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.

A 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:

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

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 A.

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%.
$P{V}_1=600\cdot a_{\overline{10}|0.05}$
$v_{5\%}^{10}=(1.05{)}^{-10}=0.61391$
$a_{\overline{10}|0.05}=\frac{1-0.61391}{0.05}=7.72173$
$P{V}_1=600\times 7.72173=4,633$

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

Discount to time 0 at 5%:
$P{V}_2=4,214\times (1.05{)}^{-10}=4,214\times 0.61391=2,587$

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

The closest answer is 7,211.