Quantitative finance in ten ideas

Ten ideas, in the order each one needs the one before it. Every section has the same shape: the idea in one sentence, the formula with numbers worked by hand, a real run where one helps, the trap, and where the idea shows up again. The real runs use the blog’s usual five tickers (SPY, AAPL, JPM, XOM, GLD, daily adjusted closes 2019 to 2023) and one simulation written from scratch; the full script is at the end of the post. The point is to be able to do each calculation on paper and to recognise the pattern when it appears in a different costume.

Two habits do most of the work in this half:

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.

the quantity being estimated or chosenrisk, spread, errora loss or bad eventreturn, the good outcome

Hedge funds and the fee that shapes them

Full post: What is quantitative finance?.

A hedge fund is a lightly regulated pool for wealthy investors, paid mostly on performance. The investor buys lower volatility, not necessarily higher return. The fee is “2 and 20”: 2% of assets every year, then 20% of the return. Both are on absolute return, not on return above a benchmark.

\key{\text{fee}} = 2\% + 0.2 \times (\good{r} - 2\%)

Fund earns	Manager takes	Investor keeps	Manager’s share of the return
4%	2 + 0.2 × 2 = 2.4%	1.6%	60%
10%	2 + 0.2 × 8 = 3.6%	6.4%	36%
−5%	2% (the 20% is never negative)	−7%	—
$15M profit on $100M	2M, then 20% of 13M = 2.6M	$10.4M = 10.4%	31%

Four strategy families: equity long/short (stock picking, the original model), arbitrage (a price relationship that should return to normal), direction or momentum (a macro view), event-driven (mergers, shocks). Leverage is borrowing to hold more than your capital: a “130/30” fund is 130% long and 30% short on 100% of capital.

Trap	Truth
The 20% is on return above a benchmark	Absolute return
No fee in a losing year	The 2% is still charged
Weights summing to 1 means no leverage	Check gross exposure, the sum of absolute weights

Returns compound

Full post: Returns compound: multiply, never add.

Returns multiply. A move from 100 to 110 is +10%, written 0.10; the growth factor is 1.10. Chain periods by multiplying growth factors, because each period starts from the last one’s end.

\good{R_{\text{cum}}} = (1 + r_1)(1 + r_2)\cdots(1 + r_n) - 1

\key{\text{CAGR}} = \left(\frac{V_{\text{end}}}{V_{\text{start}}}\right)^{1/\text{years}} - 1

Annualising: returns use $n$ (compound), volatility uses $\sqrt{n}$ . Daily $n = 252$ (about 260 weekdays minus 8 holidays), monthly $n = 12$ . Memorise $\sqrt{252} \approx 15.87$ and $\sqrt{12} \approx 3.46$ .

Case	Working	Result
+10% then −5%	1.10 × 0.95 − 1	4.5%, not 5%
+50% then −50%	1.5 × 0.5 − 1	−25%, not 0
$100M to $121M in 2 years	√1.21 − 1	10% a year, not 21/2
+30% then −20%: arithmetic vs geometric	(30 − 20)/2 vs √(1.3 × 0.8) − 1	5% vs 1.98%
1% a month, annualised	1.01¹² − 1	12.68%
1.2% daily volatility, annualised	1.2 × 15.87	19.05%
Lost 30%: gain needed to recover	1/0.70 − 1	42.86%
60% at +10%, 40% at −5%	6 − 2	4% (weighted average)
Rule of 72 at 6%	72/6	doubles in about 12 years
Down 11% when the index is down 30%	−11 − (−30)	+19 points relative: a loss that beat its benchmark

Geometric mean is at most the arithmetic mean, equal only when every return is identical; the gap is volatility drag, about $\risk{\sigma^2}/2$ . Order of returns does not matter unless money goes in or out along the way. A short position has a negative weight, so a falling price gives a positive contribution.

Trap	Truth
Add returns	Multiply growth factors
CAGR = total ÷ years	Take the root
Annualise volatility × 252	× √252
A short of a falling stock hurts	Negative weight × negative return is positive
A negative return cannot beat the benchmark	It can

Risk, and three ratios that divide by it

Full post: Sharpe, Sortino and Information ratio.

Return without risk is half a number. Each ratio divides a reward by a different kind of risk.

