---
title: "P&L and the Accounting of a Position"
book: "Markets I: The Ecosystem and Exchange-Traded Markets"
subject: quant
language: en
chapter: 7
exercises: 8
source: https://one-course.com/books/quant/1/en/chapter/7-p-l-and-the-accounting-of-a-position
---

# Chapter 7 — P&L and the Accounting of a Position

At 16:05 the trader tells her desk head she made forty thousand dollars. At 17:30 the risk report says thirty-one. The next morning the finance department books twenty-seven. Nobody is lying and nobody has made a mistake: the three numbers use the same six trades and three different answers to the questions “at what price is the [position](#def-m1-pnl-and-positions-position) worth what?” and “what did the day cost?”. A trading firm lives and pays its people on this number. This chapter defines it, shows which part of it is arithmetic and which part is convention, and builds the component that computes it.

## 7.1 Positions and exposure

**Definition 7.1 (Position).**

A *position* in an instrument is the signed quantity held: positive (*long*) if owned, negative (*short*) if owed. It changes only through fills, corporate actions and transfers; it is the sum of the signed quantities of all of them since inception.

**Definition 7.2 (Notional, gross and net exposure).**

The *notional* of a [position](#def-m1-pnl-and-positions-position) $q_i$ marked at price $p_i$ is $|q_i|\,p_i$ (times the contract multiplier, for a derivative). For a book, the *gross exposure* is $\sum_i |q_i| p_i$ and the *net exposure* is $\sum_i q_i p_i$.

**Example 7.3 (A long–short book).**

Long $300 million and short $200 million on $100 million of equity: gross $500 million, or $5\times$ the equity, the measure of how much can go wrong; net $100 million, or $1\times$, the measure of how much market direction is being taken. Limits are set on both, and on each [position](#def-m1-pnl-and-positions-position) separately.

## 7.2 Marking to market

**Definition 7.4 (Mark-to-market).**

To *mark to market* a [position](#def-m1-pnl-and-positions-position) is to value it at a current market price, the *mark*, instead of at what was paid for it. The profit and loss of a book over a period is the change in its marked value plus the net cash it produced.

**Proposition 7.5 (The P&L identity).**

Start flat. After fills $k = 1,\dots,n$ with sides $\epsilon_k = \pm1$ (buy $+1$), quantities $n_k$, prices $p_k$ and fees $f_k$, the [position](#def-m1-pnl-and-positions-position) is $q =
\sum_k \epsilon_k n_k$, the cash is $c = -\sum_k \epsilon_k n_k p_k$, and the profit and loss at mark $m$ is

$$
\pnl(m) \;=\; c + q\,m - \sum_k f_k .
$$

It depends on the fills only through $(q, c, \sum f_k)$: on no accounting convention, and not on the order of the fills.

**Proof.** The book holds cash $c - \sum f_k$ and $q$ units worth $m$ each; it started with nothing. ∎

This is the exact part. Everything is an integer if prices are in ticks or ledger units ([Section 1.6](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#bld-m1-what-a-trading-firm-does-ledger)), and a P&L system that cannot reproduce it to the unit has lost a trade. The *split* of that number is where conventions enter.

**Definition 7.6 (Average cost, realised and unrealised P&L).**

Under the *average cost* convention an open [position](#def-m1-pnl-and-positions-position) carries the quantity-weighted average price $\bar p$ of the fills that built it. A fill that reduces the [position](#def-m1-pnl-and-positions-position) by $n$ units at price $p$ *realises* $\pm n(p - \bar p)$ (plus for a long, minus for a short) and leaves $\bar p$ unchanged; a fill that increases it updates $\bar p$; a fill that flips it closes the old [position](#def-m1-pnl-and-positions-position) entirely and opens the remainder at $p$. The *realised P&L* is the sum of these amounts; the *unrealised P&L* is $q(m - \bar
p)$.

