Free SOA Exam FM (Financial Mathematics) Annuities and Non-Contingent Cash Flows Practice Questions

E is correct.

The **term** of an annuity is the duration of time from when payments begin to when they end, typically measured by the number of payment periods.

B is correct.

The payments are 24 quarterly payments of $250 at the end of each quarter, and the 8% is quoted as an EFFECTIVE ANNUAL rate. The rate must therefore be converted geometrically to a quarterly effective rate before any annuity factor is used.

Convert the rate:

$$j = (1.08)^{1/4} - 1 = 1.0194265 - 1 = 0.0194265$$

Count the payments: $n = 6 \times 4 = 24$.

Discount factor over the full term:

$$(1+j)^{24} = (1.08)^{6} = 1.586874 \qquad v^{24} = \frac{1}{1.586874} = 0.630170$$

Annuity-immediate factor:

$$a_{\overline{24}|j} = \frac{1 - v^{24}}{j} = \frac{1 - 0.630170}{0.0194265} = \frac{0.369830}{0.0194265} = 19.03737$$

Present value:

$$PV = 250 \cdot a_{\overline{24}|j} = 250 \times 19.03737 = 4{,}759.34$$

Check with the m-thly annuity form. The quarterly payments total $1,000 per year, and

$$i^{(4)} = 4\left[(1.08)^{1/4} - 1\right] = 4(0.0194265) = 0.0777062$$

$$a_{\overline{6}|0.08} = \frac{1 - (1.08)^{-6}}{0.08} = \frac{0.369830}{0.08} = 4.622880$$

$$PV = 1{,}000 \cdot a_{\overline{6}|0.08}^{(4)} = 1{,}000 \cdot \frac{i}{i^{(4)}} \cdot a_{\overline{6}|0.08} = 1{,}000 \times \frac{0.08}{0.0777062} \times 4.622880 = 4{,}759.34$$

Both routes give $PV \approx 4{,}759$, so the answer is B.

Traps to avoid. Setting the quarterly rate to $0.08/4 = 0.02$ instead of taking the fourth root gives $250 \times a_{\overline{24}|0.02} = 4{,}728$. Valuing $1,000 once a year with no m-thly adjustment gives $1{,}000 \times a_{\overline{6}|0.08} = 4{,}623$, which prices annual rather than quarterly cash flows. Using an annuity-due factor gives $1{,}000 \times \ddot{a}_{\overline{6}|0.08} = 4{,}993$, stretching the term to seven years gives $5{,}206$, and accumulating instead of discounting gives $1{,}000 \times s_{\overline{6}|0.08} = 7{,}336$.

E is correct.

Split into two parts:

Part 1: Payments at times 1-10, valued at 5%.
$PV_1 = 600 \cdot a_{\overline{10}|0.05}$
$v^{10}_{5\%} = (1.05)^{-10} = 0.61391$
$a_{\overline{10}|0.05} = \frac{1 - 0.61391}{0.05} = 7.72173$
$PV_1 = 600 \times 7.72173 = 4{,}633$

Part 2: Payments at times 11-20. First find their value at time 10:
$PV_{10} = 600 \cdot a_{\overline{10}|0.07}$
$v^{10}_{7\%} = (1.07)^{-10} = 0.50835$
$a_{\overline{10}|0.07} = \frac{1 - 0.50835}{0.07} = 7.02358$
$PV_{10} = 600 \times 7.02358 = 4{,}214$

Discount to time 0 at 5%:
$PV_2 = 4{,}214 \times (1.05)^{-10} = 4{,}214 \times 0.61391 = 2{,}587$

Total: $PV = 4{,}633 + 2{,}587 = 7{,}220$

The closest answer is 7,211.