Betty and Bob buy a 20-year bond with a coupon rate of 7% per annum payable semiannually and a face and redemption value of $1000 and which is priced to yield 10% per annum compounded semiannually.find the Macaulay duration.

To calculate the **Macaulay duration** of the bond, we first need to understand the necessary details of the bond:

- **Coupon rate (C)**: 7% per annum, which is payable semiannually. This gives a semiannual coupon rate of $ \frac{7\%}{2} = 3.5\% $.
- **Face and redemption value (F)**: \$1000.
- **Yield (YTM)**: 10% per annum compounded semiannually. This gives a semiannual yield of $ \frac{10\%}{2} = 5\% $.
- **Term (n)**: 20 years, but since payments are made semiannually, the number of periods is $ 20 \times 2 = 40 $ semiannual periods.

### Step 1: Cash Flows

For each period, Betty and Bob will receive the semiannual coupon payments, and at the final period, they will also receive the redemption value of the bond.

- **Coupon payment**: $ C = 1000 \times 3.5\% = 35 $ (every 6 months).
- **Final cash flow** at maturity (end of 40th period): \$1000 (redemption) + \$35 (last coupon) = \$1035.

### Step 2: Present Value of Cash Flows

We need to discount each cash flow to the present using the yield (5% semiannual). The present value of each coupon payment is calculated as:

\[
PV(CF_t) = \frac{C}{(1 + y)^t}
\]
Where:
- $ C $ is the coupon payment.
- $ y $ is the semiannual yield (5% or 0.05).
- $ t $ is the time period.

The present value of the final payment is:

\[
PV(Final) = \frac{1000 + 35}{(1 + 0.05)^{40}}
\]

### Step 3: Calculate Macaulay Duration

The **Macaulay duration** is a weighted average of the time periods, where the weights are the present values of the cash flows. It is given by:

\[
D = \frac{\sum_{t=1}^{40} t \cdot PV(CF_t)}{Price}
\]

Where:
- $ t $ represents each period.
- $ PV(CF_t) $ is the present value of the cash flows for each period.
- $ Price $ is the current bond price, which can be calculated by summing the present values of all cash flows.

Let me calculate this for you now.The **Macaulay duration** of the bond is approximately **19.42 periods**. Since the bond has semiannual payments, this means the Macaulay duration in years is around **9.71 years** (since $ 19.42 \div 2 = 9.71 $).

Would you like further details on any part of this calculation, or have any other questions?

Here are 5 related questions to expand on this concept:

1. How does the Macaulay duration change if the yield to maturity increases?
2. What is the relationship between Macaulay duration and bond price volatility?
3. How does duration differ between bonds with different coupon rates but the same maturity?
4. What is the difference between Macaulay duration and modified duration?
5. How would Macaulay duration be affected if the bond had annual payments instead of semiannual?

**Tip**: A bond's Macaulay duration decreases as the yield increases, meaning that higher yields shorten the effective life of the bond from an investment perspective.

Betty and Bob buy a 20-year bond with a coupon rate of 7% per annum payable semiannually and a face and redemption value of $1000 and which is priced to yield 10% per annum compounded semiannually. Find the Macaulay duration.

This detailed guide explains how to calculate the Macaulay duration of a 20-year bond with a 7% coupon rate and a yield of 10% compounded semiannually. Learn how to discount cash flows and use the Macaulay duration formula in this step-by-step solution.

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