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11-CS-1 Engineering Economics · December 2016

Question 1 of 5: Loan Effective Rates and Payments

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Notes on this paper

National Exams — December 2016 — 11-CS-1 Engineering Economics. Open book; non-communicating calculator permitted. Any four of the five questions constitute a complete paper; all questions are of equal value. Fully worked solutions to all five questions follow; standard compound-interest factors are used and minor rounding is immaterial.

Question 1: Loan Effective Rates and Payments (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

With a nominal rate $r$ compounded $m$ times a year, the effective annual rate is $i_a = (1+r/m)^m - 1$. Carrying at least the six decimals the "2-decimal accuracy" hint demands:

$$\text{F1:}\ \left(1+\tfrac{0.10}{365}\right)^{365}-1 = 0.105156 = \boxed{10.52\%}$$
$$\text{F2:}\ \left(1+\tfrac{0.101}{4}\right)^{4}-1 = 0.104890 = \boxed{10.49\%}$$
$$\text{F3:}\ \left(1+\tfrac{0.102}{2}\right)^{2}-1 = 0.104601 = \boxed{10.46\%}$$

The three effective rates span only 0.0555 of a percentage point, so the ranking survives only if the rates are kept to at least two decimals — rounding each to a whole percent would tie all three at 10%.

(b) Annual Payment on a $400,000 Loan (7 years)

The loan is repaid by seven equal end-of-year payments, so $A = 400{,}000\,(A/P,i_a,7)$ with

$$(A/P,i_a,7) = \frac{i_a(1+i_a)^7}{(1+i_a)^7-1}$$

evaluated at each lender's own effective annual rate (not the nominal rate — the nominal rate is not an annual interest rate at all once compounding is more frequent than yearly):

Lender$i_a$$(A/P,i_a,7)$Annual payment on $400,000
F1 — 10% daily0.1051560.208905$\boxed{\$83{,}562}$
F2 — 10.1% quarterly0.1048900.208724$\boxed{\$83{,}490}$
F3 — 10.2% semi-annual0.1046010.208527$\boxed{\$83{,}411}$

Worked in full for F1: $(1.105156)^7 = 2.013560$, so $(A/P) = \dfrac{0.105156(2.013560)}{1.013560} = 0.208905$ and $A = 400{,}000(0.208905) = \$83{,}562$/yr.

(c) Which to Prefer

A borrower prefers the lowest effective annual rate, and because the same 7-year repayment pattern applies to all three quotes, that is also the lowest annual payment: choose F3 (10.46%, $83,411/yr). It saves $151/yr against F1 and $79/yr against F2 — about $1,058 over the life of the loan.

Note the ranking is the reverse of the nominal rates: F3 quotes the highest nominal rate (10.2%) yet is the cheapest, because it compounds only twice a year while F1's 10% compounds 365 times. Nominal rates quoted at different compounding frequencies are simply not comparable.

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