---
title: "European and Japanese Government Bonds"
book: "Markets II: Rates, FX and Credit"
subject: quant
language: en
chapter: 7
exercises: 8
source: https://one-course.com/books/quant/2/en/chapter/7-european-and-japanese-government-bonds
---

# Chapter 7 — European and Japanese Government Bonds

On Wednesday 28 September 2022 the yield of the thirty-year UK government bond moved within a range of 127 basis points in a single day, more than its range over a whole year in all but four of the previous twenty-seven. In the days before, it had risen by 160 basis points, having started the year near 1.2%. Pension funds that had borrowed against their [gilts](#def-m2-european-and-japanese-government-bonds-ldi) to match their long liabilities were being asked for collateral faster than they could raise it, and their managers were telling the Bank of England that, at the prevailing yields, some of their funds would be worth less than nothing by the next morning. That day the Bank announced temporary purchases of long [gilts](#def-m2-european-and-japanese-government-bonds-ldi) to restore orderly market conditions; by 14 October it had bought 19.3 billion pounds’ worth. The euro area, Japan and the United Kingdom each run their government bond markets differently, and each has, in the last fifteen years, had to decide what its central bank owes the market in a crisis. This chapter compares them.

## 7.1 The euro area: one currency, many issuers

Twenty-one governments borrow in euros since Bulgaria joined on 1 January 2026, and none of them controls the central bank that issues it. The same currency therefore trades at many yields, and the differences are the market’s measure of each government’s credit and of the risk that the currency union itself might change.

**Definition 7.1 (Sovereign spread and redenomination risk).**

A *sovereign spread* is the difference between the yield of one government’s bond and that of a benchmark government of the same currency and maturity, in the euro area usually Germany. *Redenomination risk* is the part of a spread that pays for the possibility that a bond is repaid in a new national currency worth less than the euro.

**Definition 7.2 (Bund and syndication).**

A *Bund* is a German federal government bond (Bundesanleihe), issued at 7, 10, 15 or 30 years with an annual coupon; the ten-year Bund’s yield is the euro area’s benchmark long rate. *Syndication* is the sale of a new bond through a group of banks that build a book of investor orders and set the price, instead of an auction; some issuers use it for new long bonds, where demand is hardest to predict.

![Ten-year spreads to Germany, monthly averages, 2007 to August 2026. The sovereign debt crisis took Italy’s to 518 basis points in November 2011 and Spain’s to 555 in July 2012; spreads fell after the ECB announced outright monetary transactions in September 2012. By August 2026 France’s spread, 82 basis points, was slightly wider than Italy’s, 81. Data: OECD, via FRED.](https://one-course.com/images/onecourse/chapters/quant-2/m2-european-and-japanese-government-bonds/fig-3c4d82e054c5.svg)

***Figure 7.1.** Ten-year spreads to Germany, monthly averages, 2007 to August 2026. The sovereign debt crisis took Italy’s to 518 basis points in November 2011 and Spain’s to 555 in July 2012; spreads fell after the ECB announced outright monetary transactions in September 2012. By August 2026 France’s spread, 82 basis points, was slightly wider than Italy’s, 81. Data: OECD, via FRED.*

**Example 7.3 (Anatomy of a widening).**

Between June and November 2011 (monthly averages) Italy’s ten-year spread to Germany widened by 326 basis points. Italy’s yield rose 224 basis points; the other 102 came from Germany’s yield *falling* by as much, as investors fled into [Bunds](#def-m2-european-and-japanese-government-bonds-bund). A spread measures relative credit, but half of a crisis widening can be the benchmark rallying: a hedge that shorts the benchmark against the spread loses on that half.

## 7.2 Spreads, backstops and the electronic dealer market

**Definition 7.4 (Central-bank backstop).**

A *central-bank backstop* is a commitment, or a standing instrument, by which a central bank will buy a government’s bonds in a market dysfunction, so that prices reflect fundamentals rather than a run. Its conditions decide what it can be used for: the conditions for activating it, the size, and whether it is announced as unlimited.

