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💸 Deposited Capital, Total PnL and Cash Pools

💡 Concept Overview

Capital Baseline = cumulative economic capital contributed since inception:

\[ \mathrm{CapitalBaseline}(t)= \sum_{\tau \leq t}\mathrm{ExternalCashFlow}(\tau) + \sum_{\tau \leq t}\mathrm{InKindCapital}(\tau) \]

Total PnL = all value generated above external contributions:

\[ \boxed{\mathrm{TotalPnL}(t) = \mathrm{NAV}(t) - \mathrm{CapitalBaseline}(t)} \]

InKindCapital covers the securities that enter or leave the scope without cash:

  • priced ADJUSTMENT rows carrying cost_basis_override (not linked to a split), at cost: the cost an incoming row adds to the position, or the average cost an outgoing row removes;
  • TRANSFER legs whose other leg is not in the scope — it sits at a broker outside the view, is missing, or was never linked — at market value on their own date: \(+\) for securities received, \(-\) for securities sent, nothing if the security cannot be valued that day.

These rows move value without cash: they are capital, not profit or loss.

🎯 What Counts

Transaction Effect on capital baseline
DEPOSIT / WITHDRAWAL (unlinked) ✅ Yes
CASH_TRANSFER / FX_CONVERSION whose other leg is outside the view, missing, or never linked (orphan) ✅ Yes, at its amount, like a deposit or a withdrawal
CASH_TRANSFER / FX_CONVERSION / TRANSFER with both legs in the view (internal pair) ❌ No. Per broker, a pair across two brokers moves capital from the sender to the receiver on the arrival day, summing to zero (Internal Transfers)
TRANSFER of securities whose other leg is outside the view, missing, or never linked (orphan) ✅ Yes, in-kind capital at market value on its date
ADJUSTMENT with cost_basis_override, not split-linked ✅ Yes, in-kind capital at cost
SPLIT-linked ADJUSTMENT ❌ No, quantity rescale only
BUY, SELL, DIVIDEND, INTEREST, FEE, TAX ❌ No
Exchange-rate move on cash held in another currency (no transaction) ❌ No: a result, part of TotalPnL; the capital flows (deposits, withdrawals, orphan cash legs) entered the baseline at the rate of their own dates

🔀 Internal Transfers in the Broker Lines

With two or more brokers in scope, the P&L chart draws one line per broker. Each broker \(b \in S\) has its own NAV, capital and PnL:

\[ \mathrm{PnL}_b(t) = \mathrm{NAV}_b(t) - C_b(t) \]
\[ \mathrm{NAV}_b(t) = \mathrm{MV}_b(t) + \mathrm{Cash}_b(t) + \mathrm{InTransit}_b(t) \]
\[ C_b(t) = \sum_{\tau \leq t}\mathrm{ExternalCashFlow}_b(\tau) + \sum_{\tau \leq t}\mathrm{InKindCapital}_b(\tau) + \sum_{\tau \leq t}\mathrm{Internal}_b(\tau) \]

\(\mathrm{InTransit}_b\) is the value in transit that left \(b\), and \(\mathrm{Internal}_b\) is \(b\)'s side of the internal transfers described below. Since \(\sum_b \mathrm{NAV}_b = \mathrm{NAV}\) and the internal flows cancel out every day, the lines add up to the scope figures:

\[ \sum_{b \in S}\mathrm{Internal}_b(t) = 0 \quad\Rightarrow\quad \sum_{b \in S} C_b(t) = \mathrm{CapitalBaseline}(t), \qquad \sum_{b \in S}\mathrm{PnL}_b(t) = \mathrm{TotalPnL}(t) \]

A broker's line is therefore the PnL earned while the value was at that broker, or travelling from it. It is not the unrealized gain of the positions the broker holds.

