Salem Nkunda Nyisingize

Finance According to Solomon

The Book of Proverbs contains an implicit economic theory β€” not an investment manual, but a set of claims about wealth, debt, haste and time, written three millennia before stochastic processes. This article takes six of those claims seriously: each is stated in the text, translated into equations, then put to the test of a simulation. The question is not whether the text is inspired; it is whether the mathematics proves it right.

I.Hasty wealth against the gathering hand

β€œWealth gotten by vanity shall be diminished: but he that gathereth by labour shall increase.”

Proverbs 13:11 (KJV)

The verse contrasts two processes, not two amounts. Let us model them. The recipient of a windfall \( W_0 \) β€” lottery, jackpot, spectacular IPO β€” typically carries two correlated burdens: a high-volatility exposure \( \sigma_A \) (the assets that produce windfalls are the ones that take them back) and a lifestyle proportional to wealth, \( c = \kappa W \). His dynamics are a GBM with corrected drift \( \mu - \kappa \). The steady gatherer, by contrast, pours a constant flow into a quiet asset: modest drift, moderate volatility, and above all an additive term \( +s\,dt \) that randomness cannot multiply.

\[ dW^{(A)} = (\mu - \kappa) W^{(A)} dt + \sigma_A W^{(A)} dB, \qquad dW^{(B)} = s\,dt + \mu W^{(B)} dt + \sigma_B W^{(B)} dB. \]

A's median grows at rate \( \mu - \kappa - \sigma_A^2/2 \): it is enough for lifestyle and variance to eat the drift for the typical path to be decreasing β€” even while the mean, inflated by a few miracles, remains flattering. The diminishing announced by the verse is not a curse: it is a negative exponent.

A $300,000 windfall vs $10,000/yr gathered β€” 100 paths of each, 30 years

Observation. At ΞΊ = 6% and Οƒ_A = 35%, the heir's median exponent is \( 0.07 - 0.06 - 0.061 < 0 \): the red median falls while the green one climbs. The corresponding empirical phenomenon is documented among lottery winners and professional athletes; the verse had compressed it into one line.

II.The price of haste

β€œThe thoughts of the diligent tend only to plenteousness; but of every one that is hasty only to want.”

Proverbs 21:5 β€” see also 28:20 (KJV)

In portfolio finance, haste has a unit of measure: turnover. Every transaction pays fees, bid-ask spreads, realized taxes. Let \( c \) denote the total annual cost of agitation. The final wealth is not reduced by \( c \): it is reduced by \( c \) compounded,

\[ \frac{W_T^{\text{hasty}}}{W_T^{\text{diligent}}} = \left(\frac{1+\mu-c}{1+\mu}\right)^{T} \;\xrightarrow[T\ \text{large}]{}\; e^{-cT/(1+\mu)}. \]

The loss is exponential in the product \( cT \): small costs, over long periods, dig large holes. At \( \mu = 7\% \) and \( c = 1\% \), forty years of compounding surrender about 31% of final wealth β€” nearly a third, paid not to the market but to friction. The verse's diligence is not slowness: it is the minimization of a cost term that compounds against you.

Share of final wealth surrendered to friction, as a function of c and T

Observation. The curve is nearly linear near zero, then deepens: it is \( 1 - e^{-cT/(1+\mu)} \). Going from 0 to 1% in costs does more damage than going from 2 to 3% adds β€” one more reason to defend the first basis points fiercely.

III.The ant's theorem

β€œGo to the ant, thou sluggard; consider her ways, and be wise: which having no guide, overseer, or ruler, provideth her meat in the summer, and gathereth her food in the harvest.”

Proverbs 6:6-8 (KJV)

The ant solves a control problem: a seasonal income \( y(t) = \bar y\,(1 + A\sin 2\pi t) \), a consumption she wants constant \( C \), and a reserve absorbing the difference:

\[ \dot R = y(t) - C, \qquad R(t) \ge 0. \]

As long as \( C \le \bar y \), the system is periodic and viable β€” provided the initial reserve covers the seasonal trough, whose depth grows with the amplitude \( A \). Now add the real danger: random famine years (harvest at 30%). Consumption smoothing then requires a buffer stock sized not for the average season, but for the worst plausible sequence. The verse's ant does exactly what buffer-stock theory does: she transfers grain from summer to winter because income is periodic and hunger is not.

Seasonal income, constant consumption, reserve β€” with random famine years

Observation. At C = 95% of mean income, a single bad year is often enough to pierce the reserve: the margin \( \bar y - C \) is the buffer's rebuilding rate, and it is too slow. The ant's wisdom is not abstinence β€” it is the sizing of the margin against the variance of income.

