---
title: "Systematic Credit and Bond-ETF Arbitrage"
book: "Strategies II: Volatility, Relative Value, Macro and the Bank Desks"
subject: quant
language: en
chapter: 19
exercises: 8
source: https://one-course.com/books/quant/9/en/chapter/19-systematic-credit-and-bond-etf-arbitrage
---

# Chapter 19 — Systematic Credit and Bond-ETF Arbitrage

A bond ETF trades every second, while most of its bonds do not trade for days. When the two disagree, authorised participants create or redeem shares, and the discount says how illiquid the bonds really are. In March 2020, Haddad, Moreira and Muir found, liquid bond ETFs traded at large discounts to their net asset values as investors sold what they could to raise cash. The same market has systematic factors. Houweling and van Zundert found size, low-risk, value and momentum premia in corporate bonds, with low correlations between them. On this chapter’s synthetic bond market, low risk earns a Sharpe ratio of 0.95, value 0.46 and momentum 0.14. In a planted stress, the ETF’s price falls 5.0% below its stale NAV, 3.0% below the bonds’ true value, and an authorised participant can profit on 28 days. The build is `firm.syscredit`.

## 19.1 Factors in credit

**Definition 19.1 (Credit factor).**

A *credit factor* is a characteristic of corporate bonds (spread against peers of similar rating and duration, recent spread change, duration or rating, issuer size) that sorts them into portfolios whose returns in excess of Treasuries differ persistently, after adjusting for their spread duration.

Houweling and van Zundert found that size, low-risk, value and momentum portfolios generated economically meaningful and statistically significant alphas in the corporate bond market. They held up after transaction costs and within liquid bonds, and combined well because their correlations were low. Measuring factors in bonds has two difficulties that equities do not share. A bond’s return is mostly its spread duration times its spread change, so every signal must be compared across similar durations. And bond prices are stale, so a signal read from last week’s quotes may already be out of date.

`firm.syscredit` builds 300 bonds with durations from one to twelve years over eight years ([Listing 19.1](#lst-s2-systematic-credit-and-bond-etf-arbitrage-factors)). Each spread moves with a market factor, an issuer trend that persists for months, a gap from fair value that reverts over a year, and noise. Expected excess returns rise with the square root of duration, so short, safe bonds earn more per unit of risk. Each factor book buys the top fifth and sells the bottom fifth of bonds by its signal, weighted so that each side has a similar spread duration. Value is the spread above the fitted spread-duration line less its three-month change, which keeps value from being a list of recent losers. Momentum is the six-month spread tightening. Low risk buys short bonds and sells long ones at equal risk.

| eight years, monthly rebalancing | Sharpe ratio | correlation with value |
| --- | --- | --- |
| value | 0.46 | 1 |
| momentum | 0.14 | $-0.09$ |
| low risk | 0.95 | $-0.03$ |
| equal-risk combination | 0.94 |  |

Low risk dominates the combination ([Figure 19.1](#fig-s2-systematic-credit-and-bond-etf-arbitrage-factors)). Momentum is weak because the value gap, which reverts, drives part of each bond’s six-month move. The two signals fight over the same spread changes. Houweling and van Zundert found both positive with real data. A model in which one reverts and the other trends at overlapping horizons gives neither much room.

![Three synthetic credit factor books and their equal-risk combination, each scaled to 10% volatility, cumulative sums of daily returns. Data: s2_syscredit.market.](https://one-course.com/images/onecourse/chapters/quant-9/s2-systematic-credit-and-bond-etf-arbitrage/fig-8e71a2f47735.svg)

***Figure 19.1.** Three synthetic [credit factor](#def-s2-systematic-credit-and-bond-etf-arbitrage-factor) books and their equal-risk combination, each scaled to 10% volatility, cumulative sums of daily returns. Data: `s2_syscredit.market`.*

## 19.2 Bond ETFs and create-redeem

**Definition 19.2 (ETF create-redeem arbitrage).**

*ETF create-redeem arbitrage* is the trade by which an authorised participant (Book 1, chapter 14) delivers a creation basket to receive new ETF shares when the shares trade above the basket’s value, or delivers shares to receive the basket when they trade below, keeping the gap less its costs.