\text{Sharpe} = \frac{\good{R} - R_f}{\risk{\sigma}}

\text{Sortino} = \frac{\good{R} - T}{\risk{\sigma_{\text{down}}}}

\text{IR} = \frac{\good{R} - R_{\text{bench}}}{\risk{\sigma_{R - R_{\text{bench}}}}}

Sharpe uses total volatility and the risk-free rate; Sortino uses only the moves below a target; the information ratio uses the gap to a benchmark divided by the tracking error. Annualise a daily Sharpe by $\sqrt{252}$ : the mean grows with $n$ , the standard deviation with $\sqrt{n}$ .

\begin{aligned} \key{\text{Sharpe}_{\text{annual}}} &= \sqrt{252}\,\frac{\bar{r}_{\text{daily}} - r_f}{\risk{s_{\text{daily}}}} \\[10pt] &= 15.87 \times \frac{0.0005}{0.012} = \key{0.66} \end{aligned}

Sharpe is unit-free. A wild asset with a high average daily return and a calm asset with a modest one can tie exactly, while one of them has three times the volatility and a far deeper drawdown. The ratio says nothing about the path, the size of the swings, or the tail.

Case	Working	Result
Fund A 12%, $R_f$ 2%, σ 20%	10/20	0.50
Fund B 9%, $R_f$ 2%, σ 8%	7/8	0.875, better despite lower return
Daily Sharpe 0.05, annualised	0.05 × 15.87	0.79
Returns 2, −1, 3, −2, target 0: downside deviation	√((0 + 1 + 0 + 4)/4)	1.12%
Portfolio 11%, benchmark 8%, tracking error 6%	3/6	IR 0.50
Manager A beats by 2 with TE 2.5; B beats by 4 with TE 8	0.80 vs 0.50	A has the better IR
Drawdown 25%: gain to recover	1/0.75 − 1	33.3%

Drawdown is the fall from the highest point so far to a later low, not start to end. It has a depth and a duration (the longest wait to regain the old peak), often from different episodes. Sharpe is unchanged by leverage, because top and bottom scale together.

Trap	Truth
Higher return means higher Sharpe	No
Sharpe uses the raw return	Excess over risk-free
Annualise Sharpe × 252	× √252
Sortino uses total σ	Downside only
Same σ means same risk	Shape and tails differ

Diversification: correlation decides what cancels

Full post: Covariance and correlation.

Portfolio risk depends on how holdings move together, not how many you hold. Correlation changes risk and never the expected return, which stays the weighted average.

\risk{\sigma_p^2} = w_A^2 \risk{\sigma_A^2} + w_B^2 \risk{\sigma_B^2} + 2\,w_A w_B\,\key{\rho}\,\risk{\sigma_A}\risk{\sigma_B}

\text{Cov}(A, B) = \key{\rho}\,\risk{\sigma_A}\risk{\sigma_B}

Only the last term carries $\key{\rho}$ . Positive correlation adds risk, negative removes it.

Case	Working	Result
50/50, both σ 20%, ρ = 0	0.25 × 0.04 + 0.25 × 0.04	σ = √0.02 = 14.1%
Same, ρ = −0.5	0.01 + 0.01 − 0.01	σ = 10%
Same, ρ = +1	0.01 + 0.01 + 0.02	σ = 20%: nothing cancels
Covariance, ρ 0.3, σ 20% and 15%	0.3 × 0.20 × 0.15	0.009
Perfect hedge, ρ = −1, σ 10% and 20%	weights ∝ 1/σ, so 2:1	2/3 and 1/3 gives zero variance
A 35% position falls 50%	0.35 × −0.50	portfolio −17.5%

Real run. Equal weights in SPY, AAPL, JPM and XOM, daily volatility of the mix against its members:

Year	SPY	AAPL	JPM	XOM	Equal-weight mix	Average pairwise correlation
2020, the crash year	2.10%	2.94%	3.42%	3.33%	$\risk{2.56\%}$	$\key{0.69}$
2021, a calm year	0.82%	1.58%	1.35%	1.86%	$\risk{1.01\%}$	$\key{0.39}$

In both years the mix sits below three of its four members. The difference is how much cancels: in 2021 the three stocks averaged 1.60% and the mix came in at 1.01%; in 2020 they averaged 3.23% and the mix only reached 2.56%, because the correlation between them had nearly doubled. In a crash correlations rise towards 1 and the cancelling stops exactly when you need it.