**Example 7.7 (Six fills).**

The trader of the opening paragraph buys 20 000 at 49.30 and 20 000 at 49.55 ($\bar p = 49.425$), sells 10 000 at 50.10 (realising $10\,000 \times
0.675 = 6\,750$), buys 10 000 at 49.70 ($\bar p = 49.494$), sells 25 000 at 50.28 (realising $19\,656$), and buys 15 000 at 50.20 ($\bar p = 49.847$). She ends long 30 000 with $26 406 realised. Under *first in, first out* the same sales are matched to the oldest purchases and realise $28 750. Marked at 50.00 the unrealised parts are $4 594 and $2 250: both conventions total $31 000, as [Proposition 7.5](#prop-m1-pnl-and-positions-identity) says they must. The split matters for tax and for performance statistics (“win rate”); it never changes what the firm is worth.

![The day of , observed at each fill and at the close, marked at the mid. Realised P&L moves only when a sale reduces the position; the last point falls because the closing mid is below the afternoon’s prices while she is still long 30 000 shares. Data: the chapter’s fill list.](https://one-course.com/images/onecourse/chapters/quant-1/m1-pnl-and-positions/fig-678d8e3dea1b.svg)

***Figure 7.1.** The day of [Example 7.7](#ex-m1-pnl-and-positions-six), observed at each fill and at the close, marked at the mid. [Realised P&L](#def-m1-pnl-and-positions-realised) moves only when a sale reduces the [position](#def-m1-pnl-and-positions-position); the last point falls because the closing mid is below the afternoon’s prices while she is still long 30 000 shares. Data: the chapter’s fill list.*

**Method 7.8 (Choosing a mark).**

1. **The last trade** is the most recent information and the worst mark: it is at the bid or at the ask, at someone else’s size, possibly minutes old.
2. **The mid** of a tight, two-sided quote is the usual intraday mark for liquid instruments.
3. **The side against you** — bid for longs, ask for shorts — values the [position](#def-m1-pnl-and-positions-position) at what closing it would fetch for a small size; for a large [position](#def-m1-pnl-and-positions-position) subtract an estimate of impact as well.
4. **The official close** (closing auction price or settlement price) is what finance, clearing houses and fund administrators use: it is public, unique and auditable.

Decide once, in writing, which mark serves which report. A trader allowed to choose the mark can choose the P&L.

**Example 7.9 (Forty, thirty-one, twenty-seven).**

The trader’s screen marks at the last print, 50.30: cash $-\$1\,469\,000$ plus $30\,000 \times 50.30$ gives $40 000. The risk system marks at the closing mid, 50.00: $31 000. The $9 000 between them is $30\,000$ shares $\times$ 30 cents of mark. Finance uses the official close, also 50.00, and subtracts $3 400 of fees and $600 of financing: $27 000 ([Figure 7.2](#fig-m1-pnl-and-positions-three)).

![One day, three numbers. Data: , computed by the chapter’s script.](https://one-course.com/images/onecourse/chapters/quant-1/m1-pnl-and-positions/fig-fdf873ce8a28.svg)

***Figure 7.2.** One day, three numbers. Data: [Example 7.9](#ex-m1-pnl-and-positions-three), computed by the chapter’s script.*

## 7.3 What a position costs and earns while it is held

**Definition 7.10 (Carry).**

The *carry* of a [position](#def-m1-pnl-and-positions-position) is the P&L it produces if prices do not move: income received (dividends, coupons, interest on short proceeds) less costs paid (financing of longs, [borrow fees](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-seclending) on shorts, dividends owed on shorts).

A complete daily P&L therefore has five lines besides price moves: commissions and exchange fees on the day’s fills; financing accrued ([Section 6.8](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#bld-m1-financing-calc)); [borrow fees](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-seclending); dividends and coupons, booked on the ex-date ([Chapter 8](https://one-course.com/books/quant/1/en/chapter/8-shares-corporate-actions-and-indices#ch-m1-shares-corporate-actions-indices)); and, for [positions](#def-m1-pnl-and-positions-position) in a foreign currency, the move of the exchange rate.

