Find the duration of a 6% coupon bond making annual coupon payments if it has three years until maturity and a yield to maturity of 6.5%. What is the duration if the yield to maturity is 10.5%?
To find the duration of a bond, you need to calculate the present value of each cash flow (coupon payment and principal payment) and then weight them by their respective time to receive those cash flows.
Let's start with the first scenario where the yield to maturity is 6.5%. Given that, we will use the following formula to calculate the present value of a bond's cash flow:
PV = C / (1 + r)^n
Where:
PV is the present value
C is the cash flow (coupon payment or principal payment)
r is the discount rate (yield to maturity)
n is the time period until the cash flow is received
In this scenario:
Coupon payment (C) = 6% of the bond's face value (also known as the coupon rate)
Discount rate (r) = 6.5% (yield to maturity)
Time period (n) = 1 year for each of the three annual coupon payments
Now, we will calculate the present value of each cash flow using the above formula and then weight them by their respective time to receive those cash flows.
1st year coupon payment:
PV1 = (0.06 * Face Value) / (1 + 0.065)^1
2nd year coupon payment:
PV2 = (0.06 * Face Value) / (1 + 0.065)^2
3rd year coupon payment:
PV3 = (0.06 * Face Value) / (1 + 0.065)^3
Next, we will calculate the weight of each cash flow by dividing the present value of each cash flow by the present value of the bond. The present value of the bond is the sum of the present values of all cash flows:
Weight of 1st year cash flow = PV1 / (PV1 + PV2 + PV3)
Weight of 2nd year cash flow = PV2 / (PV1 + PV2 + PV3)
Weight of 3rd year cash flow = PV3 / (PV1 + PV2 + PV3)
Finally, we will calculate the duration of the bond by multiplying the time period of each cash flow by its corresponding weight and summing them up:
Duration = (1 * Weight of 1st year cash flow) + (2 * Weight of 2nd year cash flow) + (3 * Weight of 3rd year cash flow)
Repeat the same process for the second scenario where the yield to maturity is 10.5%.
Note: The face value of the bond is not provided in the question, so you would need that information to compute the actual duration.