Specific risk (one company’s news) diversifies away; systematic risk (the whole economy) stays. With many assets, portfolio variance tends to the average covariance: the floor is $\sqrt{\key{\rho}}\,\risk{\sigma}$ , and it is the same mathematics as the floor on a bagged ensemble’s variance in the machine-learning half.

Trap	Truth
Correlation changes expected return	Only the variance
Zero correlation means no joint losses	No
More holdings means diversified	Not if they share a driver
Calm-period correlations hold in a crash	They rise

Portfolio optimisation takes its inputs literally

Full post: Portfolio optimisation.

Mean-variance optimisation picks weights from estimated returns and covariances. The maths is exact; the inputs are noisy; constraints make the answer holdable.

An optimiser has four parts, and naming them is most of the understanding:

Part	In a Sharpe-maximising optimiser	Why it is there
Decision variables	one weight per asset	the thing being chosen
Objective	the negative of the annualised Sharpe ratio	a minimiser maximises by minimising the negative
Constraints	weights add up to 1	fully invested
Bounds	each weight between 0 and 1	long only, no leverage
Search procedure	a gradient-based local search from equal weights	it moves downhill until no small change helps

The prices are first rescaled so every asset starts at 1.0, so that the weights mean shares of money rather than shares of a price. A local search can stop at a nearby optimum rather than the global one, and with the Sharpe objective it will happily push to a corner of the bounds.

Real run. SciPy’s SLSQP on the five tickers, weights chosen on one period and then run on the next, against the 20%-each portfolio:

Chosen on	Weights it chose	Sharpe, same period	Sharpe, next period	Equal weights, next period
2019	57% GLD, 25% JPM, 18% AAPL	3.82	1.17	0.61
2020	64% GLD, 36% AAPL	1.70	0.76	1.91
2021	80% SPY, 14% XOM, 5% AAPL	2.24	−0.10	0.29
2022	100% XOM	1.88	$\bad{0.00}$	1.93
2019 to 2021	58% GLD, 42% AAPL	1.76	0.42	0.64

In returns, the optimised weights made +25.6% against +15.1% for equal weights in 2020, then lost every later comparison: +9.9% against +29.6% in 2021, −4.9% against +4.0% in 2022, −2.9% against +24.0% in 2023, and +10.8% against +20.8% over 2022 to 2023 together.

Three things to see. In 2022, the one year where a single asset had a strongly positive Sharpe (XOM at 1.88 while SPY and AAPL were negative), the optimiser went all in on that one asset, because any diversification would have lowered the in-sample ratio; the next year it earned nothing while equal weights made 24%. The in-sample Sharpe fell by at least half out of sample in every window. And equal weights beat the optimised weights in four of the five. This is the “error maximiser” problem: small errors in expected returns become extreme weights. Remedies: constraints, shrinking the estimates towards something plain, resampling, or dropping the return forecasts altogether.

Terms: the global minimum variance portfolio needs only covariances, no return forecasts, so it is the most robust; the tangency portfolio has the highest Sharpe, where the line from $R_f$ touches the frontier; the capital allocation line mixes it with cash, and risk appetite moves you along the line.

Case	Working	Result
Weights 1.20, 0.30, −0.50	sum; sum of absolute values	net 100%, gross 200%
A: 10%, σ 16%; B: 8%, σ 8%; $R_f$ 2%	8/16 vs 6/8	0.50 vs 0.75: B
X (9%, σ 10%) vs Y (13%, σ 20%) at $R_f$ 1% then 5%	0.80 vs 0.60; 0.40 vs 0.40	the risk-free rate can change the ranking
Two uncorrelated assets, σ 10% and 20%, weights 80/20	0.64 × 0.01 + 0.04 × 0.04	σ = 8.9%, below either
Turnover: A 50 to 20, B 50 to 80	½(30 + 30)	30%
Tangency 9%, σ 12%, $R_f$ 1%; want σ 6%	half in cash	expect 5%

Trap	Truth
Minimum variance maximises return	No
Adding a constraint improves the modelled optimum	It never can; it can improve robustness
In-sample optimal means future optimal	No
The risk-free rate does not change rankings	It can

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

Full post: What a trade really costs.

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.

