A court has ordered Security Enterprises to pay 200000 in two years and 500000 in five years. In order to meet this important liability, they wish to invest in a combination of two-year 10% par-value bonds with annual coupons and five-year zero-coupon bonds. Each of these is sold to yield an annual effective yield of 4%. how much of each type of bond should be purchased so that the present value and duration conditions of Redington immunization are satisfied at the current 4% rate? Is the convexity condition also satisfied?
yea
To solve this problem using Redington immunization, we need to find the quantities of two-year par-value bonds and five-year zero-coupon bonds that will satisfy the present value and duration conditions.
Let's denote the quantity of two-year par-value bonds as x and the quantity of five-year zero-coupon bonds as y.
The present value condition requires that the present value of the cash flows from the bond investments matches the present value of the liability.
The present value of the liability can be calculated as follows:
PV_liability = 200,000 / (1 + 0.04)^2 + 500,000 / (1 + 0.04)^5
= 200,000 / 1.0816 + 500,000 / 1.2176
= 185,017.78 + 410,246.47
= 595,264.25
Now let's calculate the present values of the bond investments. Since the par-value bonds have annual coupons, we need to calculate the present value of the annual coupon payments and the present value of the bond's face value.
The present value of the coupon payments for the two-year par-value bonds can be calculated as follows:
PV_coupon_two_year = (x * 0.10 * 100,000) / (1 + 0.04) + (x * 0.10 * 100,000) / (1 + 0.04)^2
= (20,000x) / 1.04 + (20,000x) / 1.0816
= 19,230.77x + 18,438.71x
= 37,669.48x
The present value of the face value for the two-year par-value bonds is:
PV_face_value_two_year = x * 100,000 / (1 + 0.04)^2
= x * 92,592.59
The present value of the five-year zero-coupon bonds is simply:
PV_zero_coupon_five_year = y * 500,000 / (1 + 0.04)^5
= y * 410,246.47
Now let's set up the present value equation:
PV_coupon_two_year + PV_face_value_two_year + PV_zero_coupon_five_year = PV_liability
37,669.48x + x * 92,592.59 + y * 410,246.47 = 595,264.25
Next, we need to satisfy the duration condition. The duration-weighted cash flows from the bond investments should match the duration of the liability.
The duration of the liability can be calculated as follows:
Duration_liability = (2 * 200,000) / (1 + 0.04)^2 + (5 * 500,000) / (1 + 0.04)^5
= 2 * 183,908.05 + 5 * 344,610.89
= 1,704,737.43
The duration of the two-year par-value bonds can be calculated as follows:
Duration_two_year = (2 * 19,230.77x) / (1 + 0.04)^2 + (4 * 18,438.71x) / (1 + 0.04)
= 45,464.41x
The duration of the five-year zero-coupon bonds is simply:
Duration_zero_coupon_five_year = 5 * y
Now let's set up the duration equation:
(45,464.41x + 5y) / (37,669.48x + x * 92,592.59 + y * 410,246.47) = Duration_liability / PV_liability
(45,464.41x + 5y) / (37,669.48x + x * 92,592.59 + y * 410,246.47) = 1,704,737.43 / 595,264.25
Now we have two equations with two unknowns:
37,669.48x + x * 92,592.59 + y * 410,246.47 = 595,264.25
(45,464.41x + 5y) / (37,669.48x + x * 92,592.59 + y * 410,246.47) = 1,704,737.43 / 595,264.25
Solving these equations will give us the quantities of two-year par-value bonds (x) and five-year zero-coupon bonds (y) that will satisfy the present value and duration conditions.
To determine if the convexity condition is satisfied, we need to calculate the convexity-weighted cash flows from the bond investments and check if it matches the convexity of the liability. However, since the problem does not provide any information about the convexity of the bonds or the liability, we cannot determine if the convexity condition is satisfied without additional information.
To answer this question, we need to calculate the amount of each type of bond that should be purchased to satisfy the present value and duration conditions of Redington immunization at the given 4% rate. We will also check whether the convexity condition is satisfied.
1. Present Value Condition:
The present value of the liability can be calculated using the formula for the present value of a future payment:
PV = FV / (1 + r)^n
For the 2-year liability of $200,000, the present value is:
PV1 = 200,000 / (1 + 0.04)^2
For the 5-year liability of $500,000, the present value is:
PV2 = 500,000 / (1 + 0.04)^5
2. Duration Condition:
The duration of a bond portfolio is a weighted average of the durations of its individual components. The duration of a bond can be calculated using the formula:
Duration = Σ [(PV of Cash Flow) * (Time to Cash Flow) / Present Value of Liability]
For the 2-year bond, the duration is:
Duration1 = [(PV1) * 2] / (PV1 + PV2)
For the 5-year zero-coupon bond, the duration is:
Duration2 = [(PV2) * 5] / (PV1 + PV2)
3. Convexity Condition:
To check the convexity condition, we need to calculate the convexity of the bond portfolio, which is also a weighted average of the convexities of its individual components.
Since the 2-year bond has annual coupons, it has some convexity, while the zero-coupon bond has no convexity. So, we only need to calculate the convexity of the 2-year bond using its formula:
Convexity1 = [(PV1) * (2)^2 * (1 + r)^-2] / (PV1 + PV2)
Now, we have the present values, durations, and convexity values for the bond portfolio.
Based on the Redington immunization principle, the amount of each type of bond that should be purchased can be determined by solving a system of equations that satisfy the present value and duration conditions. This process involves adjusting the weights of the bonds until the conditions are met.
If you provide the current prices (par values) of the bonds, I can help you with the calculations.