📐 The Internal Flow

An internal pair is a linked CASH_TRANSFER, FX_CONVERSION or TRANSFER whose two legs are both in \(S\), on two different brokers: the sender \(A\) (the leg going out) and the receiver \(B\) (the leg coming in). It produces one flow, on the arrival day \(t_{arr}\), the date of the later leg:

\[ \mathrm{Internal}_A(t_{arr}) = -X, \qquad \mathrm{Internal}_B(t_{arr}) = +X, \qquad X = s_A \cdot V(t_{arr}) \]

\(s_A\) is the sender's ownership share and \(V(t_{arr})\) the full value the arrival brings to \(B\) that day, valued exactly as the NAV values it:

Pair \(V(t_{arr})\)
CASH_TRANSFER The amount received, converted at \(\mathrm{fx}(c, C^*, t_{arr})\)
FX_CONVERSION across brokers The received leg, converted the same way: \(-1{,}000\) USD at \(A\) → \(+900\) EUR at \(B\) gives \(V = 900\) EUR, and the conversion cost stays in \(A\)'s line
TRANSFER of securities \(\mathrm{MV}\) of the quantity received at the valuation price of \(t_{arr}\), the last known price carried forward if there is none that day

With equal shares, \(X\) is exactly what the arrival adds to \(\mathrm{NAV}_B\), so neither line jumps. The scope's \(\mathrm{CapitalBaseline}\) is untouched, because an internal move is not external capital: NAV, CapitalBaseline, TotalPnL and its candles, ROI, TWRR, MWRR and cash are the same as without the flow. Only the split of the baseline among the brokers changes.

✈️ Value in Transit

When the legs fall on different days, the value travels from the departure day \(t_{dep}\) to the day before \(t_{arr}\). Meanwhile it is part of \(\mathrm{InTransit}_A\), at the sender's share \(s_A\), valued every day: cash at that day's exchange rate, securities at that day's valuation price (at the cost recorded on the transfer if the security has no price at all). No flow happens while the value travels, so its price and exchange-rate moves stay in \(A\)'s line, the arrival day's own move included; \(B\)'s line starts from the value received.

For example, \(A\) receives a deposit of 1,000 USD on day 0 at 0.90 EUR per USD (\(C_A = 900\)) and sends the dollars on day 1; \(B\) receives them on day 4. With the rate at 0.90, 0.91, 0.92, 0.93 on days 1–4, \(A\)'s line reads 0, +10, +20, +30 and \(B\)'s stays at 0. On day 4, \(X = 930\) and \(C_A = 900 - 930 = -30\). Once arrived, the dollars are \(B\)'s cash, revalued every day, so from day 5 \(B\)'s line moves with the rate: at 0.94, it reads +10.

🧩 Special Cases

  • Different ownership shares. \(X\) uses \(s_A\), like the value in transit, while the arrival adds \(s_B \cdot V(t_{arr})\) to \(\mathrm{NAV}_B\). \(B\)'s line keeps a residual step of \((s_B - s_A)\,V(t_{arr})\) on the arrival day, and TotalPnL shows the same step, with or without the flow.
  • Value unknown on the arrival day. If \(V(t_{arr})\) cannot be computed — the security has no observation on or before \(t_{arr}\) (\(\operatorname{mark} = \varnothing\)), or an exchange rate is missing — then \(X = 0\), which is what the arrival adds to \(\mathrm{NAV}_B\). Nothing is guessed: the jump stays, and a security's value appears in the line of the broker holding it from its first price.
  • Fees. They are not part of \(X\). A FEE row of 5 recorded at the sender, or the same 5 as a gap between the legs (\(-1{,}000\) sent, \(+995\) received, so \(X = 995\)), stays in \(A\)'s line as a loss of 5.
  • Same-broker pairs. Two legs on the same broker, such as a currency conversion inside one account, need no flow: the value never leaves the line.
  • ADJUSTMENT pairs. They are not internal pairs: each leg stays in-kind capital at cost (see the table above). The sender's capital drops by the average cost removed and the receiver's rises by the cost added, so the receiving line takes over the gain accrued at the sender. With the setup of the example below and both legs on day 2 at a cost of 10, the lines read \(A = 0\), \(B = +20\); a TRANSFER pair gives \(A = +20\), \(B = 0\).

🔍 Filtered Views

In a view with only some brokers, a TRANSFER leg whose other leg is outside \(S\) — or missing, or never linked — cannot form an internal pair. It is in-kind capital at market value on its own date: \(+\mathrm{MV}\) for securities received, \(-\mathrm{MV}\) for securities sent, at that broker's share, and nothing if the security cannot be valued that day. Unlike an internal flow, it moves the view's CapitalBaseline, so the view's TotalPnL, ROI, TWRR and MWRR treat it as capital, like an in-kind ADJUSTMENT. A CASH_TRANSFER or FX_CONVERSION leg in the same position is external cash at its amount, like a deposit or a withdrawal.