A simple cost model that a simulator can apply to every order: a fixed commission per order and an impact that moves the price against you by a fraction of the trade. With a $10 commission and 0.5% impact:

Order	Price paid or received	Cash change
Buy 1,000 at $100	100 × 1.005 = $100.50	−100,500 − 10 = −$100,510
Sell 1,000 at $100	100 × 0.995 = $99.50	+99,500 − 10 = +$99,490
Round trip at an unchanged price		−$1,020, about 1%

A rule that trades daily pays that 1% round trip two hundred and fifty times a year. Costs can flip a strategy’s sign.

Case	Working	Result
Bid 49.90, ask 50.10	spread; mid	$0.20; $50.00
$0.04 spread on $20 vs $0.20 on $200	0.04/20 vs 0.20/200	0.20% vs 0.10%: the “small” one is wider
Decide at 40.00, fill at 40.12, 10,000 shares	0.12 × 10,000	$1,200 slippage
Walk the book: 100 at 99.95, 50 at 99.90	weighted	average $99.933
Short at 50, cover at 42, 1,000 shares	8 × 1,000	$8,000 gross
Short gain 1,000, dividend owed 200, borrow fee 150		$650 net
1M shares, daily volume 2M		50% participation: split it

Short selling: borrow, sell, buy back later. Maximum gain is the price to zero; the loss is unlimited; the short seller pays the dividends and the borrow fee. Faster execution trades timing risk for impact.

Trap	Truth
Buyer trades at the bid	At the ask
Smaller dollar spread means more liquid	Compare relative spread
Zero commission means zero cost	Spread and impact remain
A limit order guarantees a fill	Price only

Valuation: price is observed, value is estimated

Full post: Value is an estimate.

A dollar later is worth less than a dollar now, because a dollar now can earn $r$ .

PV = \frac{FV}{(1 + r)^n}

P_0 = \frac{D_1}{r - g} \quad (\text{Gordon growth, needs } r > g)

Case	Working	Result
$121M in 2 years at 10%	121/1.21	$100M; at 12%, $96.46M
$1 a year forever at 8%	1/0.08	$12.50
$D_1$ $3, $r$ 10%, $g$ 5%	3/0.05	$60
$D_1$ $2, $g$ 4% vs 5%	2/0.06 vs 2/0.05	$33.33 vs $40: one point of growth is 20% of value
Assets 500M, liabilities 320M, 30M shares, price $9	equity 180M, $6 a share	P/B = 1.5
Scenarios 25% $40, 50% $60, 25% $90	10 + 30 + 22.5	expected $62.50
Margin of safety at price $50	(62.5 − 50)/62.5	20%, divide by value

Terminal value is most of a discounted cash flow, so the long-run growth guess drives the answer. Book value is accounting, not the present value of profits; P/B above 1 is normal. Being right on value is not the same as a good trade: path, carry and margin calls can force you out first.

Trap	Truth
A higher discount rate raises value	Lowers it
Gordon works when $g \ge r$	No
Margin of safety divides by price	By value

Beta is a slope, alpha is what is left

Full post: Beta and alpha.

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. Alpha is the intercept: what is left after paying for that exposure. This is linear regression from the other half of the material, with the market as the one feature.

\good{r_i} = \alpha + \key{\beta}\,r_m + \varepsilon

\key{\beta} = \frac{\text{Cov}(r_i, r_m)}{\text{Var}(r_m)} = \rho\,\frac{\risk{\sigma_i}}{\risk{\sigma_m}}

CAPM says the return you deserve for your beta is $R_f + \key{\beta}(R_m - R_f)$ , and $\alpha = R_{\text{actual}} -$ that. In the simpler form used for daily data there is no risk-free term: $r_i = \key{\beta}\,r_m + \alpha$ . CAPM’s claim is that expected alpha is zero: all stocks together are the market, so positive alphas are balanced by negative ones.

Case	Working	Result
Cov 0.018, market variance 0.012	0.018/0.012	β = 1.5
ρ 0.9, $\sigma_i$ 8%, $\sigma_m$ 16%	0.9 × 0.5	β = 0.45: tight tracking, small beta
ρ 0.2, $\sigma_i$ 30%, $\sigma_m$ 15%	0.2 × 2	β = 0.40: high volatility, low beta
$R_f$ 2%, β 1.2, market 10%, realised 14%	2 + 1.2 × 8 = 11.6	α = 2.4%
β 1.5, market excess 10%, realised 17%	2 + 15 = 17	α = 0: exposure, not skill
60% at β 1.4, 40% at β 0.5	0.84 + 0.20	portfolio β 1.04
Hedge $10M at β 1.3 with a β-1 fund	10 × 1.3	short $13M
Alpha 1.0%, standard error 0.8%	1.0/0.8	t = 1.25, not significant

Dollar neutral is not market neutral. Long $50M at β 1.4 and short $50M at β 1.0 nets to zero dollars but +20 beta-dollars. Beta neutral means shorting $70M. Long a stock at β 0.77 against the index: dollars 56% long and 44% short make the beta-dollars cancel ( $0.77 \times 0.56 \approx 0.44$ ).

