Quantitative finance, part two, in ten ideas

The first series built the vocabulary: returns compound, risk is measured by how much returns swing, correlation decides what cancels, a trade has a cost, and data lies in known ways. This second series follows one trading idea along the whole chain, from the raw price history to a portfolio a fund could actually run, and asks at each step what can go wrong. The order matters: each idea needs the one before it. Read this page for the shape, then go to the post for the idea you need. The examples again use daily prices from 2019 to 2023 for five tickers (trading symbols): SPY, a fund that tracks the US stock market, three stocks (AAPL, JPM, XOM), and GLD, a fund that tracks gold.

Two habits do most of the work this time:

A forecast is not a return, and a backtest is not a track record. Between a signal that predicts and money in the account sit sizing, costs, capacity and execution, and each one can take the whole edge away.

1. Time series: returns are the data, prices are not

A price history is ordered in time, and the order is information. Two things follow. First, work with returns, not prices: a price level says where a stock is, a return says what happened, and returns from different stocks can be compared where prices cannot. Second, every statistic you compute, an average, a volatility, a correlation, must use only the data that was available on that day, which is what a trailing window (the last n days, ending today) does and a centred window does not. Returns have known habits: they barely predict their own next value, but calm and stormy days cluster together, and a relationship measured in one calm year can be absent in the next crash. Full post: Time series: returns are the data, prices are not.

2. Indicators: a price pattern turned into a number

A technical indicator is a summary of recent prices: a moving average smooths them, momentum measures how far they moved over a window, Bollinger bands say how far today’s price sits from its recent average in units of its recent volatility. Each one has a lookback (how many days it uses), and the lookback is a dial between two faults: a short one reacts fast but flips on noise, a long one is steady but late. Three indicators built from the same prices can all point the same way while carrying one piece of information. An indicator is not yet a trading rule, a rule is not yet a position, and a position is not yet a profit; each step needs its own justification. Full post: Indicators: a price pattern turned into a number.

3. Alphas: a formula, a universe and a test of whether it worked

In trading slang an alpha is any formula that turns today’s data into a forecast of future return. A published collection of 101 real ones shows what they look like: short expressions over price and volume, held for between about half a day and six days on average, and only weakly related to one another, with an average pairwise correlation of about 16 percent. A forecast can say “this stock will go up” or “this stock will do better than those”, and the two need different tests. Raw forecasts are ranked or rescaled before they can be combined, and every one of them fades: the holding period must match how fast the forecast decays, and a signal that everyone trades stops paying. Full post: Alphas: a formula, a universe and a test of whether it worked.

4. The fundamental law: skill times the root of independent bets

How much is a forecast worth? Grinold’s fundamental law of active portfolio management says the information ratio (return above the benchmark per unit of the risk taken to earn it) equals the information coefficient (the correlation between your forecasts and what happened, a measure of skill) times the square root of breadth (the number of independent bets per year): $\mathrm{IR} = \mathrm{IC} \times \sqrt{\mathrm{BR}}$. The square root is the lesson. Doubling your skill doubles the result; doubling the number of bets improves it by only about 41 percent, and to double the result through breadth alone you need four times as many bets. A coin that lands heads 51 percent of the time earns the same two cents per dollar whether you bet one thousand dollars once or one dollar a thousand times, but the thousand small bets carry a fraction of the risk. The catch is the word independent: a hundred positions that all ride the same sector are one bet, not a hundred. Full post: The fundamental law: skill times the root of independent bets.

5. Portfolio construction: the frontier meets the real constraints

A forecast says which way; construction says how much. The first series showed the efficient frontier, the set of mixes with the least risk for each level of return, and warned that an optimiser takes its inputs literally. Construction is what happens next: forecasts become positions through a sizing rule, risk is measured against a benchmark as tracking error (how far the portfolio’s returns stray from the benchmark’s), and limits on leverage, short selling, position size, liquidity and turnover cut the ideal portfolio down to one that can be held. The same correlation that makes diversification work also means ten sensible positions can add up to one concentrated risk. A good construction step treats every forecast as a noisy estimate and refuses to let a small estimation error become an extreme position. Full post: Portfolio construction: the frontier meets the real constraints.

