Question 4 of 6: Interest and Investment Costs

Question 4: Interest and Investment Costs (20 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.

Given. Principal $P = \$21{,}000$; interest rate per month $i_m = 2.4\% = 0.024$; term $= 3$ years $= 36$ months; no intermediate payments.

Find. (i) amount due under simple interest; (ii) amount due under monthly compounding; (iii) the nominal and effective annual rates for monthly compounding.

Approach. Apply the simple-interest law for (i), the monthly-compound-interest law for (ii), and the nominal/effective conversion for (iii); the difference between (i) and (ii) is precisely the interest-on-interest that compounding adds.

1. (i) Simple interest over 36 months. Simple interest accrues only on the original principal, so $$F_\text{simple} = P\,(1 + i_m\,N) = \$21{,}000\,[\,1 + (0.024)(36)\,] = \$21{,}000(1.864) = \boxed{\$39{,}144}$$ The accrued simple interest is $\$18{,}144$.
2. (ii) Compound interest, compounded monthly. Now interest is charged on the growing balance: $$F_\text{comp} = P\,(1 + i_m)^{N} = \$21{,}000\,(1.024)^{36} = \$21{,}000(2.34854) = \boxed{\$49{,}319}$$ Compounding adds $\$49{,}319-\$39{,}144 = \$10{,}175$ more than simple interest over the same three years — the cost of leaving interest unpaid.
3. (iii) Nominal annual rate. The nominal rate simply annualises the periodic rate by the number of periods per year: $$i_\text{nom} = m\,i_m = 12(0.024) = \boxed{28.8\%}$$
4. (iii) Effective annual rate. The effective rate accounts for the intra-year compounding: $$i_\text{eff} = (1 + i_m)^{m} - 1 = (1.024)^{12} - 1 = 1.32922 - 1 = \boxed{32.92\%}$$ The effective rate exceeds the nominal rate because the 2.4 % is itself compounded twelve times a year.