The ECB has two. Outright monetary transactions, announced in September 2012, have no ex ante limit but are conditional on the country having an adjustment programme with the European Stability Mechanism: the promise is as large as needed, and its price is a programme. The transmission protection instrument of July 2022 is conditional instead on four criteria of fiscal and macroeconomic soundness, and aims at “unwarranted, disorderly market dynamics that pose a serious threat to the transmission of monetary policy”, purchases concentrating on maturities of one to ten years. A spread that widens because of fundamentals is outside both; a spread that widens because of a run is, in principle, inside. Deciding which is which is the ECB’s call.

The secondary market is organised like the US one in two tiers. Among dealers, European government bonds trade on electronic platforms, such as MTS’s inter-dealer markets, one for each country; clients trade with dealers by request for quote on [dealer-to-client platforms](https://one-course.com/books/quant/2/en/chapter/4-the-treasury-market#def-m2-the-treasury-market-idb), such as MTS’s own BondVision ([Chapter 22](https://one-course.com/books/quant/2/en/chapter/22-how-bonds-trade#ch-m2-how-bonds-trade)).

**As of September 2026 — Ten-year yields, August 2026 averages.**

Germany 3.18%, Spain 3.63%, Italy 3.99%, France 4.00%; Japan 2.94%; United Kingdom 4.99%. Spreads to Germany: Spain 45, Italy 81, France 82 basis points.

## 7.3 Japan: yield-curve control and its exit

**Definition 7.5 (Yield-curve control).**

*Yield-curve control* is a policy in which the central bank sets a target for a long-term government bond yield and buys, in whatever amount necessary, to hold it there, in addition to setting the short rate.

The Bank of Japan introduced it in September 2016, committing to buy government bonds so that ten-year yields would stay “more or less at the current level (around zero percent)”. A yield target inverts the usual relation between price and quantity: the central bank no longer chooses how much it buys, the market does, by selling to it at the target. As yields rose around the world, defending the target meant buying whatever was offered; on 20 December 2022 the Bank widened the band around it from a quarter to half a percentage point and offered to buy ten-year bonds at 0.5% every business day. On 19 March 2024 it declared that [yield-curve control](#def-m2-european-and-japanese-government-bonds-ycc) and the negative rate had “fulfilled their roles” and returned to steering the short rate alone. The ten-year yield, near zero for most of the control period, averaged 2.94% in August 2026 ([Figure 7.2](#fig-m2-european-and-japanese-government-bonds-jgb)).

![The ten-year Japanese government bond yield, monthly averages, 2007 to August 2026. Under yield-curve control (shaded) the Bank held it near its target; after the exit it rose to the highest levels of the period. Data: OECD, via FRED; dates of the policy from the Bank of Japan’s statements.](https://one-course.com/images/onecourse/chapters/quant-2/m2-european-and-japanese-government-bonds/fig-75ed070cc25a.svg)

***Figure 7.2.** The ten-year Japanese government bond yield, monthly averages, 2007 to August 2026. Under [yield-curve control](#def-m2-european-and-japanese-government-bonds-ycc) (shaded) the Bank held it near its target; after the exit it rose to the highest levels of the period. Data: OECD, via FRED; dates of the policy from the Bank of Japan’s statements.*

## 7.4 The United Kingdom and the 2022 gilt crisis

**Definition 7.6 (Gilt and liability-driven investment).**

A *gilt* is a UK government bond; conventional gilts pay semiannual coupons, index-linked gilts pay coupons and principal indexed to inflation. *Liability-driven investment* (LDI) is a strategy in which a defined-benefit pension fund hedges the interest-rate and inflation sensitivity of its liabilities with long gilts and swaps, often with leverage from repo and derivatives, so that it can hedge more of its liabilities than it holds in bonds while keeping other, higher-returning assets.