Trap	Truth
Beta equals correlation	β = ρ σ_i/σ_m
High volatility means high beta	Not if correlation is low
Beating the market means alpha	Not if beta explains it
Dollar neutral means market neutral	Beta neutral is

Prices are hard to beat: efficiency, random walks and a roulette wheel

Full post: Why prices are hard to beat.

Efficiency is about what information prices already contain. Weak form: past prices and volume are useless, so technical analysis fails. Semi-strong: public news is priced immediately, so fundamental analysis fails too. Strong: even insiders cannot win. Measure efficiency by how fast prices react; big liquid markets are the most efficient.

A random walk has tomorrow’s price equal to today’s plus an unpredictable surprise, so the best forecast of tomorrow is today. Streaks still happen by chance.

The roulette wheel, a strategy that looks like it works. The cleanest picture of “a strategy that wins almost every time on a losing game” is a betting rule, not a stock. European roulette has 37 pockets, 18 of them red, and a bet on red pays even money, so each spin is a bet with $p = 18/37 = 0.4865$ and expected value $\bad{-2.7}$ cents per dollar. The doubling rule: bet $1 on red, double after every loss, go back to $1 after a win, stop at +$50.

Each win recovers all previous losses plus $1, so the only way to lose is a run of losses the bankroll cannot cover. With no limit on the bankroll every play eventually reaches +$50 (in 2,000 simulated plays, every one did, median 102 spins), and the rule looks like a money machine. The expected value of each spin is still negative. It shows up as the dips along the way and as the size of the bet needed to climb out of them: the worst dip across those 2,000 plays was about $\bad{-\usd 2{,}100{,}000}$ , twenty-one losses in a row, with a $2,097,152 bet pending to win $1 back.

Losses in a row	Probability	Already lost	Next bet
5	$0.5135^5 = 3.6\%$	$31	$32
7	$0.5135^7 = 0.94\%$	$127	$128
9	$0.5135^9 = 0.25\%$	$511	$512

Now give the player a bankroll. A bankroll of $127 survives six losses in a row and busts on the seventh. Reaching +$50 needs about 50 winning cycles, and each cycle busts with probability 0.94%:

P(\text{reach } +50) \approx (1 - 0.0094)^{50} = \good{0.62}

\begin{aligned} E[\text{final}] &\approx 0.62 \times 50 - 0.38 \times 127 \\ &= 31 - 48 = \bad{-\usd 17} \end{aligned}

The simulation, 2,000 plays per bankroll:

Bankroll	Reach +50	Lose it all	Mean final	Median final
$31	32.1%	67.9%	−$5.0	−$31
$63	47.9%	52.0%	−$8.8	−$63
$127	67.8%	32.1%	$\bad{-\usd 6.9}$	+$50
$255	78.5%	21.4%	−$15.4	+$50
$511	87.6%	12.4%	−$19.6	+$50
$1,023	94.8%	5.1%	−$5.9	+$50
Unlimited	100%	0%	+$50	+$50

Three things in that table. A bigger bankroll raises the hit rate, from 32% to 95%, and the median flips from a total loss to +$50 somewhere around $100. The mean stays negative the whole way, because each doubling of the bankroll makes the rare loss twice as big while only roughly halving its probability; the expected value of the game never changed, and the simulated means wobble between −5 and −20 dollars because a handful of busts decide them. And the median says +$50 for every bankroll above $127 while the mean says you are losing. Median hides the tail; mean shows it. That is the whole lesson of drawdown and of Sortino in one table, and it is why a strategy with a high hit rate can still lose money. The “unlimited” row is the only one with a positive mean, and it is bought with bets of two million dollars.

Arbitrage proper means identical payoffs with both legs at once; high correlation is not identical payoff. A z-score measures how unusual a spread is, not a promise that it converges. Four limits to arbitrage: costs and carry, capital and margin calls, crowding, borrow cost and recall.

