---
title: "Financing: Repo, Securities Lending and Prime Brokerage"
book: "Markets I: The Ecosystem and Exchange-Traded Markets"
subject: quant
language: en
chapter: 6
exercises: 8
source: https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage
---

# Chapter 6 — Financing: Repo, Securities Lending and Prime Brokerage

A fund owns five billion dollars of stock and has one billion of its own money. The other four are borrowed, and the loan is renewed every night by lenders who can decline to renew it. On an ordinary day nobody thinks about this. On the day the stocks fall four percent, a fifth of the fund’s money is gone, its lenders want more collateral by the afternoon, and the only way to find it is to sell $800 million of the same stocks into the same falling market. Almost every position in professional finance is carried with somebody else’s money. This chapter is about where that money comes from, what it costs, the three contracts through which it is lent — and what happens when it leaves.

## 6.1 Leverage

**Definition 6.1 (Leverage).**

The *leverage* of a portfolio is the ratio $L = A/E$ of the assets it holds to the equity (own funds) behind them. The remainder, $A - E$, is financed.

**Proposition 6.2 (Leverage arithmetic).**

Let the assets return $r_A$ over a period and the financing cost $r_f$. Then

1. the return on equity is $r_E = L\,r_A - (L-1)\,r_f$ ;
2. the equity is wiped out when the assets fall by $1/L$ ;
3. to restore the [leverage](#def-m1-financing-leverage) ratio after the assets fall by a fraction $x$ (with $x < 1/L$ ), the portfolio must sell assets worth $(L-1)\,x\,A$ .

**Proof.** (1) The equity earns $A r_A - (A-E) r_f$; divide by $E$. (2) The loss $xA$ equals $E$ when $x = E/A$. (3) After the fall assets are $A(1-x)$ and equity is $E - xA$. Assets consistent with [leverage](#def-m1-financing-leverage) $L$ are $L(E - xA) = A - LxA$. The excess to sell is $A(1-x) - A + LxA = (L-1)xA$. ∎

**Example 6.3 (The fund of the opening paragraph).**

$A = \$5$ billion, $E = \$1$ billion, $L = 5$. With $r_f = 4\%$, a $+10\%$ year on the assets is $+34\%$ on the equity; a $-10\%$ year is $-66\%$. A fall of 20% ends the fund. After a fall of 4% the fund has lost $200 million, a fifth of its equity, and must sell $4 \times 0.04 \times 5 =
\$0.8$ billion to be five times leveraged again: *four dollars of selling for each dollar lost*.

![Return on equity against the return of the assets, with financing at 4%. The dashed line is total loss: at L = 8 it is reached when the assets fall 9% — less than 1/L = 12.5\%, because a year of financing cost is owed as well. Data: , computed by the chapter’s script.](https://one-course.com/images/onecourse/chapters/quant-1/m1-financing/fig-0e513c229a0b.svg)

***Figure 6.1.** Return on equity against the return of the assets, with financing at 4%. The dashed line is total loss: at $L = 8$ it is reached when the assets fall 9% — less than $1/L = 12.5\%$, because a year of financing cost is owed as well. Data: [Proposition 6.2](#prop-m1-financing-roe), computed by the chapter’s script.*

Who lends the $A - E$? Three contracts do almost all of the work: the repo, the securities loan, and the swap. All three are *secured*: the lender holds collateral worth more than the loan, and that excess is the quantity to watch.

## 6.2 Repo

**Definition 6.4 (Repurchase agreement and haircut).**

In a *repurchase agreement* (repo) one party sells a security for cash and agrees at the same time to buy it back at a fixed later date at a fixed higher price. Economically it is a cash loan secured by the security; the price difference is the interest, quoted as the *repo rate*. The *haircut* $h$ is the fraction by which the cash lent falls short of the security’s market value: a security worth 100 raises $100(1-h)$.

![The two legs of a repo. If the borrower fails to repurchase, the lender owns a bond worth 100 against a claim of 98: the haircut is its protection against a fall in the bond’s price while it sells.](https://one-course.com/images/onecourse/chapters/quant-1/m1-financing/fig-cd481e4c9d28.svg)

***Figure 6.2.** The two legs of a repo. If the borrower fails to repurchase, the lender owns a bond worth 100 against a claim of 98: the [haircut](#def-m1-financing-repo) is its protection against a fall in the bond’s price while it sells.*

**Proposition 6.5 (Haircuts bound leverage).**

A portfolio financed entirely by repo at [haircut](#def-m1-financing-repo) $h$ can reach a [leverage](#def-m1-financing-leverage) of at most $1/h$. If the [haircut](#def-m1-financing-repo) rises from $h$ to $h'$ with no change in prices, a portfolio at maximum [leverage](#def-m1-financing-leverage) must sell a fraction $1 - h/h'$ of its assets.

