Quantitative finance in ten ideas

This is the map of the series: the core ideas of quantitative finance, one short section each, in the order each one needs the one before it. Each section says what the idea is and why it matters, then links to the post that works it through with formulas, worked numbers and code. The examples use daily prices from 2020 to 2025 for five tickers (trading symbols): SPY, a fund that tracks the US stock market, three stocks (NVDA, GS, XOM), and GLD, a fund that tracks gold. Read this page to see the whole shape, then go to the post for the idea you need.

Three things do most of the work:

Returns multiply, they do not add. Risk grows with the square root of time, not with time. And when testing a strategy on past data (a backtest), ask of every number when it was known.

Before the ideas: who pays for quantitative finance

Quantitative finance is easiest to picture through the people who pay for it. A hedge fund is a lightly regulated pool of money whose manager is paid mostly on performance, classically “2 and 20”: 2% of the money every year, plus 20% of the gain. On a 10% year that is 4 of the 10 points, 40% of the gain, and the 2% is charged in losing years too. A fee that large only makes sense if the fund offers something an index cannot, usually steadier returns rather than higher ones. That incentive shapes everything the fund does. Full post: What is quantitative finance?.

1. Returns compound

Returns multiply, they do not add, because each period starts from where the last one ended. Go up 10% and then down 10% and you do not break even: 100 becomes 110, then 99. The same fact explains why losses hurt more than they look: after a 50% fall you need a 100% gain to get back. Risk, measured as volatility (how much returns swing), grows more slowly than time, with its square root. A year has about 252 trading days, and the square root of 252 is about 16, so a year’s volatility is about 16 times a day’s, not 252 times. Full post: Returns compound: multiply, never add.

2. Risk, and three ratios that divide by it

A return means little until you know how much risk was taken to earn it. The Sharpe ratio divides the return above what a risk-free deposit pays by how much the returns swing. Sortino counts only the downward swings, because investors do not mind surprises upward. The information ratio measures against a benchmark instead, for a manager whose job is to beat an index. Drawdown covers what the ratios miss: how far the money fell from its peak, and how long it took to recover. Full post: Sharpe, Sortino and information ratio.

3. Diversification: correlation decides what cancels

Owning many assets lowers risk only if they do not all move together. Correlation measures how much they move together: the less they do, the more one’s bad day is offset by another’s good day. Diversification changes the risk, never the expected return (the average you expect to earn), which stays the weighted average of the holdings. Its weakness is timing: in the 2020 crash the four equity tickers (SPY, NVDA, GS, XOM) suddenly moved together, so the protection shrank exactly when it was needed. Full post: Covariance and correlation.

4. Beta is a slope, alpha is what is left

Plot a stock’s daily returns against the market’s (here, SPY’s) and fit a straight line through them. Beta is the slope: how much of the market’s move the stock carries. NVDA’s beta against SPY was about 1.80, so on a day the market moved 1%, NVDA tended to move about 1.80%. Alpha is what is left over once that market exposure is accounted for, the part that might be skill. Knowing beta also tells you how much of the market to sell to hedge a position. Full post: Beta and alpha.

5. Portfolio optimisation takes its inputs literally

An optimiser picks the mix of assets with the best return for the risk, using estimates of each asset’s return and how the assets move together. The maths is exact; the estimates are not. The optimiser cannot tell a real edge from a lucky stretch, so it piles into whatever happened to look best in the data it saw, and the “optimal” mix often disappoints on the next period. In this series’ data it put 100% into NVDA, and NVDA’s 2024 run then paid it +320.6% against +118.2% for equal weights; in three of the five single years the chosen mix lost to equal weights. Limits such as “no short selling” (no betting on a fall by selling borrowed shares) and “no more than a set share in one asset” are what turn its answer into a portfolio you would actually hold. Full post: Portfolio optimisation.

6. Valuation: price is observed, value is estimated

The price of a share is a fact you can look up; its value is an estimate of all the cash it will pay in the future, each payment shrunk because money later is worth less than money now. That estimate is very sensitive: small changes in the assumed growth rate or the rate used to shrink future cash move the value a lot. This is why two careful analysts can disagree, and why investors want a margin of safety, a price well below their estimate of value. Full post: Value is an estimate.

7. Prices are hard to beat

Prices react quickly to information, so a pattern everyone can see is usually already in the price. Small patterns do exist: SPY tended to bounce slightly after a down day. But a rule that traded on that bounce looked good before costs and lost nearly everything after them. The deeper trap is a strategy that wins small almost every time and loses everything occasionally: it feels like skill for a long while, and its average is still a loss. Full post: Why prices are hard to beat.

8. Execution: every trade has a cost

A trading idea earns money only after the costs of trading it. Buyers pay the higher price sellers ask, sellers get the lower price buyers bid, large orders push the price against themselves, and short sellers pay to borrow shares. Each cost is small, but a rule that trades often pays it hundreds of times. In the series, a rule that traded almost every day made 6.9% over 2020 to 2025 before costs; at 0.5% per trade it lost 97.5%. A rule that traded only 11 times in those years kept most of its gain. Full post: What a trade really costs.

9. Data lies in known ways

Many impressive backtests are data errors, and the common ones have names. Survivorship: the companies that failed were dropped from the data, so the past looks rosier. Look-ahead: the test used a number before it was known. Multiple testing: try enough random signals and some will look like real ones by luck alone. Unadjusted prices: a stock split looks like a crash. Each has a known fix, and the post shows them one by one. Full post: Data that lies.

10. A strategy is a chain, judged net at the end

A strategy is a chain of steps: an idea, the data, the trading signal, how much to trade, how the trades are executed, what they cost, and how the result is judged. A mistake in any step can quietly break the result, so judge the strategy by its final profit after costs, compared with simply holding the market, not by how good the signal looked at the start. The Engineering series builds this chain in Python. Full post: Backtesting in Python.