A leg loses its link when the broker holding the other leg is deleted. Such an orphan follows the same rules in every view, the full one included.

Comparing a broker's own view with the combined view:

  • The receiver alone matches its combined line. In \(B\)'s own view the received securities are capital at their market value on \(t_{arr}\), which is the \(X\) of the combined view. This holds with equal shares and a price that both views can see: a price that comes only from trades at \(A\) is not visible in \(B\)'s own view.
  • The sender alone loses the moves in transit. In \(A\)'s own view its capital drops on the departure day by \(s_A \cdot V(t_{dep})\), while in the combined view the value stays with \(A\) until it arrives.

Writing \(\mathrm{PnL}_A^{(S)}\) for \(A\)'s line in the view of scope \(S\), from the departure day on:

\[ \mathrm{PnL}_A^{(A)}(t) = \mathrm{PnL}_A^{(A,B)}(t) - s_A\bigl[V(\min(t, t_{arr})) - V(t_{dep})\bigr] \]

📦 Example — Securities in Transit

All amounts in EUR, shares 100%. \(A\) receives a deposit of 100 and buys 10 units at 10 on day 0. The price is 10, 11, 12, 13, 14, 15 on days 0–5. \(A\) sends the 10 units on day 1, and \(B\) receives them on day 3.

Combined view (both brokers in \(S\)):

Day Price \(\mathrm{NAV}_A\) \(C_A\) \(\mathrm{PnL}_A\) \(\mathrm{NAV}_B\) \(C_B\) \(\mathrm{PnL}_B\) TotalPnL
0 10 100 100 0 0 0 0 0
1 11 110 (in transit) 100 +10 0 0 0 +10
2 12 120 (in transit) 100 +20 0 0 0 +20
3 13 0 −30 +30 130 130 0 +30
4 14 0 −30 +30 140 130 +10 +40
5 15 0 −30 +30 150 130 +20 +50

On day 3 the flow \(X = 10 \times 13 = 130\) moves capital from \(A\) to \(B\). \(A\) keeps the +30 earned while the units were at \(A\) or travelling (10 → 13), and \(B\) earns from 13 on. \(C_A\) turns negative because \(A\) passed on more value (130) than the capital it took in (100): the difference is its gain. TotalPnL is the same with or without the flow; without it, on day 3 \(A\)'s line would drop to \(0 - 100 = -100\) and \(B\)'s would jump to \(130 - 0 = +130\).

Single-broker views:

Day \(B\) alone: \(C_B\) \(B\) alone: \(\mathrm{PnL}_B\) \(A\) alone: \(C_A\) \(A\) alone: \(\mathrm{PnL}_A\)
0 — — 100 0
1 — — −10 +10
2 — — −10 +10
3 130 0 −10 +10
4 130 +10 −10 +10
5 130 +20 −10 +10

\(B\)'s own view starts at its first transaction and matches \(B\) in the combined view. In \(A\)'s own view the units leave on day 1 at \(10 \times 11 = 110\), so \(C_A = 100 - 110 = -10\) and the line stays at +10: the move from 11 to 13 while travelling is not in it, and \(+30 - (130 - 110) = +10\). Counted as a gain or a loss instead of capital, the same transfer would put \(A\) alone at −100 from day 1 and \(B\) alone at +130 on day 3.

💶 Example — Cash Between Two Brokers

\(A\) receives a deposit of 5,000. Later, \(A\) sends 1,000 to \(B\), both legs on the same day.

Case \(\mathrm{NAV}_A\) \(C_A\) \(\mathrm{PnL}_A\) \(\mathrm{NAV}_B\) \(C_B\) \(\mathrm{PnL}_B\) TotalPnL
After the deposit 5,000 5,000 0 0 0 0 0
After the transfer 4,000 4,000 0 1,000 1,000 0 0
\(B\) receives 995 (fee 5) 4,000 4,005 −5 995 995 0 −5
\(B\) owned at 50% 4,000 4,000 0 500 1,000 −500 −500
Without the flow 4,000 5,000 −1,000 1,000 0 +1,000 0

The last row is the same transfer read without the internal flow: the move looks like a loss for \(A\) and a gain for \(B\). With \(B\) owned at 50%, the step of −500 is the residual \((s_B - s_A)\,V = (0.5 - 1) \times 1{,}000\), and TotalPnL shows it too.

