Interest and Investment Costs: 23-Chem-A5 Chemical Plant Design and Economics, December 2013, Q4

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.

(i) Nominal versus effective interest rate

The nominal annual interest rate is the stated or "quoted" yearly rate, obtained by simply multiplying the rate per compounding period by the number of periods in a year — it ignores the interest-on-interest that occurs when compounding happens more than once a year. The effective annual interest rate is the rate that actually applies over a year once that intra-year compounding is accounted for; it is the true annual growth of the debt. They are related by $i_\text{eff}=\left(1+\dfrac{i_\text{nom}}{m}\right)^{m}-1$, where $m$ is the number of compounding periods per year. The two are equal only when compounding is annual ($m=1$); for any $m>1$ the effective rate exceeds the nominal rate. In part (iii) the 5.2% is already stated as an effective annual rate, so it is applied directly once per year.

Given. Principal $P=\$18{,}000$; first term at simple interest $i_s=3.9\%$ for $n_1=3$ yr; nothing repaid; then extended $n_2=3$ yr at effective compound $i_c=5.2\%$/yr.

Find. (ii) the amount owed after the first 3 years, and (iii) the total owed at the end of the full 6-year period.

Approach. Apply the simple-interest law over the first term, then compound that amount forward over the three-year extension at the effective annual rate.

(ii) Amount owed after 3 years — simple interest. Under simple interest the interest is charged only on the original principal each year, so $$F_3=P\,(1+i_s\,n_1)=\$18{,}000\,[\,1+(0.039)(3)\,]=\$18{,}000(1.117)=\boxed{\$20{,}106}$$ The debt after three years is $20,106 (that is $2,106 of accrued interest on the $18,000 principal).
(iii) Compound the balance over the 3-year extension. The unpaid balance $F_3$ now grows at the effective compound rate, so interest is earned on interest: $$F_6=F_3\,(1+i_c)^{n_2}=\$20{,}106\,(1.052)^3$$
Evaluate. The growth factor is $(1.052)^3=1.164253$, hence $$F_6=\$20{,}106\,(1.164253)=\boxed{\$23{,}408.46}$$ The total amount owed at the end of the six-year loan period is about $23,408.

Quantity	Value
Accrued simple interest (yr 1–3)	$2,106
Amount owed after first 3 years	$20,106.00
Compound growth factor $(1.052)^3$	1.164253
Total owed at end of 6 years	$23,408.46

Concept & Theory

Topic. Simple versus compound interest and the nominal/effective distinction in loan accounting.

Key relations. Simple interest: $F=P(1+in)$; compound interest: $F=P(1+i)^n$; nominal-to-effective: $i_\text{eff}=(1+i_\text{nom}/m)^m-1$.

Why this works. Simple interest charges only on the original principal, so the debt grows linearly; compound interest charges on the accumulated balance, so it grows geometrically. Because the first term here is explicitly simple and the extension explicitly compound (and stated as an effective rate), each law is applied over its own term and the results chained.

Common pitfall. Compounding the first three years when the problem says simple interest, or treating the 5.2% as nominal and compounding it more than once a year — it is already an effective annual rate, applied once per year. A quick check: had all six years been at 3.9% simple, the debt would be only $\$18{,}000(1+0.039\times6)=\$22{,}212$, less than the compound result — as expected.

Source. Peters, Timmerhaus & West, Ch. 7 "Interest and Investment Costs" (simple, compound, nominal, and effective interest).

Question 4 of 6: Interest and Investment Costs

Question 4: Interest and Investment Costs (20 marks)

(i) Nominal versus effective interest rate