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

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

Nominal from effective

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

Nominal discount

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

The chain that links them

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

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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