Final answer:
Eugene purchases a 15-year, $10,000 par value bond with 5% semi-annual coupons for $9,500. He reinvests the coupon payments at an annual nominal interest rate of 7% compounded semi-annually. The annual nominal yield compounded semi-annually that Eugene earns over the 15-year investment is approximately 63.16%.
Step-by-step explanation:
Eugene purchases a 15-year, $10,000 par value bond with 5% semi-annual coupons for $9,500. He is able to reinvest the coupon payments at an annual nominal interest rate of 7% compounded semi-annually. To calculate the annual nominal yield compounded semi-annually that Eugene earns over the 15-year investment, we need to find the present value of the bond, which is the price Eugene paid for it. Then, we can use the present value and the future value of the bond (which is the sum of the face value and all the coupon payments) to calculate the nominal yield compounded semi-annually.
First, we find the present value of the bond using the formula:
PV = C/(1+r/2) + C/(1+r/2)^2 + ... + C/(1+r/2)^n + F/(1+r/2)^n
Where PV is the present value, C is the coupon payment, r is the interest rate per period (compounded semi-annually), n is the number of periods, and F is the face value of the bond. In this case, C = $500 (5% of $10,000), r = 0.035 (7% divided by 2), n = 30 (15 years multiplied by 2), and F = $10,000.
Plugging in the values:
PV = 500/(1+0.035/2) + 500/(1+0.035/2)^2 + ... + 500/(1+0.035/2)^30 + 10000/(1+0.035/2)^30
Solving this equation will give the present value of the bond, which is $9,500 (the price Eugene paid).
Next, we can calculate the future value of the bond at the end of the 15-year investment period. The future value includes the face value of the bond ($10,000) and the sum of all the coupon payments. The coupon payments can be calculated using the formula:
C = F * r/2
Where C is the coupon payment, F is the face value of the bond, and r is the interest rate per period (compounded semi-annually). In this case, C = $500 (5% of $10,000) and r = 0.035 (7% divided by 2).
So the sum of all the coupon payments is:
(500 * 30) + 10000 = $25,000
Finally, we can use the present value and the future value to calculate the nominal yield compounded semi-annually:
Yield = (Future Value - Present Value) / Present Value
Yield = (25000 - 9500) / 9500
Yield = 1.6316
So the annual nominal yield compounded semi-annually that Eugene earns over the 15-year investment is approximately 63.16%.