Case	Working	Result
Same stock at 50.00 and 50.40, costs 0.10	0.40 − 0.10	$0.30 riskless
Spread 5, mean 0, σ 2	5/2	z = 2.5: short the rich leg, long the cheap one
Pair: long 100k at +5%, short 100k at +8%	5k − 8k	−$3k: dollar neutral is not safe
Anomaly 6% gross, 7% costs		−1%: real but untradeable
Merger: offer 50, price 48, 90 days	4.2% × 365/90	about 16.9% annualised

Trap	Truth
A profitable backtest disproves weak-form efficiency	Needs out-of-sample and costs
A random walk has no streaks	Streaks by chance
z = 2.5 means it will converge	No
High hit rate means profit	Sizes matter; the roulette table

Data lies in known ways

Full post: Data that lies.

Most “alpha” in a backtest is a data bug. Know each bug and its fix.

Bug	What goes wrong	Fix
Look-ahead	uses data not yet known	release timestamps, a realistic lag, `.shift(1)`
Label overlap	forward-looking labels near the split peek into the test period	purge rows near the boundary, add an embargo gap
Survivorship	today’s index members used for the past	a point-in-time universe, including the dead names
Data snooping	test 1,000 ideas, keep the lucky ones	count the trials; fresh data
Revisions	restated figures that were unknown at the time	as-first-released data
Splits, dividends	unadjusted prices show a fake −50% or miss the cash	adjusted close, total return
Stale prices	forward-filled gaps become fake zero returns then a jump	flag no-trade days

Three habits that handle most of the table. Read the adjusted close, which folds splits and dividends into earlier prices (a 2-for-1 split halves every earlier price; a dividend trims them by its yield). Fill price gaps forward first, then backward only where nothing came before, so a trading halt repeats the last known price. Keep the dead tickers in the universe; a bankrupt company’s symbol often ends in Q, and a price folder that still has those files is what a point-in-time universe looks like on disk. Survivor-only databases flatter cheap-stock and mean-reversion strategies most: one published check found a strategy returning −42% on full data and +388% on survivors only.

Case	Working	Result
1,000 useless signals tested at 5%	1,000 × 0.05	about 50 look significant by luck
Bonferroni for 100 tests at 5%	0.05/100	each must pass 0.05%
$50 to $48 with a $2 dividend	(48 + 2 − 50)/50	total return 0%, price-only says −4%
Last print $4.00, holders get $0.50	(0.5 − 4)/4	−87.5% delisting return

A new factor should clear a t-statistic of about 3, not 2, given how many have been tried. If test results guide your design, the holdout has become training data.

Trap	Truth
More survivor data fixes survivorship	No
50 hits out of 1,000 are 50 real signals	Expected by chance
Revised data is better for backtests	Unknowable then

A strategy is a chain, judged net at the end

Full post: Backtesting in Python.

Hypothesis, data, signal, sizing, execution, costs, evaluation. Every link can silently break it, so judge the net result at the end, not the signal at the start.

Case	Working	Result
Edge 0.01%, cost 0.02%		do not trade
3 wins of $10k at +1%, 1 loss of $100k at −1%	300 − 1,000	−$700 despite a 75% hit rate
40% daily turnover on $10M at 0.10%, $3k gross a day	0.4 × 10M × 0.001 = 4k	−$1,000 a day
Same 2% forecast, volatility 10% vs 40%	size ∝ forecast/σ	4:1, the riskier gets less
Edge 0.50%, impact 0.10 / 0.30 / 0.70%		0.40 / 0.20 / −0.20%: capacity binds
$120M long + $120M short on $40M equity	240/40	6× gross; net zero is not low risk
$30M exposure, $10M equity, 2% move	0.02 × 30/10	6% of equity

The chain in one line: the optimiser above scored 1.76 on the years it was fitted to and 0.42 on the two that followed, and a cost model then takes about 1% per round trip off whatever that strategy trades. A benchmark must match the mandate: long-only equity against a broad equity index, not against cash, because beating cash in a bull market is just beta.

When live results lag the backtest, check in this order: bugs, trades that differ from the backtest, costs, illiquid names, data snooping, regime change.

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 $0.5135^k$ .

The script

Every number and chart above comes from one script, which runs in a few seconds with uv run (the dependencies are declared at the top of the file). It uses the blog’s shared price file and two small helpers for loading prices and drawing charts.

