Rate Conversions | Free Exam FM Lesson

Three institutions quote 7.2% semi-annual, 7.1% daily, and 7.15% effective annual. Without converting to a common basis, you cannot compare them.

$i$ , effective annual interest rate
$d$ , effective annual discount rate
$i^{(m)}$ , nominal interest rate convertible $m$ -thly
$d^{(m)}$ , nominal discount rate convertible $m$ -thly
$δ$ , force of interest

The conversion strategy: go through $1 + i$ as a hub. Two steps, no ambiguity.

$1 + i = (1 + \frac{i ^{(m)}}{m})^{m}$

Reverse direction:

$i^{(m)} = m [(1 + i)^{1/ m} - 1]$

$d = \frac{i}{1 + i}, i = \frac{d}{1 - d}, v = 1 - d$

HIGH-FREQUENCY: The relationship $d = i v$ appears constantly. Know it in all three forms.

$1 - d = (1 - \frac{d ^{(m)}}{m})^{m}, 1 + i = (1 - \frac{d ^{(m)}}{m})^{- m}$

Reverse:

$d^{(m)} = m [1 - v^{1/ m}]$

$δ = ln (1 + i), i = e^{δ} - 1$

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Common mistakes

Confusing the $i^{(m)}$ and $d^{(m)}$ formulas. Nominal interest uses $(1 + i^{(m)} / m)^{m}$ . Nominal discount uses $(1 - d^{(m)} / m)^{m}$ . Swapping the sign (using plus for $d^{(m)}$ ) gives the wrong accumulation factor. Trap: 5.536% instead of 5.561%.
Violating the ordering. If $i = 0.08$ and you compute $i^{(4)} = 0.0824 > i$ , something is wrong. The ordering $i^{(m)} < i$ must always hold.
Forgetting to multiply by $m$ . To find $i^{(12)}$ from $i = 0.06$ : the monthly rate is $(1.06)^{1/12} - 1 = 0.004868$ , then $i^{(12)} = 12 \times 0.004868 = 0.058416$ . Reporting just 0.4868% (the monthly rate) is a common slip. Trap: 0.487%.

Bottom line

Two-step algorithm: convert any rate to $1 + i$ first, then to the target. Works for all five rate types and every pair.
Ordering: $d < d^{(m)} < δ < i^{(m)} < i$ for all $m \geq 1$ . Instant sanity check.
Key identity: $d = i v = i / (1 + i)$ . The discount rate is always less than the interest rate.
Convergence: as $m \to \infty$ , both $i^{(m)}$ and $d^{(m)} \to δ = ln (1 + i)$ .

Exam shortcut

Write the two-step approach: (1) find $1 + i$ , (2) convert to target. Then check the ordering. These checks take 5 seconds. "Through the hub" ( $1 + i$ is the hub. Ordering: "d D delta I i") all increasing left to right. "Discount divides", $d = i / (1 + i)$ always makes $d < i$ .

The full lesson (about 2,148 words, 14 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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