📊 Three-Pool Cash Model

The Growth Chart decomposes current cash into two visible aggregates plus a hidden global tracker:

\[ \mathrm{Cash}(t) \approx \sum_b K_b(t) + \sum_b R_b(t) \]
Pool Scope Meaning
\(K_b\) Per-broker External capital still at broker \(b\)
\(R_b\) Per-broker Generated returns still at broker \(b\)
\(W\) Global Returns that left the system (restorable on re-deposit)

Key properties

  • \(\mathrm{CapitalBaseline}\) = historical sum of cash and in-kind capital flows. \(\sum K_b\) = how much current cash is external capital. They diverge after BUY/SELL.
  • A BUY on broker \(b_1\) only consumes \(R_{b_1}\), never \(R_{b_2}\).
  • Cash transfers between brokers move \(R\) and \(K\) from source to destination without touching \(W\).
  • An exchange result on cash a broker holds in another currency goes to \(R_b\); a loss larger than \(R_b\) comes out of \(K_b\), as a FEE does. \(\mathrm{CapitalBaseline}\) does not move (Portfolio Engine — §6).

🔗 Full per-broker update rules: Portfolio Engine — §6 Three-Pool Model

📝 Worked Examples

🧾 A — Deposit → Buy → Sell in Gain

Step Tx \(K\) \(R\) Cash
1 DEPOSIT €1,000 1,000 0 1,000
2 BUY €1,000 0 0 0
3 SELL P=€1,200, C=€1,000 1,000 200 1,200

TotalPnL = 1,200 − 1,000 = +€200 ✓

💸 B — Dividend then Withdrawal

Step Tx \(K\) \(R\) \(W\) Cash
1 DEPOSIT €1,000 1,000 0 0 1,000
2 DIVIDEND €50 1,000 50 0 1,050
3 WITHDRAWAL €100 (K first) 900 50 0 950
4 WITHDRAWAL €950 (K=900→0, R=50→0, W+=50) 0 0 50 0
5 RE-DEPOSIT €30 (restore min(30,W=50)=30) 0 30 20 30

After step 5: Cash=30, K=0, R=30 ✓ (returns restored from W)

🧪 C — Full Sell Regression

Step Tx \(K\) \(R\) Cash
1 DEPOSIT €1,000, BUY 1@€1,000 0 0 0
2 SELL 1@€1,005 (C=1000, G=5) 1,000 5 1,005

Capital correctly returns to \(K\); only €5 gain to \(R\). Not all €1,005 to \(R\).

💱 D — Dollars Held While the Rate Moves

One broker, report currency EUR, rates in EUR per USD; \(\varepsilon\) is the exchange result on cash.

Step Event \(\varepsilon\) \(K\) \(R\) Cash TotalPnL
1 DEPOSIT 1,000 USD at 0.90 — 900 0 900 0
2 Rate rises to 0.95 +50 900 50 950 +50
3 Rate falls to 0.85 (\(R\) = 50 − 100, \(K\) absorbs −50) −100 850 0 850 −50
4 BUY for 1,000 USD at 0.85 (€850) — 0 0 0 MV − 900

The CapitalBaseline stays €900 throughout: the exchange results are profit and loss, not capital. After step 4, TotalPnL is the security's market value MV less the €900 deposited — −50 on the purchase day if MV equals the €850 paid.

⚙️ Implementation

The 3-pool model runs in a single per-transaction loop (event-driven, not daily-delta):

  1. Take the transaction's step of its position's average-cost pool, computed before the replay — for a SELL, the cost it removes (\(C\) = pre-sale WAC × quantity sold, at historical rates)
  2. Update K/R/W per transaction type rules, with that \(C\) for a SELL

On each day the exchange rate of a foreign currency held as cash changes, each broker's exchange result on that cash is also booked into its pools, after the day's cash movements and before the day's transactions (§6).

The ROI/TWRR/MWRR input series is derived from day-over-day changes in cumulative_external_cash_flow, the capital baseline. It is not derived from the cash-only external_cash_flow field.

🔗 See Portfolio Engine — §6 for all formal rules.