For a bond ETF the basket is the hard part. Koont, Ma, Pastor and Zeng found that corporate bond ETFs choose creation and redemption baskets that hold cash and only a subset of the index bonds, the more so when the bonds are illiquid. They adjust the baskets to correct imbalances while still allowing arbitrage. Basket inclusion made bonds more liquid in general but less liquid when creations and redemptions were most unbalanced, as in the COVID-19 crisis.

**Definition 19.3 (ETF discount).**

An *ETF discount* is the amount by which an ETF’s market price is below its published net asset value; for a bond ETF part of it is stale NAV, computed from bond prices that have not traded, and part is selling pressure on the shares that the creation-redemption mechanism has not absorbed.

## 19.3 Portfolio trades

A portfolio trade (Book 2, chapter 22) buys or sells a whole list of bonds at once, often an ETF’s basket, at a single negotiated price. Dealers price it against the ETF and the bonds’ quotes, and can hedge it with the ETF itself. When the ETF trades at a discount, a client selling a portfolio of bonds pays for the dealer’s cost of redeeming or holding them. Providing liquidity to portfolio trades is a way of being the authorised participant’s counterparty without the creation unit.

## 19.4 Discounts in stress

The synthetic ETF holds all 300 bonds. Its NAV uses the bonds’ last traded prices; each bond trades on a quarter of days. Its market price follows the bonds’ true value. Six years in, a planted stress widens market spreads by 150 basis points over three weeks, and the basket loses 9.8%. For the same weeks the ETF price falls a further 3% below true value, as investors sell ETF shares for cash, and recovers over two months ([Figure 19.2](#fig-s2-systematic-credit-and-bond-etf-arbitrage-etf)).

In normal times the discount averages 2.7 basis points with a standard deviation of 21.6. At the stress’s worst, fourteen trading days after it began, the ETF traded 504 basis points below NAV. Of that, the stale NAV stood 215 basis points above the bonds’ true value, and the price stood 300 basis points below it. Only the second part is an arbitrage. An authorised participant who buys shares, redeems them and sells the bonds at their true value pays a cost of 1.5% in the stress (0.3% normally). The trade was profitable on 28 days, by up to 1.63% of the value redeemed, and on no day outside the stress.

![The synthetic bond ETF through a three-week stress: the true value of its 300 bonds, the NAV computed from their last traded prices, and the ETF’s market price, which falls below both. Data: s2_syscredit.market.](https://one-course.com/images/onecourse/chapters/quant-9/s2-systematic-credit-and-bond-etf-arbitrage/fig-5d8cfe418f3a.svg)

***Figure 19.2.** The synthetic bond ETF through a three-week stress: the true value of its 300 bonds, the NAV computed from their last traded prices, and the ETF’s market price, which falls below both. Data: `s2_syscredit.market`.*

Haddad, Moreira and Muir read the March 2020 discounts the same way. Investors tried to sell their safer, more liquid holdings to raise cash, and the discounts were larger for Treasury, municipal and investment-grade ETFs than for high-yield ones. The disruptions reversed almost as fast as they appeared, after the Federal Reserve announced purchases of corporate bonds on 23 March and 9 April 2020. Which price was right? The NAV was stale, the ETF price was pushed, and the bonds’ true value was the one nobody observed.

## 19.5 Strategy files

**Strategy file 19.1 — Credit value and momentum.**

**Who pays you, and why.** Investors constrained by ratings and benchmarks, who leave cheap bonds cheap and chase recent winners slowly.

**Instruments and venues.** Corporate bonds; CDS as a liquid substitute.

**Signal.** Spread against peers of similar rating and duration; spread momentum; hedged to duration.

**Sizing and execution.** Long-short or tilts against a benchmark; monthly.

**Costs.** Bond bid-ask, large for small and old issues.

**How it dies.** Costs; value that is default risk; momentum crashes at turning points.

**Horizon, capacity, infrastructure.** Months; a bond database with stale-price flags.

**Backtest honestly.** Traded prices, not matrix prices; costs by liquidity bucket.

**Sources.** Houweling and van Zundert (2017); this chapter: value 0.46, momentum 0.14.

**Strategy file 19.2 — Low-risk credit.**

**Who pays you, and why.** Investors who reach for yield in long, risky bonds and leave short, safe ones cheap per unit of risk.