Leverage is what made the hedge fragile. A fund that holds [gilts](#def-m2-european-and-japanese-government-bonds-ldi) worth twice its capital, borrowing the other half in repo, loses twice as fast when [gilts](#def-m2-european-and-japanese-government-bonds-ldi) fall. Its manager keeps a cushion of capital against losses and asks the pension fund for more when it runs low; for pooled funds, with many small pension schemes behind them, that request takes days. In late September 2022 the yields moved in hours. The funds’ only fast source of cash was their [gilts](#def-m2-european-and-japanese-government-bonds-ldi): selling them pushed yields higher, which called for more cash, which forced more sales ([Figure 7.3](#fig-m2-european-and-japanese-government-bonds-loop)). On 26 September managers told the Bank that, at face value, they would have to sell at least 50 billion pounds of long [gilts](#def-m2-european-and-japanese-government-bonds-ldi) in a short time, against average trading of 12 billion a day in those maturities.

![The collateral spiral of September 2022. Pension-fund LDI strategies borrowed against long gilts; rising yields eroded their cushions, calls for collateral could be met fastest by selling gilts, and the selling raised yields further. A buyer that could absorb the sales without needing a return, the central bank, stopped the loop.](https://one-course.com/images/onecourse/chapters/quant-2/m2-european-and-japanese-government-bonds/fig-c574e00454f8.svg)

***Figure 7.3.** The collateral spiral of September 2022. Pension-fund LDI strategies borrowed against long [gilts](#def-m2-european-and-japanese-government-bonds-ldi); rising yields eroded their cushions, calls for collateral could be met fastest by selling [gilts](#def-m2-european-and-japanese-government-bonds-ldi), and the selling raised yields further. A buyer that could absorb the sales without needing a return, the central bank, stopped the loop.*

The Bank’s purchases were targeted and temporary: long [gilts](#def-m2-european-and-japanese-government-bonds-ldi) only, from 28 September to 14 October, 12.1 billion of conventional and 7.2 billion of index-linked [gilts](#def-m2-european-and-japanese-government-bonds-ldi), sold back to the market between November 2022 and January 2023. The Financial Policy Committee then set a standard: LDI funds should be able to withstand a rise of about 250 basis points in [gilt](#def-m2-european-and-japanese-government-bonds-ldi) yields without forced sales, against the 100 basis points that earlier stress tests had assumed.

**Proposition 7.7 (How far a levered bond fund can fall).**

A fund holds bonds worth $G$ with [modified duration](https://one-course.com/books/quant/2/en/chapter/3-government-bonds#def-m2-government-bonds-duration) $D$ and convexity $\mathcal C$, financed by repo $B = \ell G$, so that its cushion is $(1-\ell)G$. Its cushion is exhausted by the parallel yield rise $\Delta y^*$ at which the bonds’ value falls to $B$; to first order $\Delta y^* \approx (1-\ell)/D$. The second-order loss $D\,\Delta y - \frac12\,\mathcal C\,(\Delta y)^2$ never exceeds $D^2/(2\mathcal C)$, so when $1 - \ell > D^2/(2\mathcal C)$ the quadratic has no root and only a full repricing gives $\Delta y^*$. After a rise leaving the bonds worth $G'$, the fund must sell $2B - G'$ of them to restore a repo share of one half without new capital.

**Proof.** The cushion is $G' - B$. To first order $G'/G = 1 - D\Delta y$; setting $G' = B$ gives the first claim. The quadratic $D x - \frac12 \mathcal C x^2$ is maximal at $x = D/\mathcal C$ with value $D^2/(2\mathcal C)$. After selling $S$ and repaying $S$ of repo, the share is $(B - S)/(G' - S) = \frac12$, so $S = 2B - G'$. ∎

For the long [gilt](#def-m2-european-and-japanese-government-bonds-ldi) of [Figure 7.4](#fig-m2-european-and-japanese-government-bonds-ldi), $D = 21.8$ and $\mathcal C = 589$: the quadratic’s largest loss is 40.5%, less than the fund’s 50% cushion, so the second-order formula would say the cushion can never be exhausted. It is exhausted at 351 basis points. Taylor expansions are for small moves; stress tests are not about small moves.

