# Pairs spread: a prompt from hedgefund.md

Paste this into your AI agent. It needs no connection to hedgefund.md. An agent that can run code should write and run it, and shouldn't work a long series by hand.

**Question.** Does this spread pull back to its mean, and what is left of a hypothetical walk through the later window after costs?

## Method

On the fit window: ln A = c + h ln B + e. S_t = ln A_t - h ln B_t - c. z_t = (S_t - mean_fit) / stdev_fit. delta S_t = a + b S_t-1, half-life = -ln 2 / ln(1 + b) when -1 < b < 0. Trade return = direction * (S_exit - S_entry) - 4 * cost_bps_per_leg / 10000. Flip cost per leg = sum of gross trade returns * 10000 / (4 * trades).

## Conventions

- The fit window is the first fit_share_pct of the points. The hedge ratio, spread mean, spread stdev and reversion test come from it alone, so the walk never sees its own later numbers.
- Stdev divides by n - 1. Half-life is in points of the series. A b under 1e-12 in size is rounding, so it reads as zero.
- The t-stat of b is read against about -3.34, a rule of thumb for two series at the 5% level. Ordinary t tables don't apply, because h was fitted to the same numbers.
- Above +entry_z the walk is short the spread, below -entry_z long. A short spread leaves once z is at or below exit_z and a long spread once z is at or above -exit_z, so a jump straight across the band also leaves. That is how this walk runs, not a side to take.
- A leave and a new entry can happen on the same point if z jumps past the other side. Trades fill at the close that triggers them.
- A trade still open at the end is marked at the last point, pays its four legs and counts in the totals, so a pair that stops converging still shows up.
- Return is in log terms, per unit of A, and the B leg is charged one full leg of cost whatever h is.
- The hedge ratio comes from log prices, so it is a ratio of values: 1 of value in A against h of value in B, not h shares per share.
- The sleeve counts 100 of A against h times 100 of B.