IV.The multitude of counsellors

β€œWhere no counsel is, the people fall: but in the multitude of counsellors there is safety.”

Proverbs 11:14 β€” see also 15:22 (KJV)

Replace "counsellors" with "independent sources of risk" and the verse becomes a diversification theorem. For \( n \) assets of volatility \( \sigma \) and average correlation \( \rho \), the equal-weighted portfolio's volatility is

\[ \sigma_p(n) = \sigma \sqrt{\rho + \frac{1-\rho}{n}} \;\xrightarrow[n \to \infty]{}\; \sigma\sqrt{\rho}. \]

Two lessons in one formula. First, the idiosyncratic risk β€” the \( (1-\rho)/n \) part β€” is eliminated for free: it is the only free lunch in finance. Second, the floor \( \sigma\sqrt{\rho} \) cannot be crossed: a hundred counsellors who read the same newspaper amount to one. The quality of the multitude is not its number, it is its decorrelation β€” the verse says "multitude," the variance answers "multitude of independent opinions."

Οƒ_p(n): portfolio volatility as a function of the number of assets

Observation. At ρ = 0.3, most of the benefit is captured by n β‰ˆ 15-20; beyond that, the curve is flat. But slide ρ toward 0.8: the floor rises and diversification becomes cosmetic. The lever is not adding lines, it is hunting for low ρ β€” counsellors who do not talk to each other.

V.The borrower, servant of the lender

β€œThe rich ruleth over the poor, and the borrower is servant to the lender.”

Proverbs 22:7 (KJV)

The verse's servitude has a precise mathematical structure: a symmetry breaking. The same equation governs positive and negative wealth, \( \dot W = s + r(W)\,W \), where \( s \) is the net saving flow β€” but the rate is not the same on the two sides of zero: \( r_+ \) (asset returns) when \( W > 0 \), \( r_- > r_+ \) (borrowing rates) when \( W < 0 \). The system has an unstable fixed point in negative territory:

\[ W^\* = -\frac{s}{r_-}. \]

That is the frontier of servitude. Above it, the saving flow dominates the interest and the trajectory escapes upward β€” compound interest serves. Below it, the interest dominates the flow and the trajectory plunges β€” the same mechanism enslaves. Two households with identical incomes, separated by a few thousand dollars of initial position, diverge exponentially: it is not a difference of virtue, it is a difference of basin of attraction.

The same saving flow, on both sides of the frontier W* = βˆ’s/rβ‚‹

Observation. Raise rβ‚‹ from 8% to 25%: the frontier W* moves toward zero and the basin of servitude widens. That is the quantitative reading of the verse: the lender does not rule by contract, he rules by exponent.

VI.The grandchildren's horizon

β€œA good man leaveth an inheritance to his children's children…”

Proverbs 13:22 (KJV)

The verse is not about generosity; it is about horizon. Thinking of grandchildren means optimizing over 90 years instead of 30 β€” and compound interest is hypersensitive to the horizon. Model a dynasty: capital compounds at rate \( r \) for a 30-year generation, then a fraction \( \varphi \) evaporates at each transmission (heirs' consumption, splits, estate costs). The per-generation factor is \( (1+r)^{30}(1-\varphi) \), and the dynasty grows if and only if

\[ \varphi \;<\; 1 - (1+r)^{-30}. \]

At \( r = 5\% \), the threshold is \( \varphi^\* \approx 77\% \): compounding forgives enormously β€” a dynasty can squander three quarters of every estate and still grow, if the remaining quarter stays invested for thirty years. Multigenerational wealth almost never fails for lack of return; it fails because \( \varphi \) exceeds the threshold β€” or because nobody thought in generations.

Three generations, 90 years: compounding between transmissions, evaporation Ο† at each bequest

Observation. Set Ο† just above the threshold: the dynasty declines in staircase steps despite thirty years of growth between bequests. The verse states the condition for dynastic growth; the formula gives its margin β€” surprisingly wide, provided one knows it exists.

Synthesis

Six claims three thousand years old, six mathematical objects that prove them right: a negative median exponent (13:11), a cost that compounds against you (21:5), a buffer-stock problem (6:6), the law \( \sigma\sqrt{\rho + (1-\rho)/n} \) (11:14), a symmetry breaking around \( W^\* = -s/r_- \) (22:7) and a dynastic growth condition (13:22). The text had neither ItΓ΄ nor Markowitz; it had watched the trajectories. The equations do not replace the wisdom β€” they measure its margin.

Iron sharpeneth iron (Proverbs 27:17): comments, objections and counterexamples welcome. Verses quoted from the King James Version (public domain); parameters are illustrative, and nothing here is financial advice.