The full script, 151 lines

# /// script
# requires-python = ">=3.11"
# dependencies = ["numpy", "pandas", "matplotlib", "scipy"]
# ///
"""Numbers and charts for the post "Quantitative finance in ten ideas".

A doubling-rule betting simulation on a European roulette wheel, and a Sharpe-maximising optimiser
(SciPy SLSQP) run on the blog's shared prices (SPY, AAPL, JPM, XOM, GLD, 2019-2023).
Run:  uv run examples/quant-finance-in-ten-ideas/ten_ideas.py
"""
import sys
from pathlib import Path

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from scipy.optimize import minimize

sys.path.insert(0, str(Path(__file__).resolve().parents[1]))
import chartstyle as cs  # noqa: E402
import market_data as md  # noqa: E402

OUT = Path(__file__).resolve().parents[2] / "public" / "posts" / "quant-finance-in-ten-ideas"
K = np.sqrt(252)


def p(label, value):
    print(f"{label:<66} {value}")


# ------------------------------------------------------------ the doubling rule ----
# European roulette: 18 of 37 pockets are red and a red bet pays even money, so the house keeps
# 2.7% of every dollar. The doubling rule: bet 1, double after every loss, back to 1 after a
# win, stop at +50.
P_WIN, TARGET, SPINS = 18 / 37, 50, 2000


def play(rng, bankroll=None):
    """One play: winnings after each round, frozen once the target is hit or the bankroll is
    gone. bankroll=None means unlimited."""
    w = np.zeros(SPINS + 1)
    bet, total = 1, 0
    for s in range(1, SPINS + 1):
        if total >= TARGET:
            w[s:] = total; break
        if bankroll is not None:
            if bankroll + total <= 0:
                w[s:] = -bankroll; break
            bet = min(bet, bankroll + total)
        if rng.random() < P_WIN:
            total += bet; bet = 1
        else:
            total -= bet; bet *= 2
        w[s] = total
    return w


rng = np.random.default_rng(42)
print("\n== the doubling rule ==")
p("P(win a spin)", f"{P_WIN:.4f}; expected value per dollar bet {P_WIN - (1 - P_WIN):+.4f}")
for k in (5, 7, 9):
    p(f"P({k} losses in a row)", f"{(1 - P_WIN) ** k:.3%}; after them: lost {2**k - 1}, next bet {2**k}")
ten = np.array([play(rng) for _ in range(10)])
bankrolls = [31, 63, 127, 255, 511, 1023, None]
sweep = []
for bk in bankrolls:
    plays = np.array([play(rng, bk) for _ in range(2000)])
    f = plays[:, -1]
    hit = (f >= TARGET).mean(); bust = (bk is not None) and (f <= -bk).mean() or 0.0
    sweep.append((bk, hit, bust, f.mean(), np.median(f), f.std()))
    p(f"bankroll {str(bk) if bk else 'unlimited':>9}: hit +50 / lose all / mean / median", f"{hit:.1%} / {bust:.1%} / {f.mean():+.1f} / {np.median(f):+.0f}")
unl = plays
p("unlimited: median spins to reach +50", f"{np.median([(row != row[-1]).sum() for row in unl]):.0f}")
p("unlimited: worst dip seen across 2,000 plays", f"{unl.min():+.0f}")

# ---------------------------------------------------------------- optimiser ----
px = md.prices()
r = md.daily_returns(px)


def sharpe_of(v):
    rr = np.diff(v) / v[:-1]
    return K * rr.mean() / rr.std(ddof=1)


def optimise(prices):
    norm = (prices / prices.iloc[0]).to_numpy()
    n = norm.shape[1]
    res = minimize(lambda w: -sharpe_of(norm @ w), np.full(n, 1 / n), method="SLSQP",
                   bounds=[(0, 1)] * n, constraints={"type": "eq", "fun": lambda w: w.sum() - 1})
    w = np.clip(res.x, 0, 1); return w / w.sum()


def perf(prices, w):
    norm = (prices / prices.iloc[0]).to_numpy()
    v = norm @ w
    return sharpe_of(v), v[-1] / v[0] - 1


