Define the prices and units
Fictional shares A and B initially cost USD 100 and 50. A is bought and borrowed B is sold short. Quantities qA and qB stay fixed; each case assumes complete executions at the same entry and exit moments. The initial price quotient 100/50 is 2.
At exit prices 110 and 60, the quotient is 11/6. Its change is (11/6)/2−1=−1/12, about −8.333333%. This is a price-ratio change, not a USD amount or a return on account equity.
| Moment | A, USD | B, USD | A/B | A/B change |
|---|---|---|---|---|
| Entry | 100.00 | 50.00 | 2 | — |
| Exit: falling ratio | 110.00 | 60.00 | 11/6 | −1/12 ≈ −8.333333% |
| Exit: flat ratio | 110.00 | 55.00 | 2 | 0% |
| Exercise exit | 108.00 | 52.00 | 27/13 | 1/26 ≈ 3.846153846% |
Four executions rather than one line
Buying A produces −qA×A0 and selling borrowed B produces +qB×B0. At closure, selling A produces +qA×A1 and buying B back produces −qB×B1. Summing these four cash flows before charges equals the sum of the two position results.
Write gross result G=qA×(A1−A0)+qB×(B0−B1). Short-sale proceeds alone are neither profit nor freely available capital. Required collateral, starting account equity and broker requirements are not calculated in this model.
Specify the example’s charges
Each of the four executions has an invented commission of USD 0.10 per share. Total commission is C=0.10×(qA+qA+qB+qB). It is not one charge for the entire pair or an extra bid–ask spread.
The hypothetical holding period is 10 calendar days. Borrow charge L=0.20×qB USD is stipulated for that entire period, not per day or per year. Distributions, taxes, FX, corporate actions, interest on proceeds and other charges are zero by assumption. Actual agreements may differ.
Result under these stated conditions is N=G−C−L. The supplied prices are execution prices; do not subtract their spread again. Availability or recall of a stock loan and forced closure are not tested here.
One price path, two quantity choices
With 10 A and 10 B, initial position values are USD 1000 and 500. A’s gain contributes 100 and B’s rise costs 100: G=0. Four commissions total 4 and the whole-period borrow charge is 2, leaving N=−6 USD.
With 10 A and 20 B, initial values are equal at USD 1000 each. A still contributes +100; B contributes −200, so G=−100. Commission 6 and borrowing 4 give N=−110 USD. Prices and the A/B line are identical; B’s quantity changed.
| Item | 10 A / 10 B | 10 A / 20 B |
|---|---|---|
| Buy A | −1000.00 | −1000.00 |
| Sell borrowed B | 500.00 | 1000.00 |
| Sell A | 1100.00 | 1100.00 |
| Buy B back | −600.00 | −1200.00 |
| Gross result | 0.00 | −100.00 |
| Four commissions | −4.00 | −6.00 |
| Borrowing for all 10 days | −2.00 | −4.00 |
| Result after stated charges | −6.00 | −110.00 |
A flat ratio can have different cash results
Now use exit prices 110/55: both prices rose 10% and A/B stayed at 2. With 10 A and 10 B, gross result is +50 USD and stated charges of 6 leave +44 USD. With 10 A and 20 B, gross is 0 and charges of 10 leave −10 USD.
A flat quotient does not establish zero cash profit for arbitrary quantities. Unequal initial values in the first choice leave exposure to a common price move. Equal initial currency values do not establish equal risk or market neutrality.
| Exit A/B | qA / qB | Gross | Commissions | Borrowing | Result |
|---|---|---|---|---|---|
| 110 / 60 | 10 / 10 | 0.00 | 4.00 | 2.00 | −6.00 |
| 110 / 60 | 10 / 20 | −100.00 | 6.00 | 4.00 | −110.00 |
| 110 / 55 | 10 / 10 | 50.00 | 4.00 | 2.00 | 44.00 |
| 110 / 55 | 10 / 20 | 0.00 | 6.00 | 4.00 | −10.00 |
The equal-initial-value special case
If X=qA×A0=qB×B0, define price returns rA=A1/A0−1 and rB=B1/B0−1. Then G=X×(rA−rB), while the ratio return is (rA−rB)/(1+rB). With positive A0, B0 and B1, the denominator is positive: before costs, gross result and ratio change have the same sign.
Thus ratio direction can relate to gross-result direction under this explicit condition. The cash magnitude, ratio change and result after charges remain different. These identities do not provide an optimal hedge or an account-equity return without further inputs.
Exercise with a complete answer
Keep entry prices 100/50, quantities 10 A and 20 B, and every charge assumption. Close at 108/52. Find the new quotient, its change, both position results and the USD outcome.
Answer: A/B=27/13; change (27/13)/2−1=1/26≈3.846153846%. A contributes +80 USD and B −40 USD, so G=40 USD. Four commissions are 1+1+2+2=6 USD; the whole-period loan charge is 4 USD. N=40−6−4=30 USD. Price returns differ by 8%−4%=4 percentage points; dividing 4% by 1.04 yields the ratio change. Neither 30 USD nor 3.846…% is a return on account capital here.
Next: examine both price charts
Use the existing A/B workflow, then compare the quotient with each instrument’s ordinary price chart. In a separate record, keep quantities, execution times and prices, all four commissions and the stock-loan terms.
Student Wyckoff provides chart analysis and saved work. An A/B chart does not create a tradable instrument, paired order or broker accounting record. This article does not test correlation, promise mean reversion or establish strategy effectiveness.
- Separate a price quotient, share quantities and currency results.
- Sum four executions and charge every commission once.
- Keep actual agreement terms separate from a chart idea.
Primary sources
Sources explain the mechanics and limitations; the editorial examples use invented numbers rather than market quotations.
Your next practical step
Use the existing workflow to open and save an A/B chart. This is a next step for visual comparison; trade decisions and cash accounting need separate inputs.
Open the A/B chart workflow ↗