![The cushion of an illustrative LDI fund, two-times levered in repo and holding a thirty-year gilt, as gilt yields rise from 3.5%. It falls to 43% of its starting value after 160 basis points, the size of the September 2022 move, and to zero after 351. Data: the chapter’s weekend problem.](https://one-course.com/images/onecourse/chapters/quant-2/m2-european-and-japanese-government-bonds/fig-487d6b14141d.svg)

***Figure 7.4.** The cushion of an illustrative LDI fund, two-times levered in repo and holding a thirty-year [gilt](#def-m2-european-and-japanese-government-bonds-ldi), as [gilt](#def-m2-european-and-japanese-government-bonds-ldi) yields rise from 3.5%. It falls to 43% of its starting value after 160 basis points, the size of the September 2022 move, and to zero after 351. Data: the chapter’s weekend problem.*

**As of September 2026 — The Bank of England’s gilt holdings.**

On 17 September 2026 the Bank paused its auctions selling [gilts](#def-m2-european-and-japanese-government-bonds-ldi) from the Asset Purchase Facility while it reviews how to continue: [gilts](#def-m2-european-and-japanese-government-bonds-ldi) maturing before 2035 (GBP 222 billion) are to be held to maturity, the longest (GBP 120 billion) held to maturity to back the note issue, and those maturing between 2035 and 2049 (GBP 146 billion at purchase) possibly sold to the government at GBP 20 billion a year, subject to a review before April 2027.

## 7.5 Tutorial: a spread monitor

**Goal.** Build euro-area spreads from monthly yields, decompose the 2011 widening, and flag unusual moves. **End state:** [Figure 7.1](#fig-m2-european-and-japanese-government-bonds-spreads) and the numbers of [Example 7.3](#ex-m2-european-and-japanese-government-bonds-2011).

1. **Spreads and their decomposition.** `def spreads (yields: dict [str , list [tuple [str , float ]]], benchmark: str ) -> dict [str , list [tuple [str , float ]]]: """Spread of every issuer to the benchmark, in basis points, on dates both have.""" base = dict (yields[benchmark]) out = {} for name, series in yields.items(): if name == benchmark: continue out[name] = [(d, (y - base[d]) * 100.0 ) for d, y in series if d in base] return out @dataclass (frozen=True ) class Decomposition : change_bp: float # change in the spread issuer_bp: float # contribution of the issuer's own yield benchmark_bp: float # contribution of the benchmark (a fall in the benchmark widens the spread) def decompose (issuer: dict [str , float ], benchmark: dict [str , float ], start: str , end: str ) -> Decomposition: di = (issuer[end] - issuer[start]) * 100.0 db = (benchmark[end] - benchmark[start]) * 100.0 return Decomposition(di - db, di, -db)` **Listing 7.1.** Spreads to a benchmark, and a change in spread split into the two yields’ moves. code/firm/sovspread/firm_sovspread.py
2. **A z-score alert** over a rolling window. `def zscore (series: list [tuple [str , float ]], window: int ) -> list [tuple [str , float ]]: """Latest value against the mean and standard deviation of the previous `window` values.""" out = [] values = [v for _, v in series] for k in range (window, len (series)): past = values[k - window:k] sd = pstdev(past) out.append((series[k][0 ], (values[k] - mean(past)) / sd if sd > 0 else 0.0 )) return out` **Listing 7.2.** Rolling z-score of a spread. code/firm/sovspread/firm_sovspread.py
3. **Run** `sovereign_demo.widening_2011()` : 326, 224 and 102 basis points; `fig_sovereign.py` writes the three charts.
4. **The LDI fund.** `def fund (shock_bp: float ) -> dict [str , float ]: """Gilt value, repo, cushion after a parallel rise of shock_bp (gilt yield only).""" p0 = GILT.dirty_price(Y0, START) face = HOLDING / p0 * 100 value = face * GILT.dirty_price(Y0 + shock_bp / 1e4 , START) / 100 repo = HOLDING * REPO_SHARE leverage = value / (value - repo) if value > repo else float (" inf " ) return {" price0 " : p0, " value " : value, " repo " : repo, " cushion " : value - repo, " cushion_pct " : (value - repo) / (HOLDING - repo), " leverage " : leverage}` **Listing 7.3.** An LDI fund’s gilts, repo and cushion after a rise in yields. code/markets-2/07-european-and-japanese-government-bonds/python/sovereign_demo.py