**Proof.** Each unit of assets raises $1-h$ of cash, so the equity needed is $hA$ and $L = A/E \le 1/h$. With equity unchanged at $E = hA$, the new maximum is $E/h' = (h/h')A$. ∎

**Example 6.6 (Two percent to four percent).**

Government bonds financed at a 2% [haircut](#def-m1-financing-repo) allow $L = 50$: this is how relative-value funds earn a living from price differences of a few basis points (One Quant Book 9). If lenders move the [haircut](#def-m1-financing-repo) to 4%, a fund at the limit must sell *half* its assets although no price has moved. [Haircuts](#def-m1-financing-repo) are set by lenders, rise when volatility rises, and are the channel through which a funding problem becomes a market problem.

## 6.3 Securities lending and the short sale

**Definition 6.7 (Short sale).**

A *short sale* is the sale of a security the seller does not own. To deliver at settlement the seller borrows the security, and later buys it back to return it: it profits if the price has fallen.

**Definition 6.8 (Securities lending, rebate rate, locate).**

In *securities lending* an owner transfers a security to a borrower against collateral, usually cash worth a little more than the security, and receives an equivalent security back on demand. The lender invests the cash and returns part of the interest to the borrower at the *rebate rate*: the difference between the market interest rate and the rebate is the *borrow fee*, the true price of the loan. A *locate* is a broker’s confirmation, obtained before a [short sale](#def-m1-financing-short), that the security can be borrowed in time for settlement; US rules require one ([Chapter 16](https://one-course.com/books/quant/1/en/chapter/16-stock-loan-and-short-selling-in-practice#ch-m1-stock-loan-and-short-selling)).

![A short sale. Shares (blue) travel from a long-term owner to the market; cash (red) travels back and stops as collateral with the lender, who pays interest on it at the rebate rate. The short seller never holds the proceeds.](https://one-course.com/images/onecourse/chapters/quant-1/m1-financing/fig-d6158538348d.svg)

***Figure 6.3.** A [short sale](#def-m1-financing-short). Shares (blue) travel from a long-term owner to the market; cash (red) travels back and stops as collateral with the lender, who pays interest on it at the [rebate rate](#def-m1-financing-seclending). The short seller never holds the proceeds.*

**Example 6.9 (The carry of a short).**

Interest rates are 4%. A widely held large stock lends at a fee of 0.25%: the rebate is 3.75%, and a short seller *earns* 3.75% a year on the proceeds. A stock in heavy demand lends at a fee of 15%: the rebate is $-11\%$, and the short seller pays 11% a year for the right to be short. In both cases the short also pays the lender every dividend the stock distributes.

## 6.4 Prime brokerage and margin

**Definition 6.10 (Prime broker and rehypothecation).**

A *prime broker* is the bank that finances a [hedge fund](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-hedge-fund)’s securities portfolio: it lends cash against longs, lends securities for shorts, settles and holds the portfolio, and reports on it. *Rehypothecation* is the prime broker’s re-use of a client’s securities as collateral for its own borrowing, which is how it funds the loans it makes.

**Definition 6.11 (Margin call).**

A *margin call* is a lender’s demand that a borrower restore the agreed excess of collateral over loan, by delivering cash or securities or by reducing positions, usually within one day. If it is not met the lender may sell the collateral.

**As of September 2026 — US margin rules in three numbers.**

The Federal Reserve’s Regulation T sets the *initial* margin on a purchase of listed stock in a margin account at 50% (and 150% for a [short sale](#def-m1-financing-short), of which the sale proceeds provide 100%). FINRA Rule 4210 sets the *maintenance* margin at 25% of market value for long listed equities. Under SEC Rule 15c3-3 a broker may use a customer’s securities up to 140% of the customer’s debit balance; securities beyond that amount must be kept in the broker’s possession or control. [Prime brokers](#def-m1-financing-pb)’ own house requirements and portfolio-margin regimes sit on top of these.

