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Exam FM: Interest Measurement & TVM Flashcards

The Time Value of Money toolkit for SOA Exam FM: accumulation and present value, effective and nominal rates of interest and discount, the i / d / v / delta relationships, rate conversions, simple vs compound interest, equivalent rates, doubling time, and real-vs-nominal (inflation) adjustments — with fully worked calculations.

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  1. Accumulation & PV
    Define the **accumulation function** $a(t)$ and the **amount function** $A(t)$ for an investment.
    $a(t)$ accumulates a single unit invested at time $0$, with $a(0)=1$. The amount function scales it by the initial deposit $k$: $A(t)=k\cdot a(t)$, so $A(0)=k$. Under compound interest $a(t)=(1+i)^{t}$; under simple interest $a(t)=1+it$.
  2. Interest & discount rates
    Define the **effective rate of interest** $i_{n}$ over the $n$-th period in terms of the amount function $A(t)$.
    $i_{n}=\frac{A(n)-A(n-1)}{A(n-1)}$ — interest earned in the period divided by the balance at its **start**. For compound interest $i_{n}=i$ is constant every period; for simple interest the effective rate declines each period because the denominator grows while the numerator stays fixed.
  3. Accumulation & PV
    What is the **discount factor** $v$, and how does it relate to the effective rate $i$?
    $v=\frac{1}{1+i}=(1+i)^{-1}$. It is the present value of $1$ paid one period from now. Discounting $n$ periods multiplies by $v^{n}=(1+i)^{-n}$. With $i=0.07$, $v=\frac{1}{1.07}\approx 0.93458$.
  4. Interest & discount rates
    Give the four key relationships connecting the effective rate of discount $d$, the interest rate $i$, and $v$.
    $d=\frac{i}{1+i}=iv$, $d=1-v$, $i=\frac{d}{1-d}$, and $1-d=v$. Also the useful identity $i-d=id$ (interest and discount differ only by the product term).
  5. Interest & discount rates
    What is the conceptual difference between the rate of **interest** $i$ and the rate of **discount** $d$?
    Interest $i$ is paid at the **end** of the period on the amount at the start (interest in arrears). Discount $d$ is the same charge taken at the **start** of the period out of the amount due at the end (interest in advance). Because $d$ is applied earlier on a smaller base, $d<i$ for any positive rate, with $d=\frac{i}{1+i}$.
  6. Accumulation & PV
    Accumulate $\$5{,}000$ for $8$ years at an annual effective interest rate of $4.5\%$.
    Use $A(8)=5000\,(1+i)^{8}$ with $i=0.045$. $1.045^{8}\approx 1.422101$. $A(8)=5000\times 1.422101\approx \$7{,}110.50$.
  7. Accumulation & PV
    Find the present value of $\$10{,}000$ payable in $6$ years at an annual effective rate of $7\%$.
    $PV=10000\,v^{6}=\frac{10000}{1.07^{6}}$. $1.07^{6}\approx 1.500730$. $PV=\frac{10000}{1.500730}\approx \$6{,}663.42$.
  8. Interest & discount rates
    Given an effective annual discount rate $d=0.05$, find the equivalent effective annual interest rate $i$.
    $i=\frac{d}{1-d}=\frac{0.05}{0.95}\approx 0.052632$, i.e. about $5.2632\%$. Check: $d=iv=0.052632\times\frac{1}{1.052632}=0.05$. ✓
  9. Interest & discount rates
    Given an effective annual interest rate $i=0.08$, find the equivalent effective annual discount rate $d$.
    $d=\frac{i}{1+i}=\frac{0.08}{1.08}\approx 0.074074$, about $7.4074\%$. Equivalently $d=1-v=1-\frac{1}{1.08}\approx 0.074074$. Both routes agree.
  10. Nominal rates
    Distinguish a **nominal** rate $i^{(m)}$ from an **effective** rate. What is the per-period rate when interest is convertible $m$-thly?
    $i^{(m)}$ is an annual **nominal** rate compounded $m$ times per year; it is a label, never applied directly. The actual rate earned each subperiod is $\frac{i^{(m)}}{m}$. The annual effective rate follows from $1+i=\left(1+\frac{i^{(m)}}{m}\right)^{m}$. The classic trap is using $i^{(m)}$ itself instead of $\frac{i^{(m)}}{m}$.
  11. Nominal rates
    A nominal annual interest rate of $6\%$ is convertible monthly ($i^{(12)}=0.06$). Find the annual effective rate $i$.
    Monthly rate $=\frac{0.06}{12}=0.005$. $1+i=(1.005)^{12}\approx 1.061678$. $i\approx 0.061678$, i.e. about $6.1678\%$ effective annually.
  12. Nominal rates
    A nominal rate $i^{(4)}=0.08$ is convertible quarterly. Find the annual effective rate $i$.