**Proposition 7.11 (Currency translation).**

A [position](#def-m1-pnl-and-positions-position) of $q$ units of a foreign asset moves from price $p_0$ to $p_1$ in its own currency while the exchange rate (base currency per unit of foreign currency) moves from $x_0$ to $x_1$. Its P&L in base currency is

$$
q\,(p_1x_1 - p_0x_0) \;=\; \underbrace{q\,(p_1-p_0)\,x_0}_{\text{price}}
 \;+\; \underbrace{q\,p_0\,(x_1-x_0)}_{\text{currency}}
 \;+\; \underbrace{q\,(p_1-p_0)(x_1-x_0)}_{\text{cross}} .
$$

**Proof.** Expand $p_1x_1 = (p_0 + \Delta p)(x_0 + \Delta x)$. ∎

**Example 7.12 (A good stock in a bad currency).**

A dollar-based fund holds 10 000 shares of a euro stock that rises from 40 to 41 while the euro falls from 1.10 to 1.08 dollars. Price: $+\$11\,000$. Currency: $-\$8\,000$. Cross: $-\$200$. Total $+\$2\,800$. A fund that wants the first number and not the second sells € 400 000 forward; the hedge covers the opening value $qp_0$, never the cross term, and must be resized as the stock moves.

## 7.4 P&L explain: a first look

**Definition 7.13 (P&L attribution).**

A *P&L attribution*, or *explain*, is a decomposition of a period’s P&L into named causes whose sum equals the total, with any remainder reported as *unexplained*.

**Method 7.14 (The daily explain of a cash book).**

With $q_0$ the [position](#def-m1-pnl-and-positions-position) at the previous close $m_0$, and $m_1$ today’s close:

1. **Carried [position](#def-m1-pnl-and-positions-position):** $q_0(m_1 - m_0)$ .
2. **New trades:** $\sum_k \epsilon_k n_k (m_1 - p_k)$ — each fill measured from its own price to the close.
3. **Fees** , **financing** , **dividends** , **currency** : each its own line.
4. **Unexplained:** total less the sum of the above. For a cash book it should be zero to the cent; anything else is a missing trade, a wrong mark or an unprocessed corporate action.

The identity behind the method is [Proposition 7.5](#prop-m1-pnl-and-positions-identity) applied between two closes: the first two lines sum to $c + q_1 m_1 - q_0 m_0$. Line 2 is where a trading desk’s skill shows: a [market maker](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-market-maker)’s new-trade P&L is its spread capture and its [adverse selection](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-adverse-selection) ([Proposition 1.12](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#prop-m1-what-a-trading-firm-does-capture)); its [carried-position](#def-m1-pnl-and-positions-position) P&L should be small and is pure risk. For derivatives the first line is itself split by risk factor, which is the subject of One Quant Book 6.

![The next day’s explain: 30 000 shares carried from 50.00 to 50.40, 10 000 sold at 50.35 and 5 000 bought at 50.10, $900 of fees and $650 of financing. Most of the day’s $11 450 was made by yesterday’s position, not by today’s decisions. Data: computed by the chapter’s script.](https://one-course.com/images/onecourse/chapters/quant-1/m1-pnl-and-positions/fig-41cbf56cd65b.svg)

***Figure 7.3.** The next day’s explain: 30 000 shares carried from 50.00 to 50.40, 10 000 sold at 50.35 and 5 000 bought at 50.10, $900 of fees and $650 of financing. Most of the day’s $11 450 was made by yesterday’s [position](#def-m1-pnl-and-positions-position), not by today’s decisions. Data: computed by the chapter’s script.*

![Where the keeper sits in the miniature firm. Fills arrive from the venue’s or broker’s drop copy, never from the strategy’s own belief about what it traded; positions and cash are reconciled every morning against the clearing broker’s statement.](https://one-course.com/images/onecourse/chapters/quant-1/m1-pnl-and-positions/fig-204e070dd9ad.svg)

***Figure 7.4.** Where the keeper sits in the miniature firm. Fills arrive from the venue’s or broker’s drop copy, never from the strategy’s own belief about what it traded; [positions](#def-m1-pnl-and-positions-position) and cash are reconciled every morning against the clearing broker’s statement.*

## 7.5 Tutorial: three numbers, one day

**Goal.** Replay the six fills, reproduce $40 000, $31 000 and $27 000, and check that the accounting convention does not matter. **End state:** Figures [7.1](#fig-m1-pnl-and-positions-day) and [7.2](#fig-m1-pnl-and-positions-three).

