Exam FMMeasurement of interestFree to read

Nominal rates convertible m-thly

A nominal rate i⁽ᵐ⁾ is an annual label for a rate applied m times a year; only the effective rate compounds, so every problem converts to effective first.

The formulas

Effective from nominal

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

Nominal from effective

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

Nominal discount

$$1-d={\left(1-\frac{d^{(m)}}{m}\right)}^m$$

The chain that links them

$${\left(1+\frac{i^{(m)}}{m}\right)}^m=1+i=(1-d{)}^{-1}={\left(1-\frac{d^{(p)}}{p}\right)}^{-p}=e^{\delta }$$

Where it comes from

1. $i^{(m)}$ is not a rate that is ever applied; $i^{(m)}/m$ is the rate applied each $1/m$ of a year.
2. Applying it $m$ times compounds to ${\left(1+i^{(m)}/m\right)}^m$, and by definition that equals $1+i$.
3. Everything else follows by taking the same accumulation factor and writing it in a different currency of rate — which is why the chain above has one value written five ways.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• The bracket (m) is a LABEL, not an exponent: divide by m to get the rate that is actually applied.
• As m grows, i⁽ᵐ⁾ falls towards δ and d⁽ᵐ⁾ rises towards δ — the force of interest is the limit from both sides.

Traps

• Compounding the nominal rate itself: (1.07)^12 rather than (1 + 0.07/12)^12.
• Comparing a nominal monthly rate with a nominal quarterly rate directly instead of converting both to effective.

Related

• Effective interest and discount rates
• Force of interest

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