Exam FM · practice
Exam FM — Interest Measurement & TVM Practice Flashcards
Thirty original SOA-style multiple-choice problems on interest measurement and time value of money — accumulation and present value, effective and nominal rates of interest and discount, force of interest (constant and time-varying), equivalent-rate conversions, simple vs compound interest, doubling time, and real-vs-nominal inflation adjustments — each with a fully worked solution.
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- Accumulation & PVA deposit of $\$4{,}200$ is made into an account earning an annual effective interest rate of $5.75\%$. Calculate the accumulated value at the end of $9$ years. (A) $\$6{,}374$ (B) $\$6{,}569$ (C) $\$6{,}947$ (D) $\$7{,}346$ (E) $\$7{,}426$**Answer: (C).** Use compound accumulation $A(9)=4200\,(1+i)^{9}$ with $i=0.0575$. $1.0575^{9}\approx 1.653954$. $A(9)=4200\times 1.653954\approx \$6{,}946.61$. Distractor (A) uses simple interest $4200(1+0.0575\times 9)=\$6{,}373.50$; (B) uses $8$ years $4200(1.0575)^{8}=\$6{,}568.89$; (D) uses $10$ years $4200(1.0575)^{10}=\$7{,}346.04$.
- Accumulation & PVA payment of $\$25{,}000$ is due at the end of $11$ years. At an annual effective interest rate of $6.25\%$, calculate its present value. (A) $\$12{,}078$ (B) $\$12{,}292$ (C) $\$12{,}833$ (D) $\$13{,}635$ (E) $\$14{,}561$**Answer: (C).** $PV=25000\,v^{11}=\dfrac{25000}{1.0625^{11}}$. $1.0625^{11}\approx 1.948132$. $PV=\dfrac{25000}{1.948132}\approx \$12{,}832.81$. Distractor (A) discounts $12$ years $25000/1.0625^{12}=\$12{,}077.94$; (B) treats $6.25\%$ as a discount rate $25000(1-0.0625)^{11}=\$12{,}292.04$; (D) discounts only $10$ years $25000/1.0625^{10}=\$13{,}634.86$.
- Interest & discount ratesAn account is governed by the amount function $A(t)=500\,(1.04)^{t}$. Calculate the effective rate of interest earned during the seventh year. (A) $3.70\%$ (B) $4.00\%$ (C) $4.16\%$ (D) $4.49\%$ (E) $5.27\%$**Answer: (B).** For compound interest the effective rate is constant every period, equal to $i$. $i_{7}=\dfrac{A(7)-A(6)}{A(6)}=\dfrac{500(1.04)^{7}-500(1.04)^{6}}{500(1.04)^{6}}=\dfrac{(1.04)^{6}\,(1.04-1)}{(1.04)^{6}}=0.04.$ So $i_{7}=4.00\%$. The distractors come from candidates who believe a growing balance makes the effective rate drift year to year — that is true under *simple* interest, not compound.
- Interest & discount ratesThe effective annual rate of discount is $d=0.045$. Calculate the equivalent effective annual rate of interest $i$. (A) $4.30\%$ (B) $4.50\%$ (C) $4.71\%$ (D) $4.95\%$ (E) $5.21\%$**Answer: (C).** $i=\dfrac{d}{1-d}=\dfrac{0.045}{1-0.045}=\dfrac{0.045}{0.955}\approx 0.047120$, i.e. $4.71\%$. Check: $d=\dfrac{i}{1+i}=\dfrac{0.047120}{1.047120}\approx 0.045$. ✓ Distractor (A) computes $i=d(1-d)$; (B) just repeats $d$; (D) uses $\dfrac{d}{1-2d}$-type slip.
- Interest & discount ratesAn investor can earn an annual effective interest rate of $i=0.085$. Calculate the equivalent annual effective rate of discount $d$. (A) $7.66\%$ (B) $7.83\%$ (C) $8.05\%$ (D) $8.50\%$ (E) $9.23\%$**Answer: (B).** $d=\dfrac{i}{1+i}=\dfrac{0.085}{1.085}\approx 0.078341$, i.e. $7.83\%$. Equivalently $d=1-v=1-\dfrac{1}{1.085}=1-0.921659=0.078341$. ✓ Distractor (D) just repeats $i$; (E) computes $d=i(1+i)$; (A) uses $\dfrac{i}{1+2i}$.
- Nominal ratesA nominal annual interest rate of $7.2\%$ is convertible monthly. Calculate the equivalent annual effective rate of interest. (A) $7.20\%$ (B) $7.44\%$ (C) $7.50\%$ (D) $7.61\%$ (E) $7.71\%$**Answer: (B).** The per-month rate is $\dfrac{i^{(12)}}{12}=\dfrac{0.072}{12}=0.006$. $1+i=(1.006)^{12}$. With $\ln(1.006)=0.0059821$, $12\times 0.0059821=0.0717850$, and $e^{0.0717850}\approx 1.074424$. $i\approx 0.074424$, i.e. $7.44\%$. Distractor (A) wrongly treats the nominal rate as effective; (D)/(E) compound with the wrong frequency.
- Nominal ratesA nominal annual interest rate $i^{(4)}=0.094$ is convertible quarterly. Calculate the annual effective rate of interest. (A) $9.40\%$ (B) $9.62\%$ (C) $9.74\%$ (D) $9.85\%$ (E) $9.91\%$**Answer: (C).** Quarterly rate $=\dfrac{0.094}{4}=0.0235$. $1+i=(1.0235)^{4}$. With $\ln(1.0235)=0.0232280$, $4\times 0.0232280=0.0929122$, and $e^{0.0929122}\approx 1.097367$. $i\approx 0.097367$, i.e. $9.74\%$. Distractor (A) treats the nominal rate as effective; (B) uses semiannual compounding; (E) uses monthly compounding $\big(1+\tfrac{0.094}{12}\big)^{12}$.
- Nominal ratesA nominal annual rate of discount $d^{(2)}=0.07$ is convertible semiannually. Calculate the equivalent annual effective rate of interest. (A) $7.00\%$ (B) $7.12\%$ (C) $7.25\%$ (D) $7.39\%$ (E) $7.53\%$**Answer: (D).** The per-half-year discount is $\dfrac{d^{(2)}}{2}=\dfrac{0.07}{2}=0.035$. $v=\Big(1-\dfrac{d^{(2)}}{2}\Big)^{2}=(0.965)^{2}=0.931225$. $1+i=\dfrac{1}{0.931225}\approx 1.073855$, so $i\approx 0.073855$, i.e. $7.39\%$. Distractor (A) treats $d^{(2)}$ as the effective discount; (C) computes $(1+0.035)^{2}-1$; (E) treats it as a nominal interest rate.