1. **The keeper’s core**, as it is written for the systems side of the firm. Quantity, cash and fees are integers; only the reported split is floating point. `void on_fill (Side side, std::int64_t qty, std::int64_t price, std::int64_t fee = 0 ) { if (qty <= 0 || price <= 0 || fee < 0 ) throw std::invalid_argument(" bad fill " ); const std::int64_t signed_qty = static_cast <int >(side) * qty; cash -= signed_qty * price; fees += fee; const bool adding = quantity == 0 || ((quantity > 0 ) == (signed_qty > 0 )); if (adding) { const std::int64_t open = std::llabs(quantity); avg_cost = (static_cast <double >(open) * avg_cost + static_cast <double >(qty * price)) / static_cast <double >(open + qty); quantity += signed_qty; return ; } const std::int64_t closing = qty < std::llabs(quantity) ? qty : std::llabs(quantity); const double direction = quantity > 0 ? 1.0 : -1.0 ; realised += direction * static_cast <double >(closing) * (static_cast <double >(price) - avg_cost); quantity += signed_qty; if (qty > closing) avg_cost = static_cast <double >(price); // flipped else if (quantity == 0 ) avg_cost = 0.0 ; } double unrealised (std::int64_t mark) const { return static_cast <double >(quantity) * (static_cast <double >(mark) - avg_cost); } std::int64_t total (std::int64_t mark) const { return cash + quantity * mark - fees; }` **Listing 7.1.** Average-cost position keeping in C++: add, reduce, flip; and the exact total. code/firm/pnl/cpp/firm_pnl.hpp
2. **Replay and mark three ways.** `def replay (fills=FILLS) -> Position: p = Position() for _, side, qty, price, _ in fills: p.on_fill(side, qty, units(price)) return p def three_numbers () -> dict [str , float ]: p = replay() return { " trader " : p.total(units(LAST_TRADE)) / U, # marks at the last print, gross " risk " : p.total(units(CLOSING_MID)) / U, # marks at the closing mid, gross " finance " : p.total(units(OFFICIAL_CLOSE)) / U - FEES - FINANCING, # official close, net }` **Listing 7.2.** The same position object, three marks and two cost lines. code/markets-1/07-pnl-and-positions/python/pnl_day.py
3. **Run the tests** : `make test-code CH=markets-1/07-pnl-and-positions` . They assert the three numbers, and that first-in-first-out gives the same total with a different split ($28 750 realised against $26 406).
4. **Stress the identity.** The build’s tests replay five thousand random fills and require the split to agree with the exact total.

**What to change next.** Mark the long at the bid, 49.99: what is the fourth number? Then add a dividend of 20 cents going ex during the day and decide in which line of the explain it belongs.

## 7.6 Build: the position and P&L keeper

**Purpose.** The component every strategy, risk check and report of the miniature firm asks: what do we hold, and what is it worth? It is the first piece built in all three languages, because it runs both inside the low-latency process (One Quant Book 13) and in the research and reporting stack.

**Interface.** `Position.on_fill(side, quantity, price, fee)`, `unrealised(mark)`, `total(mark)`; `Book.on_fill(symbol, …)`, `total(marks)`, `gross_exposure(marks)`, `net_exposure(marks)`. Prices, cash and fees are integers in ledger units.

**Rules.** `total` is computed from quantity, cash and fees only, and is exact. The realised and unrealised split follows [Definition 7.6](#def-m1-pnl-and-positions-realised). Non-positive quantities or prices and negative fees are rejected. No allocation and no exception on the fill path in the C++ and Rust versions other than for rejected input.

**Acceptance tests.** `code/firm/pnl/tests/`, `cpp/` and `rust/`: round trip; [average cost](#def-m1-pnl-and-positions-realised) and partial close; flip; five thousand random fills with split equal to total; rejections. The three implementations must agree on the same fill file.

