New on FinoLib · Volume II: Capital Markets
My latest article is now live on FinoLib, the free finance library I write: Monte Carlo Simulation and Factor Investing (Fama-French), from Volume II: Capital Markets. Read the summary below for the substance of the topic, then continue to FinoLib for the complete treatment.
5.7 Monte Carlo Simulation — Modeling Many Futures
The idea in plain words
Monte Carlo simulation is like rolling the dice thousands of times to see the range of possible futures. You choose a model, generate thousands of random scenarios from it, and read VaR, expected shortfall or any other statistic from the results. With bell-curve inputs it simply reproduces the formula, so its value comes from testing what the formula can’t handle: fat tails, stressed correlations, options and rules that depend on the path taken. More paths give more precision, but no number of paths can fix a wrong model.
Key points
■For a $1 million 60/40 portfolio, normal inputs give a 1-year 99% VaR of $192,000, matching the $191,600 formula.
■Fat tails plus a 0.88 correlation raise 97.5% expected shortfall from $194,000 to $317,000.
■VaR estimate error fell from about $14,600 to $1,500 as paths rose from 1,000 to 100,000.
A closer look
Monte Carlo simulation removes the variance-covariance method’s rigid assumptions by generating thousands, or millions, of random possible future paths for a portfolio’s returns, drawn from a specified statistical distribution, rather than assuming a single tidy mathematical formula. Running the portfolio through every simulated path produces an entire distribution of possible outcomes, from which VaR, Expected Shortfall, and any other risk statistic can be read directly, rather than assumed analytically.
Decision rule
Use Monte Carlo when the portfolio holds options or path-dependent features, or when you want to impose fat tails or stressed correlations; for plain linear portfolios, historical simulation is simpler and easier to audit. Run at least 10,000 paths for a 99% VaR, rerun with a new seed, and add paths if the estimate moves by more than about 1%. Always run one normal-input case and reconcile it to the analytic formula before trusting the others. Do not chase simulation precision when the inputs themselves are uncertain by far more.
The costliest mistake
Believing a simulation has captured tail risk because it ran 100,000 paths. If the inputs are normal with average correlations, the output is the normal answer again: $194,000 of 97.5% expected shortfall on $1 million in the example. Fat tails with a stress correlation give $317,000, so the normal-input run understates the tail by $123,000, or 39% of the stressed figure. The number of paths controls sampling error, not model error. Choose distributions and correlations for the bad states first, then count paths.
Recap
Monte Carlo simulation reads risk statistics off thousands of simulated scenarios; with normal inputs it reproduces the formula, and its value lies in testing fat tails, stressed correlations and nonlinear positions. More paths reduce sampling error by the square root of their number, but only better assumptions reduce model error.
5.8 Factor Investing — Fama-French and Beyond
The idea in plain words
A factor model explains a fund’s returns using several systematic exposures instead of just the market. In 1992 and 1993, Fama and French added size and value, and later researchers added momentum, profitability and investment. Regressing a fund’s returns on the factors tells a tilt apart from real skill: a small-value portfolio showing a 3.99% alpha under the simple model has only 0.44% left once size and value are counted. Factor premiums are not guaranteed; value lost 1.31% a year from 2007 to August 2026.
Key points
■Return attribution = Σ loading × factor return + alpha.
■For the small-value portfolio: market 7.21%, size 1.56%, value 2.51%, alpha 0.44% a year.
■Smart beta funds sell factor exposure at index-like cost.
Decision rule
Before paying for alpha, regress at least 36 and preferably 60 monthly returns on the market, size, value and momentum factors. If alpha’s t-statistic is below about 2, treat the record as factor exposure and compare the fee with that of a factor fund. Do not assume any factor premium will be earned over your own horizon: plan as if it could be zero for a decade, as value was from 2007 to 2026.
The costliest mistake
Paying active fees for factor exposure. In the small-value example, a CAPM comparison credits 3.99% a year of alpha, but 3.55 points of it is size and value exposure, leaving 0.44%. A 1.00% fee on $10 million costs $100,000 a year, more than twice the $44,000 the true alpha would add. Run the factor regression before the fee negotiation, not after.
What the full article covers
In the full article I take you through these sections:
■5.7 Monte Carlo Simulation — Modeling Many Futures
■5.8 Factor Investing — Fama-French and Beyond
It includes worked examples, a look under the hood at the mechanics, a clear decision rule, the one mistake that costs people the most, a recap, a reveal-on-click quiz and the official sources behind every figure.
Volume II: Capital Markets
Read the full article on FinoLib
The complete article, Monte Carlo Simulation and Factor Investing (Fama-French), with every worked example, diagram, quiz and source, is free to read in Volume II, Capital Markets on finolib.com.
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