Compound Interest • Bettering Humanity

Compound interest is the most powerful force in the universe. — Albert Einstein

Compound interest is man’s greatest invention. — Albert Einstein

Compound interest is is the eighth wonder of the world. — Albert Einstein

I never said any of the above. — Albert Einstein

Force of Interest

Denote the value at time $t$ as $V$, and the initial value as $V_0$. The accumulation function $a$ is the ratio of the two values \[a = \frac{V}{V_0}.\]

The force of interest is \[\delta = \frac{d\left(\ln a\right)}{dt},\] and conversely, \[a = e^{\int_0^t \delta dt}.\]

For compount interest with a constant interest rate $r$, the force of interest is a constant \[\delta = \ln \left(1 + r\right),\] and \[V = V_0 e^{\delta t} = V_0 \left(1 + r\right)^t.\]

Automatic Investment Plan (定投)

Suppose

You start with an initial value of $V_0$,
You add more money at a constant rate $\mu$,
Your value compounds with a constant force of interest $\delta$.

At any time $t$, the differential of the value is \[\frac{dV}{dt} = \mu + \delta V.\]

Separate $V$ and $t$ and we get \[\frac{dV}{\mu + \delta V} = dt.\]

Integrate both sides and we have \[\int_{V_0}^V \frac{dV}{\mu + \delta V} = \int_0^t dt,\] \[\left.\frac{\ln \left(\mu + \delta V\right)}{\delta}\right|_{V_0}^V = \left.t\right|_0^t.\]

Solve for $V$ and we get \[V = V_0 e^{\delta t} + \frac{\mu}{\delta} \left(e^{\delta t} - 1\right).\]

As an example, we start with $\$1,000$, we invest money we save from every paycheck, averaging $\$100,000$ anually, the annual rate of return is $6\%$, and we keep doing this for $30$ years. So \[V_0 = 1,000,\] \[\mu = 100,000 \, \mathrm{a}^{-1},\] \[\delta = \ln (1 + 0.06) \, \mathrm{a}^{-1} = 0.05827 \, \mathrm{a}^{-1},\] \[t = 30 \, \mathrm{a}.\]

Plug them into the equation and we get \[V = 8,146,433.\]

We end up with more than $8$ million dollars.