**Stretch.** Add first-in-first-out lots behind the same interface and a switch per account; post each day’s explain lines to the ledger of [Chapter 1](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#ch-m1-what-a-trading-firm-does).

Sources and further reading

- L. Harris, *Trading and Exchanges* , Oxford University Press, 2003, chapter 21 (performance evaluation).
- R. Grinold and R. Kahn, *Active Portfolio Management* , 2nd ed., McGraw-Hill, 2000, chapter 17 (performance analysis).
- International Accounting Standards Board, *IFRS 13 Fair Value Measurement* — the accounting standard behind the choice of marks.

## 7.7 Exercises

**Exercise 7.1 ★.**

A book is long 40 000 shares at $25, long 10 000 at $120 and short 30 000 at $50, on equity of $2 million. Give its gross and [net exposure](#def-m1-pnl-and-positions-exposure) in dollars and as multiples of equity.

**Solution of Exercise 7.1.**

Gross $1.0 + 1.2 + 1.5 = \$3.7$ million, $1.85\times$ equity; net $2.2 - 1.5
= \$0.7$ million, $0.35\times$.

**Exercise 7.2 ★.**

Starting flat: buy 500 at 20.00, buy 300 at 20.40, sell 600 at 20.50. Give the [average cost](#def-m1-pnl-and-positions-realised) after each fill, the [realised P&L](#def-m1-pnl-and-positions-realised), and the [unrealised P&L](#def-m1-pnl-and-positions-realised) at a mark of 20.30.

**Solution of Exercise 7.2.**

After the first buy $\bar p = 20.00$; after the second $\bar p =
(500\times20.00 + 300\times20.40)/800 = 20.15$. The sale realises $600\times
0.35 = \$210$ and leaves 200 shares at $\bar p = 20.15$. Unrealised at 20.30: $200 \times 0.15 = \$30$.

**Exercise 7.3 ★.**

For the fills of the previous exercise compute the cash, and check [Proposition 7.5](#prop-m1-pnl-and-positions-identity) at the mark 20.30.

**Solution of Exercise 7.3.**

Cash $-10\,000 - 6\,120 + 12\,300 = -\$3\,820$; $-3\,820 + 200\times20.30 =
\$240 = 210 + 30$.

**Exercise 7.4 ★★.**

Starting flat: sell 1 000 at 80.00, sell 1 000 at 81.00, buy 3 000 at 79.50. Track the [position](#def-m1-pnl-and-positions-position), the [average cost](#def-m1-pnl-and-positions-realised) and the [realised P&L](#def-m1-pnl-and-positions-realised) through the flip, and give the total at a mark of 79.80.

**Solution of Exercise 7.4.**

Short 1 000 at 80.00; short 2 000 at $\bar p = 80.50$. The purchase of 3 000 closes the 2 000, realising $2\,000\times(80.50-79.50) = \$2\,000$, and opens a long of 1 000 at 79.50. At 79.80 the unrealised is $300 and the total $2 300.

**Exercise 7.5 ★★.**

A sterling-based fund holds 50 000 shares of a US stock. Over a month the stock goes from $200 to $190 and the dollar from 0.80 to 0.84 pounds. Compute the price, currency and cross terms and the total in pounds.

**Solution of Exercise 7.5.**

Price: $50\,000\times(-10)\times0.80 = -\pounds400\,000$. Currency: $50\,000\times200\times0.04 = +\pounds400\,000$. Cross: $50\,000\times(-10)\times0.04 = -\pounds20\,000$. Total $-\pounds20\,000$: the two large terms cancel and the cross term is the result.

**Exercise 7.6 ★★.**

Yesterday’s close was 30.00 with a [position](#def-m1-pnl-and-positions-position) of $-20\,000$ shares. Today the desk buys 8 000 at 29.60 and sells 5 000 at 29.90; the close is 29.70; fees are $260, the [borrow fee](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-seclending) accrued is $120 and the short proceeds earned $65 of interest. Produce the explain and the closing [position](#def-m1-pnl-and-positions-position).

