The Arithmetic of Ruin
Seven mathematical laws of financial survival. The previous article (the dynamics of accumulation) studied the drift of a wealth process — its growth. This one studies the other, more neglected half of the problem: the absorbing barrier at zero. Because a stochastic process can have a magnificent drift and die anyway.
Formally: let \( W_t \) be an agent's wealth. Accumulation cares about \( \mathbb{E}[W_t] \). Survival cares about \( \mathbb{P}\!\left(\inf_{s \le t} W_s \le 0\right) \) — the probability that at some point the trajectory touches zero. Zero is absorbing: nothing compounds from nothing. All the expectation in the world will not resurrect an absorbed path.
The seven laws below are as old as lending at interest. What is less old is that today we can prove them — and simulate them. Each section states a law, formalizes it, then lets it run before your eyes.
I.The asymmetry of losses
“Gold flees the one who forces it toward impossible gains.”
A loss and a gain of equal size are not symmetric. After a relative loss \( L \), the gain required to return to the starting point is
\[ R(L) = \frac{L}{1-L}, \]
a convex function that blows up near 1: a 10% loss is repaired with 11%; a 50% loss demands 100%; a 90% loss demands 900%. The convexity of \( R \) is the first mathematical reason to be defensive: the cost of a mistake grows faster than its size. And the repair time follows: at an annual return \( r \), it takes \( t = -\ln(1-L)/\ln(1+r) \) years to fill the hole.
The repair curve \( R(L) = L/(1-L) \)
II.The poison of variance
“A small gain that always returns is worth more than a large one that slips away.”
Two assets can have the same arithmetic mean return and opposite fates. For compounded returns, what matters is the geometric mean, and the AM–GM inequality levies a tax:
\[ g \;\approx\; \mu \;-\; \frac{\sigma^2}{2}. \]
Volatility is a drag — quadratic friction. At \( \mu = 8\% \), an asset with \( \sigma = 40\% \) has median growth \( g \approx 0\% \): the mean \( \mathbb{E}[W_t] \) rises (pulled by a handful of miraculous paths) while the median stalls. The gap between mean and median is the gap between what the lottery promises in expectation and what happens to you — the single trajectory you will actually live. This is the heart of the ergodicity problem in finance.
200 paths, 30 years — mean vs median (log scale)
III.Gambler's ruin
“Even the surest hand loses if it stakes too large.”
The classic random walk: initial capital \( a \), target \( N \), unit stake, win probability \( p \), \( q = 1-p \). The probability of being absorbed at zero before reaching \( N \) is
\[ \psi(a) = \frac{(q/p)^a - (q/p)^N}{1 - (q/p)^N}, \qquad \psi(a) = 1 - \frac{a}{N} \ \text{ if } p = \tfrac12. \]
Two readings. First, with an unfavorable game (\( p < \tfrac12 \)), ruin becomes exponentially certain as the ambition \( N \) grows — the casino theorem. Second, more subtle: even with an edge (\( p > \tfrac12 \)), ruin remains possible, and its probability depends on the granularity of the stake. A statistical edge without position sizing is not a strategy; it is deferred ruin. (Optimal sizing — Kelly — was covered in the previous article; here we watch what happens when it is ignored.)
300 simulated walks, absorption at 0 or at N
IV.The fixed point of debt
“Repay first; one fifth of what you earn is enough — but every month.”
A debt at monthly rate \( i \), repaid with a constant payment \( M \), follows the affine recurrence
\[ D_{t+1} = (1+i)\,D_t - M, \]
whose fixed point is \( D^\* = M/i \). This fixed point is unstable (the slope \( 1+i > 1 \)): below it, the debt collapses to zero in finite time; above it, it diverges exponentially. The whole theory of over-indebtedness fits in that dichotomy. And the credit-card “minimum payment” is calibrated to keep you in the neighborhood of the fixed point — where the debt is nearly motionless and the interest nearly perpetual. The time to freedom, when \( M > iD_0 \), is \( n = -\ln(1 - iD_0/M)/\ln(1+i) \): brutally nonlinear in \( M \), which is why a slightly larger repayment effort changes everything.
Debt trajectories and the unstable fixed point D* = M/i
V.The equation of an impossible promise
“The return that outruns the world is borrowed from someone.”
A scheme that promises a rate \( \rho \) while producing nothing must pay the old with the money of the new. Let \( L_t \) be the liabilities (promised balances), \( C_t \) the real cash, \( n_t = n_0 e^{\gamma t} \) the incoming deposits and \( w \) the withdrawal rate. In continuous time:
\[ \dot L = \rho L + n(t) - wL, \qquad \dot C = n(t) - wL. \]
Liabilities grow at rate \( \rho - w \) plus the inflows; the cash receives only the inflows. The survival condition is essentially \( \gamma \gtrsim \rho - w \): you must recruit at an exponential rate comparable to the promised return. Since no population of savers grows exponentially forever, collapse is not a risk of the model — it is a theorem of it. The only free parameter is the date. Meanwhile the honesty ratio \( C_t / L_t \) declines from day one: it is the statistic the scheme must hide, and the first one an auditor asks for.
Promised liabilities vs real cash (log scale) — collapse as a theorem
VI.The cushion theorem
“The first share of all that enters your hand is yours. Keep it.”
Here is the most beautiful result on this page, borrowed from actuarial ruin theory. Model your finances the way an insurer models itself: an initial surplus \( u \) (the emergency fund), a premium \( c \) (your monthly saving), and claims arriving as a Poisson process of intensity \( \lambda \) with i.i.d. costs of mean \( m \) — the transmission that dies, the dental emergency, the contract that falls through. This is the Cramér–Lundberg model:
\[ U(t) = u + ct - \sum_{k=1}^{N(t)} X_k. \]
Write the premium with a safety loading \( \theta \): \( c = (1+\theta)\lambda m \) — you save \( \theta \) more than the average cost of the shocks. For exponential claim sizes, the infinite-horizon ruin probability has a closed form:
\[ \psi(u) = \frac{1}{1+\theta}\, e^{-Ru}, \qquad R = \frac{\theta}{m(1+\theta)}. \]
Read the structure carefully: safety is exponential in the cushion \( u \), and the coefficient \( R \) — the speed of that exponential — is driven by \( \theta \), that is, by your saving rate. Every month of expenses added to the cushion multiplies the survival probability by the same factor. Few financial acts offer guaranteed exponential returns; building a reserve is one of them.
ψ(u): ruin probability as a function of the cushion (in months of expenses)
Synthesis: the seven laws
| Law | Statement | Mathematical object |
|---|---|---|
| I | Protect the principal before growing it. | Convexity of \( R(L)=L/(1-L) \) |
| II | Prefer the steady return to the brilliant one. | Drag \( g \approx \mu - \sigma^2/2 \), non-ergodicity |
| III | An edge without sizing is deferred ruin. | Absorption probability \( \psi(a) \) |
| IV | Pay your debt above its fixed point, or it owns you. | Unstable fixed point \( D^\*=M/i \) |
| V | A return that beats real growth is borrowed from the next victims. | Collapse ODE, ratio \( C/L \) |
| VI | Keep a share of every income first: the reserve buys exponential safety. | \( \psi(u)=\tfrac{1}{1+\theta}e^{-Ru} \) |
| VII | The best parameter to optimize is your capacity to earn. | \( R \) increasing in \( \theta \), hence in \( c \) |
Postscript — None of these laws is new. They are said to have circulated long ago, pressed into clay, among the lenders on the banks of the Euphrates. All they lacked were the exponentials.