1.The master equation

Everything starts from a first-order linear ODE. With income \( Y \), a saving rate \( s \) and a return \( r \), wealth follows

\[ \frac{dW}{dt} = sY + rW, \qquad W(t) = \left(W_0 + \frac{sY}{r}\right)e^{rt} - \frac{sY}{r}. \]

Two forces, two regimes. Early on, the \( sY \) term dominates: wealth grows linearly, at the pace of effort. Then \( rW \) takes over and growth becomes exponential: the capital works harder than you do. The switching point — the moment interest exceeds contributions — arrives when \( W = sY/r \), which comes earlier the larger \( r \) is. The doubling time of the exponential regime is \( \ln 2 / r \): the rule of 72 is a Taylor expansion that made a career.

2.The time to freedom

Define financial independence as the moment when capital can fund current expenses at a withdrawal rate \( w \) (classically 4%, i.e. a multiple \( 1/w = 25 \) of annual spending). With spending \( (1-s)Y \), the target is \( W^\dagger = 25(1-s)Y \), and the master equation yields the time to reach it:

\[ T(s) = \frac{\ln\!\left(1 + \dfrac{25\,r\,(1-s)}{s}\right)}{\ln(1+r)}. \]

The structure of \( T(s) \) is remarkable: the saving rate enters as the ratio \( (1-s)/s \) — it acts twice, accelerating accumulation and lowering the target. That is why \( T \) is hyperbolically sensitive to \( s \) when \( s \) is small: going from 10% to 15% saving buys back more years than going from 40% to 45%. The return \( r \) only acts through a logarithmic denominator: at short horizons, saving beats returns.

3.Multiplicative randomness

The real return is not a constant; it is a process. The canonical model is geometric Brownian motion:

\[ dW_t = \mu W_t\,dt + \sigma W_t\,dB_t \quad\Longrightarrow\quad W_t = W_0\, e^{\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma B_t}. \]

All the structure lives in the exponent. The median path grows at rate \( \mu - \sigma^2/2 \) — not \( \mu \). The mean \( \mathbb{E}[W_t] = W_0 e^{\mu t} \) grows faster, but it is carried by a minority of extreme paths you will probably not live. On a log scale the bundle becomes a cone of straight lines: the median slope is \( \mu - \sigma^2/2 \), the width grows as \( \sigma\sqrt{t} \). That is the entire geometry of the long run in two parameters.

4.The Kelly criterion

What fraction \( f \) of capital should be exposed to a risky asset with premium \( \mu - r \) and volatility \( \sigma \)? If the objective is to maximize the long-run logarithmic growth rate, the answer is a one-variable optimization:

\[ g(f) = r + f(\mu - r) - \tfrac{1}{2}f^2\sigma^2, \qquad f^\* = \frac{\mu - r}{\sigma^2}. \]

A parabola. Three zones. Left of \( f^\* \): growth left on the table. Right of it: every additional unit of risk destroys growth — and at \( f = 2f^\* \), the premium is entirely handed back to the variance: you carry all the risk for zero excess growth. The parabola is flat at the top and steep on the right: the cost of under-exposure is small, the cost of over-exposure is everything. Hence the practice of "fractional Kelly": target \( f^\*/2 \), because \( \mu \) and \( \sigma \) are estimates, and the asymmetry of the parabola punishes optimism.

5.The accumulator's harmonic mean

One discrete arithmetic fact deserves its place among these continuous theorems. Whoever invests a fixed amount at regular intervals acquires shares at the average price

\[ \bar P_{\text{cost}} = \frac{\sum_t c}{\sum_t c/P_t} = \text{harmonic mean of } P_t \;\le\; \text{arithmetic mean}, \]

with equality only if the price never moves. The constant contribution mechanically buys more shares when the price falls: the convexity of \( 1/P \) works for the regular buyer. Be careful about what this result says — and does not say. It does not say that spreading out an available lump sum beats investing it immediately (with positive drift, the opposite holds on average). It says that for someone accumulating out of income — the actual case of the saver — the path's volatility is not the enemy: it lowers the cost basis.

6.The human capital lifecycle

A young graduate's largest asset appears on no statement: it is the present value of future earnings,

\[ H(t) = \int_t^{R} y(u)\, e^{-r(u-t)}\, du, \]

where \( y(u) \) is the earnings profile and \( R \) the retirement age. At 25, \( H \) is measured in discounted millions and \( W \approx 0 \); one's entire financial life consists of converting one into the other before \( H \) runs out. This reading changes decisions: education is an investment in \( y(u) \) — hence in \( H \) — whose return compounds over a whole career; and a young person's financial portfolio can carry more risk, because their dominant asset, \( H \), behaves like a bond indexed on their labor.

Synthesis

One story runs through these six objects: wealth is a process \( dW = (\text{control flow})\,dt + (\text{chosen exposure})\,dB \), and the levers do not act at the same time. Early: \( s \) and \( H \) — saving and the capacity to earn. Late: \( r \), \( \sigma \) and time. The Kelly criterion bounds the exposure, the harmonic mean consoles the bumpy path, and the crossing \( H = W \) dates the midpoint of a financial life. There remains the symmetric question — not how the process grows, but how it dies: that is the arithmetic of ruin.

Default parameters are illustrative, not prescriptive; returns are real (net of inflation) and taxes are left as an exercise for the reader.