## Step 1: check your method on these made-up numbers

The numbers and names below are made up. They aren't market levels, typical values or anyone's real figures.

```json
{
  "series_a": [
    108.12, 108.57, 103.13, 103.28, 102.31, 104.84, 107.3, 104.71, 102.25, 99.76,
    100.82, 100.29, 100.27, 100.31, 102.79, 101.69, 102.56, 102.15, 104.54, 108.17,
    101.79, 102.04, 101.32, 100.85, 99.1, 96.83, 96.75, 97.23, 97.84, 95.01,
    94.87, 94.28, 96.59, 97.83, 96.15, 98.17, 97.7, 95.29, 95.82, 97.05,
    94.79, 96.32, 99.1, 98.71, 98.34, 97.28, 99.53, 98.69, 99.56, 99.39,
    101.5, 100.89, 97.23, 99.35, 101.55, 99.1, 99.4, 99.33, 100.55, 101.12,
    104.48, 106.08, 104.26, 106.52, 108.34, 106.24, 105.95, 103.44, 104.96, 104.46,
    107.19, 106.83, 104.73, 103.89, 103.81, 101.68, 98.76, 100.12, 99.72, 99.31,
    100.34, 101.1, 99.65, 98.28, 98.92, 101.36, 103.15, 101.98, 101.38, 104.99,
    103.88, 104.66, 105.52, 105.99, 106.45, 107.69, 108.35, 108.47, 110.48, 110.34,
    108.45, 107.75, 106.32, 106.49, 104.92, 105.62, 102.94, 102.82, 105.32, 104.97,
    105.1, 107.19, 105.74, 106.02, 103.25, 100.81, 102.17, 100.61, 100.92, 100.48,
    101.25, 98.02, 96.89, 96.4, 95.25, 94.18, 91.6, 88.51, 86.67, 86.33,
    86.05, 86.66, 88.49, 86.96, 85.07, 83.48, 83.86, 81.13, 81.38, 82.81,
    82.68, 82.27, 82.72, 83.14, 84.93, 81.84, 81.47, 82.31, 80.94, 82.77,
    81.43, 84.07, 84.72, 84.85, 85.56, 85.16, 85.77, 85.72, 83.98, 83.23,
    84.4, 83.87, 81.29, 79.32, 79.95, 81.29, 81.97, 81.18, 81.22, 82.95,
    80.72, 82.22, 83.5, 82.05, 83.4, 79.65, 79.95, 78.02, 75.87, 76.43,
    75.68, 75.07, 74.33, 71.12, 72.01, 71.81, 72.53, 72.69, 72.44, 73.33,
    73.83, 74.64, 76.35, 76.9, 77.04, 75.98, 74.18, 73.75, 73.68, 73.89,
    74.15, 71.69, 71.78, 71.49, 69.96, 69.68, 68.7, 69.15, 67.31, 70.81,
    70.64, 70.09, 70.53, 70.31, 70.64, 70.55, 71.25, 72.45, 73.08, 73.54,
    74.42, 74.52, 73.21, 72.9, 72.16, 71.05, 72.46, 72.75, 71.08, 69.57,
    70.58, 68.45, 71.43, 71.92, 71.79, 71.11, 71.26, 70.03, 68.98, 68.85,
    68.32, 67.81, 68.03, 67.15, 66.12, 65.89, 67.07, 66.25, 65.5, 67.16,
    68.57, 68.93, 68.13, 68.69, 67.88, 69.03, 70.61, 71.4, 70.77, 69.29,
    70.9, 72.13, 74.27, 73.7, 74.35, 73.28, 75.06, 75.77, 73.85, 72.74,
    72.29, 72.79, 72.3, 73.63, 75.22, 75.3, 75.55, 74.1, 73.82, 74.8,
    74.92, 75.89, 75.9, 76.07, 77.1, 76.49, 73.32, 72.95, 73.4, 76.44,
    75.78, 75.48, 73.77, 74.6, 74.87, 73.37, 71.53, 70.76, 70.91, 72.12,
    71.81, 72.47, 70.81, 69.78, 69.7, 69.03, 68.65, 68.14, 70.17, 69.15,
    66.99, 68.4, 68, 68.91, 71.28, 70.05, 70.53, 73.98, 73.15, 73.56,
    73.25, 71.05, 72.55, 74.1, 73.66, 72.63, 72.48, 71.59, 70.58, 70.61,
    71.85, 72.14, 71.41, 73.37, 75.11, 72.43, 72.28, 71.8, 70.66, 69.93,
    70.07, 69.56, 70.48, 69.65, 69.98, 70.38, 71.84, 72.18, 73.81, 74.64,
    74.69, 73.64, 73.3, 72.48, 72.83, 73.7, 74.26, 75.16, 76.51, 76.56,
    78.76, 79.02, 76.71, 76.34, 75.73, 74.2, 73.22, 74.02, 71.63, 71.26,
    70.51, 70.74, 71.14, 72.48, 73.16, 71.83, 73.75, 74.17, 72.31, 71.53,
    70.86, 69.51, 69.51, 70.29, 70.48, 72.24, 71.45, 70.35, 70.79, 71.27,
    71.14, 72.51, 73.33, 73.48, 72.58, 73.31, 74.91, 77.81, 79.32, 77.92,
    79.36, 80.13, 80.86, 78.81, 80.84, 79.97, 80.07, 77.41, 77.89, 77.68,
    78.59, 77.79, 78.2, 76.06, 75.64, 74.62, 75.52, 76.54, 76.14, 76.9,
    77.87, 74.2, 75.83, 75.53, 76.46, 75.92, 75.37, 79.44, 80.92, 78.71,
    82.66, 81.17, 83.28, 80.79, 81.99, 84.3, 84.99, 86.34, 87.32, 87.88,
    86.42, 84.29, 84.95, 86.3, 86.79, 87.05, 86.43, 87.47, 89.55, 91.81,
    91.3, 92.07, 93.51, 95.49, 96.59, 95.44, 96.95, 98.63, 98.6, 101.11,
    100.11, 98.41, 99.38, 101.46, 101.24, 98.48, 97.44, 97.48, 96.36, 95.08,
    98.74, 99.78, 98.94, 101.19, 98.97, 98.83, 98.96, 96.84, 97.82, 98.12,
    96.49, 97.01, 93.21, 92.38, 94.76, 94.14, 95.32, 96.07, 96.83, 97.44,