**Solution of Exercise 7.6.**

Carried: $-20\,000\times(29.70-30.00) = +\$6\,000$. New trades: $8\,000\times0.10 + (-5\,000)\times(-0.20) = +\$1\,800$. Fees $-260$; borrow $-120$; interest on proceeds $+65$. Total $+\$7\,485$. Closing [position](#def-m1-pnl-and-positions-position) $-17\,000$.

**Exercise 7.7 ★★★.**

*Coding.* Implement first-in-first-out for [positions](#def-m1-pnl-and-positions-position) that may be short as well as long (the chapter’s `fifo` handles a long-only day). Test it on the fills of exercise [7.4](#exo-m1-pnl-and-positions-4) and report the [realised P&L](#def-m1-pnl-and-positions-realised); compare with [average cost](#def-m1-pnl-and-positions-realised).

**Solution of Exercise 7.7.**

Keep a queue of signed lots; a fill of opposite sign consumes lots from the front, realising $\pm n(p - p_{\text{lot}})$ with the sign of the lot, and any remainder becomes a new lot. On exercise 4’s fills: $1\,000\times0.50 +
1\,000\times1.50 = \$2\,000$, the same as [average cost](#def-m1-pnl-and-positions-realised). The two conventions agree whenever a [position](#def-m1-pnl-and-positions-position) is closed *entirely*, since then every lot is consumed; they differ only on partial closes.

**Exercise 7.8 ★★★.**

*Find the flaw.* A desk reports a win rate of 78% “on closed trades” and a positive [realised P&L](#def-m1-pnl-and-positions-realised) for eleven consecutive months; its total P&L over the period is negative. Explain how both can be true, which convention makes it easy, and which single report would have revealed it in the first month.

**Solution of Exercise 7.8.**

The desk closes its winners and keeps its losers open: each closed trade is a gain, [realised P&L](#def-m1-pnl-and-positions-realised) is positive every month, and the losses accumulate in the unrealised line, which the report omits. Lot-level accounting with a free choice of which lot to close makes this easiest, but any convention permits it. [Proposition 7.5](#prop-m1-pnl-and-positions-identity) is immune: the *total* P&L marked at an independent price, reported daily, shows the loss from the first month. [Realised P&L](#def-m1-pnl-and-positions-realised) and win rate are descriptions of when a trader chose to close, not of how much was made.

## 7.8 Problem: Three Numbers, One Day

**Problem 7.1.**

Weekend problem — reconciling a desk’s P&L

You are the desk’s new quant. On your first morning three P&L figures for yesterday are on the head of desk’s table and he wants one. The desk trades one stock. It started the day long 12 000 shares; the previous close was 84.00. Yesterday’s fills, in order: sell 7 000 at 84.60; sell 9 000 at 84.90; buy 6 000 at 84.20; buy 10 000 at 83.70. The last trade of the day printed at 83.55, the closing quote was 83.70 bid, 83.80 ask, and the closing auction price was 83.76. Fees were 0.12 cent a share; financing and borrow cost $310.

**Part I — [Position](#def-m1-pnl-and-positions-position) and cash.**

1. Give the [position](#def-m1-pnl-and-positions-position) after each fill and at the close.
2. Give the cash produced by the day’s fills.
3. Compute the fees.
4. At which moments was the desk short, and how many shares?

**Part II — Three marks.**

5. Give the day’s gross P&L marked at the last trade.
6. At the closing mid.
7. At the closing auction price.
8. At the price that would be obtained by closing the [position](#def-m1-pnl-and-positions-position) against the quote.
9. Which mark would you use for the trader’s bonus, and why?

**Part III — Explain** (use the auction price).

10. Compute the [carried-position](#def-m1-pnl-and-positions-position) line.
11. Compute the new-trades line, fill by fill.
12. Add fees and financing and give the net total.
13. Check that the unexplained is zero.
14. The trader says “I made money trading and lost it on the overnight [position](#def-m1-pnl-and-positions-position) ”. Is that what the explain says?