**Proposition 6.12 (Where the margin call is).**

A long position bought at $P_0$ with [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) ratio $m_0$ (equity over value) receives a maintenance call at maintenance ratio $m$ when the price falls below

$$
P^\star = P_0\,\frac{1-m_0}{1-m}.
$$

**Proof.** The loan is $P_0(1-m_0)$ and does not change with the price. At price $P$ the equity ratio is $(P - P_0(1-m_0))/P$; set it equal to $m$. ∎

**Example 6.13 (Fifty and twenty-five).**

With $m_0 = 50\%$ and $m = 25\%$, $P^\star = \tfrac23 P_0$: the call comes after a fall of a third. A professional client margined by its [prime broker](#def-m1-financing-pb) at $m_0 = 15\%$ and $m = 10\%$ is called at $0.944\,P_0$, after a fall of 5.6%.

## 6.5 Synthetic financing

**Definition 6.14 (Total return swap).**

In an equity *total return swap* the dealer pays the client the total return (price change and dividends) of a stock on a notional amount, and the client pays a financing rate plus a spread on the same notional. The client has the economics of owning the stock financed by the dealer, who holds the actual shares as its hedge; the client posts margin agreed in the contract.

The swap is financing in another wrapper, and the wrapper matters: the client appears on no shareholder register; the margin is whatever the two parties negotiated, not what Regulation T prescribes; and a client using several dealers shows each of them only its own slice.

**Example 6.15 (Archegos).**

In March 2021 a family office, Archegos Capital Management, defaulted on [margin calls](#def-m1-financing-margincall) from its dealers. It had built concentrated positions in a handful of stocks through [total return swaps](#def-m1-financing-trs) with several banks, positions which the SEC’s complaint later put at $36 billion. One of those banks, Credit Suisse, lost close to $5.5 billion. The report its board commissioned found that the bank had agreed to a swap margin of 7.5% — [leverage](#def-m1-financing-leverage) above thirteen — that the margin was *static*, fixed on the price at which each swap was opened, so that as the stocks rose the average margin held fell to 6.9% of current value, and that a move to dynamic margining had not been given priority. When the stocks fell, every dealer held the same shares as its hedge, and each had to sell them into the others’ selling.

## 6.6 When financing runs

Combine [Proposition 6.2](#prop-m1-financing-roe)(3) with the fact that large sales move prices ([Method 2.6](https://one-course.com/books/quant/1/en/chapter/2-the-sell-side#met-m1-the-sell-side-sqrt)) and the mechanism of every leveraged crisis appears: a fall forces sales, sales cause a further fall, which forces more sales.

![A 5% shock, then rounds of selling to restore leverage, each sale moving the price by 10% per unit of assets sold (an illustrative impact). At L = 8 the market ends twice as far down as the shock that started it. Data: the chapter’s script.](https://one-course.com/images/onecourse/chapters/quant-1/m1-financing/fig-422b99e5283c.svg)

***Figure 6.4.** A 5% shock, then rounds of selling to restore [leverage](#def-m1-financing-leverage), each sale moving the price by 10% per unit of assets sold (an illustrative impact). At $L = 8$ the market ends twice as far down as the shock that started it. Data: the chapter’s script.*

**Remark 6.16 (Three accelerants).**

The toy spiral of [Figure 6.4](#fig-m1-financing-spiral) converges. Real ones are made worse by three things it leaves out. Lenders raise [haircuts](#def-m1-financing-repo) ([Proposition 6.5](#prop-m1-financing-haircut)) exactly when prices fall. Funds holding the same positions are hit together, so the “market” absorbing the sales is itself selling. And lenders who fear for a borrower stop renewing overnight loans altogether, turning a [margin call](#def-m1-financing-margincall) into a run. The quantitative-fund losses of August 2007, the repo run of 2008 and the 2021 episode above are the three cases this series returns to.

## 6.7 Tutorial: what a long–short book costs to carry

**Goal.** Compute the annual financing cost of a long–short equity book, first with a cash prime-brokerage account, then with swaps. **End state:** two numbers, $2.2 million and $2.7 million, and the reason for the difference.

