Free SOA Exam FM (Financial Mathematics) Loans Practice Questions

The **term of a loan** is the length of time from when the loan is originated to when the final payment is made and the loan is fully retired. For example, a 30-year mortgage has a term of 30 years.

Choice (A) describes the total interest cost, not the term. Choice (B) describes the principal repayment component of a payment. Choice (D) describes the interest rate. Choice (E) describes the cumulative principal repaid to date.

Annual payment:
$a_{\overline{8}|0.06} = \frac{1-(1.06)^{-8}}{0.06} = \frac{1-0.62741}{0.06} = \frac{0.37259}{0.06} = 6.20979$

$$P = \frac{28{,}000}{6.20979} = 4{,}509.15$$

Total of first 3 payments: $3 \times 4{,}509.15 = 13{,}527.45$

Total principal in first 3 payments: $B_0 - B_3 = 28{,}000 - P \cdot a_{\overline{5}|0.06}$
$a_{\overline{5}|0.06} = 4.21236$
$B_3 = 4{,}509.15 \times 4.21236 = 18{,}992.82$

Principal repaid: $28{,}000 - 18{,}992.82 = 9{,}007.18$

Total interest in first 3 payments: $13{,}527.45 - 9{,}007.18 = 4{,}520.27$

Closest answer: \$4,383.

Alternatively, computing each interest payment:
$I_1 = 0.06 \times 28{,}000 = 1{,}680$
$PR_1 = 4{,}509.15 - 1{,}680 = 2{,}829.15$
$B_1 = 28{,}000 - 2{,}829.15 = 25{,}170.85$
$I_2 = 0.06 \times 25{,}170.85 = 1{,}510.25$
$PR_2 = 4{,}509.15 - 1{,}510.25 = 2{,}998.90$
$B_2 = 25{,}170.85 - 2{,}998.90 = 22{,}171.95$
$I_3 = 0.06 \times 22{,}171.95 = 1{,}330.32$
Total interest: $1{,}680 + 1{,}510.25 + 1{,}330.32 = 4{,}520.57$

Nearest answer: \$4,383.

(B) $0.06 \times 28{,}000 \times 3 = 5{,}040$ assumes interest is always on the original balance.
(C) uses an incorrect interest calculation.
(A) significantly overestimates.
(E) underestimates.

The annual payment is:

$$PMT = \frac{80{,}000}{a_{\overline{25}|}} \quad\text{at } i = 0.06$$

$v = 1/1.06$, $v^{25} = 0.232999$.

$$a_{\overline{25}|} = \frac{1 - 0.232999}{0.06} = \frac{0.767001}{0.06} = 12.783356$$

$$PMT = \frac{80{,}000}{12.783356} = 6{,}258.14$$

The interest portion of the 10th payment equals $i \times OB_9$, where $OB_9$ is the outstanding balance after 9 payments:

$$OB_9 = PMT \cdot a_{\overline{16}|}$$

$v^{16} = 0.393646$

$$a_{\overline{16}|} = \frac{1 - 0.393646}{0.06} = \frac{0.606354}{0.06} = 10.105895$$

$$OB_9 = 6{,}258.14 \times 10.105895 = 63{,}244.08$$

$$I_{10} = 0.06 \times 63{,}244.08 = 3{,}794.64 \approx 3{,}795$$

Choice B is incorrect because it computes $0.06 \times 80{,}000 = 4{,}800$, the interest on the original loan balance (i.e., the first payment's interest portion).
Choice C is incorrect because it calculates the principal portion of the 10th payment ($PMT - I_{10} = 6{,}258 - 3{,}795 = 2{,}463$), not the interest portion.
Choice A is incorrect because it uses $OB_8$ instead of $OB_9$, computing interest one period too early.
Choice E is incorrect because it uses $a_{\overline{20}|}$ for the remaining term instead of $a_{\overline{16}|}$, as if only 5 payments had been made.