    Quarterly rate $=\frac{0.08}{4}=0.02$. $1+i=(1.02)^{4}\approx 1.082432$. $i\approx 0.082432$, about $8.2432\%$.
  13. Nominal rates
    Write the conversion identity for a **nominal discount** rate $d^{(m)}$ convertible $m$-thly, and relate it to $i$ and $v$.
    $\left(1-\frac{d^{(m)}}{m}\right)^{m}=1-d=v=\frac{1}{1+i}$. So $1+i=\left(1-\frac{d^{(m)}}{m}\right)^{-m}$. The per-period discount actually applied is $\frac{d^{(m)}}{m}$, taken in advance each subperiod.
  14. Nominal rates
    A nominal discount rate $d^{(2)}=0.06$ is convertible semiannually. Find the annual effective interest rate $i$.
    Per-half-year discount $=\frac{0.06}{2}=0.03$. $v=(1-0.03)^{2}=0.97^{2}=0.9409$. $1+i=\frac{1}{0.9409}\approx 1.062812$, so $i\approx 6.2812\%$.
  15. Equivalent rates
    State the **unifying chain** linking $i^{(m)}$, $i$, $d$, $d^{(m)}$, and the force of interest $\delta$.
    $\left(1+\frac{i^{(m)}}{m}\right)^{m}=1+i=(1-d)^{-1}=\left(1-\frac{d^{(m)}}{m}\right)^{-m}=e^{\delta}$. Every equivalent-rate question converts a given quantity to $(1+i)$ — the common hub — then back out to the target form.
  16. Force of interest
    Define the **force of interest** $\delta$ for a constant-rate investment and relate it to $i$ and $v$.
    $\delta$ is the instantaneous (continuously compounded) rate, with $\delta=\ln(1+i)$, equivalently $1+i=e^{\delta}$ and $v=e^{-\delta}$. Accumulating a unit for time $t$ gives $e^{\delta t}$; discounting gives $e^{-\delta t}$.
  17. Force of interest
    An account earns an annual effective rate of $7\%$. Find the equivalent **force of interest** $\delta$.
    $\delta=\ln(1+i)=\ln(1.07)\approx 0.067659$, i.e. about $6.7659\%$. Note $\delta<i$: continuous compounding needs a slightly lower instantaneous rate to reach the same annual growth.
  18. Force of interest
    Invest $\$3{,}000$ for $7$ years under a constant force of interest $\delta=0.045$. Find the accumulated value.
    Accumulation factor $=e^{\delta t}=e^{0.045\times 7}=e^{0.315}\approx 1.370260$. $AV=3000\times 1.370260\approx \$4{,}110.78$.
  19. Simple vs compound
    Contrast **simple** and **compound** interest accumulation. When does simple interest accumulate to *more* than compound?
    Simple: $a(t)=1+it$ (interest only on principal). Compound: $a(t)=(1+i)^{t}$ (interest on interest). For $0<t<1$ simple interest gives the larger value; at $t=1$ they coincide; for $t>1$ compound dominates and the gap widens. Simple interest is normally used only for accumulation, not discounting, unless a problem states otherwise.
  20. Simple vs compound
    Invest $\$1{,}000$ for $3$ years. Compare the accumulated value under simple interest of $5\%$ versus compound interest of $5\%$.
    Simple: $1000(1+0.05\times 3)=1000\times 1.15=\$1{,}150.00$. Compound: $1000\,(1.05)^{3}=1000\times 1.157625=\$1{,}157.63$. Compound earns $\$7.63$ more — the interest-on-interest from the first two years.
  21. Simple vs compound
    An investor lends $\$800$ at a **simple** interest rate of $6\%$ per year. What is the value after $5$ years, and what is the effective rate earned in year $5$?
    Value $=800(1+0.06\times 5)=800\times 1.30=\$1{,}040.00$. Interest each year is a flat $800\times 0.06=\$48$. The effective rate in year $5$ is $\frac{48}{800(1+0.06\times4)}=\frac{48}{992}\approx 0.04839$, i.e. $4.839\%$ — lower than $6\%$ because the base has grown.
  22. Equivalent rates
    What does it mean for two interest/discount rates to be **equivalent**, and what is the general strategy to convert between any two?
    Two rates are equivalent if they produce the **same accumulated value** over the same period. Strategy: convert the given rate to the annual effective $(1+i)$ hub, then convert $(1+i)$ to the target form using the unifying chain. Never add or average nominal rates — always work through accumulation factors.
  23. Equivalent rates
    Find the nominal rate $i^{(2)}$ convertible semiannually that is equivalent to an annual effective rate of $10\%$.
    Per-half-year rate $=(1+i)^{1/2}-1=1.10^{0.5}-1\approx 0.048809$. $i^{(2)}=2\times 0.048809\approx 0.097618$, i.e. about $9.7618\%$ convertible semiannually.
