Final answer:
To choose the best borrowing option, we need to compare the effective annual rates (EAR) of each. After calculating the EAR for each rate, the preferred option is the one with the lowest EAR, which is the nominal rate compounded quarterly (d(4) at 4.08%).
Step-by-step explanation:
The question at hand involves choosing between different interest rate options for borrowing money. The options are a nominal rate compounded semiannually (i(2) = 6%), a nominal rate compounded quarterly (d(4) = 4%), and a force of interest (δ) of 9%. To determine the preferred option, we need to compare the effective annual rates (EAR) for each option, as they reflect the actual annual cost of borrowing when compounding is taken into account.
Effective annual rate for i(2): Since the nominal rate is compounded semiannually, the EAR can be calculated using the formula EAR = (1 + i/n)^(n) - 1, where i is the nominal rate and n is the number of compounding periods per year. Therefore, for i(2) = 6%, EAR = (1 + 0.06/2)^(2) - 1 = 6.09%.
Effective annual rate for d(4): The nominal rate compounded quarterly is d(4) = 4%, with d representing the nominal discount rate. The EAR for this option can be calculated as EAR = (1 - d/n)^(-n) - 1. Therefore, EAR = (1 - 0.04/4)^(-4) - 1 = 4.08%.
Effective annual rate for force of interest: With a force of interest at δ = 9%, the EAR is calculated as EAR = e^(δ) - 1. Therefore, the EAR for the force of interest option is e^(0.09) - 1 = 9.42%.
In conclusion, the preferred option would be the one with the lowest EAR, which is the nominal rate compounded quarterly, d(4) at 4.08%.