Solving for Unknown TVM Variables

Three reserve questions (how much now, what rate, how many years) each has one unknown. A wrong answer means over-reserving or under-reserving by hundreds of thousands.

Every single-payment TVM problem involves:

$$FV = PV \cdot (1 + i)^n$$

Given any three, solve for the fourth.

Solving for FV: direct multiplication. No algebra needed.

$$PV = FV \cdot v^n$$

Think "multiply by $v^n$" rather than "divide by $(1+i)^n$." Same result, faster notation.

$$i = \left(\frac{FV}{PV}\right)^{1/n} - 1$$

This gives the annual effective rate. Convert to nominal or force of interest as needed.

$$n = \frac{\ln(FV/PV)}{\ln(1+i)}$$

If $n$ is not an integer, "how many complete years" means round up.

HIGH-FREQUENCY: Solving for $n$ via logarithms is a core FM skill. Practice until automatic.

Accumulation from $t_1$ to $t_2$:

$$a(t_1, t_2) = e^{\int_{t_1}^{t_2} \delta(s)\,ds}$$

To find the time $T$ when a deposit reaches a target:

Common mistakes

• Forgetting the $n$-th root when solving for $i$. Given $PV = 10000$, $FV = 15000$, $n = 8$: the correct rate is $1.5^{1/8} - 1 = 5.158\%$. Computing $(0.5)/8 = 6.25\%$ gives the simple interest rate. Trap: 6.25%.
• Integrating over the wrong interval for a delayed deposit. A deposit at time 3 accumulated to time 8 requires $\int_3^8 \delta(t)\,dt$, not $\int_0^8$. Using the full interval gives $50000 \times e^{0.368} = 72242$ instead of $64847$. Trap: $216,726 total.
• Rounding intermediate values too early. Carrying only 2 decimal places in the accumulation factor, then multiplying by $100,000, magnifies the error by nearly $500. Keep at least 4 decimal places.

Bottom line

• Four TVM variables: PV, FV, $i$, $n$. Given any three, solve the fourth via $FV = PV(1+i)^n$.
• Solve for $i$: $i = (FV/PV)^{1/n} - 1$. Take the $n$-th root; never divide the ratio by $n$.
• Solve for $n$: $n = \ln(FV/PV)/\ln(1+i)$. This is the only variable requiring logarithms.
• Solve for PV: multiply $FV$ by the discount factor $v^n$.

Exam shortcut

Convert any nominal rate or force of interest before doing anything else. For solve-for-$n$, go straight to log-over-log. For multiple deposits under variable force, compute each accumulation factor separately. "PV is FV times v-to-the-n": discount factor moves money backward. "Log over log gives time." Solve-for-$i$: ratio, root, subtract 1.

The full lesson (about 2,666 words, 18 min read) adds 9 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 1b

Browse all free Exam FM lessons or jump into free Exam FM practice questions.