  24. Equivalent rates
    An annual effective rate is $6\%$. Find the equivalent nominal discount rate $d^{(12)}$ convertible monthly.
    $v=\frac{1}{1.06}$, and $\left(1-\frac{d^{(12)}}{12}\right)^{12}=v$. Monthly discount $=1-v^{1/12}=1-1.06^{-1/12}\approx 1-0.995157=0.0048439$. $d^{(12)}=12\times 0.0048439\approx 0.058128$, about $5.8128\%$.
  25. Equivalent rates
    A nominal rate $i^{(12)}=0.09$ is convertible monthly. Find the equivalent nominal rate $i^{(4)}$ convertible quarterly.
    Annual accumulation factor $=(1+\frac{0.09}{12})^{12}=(1.0075)^{12}\approx 1.093807$. Quarterly rate $=1.093807^{1/4}-1\approx 0.022669$. $i^{(4)}=4\times 0.022669\approx 0.090677$, about $9.0677\%$.
  26. Doubling time
    What is **doubling time** under compound interest, and how does the **Rule of 72** approximate it?
    Exact doubling time solves $(1+i)^{n}=2$, so $n=\frac{\ln 2}{\ln(1+i)}$. The Rule of 72 estimates it as $n\approx \frac{72}{100\,i}=\frac{72}{\text{rate in percent}}$ — a quick mental approximation that is accurate for typical rates near $8\%$.
  27. Doubling time
    At an annual effective rate of $9\%$, compare the exact doubling time with the Rule of 72 estimate.
    Exact: $n=\frac{\ln 2}{\ln 1.09}=\frac{0.693147}{0.086178}\approx 8.043$ years. Rule of 72: $\frac{72}{9}=8$ years. The approximation is excellent — within about half a month of the exact value.
  28. Doubling time
    Express doubling time directly in terms of the **force of interest** $\delta$. Apply it to an account with $i=5\%$.
    Continuous accumulation $e^{\delta n}=2$ gives $n=\frac{\ln 2}{\delta}$. With $i=0.05$, $\delta=\ln 1.05\approx 0.048790$, so $n=\frac{0.693147}{0.048790}\approx 14.207$ years.
  29. Doubling time
    How long until $\$1{,}000$ grows to $\$1{,}500$ at an annual effective rate of $6\%$?
    Solve $1000(1.06)^{n}=1500\Rightarrow 1.06^{n}=1.5$. $n=\frac{\ln 1.5}{\ln 1.06}=\frac{0.405465}{0.058269}\approx 6.959$ years.
  30. Real vs nominal
    State the relationship between the **real** rate $i_{\text{real}}$, the **nominal** rate $i_{\text{nom}}$, and the inflation rate $r$.
    $1+i_{\text{real}}=\frac{1+i_{\text{nom}}}{1+r}$, so $i_{\text{real}}=\frac{1+i_{\text{nom}}}{1+r}-1=\frac{i_{\text{nom}}-r}{1+r}$. The quick approximation $i_{\text{real}}\approx i_{\text{nom}}-r$ understates the gap; use the exact ratio on the exam. (Here "nominal" means the actual money rate, not a compounding label.)
  31. Real vs nominal
    An investment earns a nominal (money) return of $8\%$ in a year with $3\%$ inflation. Find the real rate of return.
    $i_{\text{real}}=\frac{1.08}{1.03}-1\approx 1.048544-1=0.048544$, i.e. about $4.85\%$. The rough subtraction $8\%-3\%=5\%$ is close but overstates; the exact figure is $4.85\%$.
  32. Real vs nominal
    An investor requires a **real** return of $5\%$ and expects inflation of $7\%$. What nominal (money) rate of return must the investment earn?
    $1+i_{\text{nom}}=(1+i_{\text{real}})(1+r)=1.05\times 1.07=1.1235$. So $i_{\text{nom}}=0.1235$, i.e. $12.35\%$ — more than the naive $5\%+7\%=12\%$ because of the cross term.
  33. Equivalent rates
    A fund grows by a force of interest $\delta=0.06$ for the first $4$ years, then earns an annual effective rate of $5\%$ for the next $3$ years. Accumulate $\$1$ over the full $7$ years.
    First segment: $e^{0.06\times 4}=e^{0.24}\approx 1.271249$. Second segment: $(1.05)^{3}\approx 1.157625$. Total $=1.271249\times 1.157625\approx 1.47163$, so $\$1$ grows to about $\$1.4716$.
  34. Equivalent rates
    A force of interest $\delta=0.05$ applies. Find the equivalent annual effective interest rate $i$ and discount rate $d$.
    $1+i=e^{\delta}=e^{0.05}\approx 1.051271$, so $i\approx 5.1271\%$. $d=1-v=1-e^{-\delta}=1-e^{-0.05}=1-0.951229\approx 0.048771$, i.e. about $4.8771\%$.