Exam FMMeasurement of interestFree to read

Force of interest

The force of interest δ(t) is the instantaneous rate of growth of an accumulation function - the continuous-compounding limit that makes every other rate a special case.

The formulas

Definition

$$\delta_t = \dfrac{a'(t)}{a(t)} = \dfrac{d}{dt}\ln a(t)$$

Accumulation

$$a(t) = \exp\left(\int_0^{t} \delta_s\, ds\right)$$

Constant force

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

Discounting

$$v^{t} = e^{-\delta t}$$

Where it comes from

1. Over a tiny interval $dt$ the fund grows by $a'(t)\,dt$, which as a PROPORTION of the fund is $\frac{a'(t)}{a(t)}dt$ - that proportion per unit time is the force.
2. $\frac{a'(t)}{a(t)} = \frac{d}{dt}\ln a(t)$, so integrating both sides from 0 to $t$ and using $a(0)=1$ gives $\ln a(t) = \int_0^t \delta_s ds$.
3. If the force is constant then $a(t) = e^{\delta t}$, and matching to $(1+i)^t$ gives $\delta = \ln(1+i)$.

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

• δ = ln(1 + i). Everything continuous is that one substitution.
• δ < d⁽ᵐ⁾ … < δ < … < i⁽ᵐ⁾ < i is the standard ordering: d < d⁽²⁾ < … < δ < … < i⁽²⁾ < i.

Traps

• Using a constant-force formula when δ(t) varies with t - then you must integrate.
• Forgetting that ∫₀ᵗ δₛ ds appears inside an exponential, not multiplied by it.

Related

• Nominal rates convertible m-thly
• Effective interest and discount rates
• Continuously payable annuities

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