \times 37.70 + 3 \times 36.60 + 4 \times 924.03 \\ &= 38.84 + 75.40 + 109.80 + 3696.12 \\ &= 392.16 \end{align*} \] ### Duration Calculation \[ D = \frac{392.16}{1037.17} = 0.378 \text{ semi-annual periods} \] ## Convert to Years \[ \text{Macaulay Duration (years)} = \frac{3.78}{2} = 1.89 \text{ years} \] ## Final Answer > **Macaulay duration of the bond is: 1.89 years.**

# Macaulay Duration Calculation for a 2-Year Bond ## Introduction Macaulay duration is a key measure of the sensitivity of a bond's price to interest rate changes, providing investors with an estimate of the bond's weighted average time to receive cash flows. It is particularly important in fixed-income investing, where understanding the time value of money and the timing of cash flows is crucial. This calculation involves determining the present value of each cash flow and then weighting these values by the time until they are received, ultimately allowing for an assessment of interest rate risk. ### Explanation The introduction clarifies the purpose and significance of Macaulay duration in bond valuation. Understanding this concept is essential before delving into the calculations, as it establishes the framework for assessing a bond's risk and return profile. ## Relevant Formulas 1. **Present Value of Cash Flows**: \[ PV(CF_t) = CF_t \times (1 + YTM)^{-t} \] - **Description**: This formula calculates the present value of each cash flow (\(CF_t\)) received at time \(t\) based on the yield to maturity (YTM). 2. **Macaulay Duration**: \[ D = \frac{\sum_{t=1}^{n} t \times PV(CF_t)}{\text{Bond Price}} \] - **Description**: This formula computes the weighted average time until cash flows are received, with weights being the present values of the cash flows. ### Explanation These formulas are fundamental in the calculation of Macaulay duration. The first formula allows for the conversion of future cash flows into present values, which are essential for determining how long an investor will have to wait until they receive their payments. The second formula aggregates these present values over time, providing a comprehensive measure of duration. ## Step-by-Step Solution ### Step 1: Given Data - **Par value (\(F\))**: \$1000 - **Coupon rate**: 8% - **Coupon payment (\(C\))**: \[ C = \frac{0.08 \times 1000}{2} = \$40 \text{ (per period)} \] - **Number of periods (\(n\))**: \[ n = 2 \text{ years} \times 2 = 4 \text{ periods} \] - **Yield to maturity (YTM)**: \[ YTM = 0.06 \text{ per year} = 0.03 \text{ per period (semi-annually)} \] ### Explanation The initial step involves clearly defining the parameters necessary for the calculations, including the coupon payments and the number of periods based on the bond's life. This information lays the groundwork for subsequent calculations. ### Step 2: Calculate Present Value of Each Cash Flow | Period (\(t\)) | Cash Flow (\(CF_t\)) | PV Factor \((1 + 0.03)^{-t}\) | PV of Cash Flow (\(CF_t \times PV\)) | |:--------------:|:-------------------:|:------------------------------:|:------------------------------------:| | 1 | \$40 | \(1.03^{-1} = 0.9709\) | \$38.84 | | 2 | \$40 | \(1.03^{-2} = 0.9426\) | \$37.70 | | 3 | \$40 | \(1.03^{-3} = 0.9151\) | \$36.60 | | 4 | \$104 | \(1.03^{-4} = 0.8885\) | \$924.03 | **Total Present Value (Bond Price)**: \[ \text{Price} = 38.84 + 37.70 + 36.60 + 924.03 = \$1037.17 \] ### Explanation Calculating the present value of each cash flow involves applying the present value formula for each cash flow at its respective period. The resulting values are then summed to determine the total price of the bond. This is crucial as it reflects the bond's market value based on current interest rates. ### Step 3: Calculate Macaulay Duration Using the formula for Macaulay duration: \[ D = \frac{\sum_{t=1}^{n} t \times PV(CF_t)}{\text{Bond Price}} \] #### Numerator Calculation \[ \begin{align*} \text{Numerator} &= 1 \times 38.84 + 2 \times 37.70 + 3 \times 36.60 + 4 \times 924.03 \\ &= 38.84 + 75.40 + 109.80 + 3696.12 \\ &= 392.16 \end{align*} \] ### Explanation This step aggregates the weighted present values of cash flows by multiplying each cash flow's present value by the time until it is received. The result forms the numerator in the duration formula, which is essential for determining how long, on average, cash flows will be received. ### Step 4: Duration Calculation \[ D = \frac{392.16}{1037.17} = 0.378 \text{ semi-annual periods} \] ### Convert to Years \[ \text{Macaulay Duration (years)} = \frac{3.78}{2} = 1.89 \text{ years} \] ### Explanation The final calculations yield the Macaulay duration by dividing the sum of the weighted present values by the bond price. Converting the semi-annual duration to years provides a more intuitive understanding of the bond's sensitivity to interest rate changes. ## Conclusion The Macaulay duration of the bond has been determined to be **1.89 years**. This value provides investors with insight into the bond's interest rate risk and cash flow timing.