**Instruments and venues.** Short-dated, higher-rated bonds, levered or against long, risky ones.

**Signal.** Duration and rating.

**Sizing and execution.** Equal risk on each side.

**Costs.** Funding for leverage.

**How it dies.** A rally in risky credit; funding costs.

**Horizon, capacity, infrastructure.** Years.

**Backtest honestly.** Leverage at realistic funding rates.

**Sources.** Houweling and van Zundert (2017); this chapter: 0.95.

**Strategy file 19.3 — ETF premium-discount arbitrage.**

**Who pays you, and why.** ETF holders who sell shares for cash faster than the bonds can be sold.

**Instruments and venues.** ETF shares; creation and redemption with the sponsor as an authorised participant; the bonds.

**Signal.** The price against the bonds’ true value, not against a stale NAV.

**Sizing and execution.** Redeem when the gap exceeds the cost of selling the basket; baskets as the sponsor sets them.

**Costs.** Selling illiquid bonds in stress.

**How it dies.** A discount that is all stale NAV; baskets changed to hold bonds that cannot be sold.

**Horizon, capacity, infrastructure.** Days; authorised-participant status and bond trading desks.

**Backtest honestly.** Estimate true value from traded prices; include basket rules.

**Sources.** Haddad, Moreira and Muir (2020); Koont, Ma, Pastor and Zeng (2022); this chapter: 28 profitable days in the stress.

**Strategy file 19.4 — Portfolio-trade liquidity provision.**

**Who pays you, and why.** Clients who pay to trade a whole list of bonds at once.

**Instruments and venues.** Portfolio trades priced against the ETF; the ETF as a hedge.

**Signal.** The list’s price against the ETF and the bonds’ quotes.

**Sizing and execution.** Hedge with the ETF at once, work out of the bonds over days.

**Costs.** The bonds’ bid-ask as they are worked out.

**How it dies.** Lists of bonds that only the dealer holds; ETF and bonds diverging.

**Horizon, capacity, infrastructure.** Days; balance sheet.

**Backtest honestly.** Execution of each bond at the prices that were available.

**Sources.** No performance figure verified.

## 19.6 Tutorial: which price is right

**Goal.** Build a bond universe with factor structure and stale prices, trade [credit factors](#def-s2-systematic-credit-and-bond-etf-arbitrage-factor), and run the ETF and its authorised participant through a stress. **End state:** the table and the two figures.

1. **Factor books**. `def factor_book (sim: dict , cfg: SysCreditConfig | None = None , name: str = " value " , every: int = 21 ) -> np.ndarray: """Long the top fifth and short the bottom fifth of bonds by the signal, duration-neutral within each side by weighting with 1 / duration, rebalanced monthly; value: spread over the fitted spread-duration line less its three-month change, so that value is not recent losers; momentum: minus the six-month spread change; low risk: minus duration (long short bonds, short long ones, equal risk).""" cfg = cfg or SysCreditConfig() ret, dur, sp = sim[" ret " ], sim[" dur " ], sim[" spread " ] T, N = ret.shape out = np.zeros(T) w = np.zeros(N) for t in range (127 , T): if (t - 127 ) % every == 0 : if name == " value " : fit = np.polyval(np.polyfit(dur, sp[t - 1 ], 1 ), dur) sig = sp[t - 1 ] - fit - (sp[t - 1 ] - sp[t - 64 ]) # cheap, less its last three months' move elif name == " momentum " : sig = -(sp[t - 1 ] - sp[t - 127 ]) else : sig = -dur q = np.quantile(sig, [0.2 , 0.8 ]) long, short = sig >= q[1 ], sig <= q[0 ] w = np.zeros(N) w[long] = (1 / dur[long]) / (1 / dur[long]).sum() w[short] = -(1 / dur[short]) / (1 / dur[short]).sum() if name == " lowrisk " : # equal risk: scale each side by duration w[long] = 1 / long.sum() / dur[long].mean() w[short] = -1 / short.sum() / dur[short].mean() out[t] = w @ ret[t] return out` **Listing 19.1.** Value, momentum and low-risk books, duration-balanced, monthly. code/firm/syscredit/firm_syscredit.py
2. **The authorised participant**. `def ap_arbitrage (sim: dict , cfg: SysCreditConfig | None = None ) -> dict : """Each day the ETF trades below the basket's true value by more than the AP's cost, the AP buys shares and redeems them, selling the bonds at true value less the cost: profit in bp of the redeemed value.""" cfg = cfg or SysCreditConfig() a, b = sim[" stress " ] cost = np.full(len (sim[" price " ]), cfg.cost_normal) cost[a:b + cfg.stress_back] = cfg.cost_stress edge = sim[" basket " ] / sim[" price " ] - 1 - cost profit = np.where(edge > 0 , edge, 0.0 ) * 1e4 discount = (sim[" price " ] / sim[" nav " ] - 1 ) * 1e4 return {" profit " : profit, " discount " : discount, " cost " : cost}` **Listing 19.2.** Redeeming at a discount beyond the cost of selling the bonds. code/firm/syscredit/firm_syscredit.py
3. **Run** `factors()` , `etf()` and `fig_syscredit.py` .