1. **The book.** $300 million long, $200 million short, $100 million of equity. Illustrative terms: interest rate 4%; the [prime broker](#def-m1-financing-pb) charges 0.50% over on debit balances and pays 0.30% under on short proceeds; the average [borrow fee](#def-m1-financing-seclending) is 0.30%.
2. **Cash account.** The fund borrows $300 - 100 = \$200$ million. `def annual_cost_cash (b: Book, rate: float , debit_spread: float , borrow_fee: float , short_credit_spread: float ) -> float : """Cash prime brokerage: pay (r + spread) on the debit, receive (r - credit spread - fee) on shorts.""" debit = max (0.0 , b.long_value - b.equity) return debit * (rate + debit_spread) - b.short_value * (rate - short_credit_spread - borrow_fee)` **Listing 6.1.** Cash prime brokerage: pay on the debit, receive on the short proceeds. code/markets-1/06-financing/python/financing_demo.py Cost: $200 \times 4.5\% - 200 \times 3.4\% = \$2.2$ million a year.
3. **Swaps.** The dealer finances the whole long notional and the fund’s equity sits in cash earning the interest rate. `def annual_cost_swap (b: Book, rate: float , long_spread: float , short_spread: float , borrow_fee: float ) -> float : """Swaps: pay (r + spread) on the long notional, receive (r - spread - fee) on the short notional; the fund's equity, posted as margin or not, earns r.""" return (b.long_value * (rate + long_spread) - b.short_value * (rate - short_spread - borrow_fee) - b.equity * rate)` **Listing 6.2.** The same book on swap. code/markets-1/06-financing/python/financing_demo.py Cost: $300 \times 4.5\% - 200 \times 3.4\% - 100 \times 4\% = \$2.7$ million.
4. **Explain the gap.** $0.5 million is the 0.50% spread paid on the $100 million of longs that the cash account funded with the fund’s own money. Synthetic financing charges the spread on everything.

**What to change next.** Put a fifth of the short book in stocks lending at a 12% fee. Then let the interest rate fall to zero: which of the two costs changes, and why does neither change much?

## 6.8 Build: the financing calculator

**Purpose.** Each night the miniature firm accrues what its positions cost to carry and posts it 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) under `financing`.

**Interface.** `FinancingTerms(debit_spread, credit_spread, day_count=360)`, a per-symbol table of [borrow fees](#def-m1-financing-seclending), and `accrue(date, positions, prices, equity, rate)` returning one signed amount per line: debit interest, short-proceeds interest, [borrow fees](#def-m1-financing-seclending).

**Rules.** Accrue on settled, not traded, positions. Interest is simple, actual days over the day count, so a Friday accrues three days. [Borrow fees](#def-m1-financing-seclending) accrue on the value of the short marked at the previous close. A missing [borrow fee](#def-m1-financing-seclending) for a short position is an error, not a zero.

**Acceptance tests.** `code/firm/financing/tests/`: reproduces the tutorial’s $2.2 million over a 360-day year; a weekend accrues three days; a missing fee raises.

**Stretch.** Add [haircuts](#def-m1-financing-repo) per asset class and a `max_financing` query: how much more can the firm buy today?

Sources and further reading

- Credit Suisse Group Special Committee of the Board of Directors, *Report on Archegos Capital Management* , 29 July 2021.
- US Securities and Exchange Commission, *SEC Charges Archegos and its Founder with Massive Market Manipulation Scheme* , press release 2022-70, 27 April 2022.
- 12 CFR 220.12 (Regulation T, supplement: margin requirements); FINRA Rule 4210; SEA Rule 15c3-3 (FINRA interpretations handbook).
- M. Brunnermeier and L. Pedersen, “Market liquidity and funding liquidity”, *Review of Financial Studies* 22 (2009).
- G. Gorton and A. Metrick, “Securitized banking and the run on repo”, *Journal of Financial Economics* 104 (2012).

## 6.9 Exercises

**Exercise 6.1 ★.**

A fund has equity of $250 million and assets of $1.5 billion. Give its [leverage](#def-m1-financing-leverage), the fall in its assets that wipes it out, and its return on equity if the assets return $+6\%$ with financing at 4%.

**Solution of Exercise 6.1.**

$L = 6$; wiped out by a fall of $1/6 = 16.7\%$; $r_E = 6 \times 6\% - 5
\times 4\% = 16\%$.

**Exercise 6.2 ★.**

A bond worth $50 million is financed in repo at a [haircut](#def-m1-financing-repo) of 3% and a [repo rate](#def-m1-financing-repo) of 4.2% for 7 days (actual/360). How much cash is raised and how much interest is paid?