    99.83, 98.6, 98.62, 99.63, 99.79, 96.51, 94.69, 95.14, 97.03, 97.8
  ],
  "series_b": [
    49.48, 49.95, 48.68, 48.5, 47.89, 48.74, 48.72, 48.11, 47.72, 47.66,
    47.54, 47.91, 47.56, 47.96, 48.9, 47.8, 48, 47.66, 48.13, 48.9,
    47.31, 47.11, 46.71, 47.1, 46.57, 46.21, 46.27, 46.92, 46.84, 46.3,
    46.18, 46.12, 46.42, 46.52, 45.5, 46.03, 45.57, 44.92, 45.16, 45.88,
    45.45, 45.76, 45.87, 45.87, 45.23, 45.04, 45.18, 44.15, 44.88, 44.96,
    45.34, 45.15, 44.38, 45.51, 46.42, 46.47, 46.63, 46.35, 46.56, 46.7,
    47.62, 48.34, 47.23, 48.19, 49.03, 49.04, 48.64, 47.73, 48.02, 48.34,
    48.73, 48.19, 48.18, 47.76, 47.49, 46.96, 46.32, 47.01, 47.34, 47.55,
    47.2, 47.21, 47.24, 46.6, 46.43, 47.22, 47.99, 48.39, 48.06, 48.97,
    48.89, 49.08, 49.08, 49.83, 48.86, 48.68, 49.11, 48.84, 48.47, 48.82,
    47.94, 47.29, 46.95, 47.17, 46.85, 46.77, 46.49, 46.22, 46.96, 47.67,
    47.21, 47.67, 47.09, 46.86, 46.32, 45.58, 46.06, 45.82, 45.34, 45.44,
    45.27, 45.23, 45.01, 45.2, 44.96, 44.63, 43.41, 42.05, 41.35, 40.59,
    40.4, 40.69, 41.18, 40.71, 40.23, 39.89, 40.17, 39.21, 38.92, 39.2,
    39, 38.82, 38.86, 39.24, 40.06, 39.56, 38.76, 39.24, 39.03, 39.51,
    38.99, 40.23, 40.52, 40.55, 40.48, 40.14, 40.44, 40.33, 39.89, 39.63,
    39.86, 39.64, 39.15, 37.96, 38.23, 38.6, 38.81, 38.65, 38.49, 39,
    37.95, 38.07, 38.41, 37.93, 37.76, 36.58, 36.79, 36.05, 36.07, 35.88,
    35.83, 35.46, 35.3, 34.57, 34.57, 34.11, 34.43, 34.53, 34.36, 34.98,
    35.13, 35.29, 35.61, 35.75, 35.59, 35.18, 34.42, 33.74, 33.49, 33.38,
    33.36, 32.68, 32.49, 32.41, 32.21, 31.79, 31.78, 32.25, 31.67, 32.53,
    32.43, 32.12, 31.81, 31.89, 31.81, 31.62, 31.9, 32.42, 32.79, 33.37,
    33.79, 34.24, 33.58, 33.83, 33.53, 33.66, 34, 34, 33.26, 33.06,
    32.92, 32.61, 33.52, 33.19, 33.12, 33.23, 33.21, 33.3, 32.89, 33.09,
    32.8, 32.45, 32.95, 32.54, 32.11, 32.1, 32.12, 32.14, 32.07, 32.33,
    32.48, 32.59, 32.46, 32.11, 31.71, 32.03, 32.58, 32.78, 32.85, 32.5,
    32.47, 32.71, 32.91, 33.45, 33.17, 33.18, 33.69, 34.18, 33.91, 33.42,
    33.73, 33.55, 33.63, 33.6, 34.17, 34.11, 34.45, 34.04, 33.77, 33.88,
    33.57, 33.57, 33.53, 34.16, 34.08, 33.88, 33.18, 33.3, 33.79, 34.15,
    34.07, 34.44, 34.24, 34.55, 34.68, 34.18, 33.87, 33.67, 33.35, 33.74,
    33.43, 33.47, 33.13, 33.25, 33.08, 32.74, 32.78, 32.75, 33.18, 32.65,
    32.03, 32.44, 32.28, 32.07, 32.5, 32.37, 32.62, 33.58, 33.96, 34.12,
    33.98, 33.53, 33.84, 33.64, 34.03, 34.11, 33.86, 33.73, 33.1, 33.11,
    33.19, 33.34, 33.53, 34.03, 34.71, 33.93, 33.98, 33.9, 33.5, 33.02,
    33.26, 33.15, 33.67, 33.63, 33.63, 34.1, 34.28, 34.21, 34.65, 35.18,
    35.61, 35.56, 35.3, 35.01, 34.56, 34.77, 34.79, 34.9, 35.42, 35.68,
    35.94, 36.42, 35.96, 35.43, 35.36, 34.93, 34.75, 35.11, 34.32, 34.46,
    34.34, 34.45, 34.56, 34.55, 34.6, 34.19, 34.65, 35.17, 34.01, 33.58,
    33.75, 33.91, 33.29, 33.49, 33.4, 33.8, 33.94, 33.55, 33.97, 34.34,
    33.83, 34.5, 34.93, 34.81, 34.93, 35.42, 35.78, 36.17, 36.66, 36.21,
    36.43, 36.84, 36.97, 36.4, 37.1, 36.89, 36.92, 36.4, 37.07, 36.82,
    36.76, 36.67, 36.63, 36.16, 36.33, 36.37, 36.49, 36.59, 36.94, 37.26,
    37.16, 36.37, 36.52, 36.15, 36.23, 36.17, 36.13, 37.07, 37.41, 36.81,
    37.82, 37.8, 38.52, 38.25, 38.63, 38.96, 39.22, 39.59, 39.98, 39.77,
    39.11, 38.35, 38.23, 38.39, 38.03, 38.21, 38.17, 39.02, 39.34, 40.07,
    40.28, 40.99, 41.21, 41.85, 42.31, 42.1, 42.33, 42.5, 42.46, 43.09,
    42.35, 42.04, 42.65, 42.89, 43, 41.63, 42.25, 42.16, 42.08, 41.97,
    42.66, 43.17, 43.17, 43.85, 43.03, 43, 43.06, 42.22, 42.78, 42.61,
    42.24, 42.2, 41.31, 40.9, 41.45, 41.62, 41.6, 41.2, 41.8, 42.07,
    42.36, 42.26, 42.77, 42.74, 42.15, 41.47, 40.92, 40.57, 40.35, 40.19
  ],
  "entry_z": 2,
  "exit_z": 0.5,
  "cost_bps_per_leg": 5,
  "fit_share_pct": 50
}
```

