Final answer:
To find the term n of Joey's bond, we calculate the present value of quarterly coupon payments and the redemption amount, considering the given yield rate. We then compare the priced difference between Joey's and Todd's bonds to solve for n, ensuring that 4n is an integer.
Step-by-step explanation:
The question involves calculating the value of two bonds with different terms but identical coupon rates and yield rates. We must determine the term n of the bond that Joey purchased, given the price difference of $233.02 with Todd's bond having a term of 2n years. Both bonds have a coupon rate of 9% convertible quarterly and a yield rate of 6% convertible quarterly.
Present Value of Coupon Payments
First, we calculate the value of the coupon payments. With par value of $2,000 and a 9% annual coupon rate convertible quarterly, each coupon payment is $2000 × 0.09 / 4 = $45. The present value of these payments over n years is calculated using the formula for the present value of an annuity:
PV = C × [(1 - (1 + r)^(-nt)) / r]
Where:
PV is present value of the annuity
C is the coupon payment per period
r is the interest rate per period
t is the number of periods per year
n is the number of years
Present Value of Redemption Amount
The present value of the redemption amount is the par value discounted back to the present using the formula:
PV = F / (1 + r)^(nt)
Where:
F is the face (or par) value of the bond
r, t, and n are as described above
After calculating these present values, we can set up an equation based on the provided information that Todd paid $233.02 more than Joey. Using that information and the provided formulas, we solve for n, ensuring that 4n is an integer as given in the question.
Yield and Price Relationship
The relationship between yield rates and bond prices is inversely related. When interest rates rise, the value of previously issued bonds with lower rates falls, and conversely, when rates fall, the value of bonds with higher coupon rates increases as seen in the given examples.