**Solution of Exercise 6.2.**

Cash raised $0.97 \times 50 = \$48.5$ million; interest $48.5\times10^6
\times 0.042 \times 7/360 = \$39\,608$.

**Exercise 6.3 ★.**

Interest rates are 4%. A short seller is short $20 million of a stock with a [borrow fee](#def-m1-financing-seclending) of 6%. What does the position earn or cost per year in financing, before dividends?

**Solution of Exercise 6.3.**

The rebate is $4\% - 6\% = -2\%$: the short costs $400 000 a year to carry.

**Exercise 6.4 ★★.**

A stock is bought at $80 with 40% [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin); the maintenance ratio is 30%. At what price is the [margin call](#def-m1-financing-margincall)? The stock falls to $60: how much cash per share restores the *initial* ratio?

**Solution of Exercise 6.4.**

Loan $48. Call at $48/(1-0.30) = \$68.57$. At $60 the equity is $12 against a required $0.40\times60 = \$24$: $12 a share.

**Exercise 6.5 ★★.**

A fund with $L = 6$ and assets of $3 billion loses 5% on its assets. Compute its new [leverage](#def-m1-financing-leverage), and the sale needed to return to $L = 6$. If its lenders now allow only $L = 4$, how much must it sell?

**Solution of Exercise 6.5.**

Assets $2.85 billion, equity $0.5 - 0.15 = \$0.35$ billion: $L = 8.14$. Back to 6: sell $5 \times 0.05 \times 3 = \$0.75$ billion. To 4: assets must be $1.4 billion, so sell $1.45 billion, half the book.

**Exercise 6.6 ★★.**

A relative-value fund runs at the maximum [leverage](#def-m1-financing-leverage) allowed by a 2.5% [haircut](#def-m1-financing-repo). Lenders raise the [haircut](#def-m1-financing-repo) to 4%. What fraction of the book must be sold? The trade earns 0.12% a year on assets above its financing cost: give the return on equity before and after.

**Solution of Exercise 6.6.**

$1 - 2.5/4 = 37.5\%$ of the book. Return on equity: $0.12\% \times 40 =
4.8\%$ before, $0.12\% \times 25 = 3.0\%$ after — and the forced sale will have moved the very spreads the fund is long.

**Exercise 6.7 ★★★.**

*Coding.* With `spiral`, find by bisection the impact per unit sold above which a 5% shock wipes out a fund with $L = 8$ within twelve rounds. Compare with the value $1/(L-1)$ and explain the relationship.

**Solution of Exercise 6.7.**

Bisection gives about 0.129 per unit sold. In a linearised spiral each round’s fall is $\lambda(L-1)$ times the previous one, where $\lambda$ is the impact: the series diverges when $\lambda \ge 1/(L-1) = 0.143$. The numerical threshold is a little lower because a divergent series is not needed: it is enough that the *sum* of the falls reaches $1/L = 12.5\%$ within twelve rounds starting from 5%.

**Exercise 6.8 ★★★.**

*Find the flaw.* A dealer’s risk report says of a swap client: “Margin held is 7.5% of notional; the largest one-day move of the portfolio in the last five years was 6%; the exposure is fully covered.” The margin is static and the client’s stocks have doubled since the swaps were opened. Identify three independent errors.

**Solution of Exercise 6.8.**

(i) Static margin on a doubled price is 3.75% of current value, not 7.5%. (ii) The relevant horizon is not one day but the time needed to liquidate a concentrated position, several days during which the dealer’s own selling moves the price. (iii) The historical worst day was observed while the client was buying; the move that matters is the one that occurs when it, and every other dealer holding the same hedge, is selling: the past distribution excludes exactly that event. (Also: five years of a rising stock say little about its left tail.)

## 6.10 Problem: A Family Office on Swap

**Problem 6.1.**

Weekend problem — leverage through five dealers

A family office has $4 billion of equity. Through [total return swaps](#def-m1-financing-trs) with five dealers, in equal shares, it holds long positions in eight stocks with a current value of $28 billion. Each dealer took a static margin of 10% of the price at which its swaps were opened; the stocks have risen 60% on average since.

**Part I — How leveraged is it?**

1. Give the [leverage](#def-m1-financing-leverage) on current values and the fall that wipes out the equity.
2. What was the notional when the swaps were opened, and how much margin do the dealers hold in total?
3. Express that margin as a percentage of current value.
4. Where is the rest of the family office’s $4 billion? Why does it matter to the dealers that they cannot see it?