**What to change next.** Estimate the bonds’ true value from the ETF and recent trades and trade the difference; let the basket omit the least liquid bonds; add a Fed-purchase announcement that closes the dislocation in a day.

## 19.7 Build: systematic credit

**Purpose.** A synthetic bond panel with [credit factors](#def-s2-systematic-credit-and-bond-etf-arbitrage-factor) and stale prices, a bond ETF with NAV and price, and create-redeem arbitrage.

**Interface.** `SysCreditConfig(…)`, `simulate_bonds(cfg)`, `factor_book(sim, cfg, name)`, `ap_arbitrage(sim, cfg)`.

**Rules.** Stale prices update only on trading days; NAV from stale prices; arbitrage against true value less cost.

**Acceptance tests.** `code/firm/syscredit/tests/`: stale prices move only on trades and NAV is their mean; no arbitrage without a dislocation; factor books are long-short and finite.

**Stretch.** Rating migration; defaults; basket choice.

Sources and further reading

- P. Houweling and J. van Zundert, “Factor investing in the corporate bond market”, *Financial Analysts Journal* 73(2), 2017.
- V. Haddad, A. Moreira and T. Muir, “When selling becomes viral: disruptions in debt markets in the COVID-19 crisis and the Fed’s response”, NBER Working Paper 27168, 2020.
- N. Koont, Y. Ma, L. Pastor and Y. Zeng, “Steering a ship in illiquid waters: active management of passive funds”, NBER Working Paper 30039, 2022.

## 19.8 Exercises

**Exercise 19.1 ★.**

An ETF trades 504 basis points below NAV; the NAV is 215 basis points above the bonds’ true value. How far is the price below true value?

**Solution of Exercise 19.1.**

About 300 basis points: the price is $0.9496$ of NAV and the NAV $1.0215$ of true value, so the price is $0.9496 \times 1.0215 = 0.970$ of true value. The two parts multiply rather than add, which is why $215 + 300$ exceeds 504.

**Exercise 19.2 ★.**

The price is 300 basis points below true value and selling the basket costs 150. What does redeeming earn?

**Solution of Exercise 19.2.**

Buying at $0.97$ of true value and selling the bonds at true value earns $1/0.97 - 1 = 3.09\%$ of the redeemed value; less the cost of 1.5% that is 159 basis points. The chapter’s best day, 163, adds the price noise.

**Exercise 19.3 ★.**

Market spreads widen 150 basis points on bonds of average spread duration 6.5. Roughly how much does the basket lose?

**Solution of Exercise 19.3.**

$1.50\% \times 6.5 = 9.75\%$, close to the simulated $9.8\%$; carry over three weeks and the spread of durations around 6.5 account for the rest.

**Exercise 19.4 ★★.**

Why must [credit factors](#def-s2-systematic-credit-and-bond-etf-arbitrage-factor) be compared across similar durations?

**Solution of Exercise 19.4.**

A bond’s excess return is mostly its spread duration times its spread change, and spreads rise with duration. A signal compared across all bonds would buy long bonds for their wide spreads and load on market credit risk, which is a beta, not a factor. Comparing against peers of similar duration and rating, and balancing the long and short sides in spread duration, isolates the characteristic.

**Exercise 19.5 ★★.**

Why were [ETF discounts](#def-s2-systematic-credit-and-bond-etf-arbitrage-discount) larger for safer bonds in March 2020?

**Solution of Exercise 19.5.**

Haddad, Moreira and Muir found that investors tried to raise cash by selling their safer and more liquid holdings. The selling fell on the ETFs of Treasuries, municipal and investment-grade bonds, which investors hold as liquid reserves, more than on high-yield ETFs.

**Exercise 19.6 ★★.**

Why does a bond ETF choose a redemption basket that is not the index?