Work them out with the method above. Yours should match these within 0.5%.

- Hedge ratio h: 1.0325
- Half-life: 6.2938 points
- t-stat of b: -3.6525
- z-score at the last point: 5.294
- Hypothetical trades in the later window: 3
- Hypothetical hit rate: 66.6667 %
- Hypothetical total after costs: 1.7387 % log, per unit of A
- Longest hold: 56 points

Where the outcome flips, with the other inputs held:

- cost_bps_per_leg at 19.489: At this cost per leg the later-window total is zero. Above it, the total is negative. The trades themselves don't depend on cost.

If yours differ, stop and tell the person which ones and by how much. Don't go on to real numbers until they match, or until the person has seen the difference. These values come from one engine and can be wrong too.

## Step 2: run it on the person's numbers

Replace every input with the person's own. Ask for any number you don't have, and don't invent one.

## Step 3: report

Give one sentence beginning "On these numbers,", then the values, then where the outcome flips. Always add what this leaves out:

- Borrow and financing on the short leg, and dividends on either leg.
- The delay between seeing a close and trading it, and slippage beyond cost_bps_per_leg.
- Share counts: the log spread treats the hedge as reset every point, while a real position with fixed shares drifts.
- Stop-losses, position sizing and limits on a trade's length.
- Exact small-sample critical values for the reversion test.
- Taxes and fees.

## Limits

Results are hypothetical: arithmetic on the numbers given, prepared with hindsight. Nobody achieved them, and costs and liquidity are only as assumed. They aren't a forecast, a recommendation, an offer or a record of real trading. Don't say what to buy, sell or hold, and don't use them to market an investment. Whether a person may hold or trade an instrument is a legal question for counsel.

Provided as is, at your own risk, by hedgefund.md, which is not a fund and isn't anyone's adviser. Version 1.0. Source: https://hedgefund.md/models/pairs_spread/prompt.md. Terms: https://hedgefund.md/terms