**Part II — The carry.**

5. The swaps cost the interest rate (4%) plus 0.45%. Give the annual financing cost in dollars.
6. The family office’s cash earns 4%. Give its net annual carry cost and express it as a percentage of equity.
7. What annual return on the stocks does it need to break even?

**Part III — The fall.**

8. The stocks fall 10% in a day. Give the loss, the remaining equity and the new [leverage](#def-m1-financing-leverage) .
9. Each dealer calls for [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) equal to the loss on its slice. How much cash is called in total? The family office pays out all its free cash, pro rata. What does each dealer receive, and what is each still owed?
10. The next day the stocks fall another 8%. Repeat question 8.
11. The family office is in default. Give the value of one dealer’s hedge shares after both falls, its total loss on the swaps over the two days, and what it holds against that loss (margin plus cash received).
12. A dealer sells its shares over the following days at an average 12% below the second day’s close. What is its final loss?
13. What would it have lost had it sold immediately at 3% below that close?

**Part IV — What should have been different.**

14. With dynamic margin of 10% of *current* value, how much margin would the dealers have held before the fall?
15. Each position is about five days of the stock’s trading volume for one dealer alone. With a daily volatility of 3%, estimate the 95% discount of [Proposition 2.8](https://one-course.com/books/quant/1/en/chapter/2-the-sell-side#prop-m1-the-sell-side-discount) for a dealer liquidating alone at 10% participation, and compare with the margin held.
16. Why is that estimate still too low in this situation?
17. What single piece of information, had the dealers shared it, would have changed their margin?
18. Why did the first dealer to sell lose least?
19. State the *named result* : the fall in the stocks that wipes out the family office’s equity, in percent.
20. In one sentence: what is the difference between [leverage](#def-m1-financing-leverage) and margin?

**Solution of Problem 6.1.**

**1.** $L = 28/4 = 7$; wiped out by a fall of 14.3%. **2.** $28/1.6 = \$17.5$ billion; margin $1.75 billion. **3.** $1.75/28 = 6.25\%$. **4.** $2.25 billion is free cash held elsewhere. No dealer can see the total position or verify that the cash is unencumbered: each margins its fifth as if it were the whole. **5.** $28 \times 4.45\% = \$1.246$ billion. **6.** $1.246 - 0.16 = \$1.086$ billion, 27.2% of equity. **7.** $1.086/28 = 3.88\%$. **8.** Loss $2.8 billion; equity $1.2 billion; [leverage](#def-m1-financing-leverage) $25.2/1.2 =
21$. **9.** $2.8 billion is called. The family office pays $2.25 billion: $450 million per dealer against $560 million owed, leaving $110 million unpaid at each. **10.** Loss $0.08 \times 25.2 = \$2.016$ billion; equity $-\$0.816$ billion: the family office is insolvent and [leverage](#def-m1-financing-leverage) has no meaning. **11.** Hedge value $23.184/5 = \$4.64$ billion. Loss owed on the swaps $560 + 403 = \$963$ million; held against it, $350 + 450 = \$800$ million. **12.** Shortfall $163 million plus $0.12 \times 4\,637 = \$556$ million of liquidation loss: $720 million. **13.** $163 + 0.03 \times 4\,637 = \$302$ million. **14.** $2.8 billion instead of $1.75 billion. **15.** $T = 50$ days; impact $0.7\times0.03\times\sqrt5 = 4.7\%$; risk $0.03\sqrt{50/3} = 12.2\%$; discount $4.7 + 1.645\times12.2 = 24.8\%$, four times the 6.25% held. **16.** Four other dealers are liquidating the same stocks at the same time, so the volume available to each is a fraction of the total and the impact adds up; and the formula’s random walk has no drift, while here the whole market knows that $23 billion is for sale. **17.** The client’s *total* position across dealers. Each would have seen a holding of twenty-five days’ volume, not five. **18.** Its sales were made before the others’ sales had moved the price; the last to sell sold into the impact of everyone else. In a common liquidation the order of exit is the whole result, which is why informal agreements to sell in an orderly way do not hold. **19.** **14.3%.** **20.** [Leverage](#def-m1-financing-leverage) measures how much the client loses per unit of price move; margin measures how much of that loss the lender has been given in advance.