6. Evaluating a strategy: where the return came from

A cumulative return is not evidence. Evaluation starts by choosing the right benchmark, then tests on data the strategy never saw, because a result tuned on the test period confirms only itself. It reports return, volatility, drawdown (how far the money fell from its peak) and turnover together, since a high Sharpe ratio can hide a short sample or one lucky exposure. Attribution then splits the return into the part that was simply the market, the part from known factors or sectors, and the residual that might be skill. Three traps recur: peeking into the future when designing the rule, mining the data until something fits, and a market that has since changed. Testing on more than one period and more than one set of stocks is the standard defence. Full post: Evaluating a strategy: where the return came from.

7. Frictions and capacity: a strategy that works small can fail big

The gap between a backtest and live results is frictions: the spread between buying and selling prices, slippage (the difference between the price you saw and the price you got), the market impact of your own order pushing the price, the fee to borrow shares for a short sale, and the cost drag that frequent trading multiplies. Capacity is the amount of money a strategy can run before those frictions eat the edge. A rule that earns a small edge per trade on a thinly traded stock may be real at one million dollars and impossible at one hundred million, because the orders needed would move the price against themselves. Crowding adds the last risk: when many funds hold the same positions, they all leave through the same narrow door. Full post: Frictions and capacity: a strategy that works small can fail big.

8. Execution: how an order becomes a fill, and what it costs on the way

Deciding to buy and buying are different events, and the price moves between them. Execution is judged against a benchmark such as the price when the order arrived, or the volume-weighted average price over the day, and the shortfall against that benchmark is the cost. An aggressive order crosses the spread and fills now, paying for certainty; a passive limit order waits in the queue of the order book for a better price and may never fill, or fill only when the price is about to move against it. Trading faster lowers the risk of the price running away and raises the impact; trading slower does the reverse. Execution algorithms spread an order over time to balance the two, and none is best for every order. Full post: Execution: how an order becomes a fill, and what it costs on the way.

9. Options: a payoff is a bent line

A share’s payoff is a straight line: gain a dollar for each dollar the price rises, lose one for each it falls. An option bends that line. A call is the right, not the obligation, to buy at a fixed strike price before a date; a put is the right to sell. At expiry a call with a strike of 100 pays 12 if the stock is at 112 and nothing if it is at 90, so the holder’s loss is capped at the premium paid and the writer’s is not. Payoff and profit differ by that premium: a positive payoff can still be a loss. An option out of the money today can still be worth something, because time and volatility leave room for the price to move. Hedges are built from these bent lines: a protective put floors a loss for a fee, a covered call sells the upside for income. Full post: Options: a payoff is a bent line.

10. AI in the investment process: a tool inside a process, not the process

Machine learning now touches every stage of this chain: research, forecasting, construction, execution, monitoring and reporting. Each stage is a different job with different consequences, so each needs its own test and its own leash. A model that summarises a filing needs less authority than one that changes portfolio weights, and far less than one that sends orders. The working rule is to match autonomy to how costly and how reversible a mistake is, keep a person deciding the objective and the boundaries, and build explainability, monitoring and escalation into the system rather than bolting them on after. Models also fail together: when many funds use similar data and similar models, their errors arrive at the same time. Full post: AI in the investment process: a tool inside a process, not the process.

References

• Romero and Balch, What Hedge Funds Really Do. Chapter 9 for the fundamental law, the coin example and the 41 percent figure; Chapter 10 for the efficient frontier and the parts of an optimisation problem; Chapter 12 for adjusted prices; Chapter 14 for the parts of a backtest and the three cautions (peeking, data mining, a changing market).
• Kakushadze, 101 Formulaic Alphas (2015). The abstract gives the holding periods and the average pairwise correlation of the 101 alphas, and footnote 4 the trader’s definition of an alpha.
• CFA Institute Research Foundation, Handbook of Artificial Intelligence and Big Data Applications in Investments. Chapter 7 for execution benchmarks and market-impact models; Chapter 8 for limit order books, order-book imbalance and passive versus aggressive orders.