Quantitative finance in ten ideas

This is the map of the series. Each idea gets one paragraph, its key number from the blog’s five tickers (SPY, AAPL, JPM, XOM, GLD, daily adjusted closes 2019 to 2023), and a link to the post that works it through with formulas, worked numbers and code. The ideas are in the order each one needs the one before it. At the end is the formula sheet for the whole series.

Two habits do most of the work:

Multiply, never add, returns. Scale volatility and Sharpe by the square root of time, never by time. And for every number in a backtest, ask when it was known.

1. Hedge funds and the fee that shapes them

A hedge fund is a lightly regulated pool paid mostly on performance, “2 and 20”: 2% of assets every year, then 20% of the return, both on absolute return, not on return above a benchmark. On a 10% year the manager takes 3.6 points, over a third of the return; in a losing year the 2% is still charged. The investor is buying lower volatility, not higher return. Full post: What is quantitative finance?.

2. Returns compound

Returns multiply. Chain periods by multiplying growth factors, because each period starts from the last one’s end: +10% then −10% leaves 0.99, a loss. A −50% drawdown needs +100% to recover. Annualise a return with a power, volatility and Sharpe with the square root of time (√252 ≈ 15.87 for daily, √12 ≈ 3.46 for monthly). Full post: Returns compound: multiply, never add.

3. Risk, and three ratios that divide by it

Return without risk is half a number. Sharpe divides excess return by total volatility, Sortino by downside volatility only, the information ratio by tracking error against a benchmark. Each answers a different question, and all three scale with √time. Drawdown is the risk the ratios miss: how far from the peak, and for how long. Full post: Sharpe, Sortino and information ratio.

4. Diversification: correlation decides what cancels

Portfolio risk depends on how holdings move together, not how many you hold. Correlation changes the risk and never the expected return, which stays the weighted average. With correlation ρ between every pair, adding assets drives risk down to a floor of ρσ², never to zero; in the 2020 crash the five tickers’ correlations jumped, and the floor rose exactly when it mattered. Full post: Covariance and correlation.

5. Portfolio optimisation takes its inputs literally

Mean-variance optimisation picks weights from estimated returns and covariances. The maths is exact, the inputs are noisy, so the optimiser piles into whatever looked best in the window it saw. Over five rolling windows the optimised portfolio beat equal weights in some and lost in others; constraints (long only, weight caps) are what make the answer holdable. Full post: Portfolio optimisation.

6. Execution: a trade pays the spread, the book and the impact

An idea becomes profit only after paying the bid-ask spread, walking the order book, moving the price, and, for shorts, borrowing. A market order guarantees execution and risks the price; a limit order guarantees the price and risks never filling. A buyer trades at the ask. At 0.5% per trade a rule that trades almost daily turned +14.5% gross into −95%. Full post: What a trade really costs.

7. Valuation: price is observed, value is estimated

A dollar later is worth less than a dollar now, because a dollar now can earn $r$ . Present value divides by $(1 + r)^n$ ; the Gordon model prices a growing dividend stream as $D_1 / (r - g)$ and blows up as $g$ approaches $r$ . Small changes in $r$ or $g$ move the value a lot, which is why value is an estimate and the margin of safety exists. Full post: Value is an estimate.

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

Plot a stock’s daily return against the market’s and fit a line. Beta is the slope, how much of the market’s move you carry (AAPL: 1.24 against SPY); alpha is the intercept, what is left after paying for that exposure. CAPM turns beta into a required return, and a hedge sized in beta-dollars removes the market from a position. Full post: Beta and alpha.

9. Prices are hard to beat

Efficiency is about what information prices already contain: weak form (past prices), semi-strong (public news), strong (even insiders). SPY’s day-to-day autocorrelation is −0.17, enough to make a “buy after a down day” rule look great before costs and lose nearly everything after them. A roulette doubling rule shows the same shape: most runs win small, the rare run loses everything, and the expected value is still negative. Full post: Why prices are hard to beat.

10. Data lies in known ways

Most “alpha” in a backtest is a data bug: survivorship (the losers were deleted), look-ahead (a number used before it was known), multiple testing (1,000 coin-flip signals produce about 50 “significant” ones), unadjusted prices (a split looks like a crash). Know each bug and its fix. Full post: Data that lies.

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

Hypothesis, data, signal, sizing, execution, costs, evaluation. Every link can silently break the chain, so judge the net result at the end against a benchmark, not the signal at the start. Full post: Backtesting in Python.

The formula sheet

Topic	Formula	Remember
Cumulative return	$\prod (1 + r) - 1$	multiply
CAGR	$(V_{\text{end}}/V_{\text{start}})^{1/T} - 1$	root, not divide
Annualise return	$(1 + r)^n - 1$	$n$ = 12 or 252
Annualise σ, Sharpe	$\times \sqrt{n}$	√252 ≈ 15.87, √12 ≈ 3.46
Recovery	$1/(1 - L) - 1$	−50% needs +100%
Sharpe	$(R - R_f)/\sigma$	total risk
Sortino	$(R - T)/\sigma_{\text{down}}$	bad risk only
Information ratio	$(R - R_b)/\text{tracking error}$	against a benchmark
Covariance	$\rho\,\sigma_A\sigma_B$	carries units
Two-asset variance	$w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\sigma_A\sigma_B$	ρ only in the cross term
Perfect hedge	$w_A/w_B = \sigma_B/\sigma_A$	inverse to volatility
Turnover	$\tfrac{1}{2}\sum \lvert \Delta w \rvert$	half
Beta	$\text{Cov}(r_i, r_m)/\text{Var}(r_m) = \rho\,\sigma_i/\sigma_m$	a slope
CAPM	$R_f + \beta(R_m - R_f)$	required return
Alpha	actual − CAPM	skill, if real
Hedge size	dollars × β_asset / β_hedge	beta-dollars
Present value	$FV/(1 + r)^n$	divide
Gordon	$D_1/(r - g)$	needs $r > g$
Margin of safety	$(V - P)/V$	divide by value
Spread, mid	ask − bid; (ask + bid)/2	buyer pays the ask
Impact cost	price × (1 ± impact)	plus commission per order
False positives	tests × α	1,000 × 5% = 50
Total return	$(P_1 + D - P_0)/P_0$	include the dividend
Gross leverage	(long + short)/equity	net 0 is not safe
2 and 20	2% + 20% × (return − 2%)	absolute return
Roulette doubling rule	bust after $k$ losses in a row, $P = 0.5135^k$	the median wins, the mean loses

The arithmetic to be fluent in

Multiply growth factors; CAGR with a root; geometric mean.
Daily to annual: σ × 15.87, Sharpe × 15.87; monthly × 3.46.
Recovery gain after a loss; drawdown from the peak.
Two-asset variance with the cross term, then the square root.
Beta, CAPM, alpha, hedge size, beta-neutral weights.
Present value, Gordon, margin of safety.
Walk the book, slippage, short profit net of dividend and fee, impact plus commission.
The roulette expected value: $p \times \text{win} - (1 - p) \times \text{loss}$ , and a run of $k$ losses as $0.5135^k$ .