## 6.11 Interview questions

**Interview question 6.1 ★ trader, researcher, bank.**

Walk me through the cash flows of a [short sale](#def-m1-financing-short), from the [locate](#def-m1-financing-seclending) to the buy-back.

**Solution of Interview question 6.1.**

Obtain a [locate](#def-m1-financing-seclending) from the [prime broker](#def-m1-financing-pb). Sell the shares in the market. At settlement the [prime broker](#def-m1-financing-pb) borrows the shares from a lender and delivers them to the buyer; the sale proceeds go to the lender as cash collateral, marked to market daily. While short: receive interest on the collateral at the [rebate rate](#def-m1-financing-seclending) (rate minus fee), pay the lender any dividend. To close: buy the shares, return them to the lender, get the collateral back; profit is the fall in price plus rebate less dividends.

*What the interviewer is looking for: that the seller never holds the proceeds, and the dividend.*

**Interview question 6.2 ★ bank, trader.**

What is a repo, and why is it considered safe for the cash lender?

**Solution of Interview question 6.2.**

A sale of a security with an agreement to repurchase it later at a higher price: a collateralised cash loan. It is safe for the lender because it holds a liquid security worth more than the loan (the [haircut](#def-m1-financing-repo)), marked to market daily, which in most jurisdictions it can sell at once if the borrower fails, without waiting for a bankruptcy court.

*What the interviewer is looking for: [haircut](#def-m1-financing-repo), daily margining and the close-out right.*

**Interview question 6.3 ★★ researcher, trader.**

A fund is leveraged four times and loses 5% on its assets. How much must it sell to keep its [leverage](#def-m1-financing-leverage) constant? Do it in your head.

**Solution of Interview question 6.3.**

Sale $= (L-1)\,x\,A = 3 \times 5\% = 15\%$ of its assets. Check: assets $100 \to 95$, equity $25 \to 20$, target assets 80, sell 15.

*What the interviewer is looking for: the $(L-1)x$ shortcut or a fast check with round numbers.*

**Interview question 6.4 ★★ researcher.**

Your backtest of a long–short strategy shows 9% a year. It ignores financing. List what you must subtract and give orders of magnitude.

**Solution of Interview question 6.4.**

Financing spread on the long debit (around half a percent on the financed part); loss of interest and credit spread on short proceeds; [borrow fees](#def-m1-financing-seclending), negligible for large liquid names and many percent for crowded shorts, which are often exactly the ones the model wants; dividends paid on shorts, if the backtest used price returns; and withholding-tax effects on long dividends. For a $3{:}2$ book on a unit of equity the first items alone are about 2% of equity a year; with hard-to-borrow shorts, several times that.

*What the interviewer is looking for: [borrow fees](#def-m1-financing-seclending) correlated with the signal.*

**Interview question 6.5 ★★ bank.**

Why would a client choose a [total return swap](#def-m1-financing-trs) over buying the shares through its [prime broker](#def-m1-financing-pb)? Why would the bank prefer one or the other?

**Solution of Interview question 6.5.**

The client: no entry on the share register and, in some jurisdictions, no disclosure of large holdings; no stamp or transaction tax where those apply; negotiated margin; access to markets where it cannot hold shares directly. The bank: a swap is a derivative on its balance sheet with different capital and netting treatment, it earns the spread on the whole notional, and it controls the hedge; against that, it carries the client’s credit risk with only the contractual margin, and a client’s total position is harder to see.

*What the interviewer is looking for: both parties’ motives, and the credit risk the dealer takes.*

**Interview question 6.6 ★★★ bank, researcher.**

You are the risk manager of a [prime broker](#def-m1-financing-pb). A client’s portfolio is concentrated in five stocks, in each of which it holds several days of volume, and you suspect it has the same positions at other banks. How do you set its margin?

**Solution of Interview question 6.6.**

On liquidation cost, not on daily volatility: margin $\approx$ impact plus a high quantile of the price move over the days needed to sell the position at a realistic participation rate, with add-ons for concentration per name and for the portfolio’s lack of diversification. Make it dynamic, on current value. Require disclosure of aggregate positions elsewhere as a condition of the credit, and cap the limit if it is refused. Stress the whole portfolio for a joint fall with correlated names. Make sure the contract allows margin to be raised on short notice — and use that right before the crisis, since during it the client cannot pay.

*What the interviewer is looking for: days-to-liquidate as the horizon, and asking for the aggregate position.*