Question
calculate duration or effective maturity on a $1000 ten-year zero coupon bond when its interest rate is 20% and cash payment of $100 per year. explain the result
Answers
GPT-4o mini
To calculate the effective duration or effective maturity of a cash flow series, we will treat the ten-year zero-coupon bond and the annual cash payments of $100 separately. However, the effective duration of such a scenario involves measuring the sensitivity of the bond's price to interest rate changes and determining the weighted average time until cash flows are received.
**Step 1: Determine the Present Value (PV) of Cash Flows**
1. The zero-coupon bond has a face value (FV) of $1,000 and matures in 10 years.
2. The market interest rate is 20%, which means we will use this to discount future cash flows.
The present value of the zero-coupon bond is calculated as:
\[
PV = \frac{FV}{(1 + r)^n} = \frac{1000}{(1 + 0.20)^{10}}
\]
Calculating that:
\[
PV = \frac{1000}{(1.20)^{10}} \approx \frac{1000}{6.191736422} \approx 161.51
\]
3. For the $100 annual cash payments, we need to calculate the present value of an annuity. The formula for the present value of an annuity is:
\[
PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r
\]
where:
- \( PMT \) = Payment per period ($100),
- \( r \) = interest rate (0.20),
- \( n \) = number of periods (10).
\[
PV = 100 \times \frac{1 - (1.20)^{-10}}{0.20}
\]
Calculating that:
\[
PV \approx 100 \times \frac{1 - 0.1615}{0.20} \approx 100 \times \frac{0.8385}{0.20} \approx 419.25
\]
4. Add the present values of the zero-coupon bond and the annuity:
\[
Total \, PV = 161.51 + 419.25 \approx 580.76
\]
**Step 2: Calculate the Effective Duration**
Effective Duration can be approximated by the weighted average time until cash flows are received, adjusted for present value contributions. However, for simplicity, you can also compute it using cash flow timing and calculating the weighted average of the cash flow present ages.
However, more commonly for zero-coupon bonds or similar structures, a modified duration calculation is used that considers how sensitive the bond's price is to a change in yield.
If we take the concept and treat the combination with cash flows, the average time until cash flows occur, weighted by the present values of each payment, would provide insight into maturity rather than strict duration.
1. **Cash flows occur:**
- Cash payments occur at end of every year for 10 years ($100 each).
- The bond matures at year 10 ($1,000).
While exact cash flow sensitivity can lead to complex calculations often involving differential present value changes, for practical purposes:
- Effective maturity can be approximated by summing the time-weighted present value contributions.
### Explanation of Results
1. The total present value combines payments from the bond and annuity effectively reflecting a yield to modifications in the 20% rate.
2. Effective maturity here would illustrate not only 10 underlying years from the bond but also transitional cash from the annual payments leading towards the bond end.
Thus, the effective maturity or duration can indeed be closer to 10 years, but with an adjustment since you're also including a series of intermediate cash flows which reduce effective duration in terms of sensitivity to interest rate changes.
This scenario portrays the interplay of long vs. short-term cash flows and provides insight into the bond's structure concerning duration within a high yield rate environment.
**Step 1: Determine the Present Value (PV) of Cash Flows**
1. The zero-coupon bond has a face value (FV) of $1,000 and matures in 10 years.
2. The market interest rate is 20%, which means we will use this to discount future cash flows.
The present value of the zero-coupon bond is calculated as:
\[
PV = \frac{FV}{(1 + r)^n} = \frac{1000}{(1 + 0.20)^{10}}
\]
Calculating that:
\[
PV = \frac{1000}{(1.20)^{10}} \approx \frac{1000}{6.191736422} \approx 161.51
\]
3. For the $100 annual cash payments, we need to calculate the present value of an annuity. The formula for the present value of an annuity is:
\[
PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r
\]
where:
- \( PMT \) = Payment per period ($100),
- \( r \) = interest rate (0.20),
- \( n \) = number of periods (10).
\[
PV = 100 \times \frac{1 - (1.20)^{-10}}{0.20}
\]
Calculating that:
\[
PV \approx 100 \times \frac{1 - 0.1615}{0.20} \approx 100 \times \frac{0.8385}{0.20} \approx 419.25
\]
4. Add the present values of the zero-coupon bond and the annuity:
\[
Total \, PV = 161.51 + 419.25 \approx 580.76
\]
**Step 2: Calculate the Effective Duration**
Effective Duration can be approximated by the weighted average time until cash flows are received, adjusted for present value contributions. However, for simplicity, you can also compute it using cash flow timing and calculating the weighted average of the cash flow present ages.
However, more commonly for zero-coupon bonds or similar structures, a modified duration calculation is used that considers how sensitive the bond's price is to a change in yield.
If we take the concept and treat the combination with cash flows, the average time until cash flows occur, weighted by the present values of each payment, would provide insight into maturity rather than strict duration.
1. **Cash flows occur:**
- Cash payments occur at end of every year for 10 years ($100 each).
- The bond matures at year 10 ($1,000).
While exact cash flow sensitivity can lead to complex calculations often involving differential present value changes, for practical purposes:
- Effective maturity can be approximated by summing the time-weighted present value contributions.
### Explanation of Results
1. The total present value combines payments from the bond and annuity effectively reflecting a yield to modifications in the 20% rate.
2. Effective maturity here would illustrate not only 10 underlying years from the bond but also transitional cash from the annual payments leading towards the bond end.
Thus, the effective maturity or duration can indeed be closer to 10 years, but with an adjustment since you're also including a series of intermediate cash flows which reduce effective duration in terms of sensitivity to interest rate changes.
This scenario portrays the interplay of long vs. short-term cash flows and provides insight into the bond's structure concerning duration within a high yield rate environment.