Bank-Loan Effective Interest Rates: 23-CS-1 Engineering Economics, December 2014, Q1

Question 1: Bank-Loan Effective Interest Rates (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Effective Annual Rates

\text{A:}\ \left(1+\tfrac{0.10}{366}\right)^{366}-1 = \boxed{10.52\%};\quad \text{B:}\ \left(1+\tfrac{0.1025}{52}\right)^{52}-1 = \boxed{10.78\%};\quad \text{C:}\ \left(1+\tfrac{0.105}{12}\right)^{12}-1 = \boxed{11.02\%}

(b) Effective Quarterly Rates

Using $i_q = (1+i_a)^{1/4}-1$:

\text{A:}\ \boxed{2.53\%};\quad \text{B:}\ \boxed{2.59\%};\quad \text{C:}\ \boxed{2.65\%}

(c) Which Bank

These are loan (borrowing) rates, so the business wants the lowest effective cost, and the comparison must be made on the effective rates of part (a), never on the quoted nominal rates. Bank A is cheapest at 10.52% effective, against 10.78% for Bank B and 11.02% for Bank C, so choose Bank A. Over a one-year $100,000 loan that is about $10,516 of interest with Bank A against $10,783 with Bank B and $11,020 with Bank C, a spread of roughly $500. The ranking happens to follow the nominal rates here only because all three compounding frequencies add a similar amount (0.52, 0.53 and 0.52 percentage points respectively); had Bank A quoted 10.5% compounded daily it would have reached 11.07% effective and lost to Bank C's monthly 10.5%, which is why nominal rates quoted on different compounding periods can never be compared directly.

(d) Break-Even Rate for Bank A

\left(1+\tfrac{r}{366}\right)^{366} = 1.10783 \;\Rightarrow\; r \approx \boxed{10.24\%\ \text{compounded daily}}

At about 10.24% compounded daily, Bank A would match Bank B's 10.78% effective annual rate.