Solving for Unknown TVM Variables | Free Exam FM Lesson

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:

$F V = P V \cdot (1 + i)^{n}$

Given any three, solve for the fourth.

Solving for FV: direct multiplication. No algebra needed.

$P V = F V \cdot v^{n}$

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

$i = (\frac{F V}{P V})^{1/ n} - 1$

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

$n = \frac{ln ( F V / P V )}{ln ( 1 + i )}$

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 $t_{2}$ :

$a (t_{1}, t_{2}) = e^{\int_{t_{1}}^{t_{2}} δ (s) d s}$

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

Read the full lesson, free →

Worked examples and practice. Free with a free account, no card.

Common mistakes

Forgetting the $n$ -th root when solving for $i$ . Given $P V = 10000$ , $F V = 15000$ , $n = 8$ : 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 $\int_{3}^{8} δ (t) d t$ , 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 $F V = P V (1 + i)^{n}$ .
Solve for $i$ : $i = (F V / P V)^{1/ n} - 1$ . Take the $n$ -th root; never divide the ratio by $n$ .
Solve for $n$ : $n = ln (F V / P V) / ln (1 + i)$ . This is the only variable requiring logarithms.
Solve for PV: multiply $F V$ 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 2,048 words, 14 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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