print("\n== optimiser: chosen on one period, run on the next ==")
windows = [("2019", "2020"), ("2020", "2021"), ("2021", "2022"), ("2022", "2023"), ("2019-2021", "2022-2023")]
rows = []
for a, b in windows:
    pa = px.loc[a.split("-")[0]:a.split("-")[-1]]
    pb = px.loc[b.split("-")[0]:b.split("-")[-1]]
    w = optimise(pa)
    eq = np.full(5, 0.2)
    s_in, _ = perf(pa, w); s_out, r_out = perf(pb, w); s_eq, r_eq = perf(pb, eq)
    singles = {t: sharpe_of((pa[t] / pa[t].iloc[0]).to_numpy()) for t in md.TICKERS}
    rows.append((a, b, w, s_in, s_out, r_out, s_eq, r_eq))
    p(f"chosen on {a}: weights", " ".join(f"{t}:{x:.0%}" for t, x in zip(md.TICKERS, w)))
    p(f"   singles' Sharpe on {a}", " ".join(f"{t}:{s:.2f}" for t, s in singles.items()))
    p(f"   Sharpe in-sample {a} / out-of-sample {b} / equal weights {b}", f"{s_in:.2f} / {s_out:.2f} / {s_eq:.2f}")
    p(f"   return on {b}: optimised / equal weights", f"{r_out:+.1%} / {r_eq:+.1%}")

print("\n== diversification: four-asset mix vs its members ==")
for yr in ["2020", "2021"]:
    sub = r.loc[yr, ["SPY", "AAPL", "JPM", "XOM"]]
    mix = sub.mean(axis=1)
    p(f"{yr}: daily sigma of each", " ".join(f"{t}:{sub[t].std():.2%}" for t in sub))
    p(f"{yr}: equal-weight mix sigma, average pairwise correlation", f"{mix.std():.2%}, {sub.corr().values[np.triu_indices(4, 1)].mean():.2f}")

# ------------------------------------------------------------------- charts ----
cs.apply()
fig, (a1, a2) = plt.subplots(2, 1, figsize=(8, 3.8 * 2))
rounds = np.arange(SPINS + 1)
cs.title(a1, "Ten plays of the doubling rule with no bankroll limit", 1)
for i in range(10):
    a1.plot(rounds, ten[i], color=cs.MAROON if i == 0 else cs.GREY, lw=1.2 if i == 0 else 0.9,
            alpha=1 if i == 0 else 0.6, label="one play (winnings after each spin)" if i == 0 else "_x")
a1.axhline(TARGET, color=cs.BLUE, lw=1, ls="--", label="the +50 target")
a1.set_xlim(0, 250); a1.set_ylim(-520, 80); a1.set_xlabel("Spin number"); a1.set_ylabel("Winnings so far (dollars)"); a1.legend()
cs.title(a2, "A bigger bankroll wins more often, and the average play still loses", 2)
xs = [bk if bk else 4095 for bk, *_ in sweep]
a2.plot(xs, [100 * h for _, h, *_ in sweep], color=cs.MAROON, marker="o", ms=4, label="plays that reach +50 (%)")
a2.plot(xs, [m for *_, m, _, _ in sweep], color=cs.BLUE, marker="o", ms=4, label="mean final winnings (dollars)")
a2.axhline(0, color=cs.GREY, lw=0.8)
a2.set_xscale("log", base=2); a2.set_xticks(xs); a2.set_xticklabels([str(bk) if bk else "no limit" for bk, *_ in sweep])
a2.set_xlabel("Bankroll (dollars, log scale)"); a2.set_ylabel("Percent of plays, or dollars"); a2.legend()
cs.save(fig, OUT / "doubling-rule.svg")

fig, ax = plt.subplots(1, 1, figsize=(8, 3.8))
cs.title(ax, "Weights chosen on 2019 to 2021, then run on 2022 to 2023, against equal weights")
a, b, w, *_ = rows[-1]
pb = px.loc["2022":"2023"]
norm = (pb / pb.iloc[0]).to_numpy()
ax.plot(pb.index, norm @ w, color=cs.MAROON, label="Sharpe-maximising weights from 2019 to 2021")
ax.plot(pb.index, norm @ np.full(5, 0.2), color=cs.BLUE, label="equal weights, 20% each")
ax.plot(pb.index, (pb["SPY"] / pb["SPY"].iloc[0]).to_numpy(), color=cs.GREY, label="SPY")
ax.set_xlabel("Date"); ax.set_ylabel("Value of 1 dollar invested at the start of 2022"); ax.legend()
cs.save(fig, OUT / "optimised-vs-equal.svg")