**Solution of Exercise 19.6.**

The index holds thousands of bonds, many of which rarely trade; delivering or receiving all of them would be costly or impossible. Koont, Ma, Pastor and Zeng found that bond ETFs choose baskets with cash and a subset of bonds, the more so for illiquid ones, and adjust them to correct imbalances between creations and redemptions while still allowing arbitrage.

**Exercise 19.7 ★★★.**

*Coding.* Rerun with `SysCreditConfig(trade_prob=0.05)`. What happens to the stale part of the discount at the stress’s worst?

**Solution of Exercise 19.7.**

With bonds trading on 5% of days, the worst discount doubles to 1 050 basis points, still fourteen trading days into the stress, and the stale part rises from 215 to 838 basis points: the NAV lags true value by about nineteen days on average, longer than the stress, so it misses most of the widening. The part that is selling pressure is unchanged, since the price follows true value, but a discount against NAV now mostly measures stale prices.

**Exercise 19.8 ★★★.**

*Find the flaw.* “The ETF trades 5% below NAV, so buying it earns 5% when the discount closes.”

**Solution of Exercise 19.8.**

The NAV is computed from stale prices; in the chapter’s stress 215 of the 504 basis points is NAV that has not caught up with falling bonds, and it closes by the NAV falling, not the price rising. Only the part against true value, 300 basis points, is a return, and only to someone who can redeem and sell the bonds, at a cost of 150 basis points in the stress.

## 19.9 Problem: Which Price Is Right

**Problem 19.1.**

Weekend problem — credit factors and bond ETFs

The chapter’s synthetic bonds and ETF and the public record.

**Part I — Factors.**

1. Define a [credit factor](#def-s2-systematic-credit-and-bond-etf-arbitrage-factor) .
2. What did Houweling and van Zundert find?
3. Describe the synthetic bonds and the three signals.
4. Give the factors’ Sharpe ratios and correlations.

**Part II — ETFs.**

5. Define create-redeem arbitrage and the [ETF discount](#def-s2-systematic-credit-and-bond-etf-arbitrage-discount) .
6. What did Koont, Ma, Pastor and Zeng find about baskets?
7. How is the synthetic NAV computed?
8. What is a portfolio trade?

**Part III — The stress.**

9. Describe the planted stress.
10. Decompose the worst discount.
11. When could the authorised participant profit?
12. What did Haddad, Moreira and Muir find?

**Part IV — The verdict.**

13. State the *named result* : the [credit factors](#def-s2-systematic-credit-and-bond-etf-arbitrage-factor) ’ Sharpe ratios and the ETF arbitrage’s return in the planted stress.
14. Which price is right, and why can no one see it?
15. Why is momentum weak here?
16. How would you estimate true value in a stress?
17. How would you backtest the ETF arbitrage honestly?
18. Which strategy file needs authorised-participant status?
19. How does this chapter relate to chapter 18?
20. In one sentence: what does a bond [ETF discount](#def-s2-systematic-credit-and-bond-etf-arbitrage-discount) measure?

