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 find the **Macaulay duration** of a bond, we first need to understand the key variables involved in the problem. The Macaulay duration is a weighted average of the times until the bond's cash flows are received, where the weights are proportional to the present value of each cash flow.

### Given Data:
- **Coupon rate** = 7% per annum, payable semiannually, which means the coupon is $ \frac{7\%}{2} = 3.5\% $ every 6 months.
- **Face value / Redemption value** = $1000.
- **Yield to maturity (YTM)** = 10% per annum compounded semiannually, so the semiannual YTM is $ \frac{10\%}{2} = 5\% $.
- **Bond maturity** = 20 years, meaning there are $ 20 \times 2 = 40 $ periods (semiannual periods).
- **Coupon payment** = $ 1000 \times 3.5\% = 35 $ dollars every 6 months.

### Step-by-Step Calculation:

1. **Determine the Present Value of each Coupon Payment**:
   Each coupon payment is discounted at the semiannual yield (5%).
   \[
   PV_{\text{coupon}} = \frac{35}{(1.05)^t}
   \]
   where $ t $ is the semiannual period.

2. **Determine the Present Value of the Face Value**:
   The face value is received at the end of the 40th period, so its present value is:
   \[
   PV_{\text{face value}} = \frac{1000}{(1.05)^{40}}
   \]

3. **Bond Price**:
   The total bond price is the sum of the present values of all the coupon payments and the face value. However, since it's priced to yield 10% compounded semiannually, this should match the bond price under this yield.

4. **Macaulay Duration**:
   The Macaulay duration is calculated as the weighted average of the times until the payments are received, weighted by the present value of the cash flows. It is given by:
   \[
   \text{Macaulay Duration} = \frac{\sum_{t=1}^{40} t \times \frac{35}{(1.05)^t} + 40 \times \frac{1000}{(1.05)^{40}}}{\text{Bond Price}}
   \]

I will now calculate each of these terms for you to determine the Macaulay duration.The **Macaulay duration** of the bond is approximately **19.28 semiannual periods**. Since the bond pays coupons semiannually, this translates to about **9.64 years**.

Would you like further details or have any other questions?

Here are five related questions you might find interesting:
1. What is the difference between Macaulay duration and modified duration?
2. How does increasing the yield to maturity affect the Macaulay duration?
3. Why is the Macaulay duration longer for bonds with lower coupon rates?
4. How is the Macaulay duration used in managing bond portfolios?
5. What impact does reinvestment risk have on duration?

**Tip**: Macaulay duration provides an estimate of the bond's sensitivity to interest rate changes, but modified duration gives a more direct measure of price change.

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.

Learn how to calculate the Macaulay duration for a 20-year bond with a 7% coupon rate, priced to yield 10% per annum compounded semiannually. This problem explores key financial concepts like present value and bond valuation. Suitable for college-level finance students.

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