**Part IV — Realised and unrealised.**

15. The opening 12 000 shares have an [average cost](#def-m1-pnl-and-positions-realised) of 82.50. Track the [average cost](#def-m1-pnl-and-positions-realised) and the [realised P&L](#def-m1-pnl-and-positions-realised) through the four fills.
16. Give the [unrealised P&L](#def-m1-pnl-and-positions-realised) at the auction price.
17. Realised plus unrealised is not the day’s P&L. What is it, and what must be subtracted to recover the day’s figure?
18. The clearing broker’s statement shows a closing [position](#def-m1-pnl-and-positions-position) of 13 000 shares. List three possible causes in order of likelihood and what you would check first.
19. State the *named result* : the single net P&L for the day that you give the head of desk, and its three largest components.
20. In one sentence, state the policy that would have prevented three numbers from reaching his table.

**Solution of Problem 7.1.**

**1.** $12\,000 \to 5\,000 \to -4\,000 \to 2\,000 \to 12\,000$. **2.** $592\,200 + 764\,100 - 505\,200 - 837\,000 = +\$14\,100$. **3.** $32\,000 \times 0.0012 = \$38.40$. **4.** Between the second and third fills, short 4 000. **5.** The [position](#def-m1-pnl-and-positions-position) is again 12 000 shares, so the gross P&L is $14\,100 + 12\,000(m - 84.00)$. At 83.55: $8 700. **6.** Mid 83.75: $11 100. **7.** Auction 83.76: $11 220. **8.** A long is closed at the bid, 83.70: $10 500. **9.** The auction price: it is public, unique and cannot be influenced by the trader’s own last small print; the $2 520 between the highest and lowest of these marks is otherwise hers to choose. **10.** $12\,000\times(83.76-84.00) = -\$2\,880$. **11.** $7\,000\times0.84 = 5\,880$; $9\,000\times1.14 = 10\,260$; $6\,000\times(-0.44) = -2\,640$; $10\,000\times0.06 = 600$: $+\$14\,100$. **12.** $-2\,880 + 14\,100 - 38.40 - 310 = \$10\,871.60$. **13.** Gross $11\,220 = -2\,880 + 14\,100$: nothing is unexplained. **14.** Yes, but look closer: nearly all the trading gain comes from the two sales made above the close, that is, from having been *less long* while the price fell. Both lines are one decision — the view that the stock would fall — and the second purchase gave part of it back. **15.** Sell 7 000: realised $7\,000\times2.10 = 14\,700$, long 5 000 at 82.50. Sell 9 000: closes 5 000 ($+12\,000$, total 26 700) and opens short 4 000 at 84.90. Buy 6 000: closes 4 000 ($+2\,800$, total 29 500) and opens long 2 000 at 84.20. Buy 10 000: long 12 000 at $\bar p = 83.783$. **16.** $12\,000\times(83.76-83.783) = -\$280$. **17.** $29\,500 - 280 = \$29\,220$ is the P&L *since the [position](#def-m1-pnl-and-positions-position) was opened*. Subtract the [unrealised P&L](#def-m1-pnl-and-positions-realised) carried in yesterday, $12\,000\times(84.00-82.50) = \$18\,000$: $11 220. **18.** A fill of 1 000 shares missing from the desk’s records (a late or manually booked execution, or a fill for another desk’s account allocated here) — check the broker’s execution list against the fill log by execution identifier. Then a corporate action or transfer processed by the broker and not by the desk. Last, an error at the broker. **19.** **$10 872**, net, at the official close: $+14\,100$ from the day’s trades, $-2\,880$ on the carried [position](#def-m1-pnl-and-positions-position), $-348$ of fees and financing. **20.** One written marking policy — official close for the daily P&L, net of all costs — with every other figure labelled as an estimate.

## 7.9 Interview questions

**Interview question 7.1 ★ trader, developer, researcher.**

What is the difference between realised and [unrealised P&L](#def-m1-pnl-and-positions-realised)? Does the choice between first-in-first-out and [average cost](#def-m1-pnl-and-positions-realised) change how much money you made?

**Solution of Interview question 7.1.**

Realised is the gain on the part of the [position](#def-m1-pnl-and-positions-position) that has been closed, measured against a cost convention; unrealised is the marked gain on what is still open. The convention moves money between the two and between tax years; it cannot change the total, which is cash plus [position](#def-m1-pnl-and-positions-position) times mark and does not know in which order anything was bought.

