Rate Conversions

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
• $\delta$, force of interest

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

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

Reverse direction:

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

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

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

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

Reverse:

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

$$\delta = \ln(1+i), \quad i = e^{\delta} - 1$$

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: $d < d^{(m)} < \delta < i^{(m)} < i$ for all $m \geq 1$. Instant sanity check.
• Key identity: $d = iv = i/(1+i)$. The discount rate is always less than the interest rate.
• Convergence: as $m \to \infty$, both $i^{(m)}$ and $d^{(m)} \to \delta = \ln(1+i)$.

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 2,809 words, 19 min read) adds 8 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 1c

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