**Solution of Problem 19.1.**

1. A characteristic of bonds, such as spread against peers, spread momentum, duration or size, that sorts them into portfolios with persistently different duration-adjusted excess returns.
2. Size, low-risk, value and momentum portfolios earned economically meaningful and statistically significant alphas; low correlations made a combination more efficient; the results held after costs and in liquid bonds.
3. 300 bonds over eight years, durations from one to twelve; spreads move with a market factor, a persistent issuer trend, a reverting gap and noise; expected excess returns rise with the square root of duration. Value is the spread above the fitted spread-duration line less its three-month change; momentum is the six-month tightening; low risk buys short bonds against long ones at equal risk.
4. Value 0.46, momentum 0.14, low risk 0.95, combined 0.94; correlations $-0.09$ (value, momentum), $-0.03$ (value, low risk) and $-0.02$ (momentum, low risk).
5. Create-redeem arbitrage: delivering a basket for new shares when shares trade rich, or shares for the basket when they trade cheap, keeping the gap less costs. The discount: price below published NAV.
6. Bond ETFs choose baskets with cash and a subset of the index, especially when bonds are illiquid, and adjust them to correct imbalances; basket inclusion helps liquidity in general but hurts it in large imbalances such as the COVID-19 crisis.
7. The mean of the bonds’ last traded values; each bond trades on a quarter of days.
8. A trade of a whole list of bonds at a single negotiated price, priced against the ETF and the bonds’ quotes and hedgeable with the ETF.
9. Six years in, market spreads widen 150 basis points over fifteen days and half of it returns over forty; the ETF price falls a further 3% below true value at the depth and recovers over forty days. The basket loses 9.8%.
10. At the worst, $-504$ basis points against NAV: stale NAV 215 above true value, price 300 below it, multiplied.
11. On 28 days, all in the stress, by up to 163 basis points of the redeemed value after a cost of 1.5%; never outside the stress, where the discount averages 2.7 basis points with a standard deviation of 21.6.
12. Liquid bond ETFs traded at large discounts to NAV in March 2020, more for Treasuries, municipal and investment-grade bonds than high yield, as investors sold safer assets for cash; the disruptions reversed after the Fed’s announcements of 23 March and 9 April.
13. Sharpe ratios of 0.95 (low risk), 0.46 (value), 0.14 (momentum) and 0.94 combined; in the stress the ETF traded 5.0% below NAV and 3.0% below true value, and the authorised participant could profit on 28 days by up to 1.63%.
14. True value is right, and no one sees it because most bonds did not trade: the NAV is stale and the price is pushed by sellers.
15. The reverting gap drives part of each six-month spread change, so momentum partly buys bonds whose gap is about to revert, against value.
16. From the ETF price, the prices of bonds that did trade and their co-movement with those that did not, and dealer quotes, with a model of how far selling pressure moves the ETF.
17. Against an estimate of true value from traded prices, not the stale NAV; with basket rules as they were and costs of selling bonds in stress.
18. The ETF premium-discount arbitrage.
19. Chapter 18’s negative basis is bonds below their CDS because funding is scarce; this chapter’s discount is an ETF below its bonds because cash is scarce; both widen in the same crises and pay whoever has balance sheet left.
20. How much the bonds’ true value has moved since they last traded, plus how much sellers of the ETF will pay for immediacy.

## 19.10 Interview questions

**Interview question 19.1 ★ trader.**

How does creation and redemption keep an ETF near its NAV?

**Solution of Interview question 19.1.**

When the shares trade above the basket’s value, an authorised participant buys the basket, delivers it for new shares and sells them; below it, the participant buys shares, redeems them for the basket and sells the basket. Both trades push the price back towards the basket’s value, within the participant’s costs.

**Interview question 19.2 ★★ researcher.**

How would you build a value factor for corporate bonds?

**Solution of Interview question 19.2.**

The spread against a fitted curve of peers of similar rating, duration and sector, possibly against a model of default risk; remove the part that is recent widening, balance the sides in spread duration, check it is not just default risk, and price it with realistic bond costs.

**Interview question 19.3 ★★ trader.**

A client wants to sell a list of 200 bonds at once. How do you price it?

**Solution of Interview question 19.3.**

Against the ETF and the bonds’ quotes: hedge the list at once with the ETF, estimate how long each bond takes to work out and what it costs, add the risk of the list diverging from the ETF over that time, and charge more for bonds the ETF does not hold.

**Interview question 19.4 ★★ risk.**

Your book holds bond ETFs against their bonds. What happens in a stress like March 2020?

**Solution of Interview question 19.4.**

The ETF falls below its bonds and the book is marked at a loss; the bonds cannot be sold at their marks, redemptions deliver baskets that may not match the book, and funding tightens. The hedge works only for whoever can hold until the discount closes.

**Interview question 19.5 ★★ developer.**

Design a fair-value estimate for bonds that have not traded today.

**Solution of Interview question 19.5.**

Start from each bond’s last trade and dealer quotes; move it with the observed moves of liquid proxies (the ETF, CDS, the bonds of the same issuer and similar bonds that did trade) weighted by its duration and rating; flag its age and confidence, and test the estimate against the next trade.

**Interview question 19.6 ★★★ researcher.**

If each bond trades on a share $p$ of days and true values follow a random walk, show that the NAV lags true value by $(1 - p)/p$ days on average, and find the stale gap after a steady move of $m$ a day.

**Solution of Interview question 19.6.**

The last trade was $k$ days ago with probability $p(1-p)^k$, so the NAV reflects true value $k$ days old, and $E[k] = \sum_k k\, p(1-p)^k =
(1-p)/p$. After a steady move of $m$ a day that has lasted longer than most bonds’ gaps between trades, the stale gap is $m\,E[k] = m(1-p)/p$: three days’ move for $p = 1/4$, nineteen for $p = 1/20$.