*What the interviewer is looking for: the invariant stated as a formula.*

**Interview question 7.2 ★ developer.**

Why should a P&L system not store prices and cash as floating-point numbers? What do you use instead?

**Solution of Interview question 7.2.**

Binary floating point cannot represent most decimal prices exactly; sums of many fills accumulate errors that depend on the order of addition, so two systems disagree by a cent and reconciliations break. Use integers in a fixed minor unit (ticks, or ten-thousandths of the currency) for prices and cash, a decimal type in reporting layers, and floating point only for derived statistics such as an [average cost](#def-m1-pnl-and-positions-realised).

*What the interviewer is looking for: order-dependence and reconciliation, not just “rounding”.*

**Interview question 7.3 ★★ trader, researcher.**

You are long 10 000 shares, the last trade is 50.30 and the market is 49.99 bid, 50.01 ask. What is your [position](#def-m1-pnl-and-positions-position) worth? What if the [position](#def-m1-pnl-and-positions-position) were two million shares?

**Solution of Interview question 7.3.**

At the mid, $500 000; to liquidate, the bid: $499 900. The last trade is irrelevant: it is stale and away from the market. For two million shares no screen price applies: the value is the mid less the expected cost of selling that size, a square-root impact of some tens of basis points at least, and a prudent mark carries that reserve.

*What the interviewer is looking for: ignoring the last print, and size-dependent valuation.*

**Interview question 7.4 ★★ developer, trader.**

Your strategy believes it is flat; the broker’s end-of-day file says it is long 400 shares. Walk through how you find out which is right, and what the system should have done the moment they diverged.

**Solution of Interview question 7.4.**

The broker’s record is the legal one. Compare execution by execution, by identifier, the broker’s list with the strategy’s fill log and the drop copy: a fill missing on our side (lost acknowledgement, a partial fill after a cancel request, a fill during a reconnect) is the usual cause. The system should build its [position](#def-m1-pnl-and-positions-position) from the drop copy, reconcile it continuously against its own belief, and on divergence stop quoting and alert: a strategy trading on a wrong [position](#def-m1-pnl-and-positions-position) hedges the wrong way.

*What the interviewer is looking for: drop copy as the source of truth and “stop trading” as the reflex.*

**Interview question 7.5 ★★ researcher, bank.**

What is a P&L explain, and what does a persistent unexplained amount tell you?

**Solution of Interview question 7.5.**

A decomposition of the P&L into causes — carried [positions](#def-m1-pnl-and-positions-position) by risk factor, new trades, [carry](#def-m1-pnl-and-positions-carry), fees, currency — that sums to the total. Unexplained P&L that is small and random is noise from approximations; a persistent or growing one means the risk model does not describe the book (a missing risk factor, wrong marks or stale parameters) or that trades are missing or misbooked. Either way the desk does not know what it is exposed to.

*What the interviewer is looking for: unexplained as a control signal, not an accounting nuisance.*

**Interview question 7.6 ★★★ developer.**

Design the data model of a [position](#def-m1-pnl-and-positions-position) keeper that must handle fills arriving out of order, busted (cancelled) trades and price corrections. What is stored, and what is derived?

**Solution of Interview question 7.6.**

Store the immutable event log: every fill, bust and correction as its own event with the execution identifier, exchange timestamp, receive timestamp and a reference to the event it amends. Derive everything else: quantity, cash and fees are sums over the log, independent of order by [Proposition 7.5](#prop-m1-pnl-and-positions-identity), so late and out-of-order fills need no special case for the total. Order-dependent quantities ([average cost](#def-m1-pnl-and-positions-realised), [realised P&L](#def-m1-pnl-and-positions-realised)) are recomputed by replaying the log in exchange-time order from the last checkpoint before the amended event. A bust is a negating event, never a deletion, so that every published figure can be reproduced as of any time.

*What the interviewer is looking for: event sourcing, and the observation that the exact total is order-free.*