**What to change next.** Run the z-score alert on Italy’s spread with a 24-month window and list the months it flags; then give the LDI fund three-times leverage and find the yield rise that exhausts it.

## 7.6 Build: the sovereign spread monitor

**Purpose.** The miniature firm’s rates desk trades euro-area spreads and needs a live view of them, of what moved each one, and of which moves are unusual.

**Interface.** `spreads(yields, benchmark)`; `decompose(issuer, benchmark, start, end)` returning `Decomposition(change_bp, issuer_bp, benchmark_bp)`; `zscore(series, window)`; `alerts(series, window, threshold)`.

**Rules.** Yields in percent, spreads in basis points; spreads only on dates both issuers have; the z-score of a value uses the window *before* it.

**Acceptance tests.** `code/firm/sovspread/tests/`: spreads on common dates; the decomposition adds up and gives the benchmark’s fall a widening sign; a jump triggers the alert and noise does not.

**Stretch.** Daily data from the issuers’ own benchmark curves; spreads at matched maturities with the bond library ([Section 3.6](https://one-course.com/books/quant/2/en/chapter/3-government-bonds#bld-m2-government-bonds-bond)); a spread in asset-swap terms ([Chapter 21](https://one-course.com/books/quant/2/en/chapter/21-corporate-bonds#ch-m2-corporate-bonds)).

Sources and further reading

- Bank of England, letter from Sir Jon Cunliffe to the Treasury Committee on LDI, 5 October 2022; *Quarterly Bulletin* 2023, “Financial stability buy/sell tools: a gilt market case study”; staff paper on LDI minimum resilience, 2023; APF market notice, 17 September 2026.
- European Central Bank, press releases on outright monetary transactions (6 September 2012) and the transmission protection instrument (21 July 2022).
- Bank of Japan, “New Framework for Strengthening Monetary Easing”, 21 September 2016; “Changes in the Monetary Policy Framework”, 19 March 2024.
- OECD, long-term interest rates (via FRED); MTS, company website.

## 7.7 Exercises

**Exercise 7.1 ★.**

With the yields of [Box 7.1](#dat-m2-european-and-japanese-government-bonds-yields), give the spreads of Spain, Italy and France to Germany in basis points.

**Solution of Exercise 7.1.**

Spain $3.633 - 3.18 = 45$, Italy $3.986 - 3.18 = 81$, France $4.00 - 3.18 = 82$ basis points.

**Exercise 7.2 ★.**

Italy’s yield rises 30 basis points and Germany’s falls 10 in a day. Give the change in the spread and its decomposition.

**Solution of Exercise 7.2.**

The spread widens by $30 - (-10) = 40$ basis points: 30 from Italy, 10 from Germany’s fall.

**Exercise 7.3 ★.**

An LDI fund holds GBP 1 billion of [gilts](#def-m2-european-and-japanese-government-bonds-ldi), half financed in repo. What is its leverage? What is its cushion after a 20% fall in [gilt](#def-m2-european-and-japanese-government-bonds-ldi) prices, and its new leverage?

**Solution of Exercise 7.3.**

[Gilts](#def-m2-european-and-japanese-government-bonds-ldi) 1 000, repo 500, cushion 500: leverage 2. After a 20% fall: [gilts](#def-m2-european-and-japanese-government-bonds-ldi) 800, repo 500, cushion 300 (60% of the original), leverage $800/300 = 2.67$. Leverage rises as the cushion shrinks: the next loss hurts more.

**Exercise 7.4 ★★.**

With [Proposition 7.7](#prop-m2-european-and-japanese-government-bonds-cushion), estimate the yield rise that exhausts the cushion of a fund with duration 21.8, convexity 589 and half its [gilts](#def-m2-european-and-japanese-government-bonds-ldi) in repo, to first order. What does the second-order formula say, and why?

**Solution of Exercise 7.4.**

First order: $0.5/21.8 = 229$ basis points. Second order: the loss $21.8\,x -
294.6\,x^2$ is largest at $x = 21.8/589 = 3.7\%$ and there equals $21.8^2/(2
\times 589) = 40.5\%$, below the 50% cushion: the formula says the cushion is never exhausted, which is false. Full repricing gives 351 basis points. The quadratic understates losses far from the starting point because the true price curve is flatter there than the parabola: convexity itself falls as yields rise.

**Exercise 7.5 ★★.**

Why did the Bank of England buy only long [gilts](#def-m2-european-and-japanese-government-bonds-ldi) in 2022, and only for a fortnight? Contrast with [quantitative easing](https://one-course.com/books/quant/2/en/chapter/1-central-banks-and-the-short-rate#def-m2-central-banks-and-the-short-rate-qe).

**Solution of Exercise 7.5.**

The problem was a dysfunction at the long end, forced selling by LDI funds, not the stance of monetary policy; the Bank bought only what the funds were selling, long conventional and index-linked [gilts](#def-m2-european-and-japanese-government-bonds-ldi), for as long as it took the funds to raise capital, and sold the [gilts](#def-m2-european-and-japanese-government-bonds-ldi) back afterwards. [Quantitative easing](https://one-course.com/books/quant/2/en/chapter/1-central-banks-and-the-short-rate#def-m2-central-banks-and-the-short-rate-qe) buys across maturities, for years, to lower yields; the Bank was at the same time raising rates and preparing to shrink its balance sheet, and a temporary, targeted operation kept the two apart.

**Exercise 7.6 ★★.**

Under [yield-curve control](#def-m2-european-and-japanese-government-bonds-ycc), the market sells ten-year bonds at the target yield. Who decides how many bonds the central bank buys? What happens to its balance sheet if the market expects the target to be abandoned?

**Solution of Exercise 7.6.**

The market: the central bank buys whatever is offered at the target. If the market expects the target to be raised or abandoned, holding bonds bought at the target means a loss when the yield jumps, so holders sell to the central bank first; its purchases, and its balance sheet, grow fastest exactly when the policy is least credible, and it takes the loss when the target goes.

**Exercise 7.7 ★★★.**

*Coding.* With `sovereign_demo`, compute the sales the LDI fund must make to restore half-leverage after rises of 100 and 160 basis points, and the cushion (as a share of the holding) needed to survive 250.

**Solution of Exercise 7.7.**

After $+100$: [gilts](#def-m2-european-and-japanese-government-bonds-ldi) worth 808.4 million, sales $2 \times 500 - 808.4 =
\text{GBP}~191.6$ million. After $+160$: [gilts](#def-m2-european-and-japanese-government-bonds-ldi) 715.6 million, sales GBP 284.4 million. To survive 250 basis points without selling, the cushion must cover the loss at 250: 39.9% of the holding (the fund’s 50% does).

**Exercise 7.8 ★★★.**

*Find the flaw.* “The LDI funds lost money in 2022 because rising yields made the pension schemes poorer.” Correct the statement using the Bank’s own account of the schemes’ balance sheets.

**Solution of Exercise 7.8.**

Rising long yields reduce the present value of a defined-benefit scheme’s liabilities more than that of its equities and short bonds: the Bank’s own illustration notes that the scheme might be better off overall. The loss was concentrated in the leveraged LDI fund, whose cushion was eroded, and the crisis was one of liquidity: collateral had to be posted in hours, while the schemes’ other assets could not be turned into cash that fast.

## 7.8 Problem: The Collateral Spiral

**Problem 7.1.**

Weekend problem — how far an LDI fund can fall

On 22 September 2022 an LDI fund holds GBP 1 billion (market value) of a thirty-year [gilt](#def-m2-european-and-japanese-government-bonds-ldi) with a 1.5% coupon, maturing 22 July 2052, at a yield of 3.50%; half is financed in repo. The fund cannot raise new capital in less than a week. The [gilt](#def-m2-european-and-japanese-government-bonds-ldi) and the fund are illustrative.

**Part I — The fund.**

1. Give the [gilt](#def-m2-european-and-japanese-government-bonds-ldi) ’s dirty price, the fund’s repo and its cushion.
2. Give the [gilt](#def-m2-european-and-japanese-government-bonds-ldi) ’s [modified duration](https://one-course.com/books/quant/2/en/chapter/3-government-bonds#def-m2-government-bonds-duration) and the fund’s DV01 per basis point.
3. What is the fund’s leverage, and its DV01 per pound of capital?
4. How many basis points of rise cost 1% of the cushion, to first order?
5. Why would a pension scheme choose such a fund rather than hold [gilts](#def-m2-european-and-japanese-government-bonds-ldi) outright?

**Part II — The shock.**

6. Yields rise 100 basis points. Give the [gilts](#def-m2-european-and-japanese-government-bonds-ldi) ’ value, the cushion, and the leverage.
7. The same for 160 basis points, the size of the September move.
8. How much must the fund sell after 100 basis points to restore half-leverage without new capital? After 160?
9. Compare the aggregate sales that managers reported with the market’s daily volume. What happens to the price of what they sell?
10. Why did the collateral calls come on swaps as well as repo?

**Part III — The limit.**

11. Give the rise that exhausts the cushion.
12. What happens to the repo lender when it is exhausted?
13. Give the cushion the fund needs to survive 250 basis points without selling.
14. Earlier stress tests used 100 basis points. What cushion did that imply?
15. Why is a percentage cushion not enough to state resilience, without the duration of what is held?

**Part IV — Judgement.**

16. Why did a central bank with an inflation problem buy long bonds?
17. Why were the purchases temporary and small against the market?
18. Why was the pension scheme itself, as opposed to the fund, not insolvent?
19. State the *named result* : the yield rise that exhausts the cushion, and the cushion after the September move.
20. In one sentence: what turned a rise in yields into a crisis?

**Solution of Problem 7.1.**

**1.** Dirty price 63.41 per 100 (the [gilt](#def-m2-european-and-japanese-government-bonds-ldi) holds GBP 1.577 billion of face); repo GBP 500 million; cushion GBP 500 million. **2.** [Modified duration](https://one-course.com/books/quant/2/en/chapter/3-government-bonds#def-m2-government-bonds-duration) 21.84; the fund’s DV01 is $10^9 \times 21.84 \times
10^{-4} = \text{GBP}~2.18$ million per basis point. **3.** Leverage 2; 0.44% of capital per basis point. **4.** 1% of the cushion is GBP 5 million: about 2.3 basis points. **5.** To hedge more of its long liabilities than its bond holdings alone would, while keeping growth assets: leverage buys duration without selling equities. **6.** [Gilts](#def-m2-european-and-japanese-government-bonds-ldi) GBP 808.4 million, cushion 308.4 million (62% of the original), leverage 2.62. **7.** [Gilts](#def-m2-european-and-japanese-government-bonds-ldi) GBP 715.6 million, cushion 215.6 million (43%), leverage 3.32. **8.** GBP 191.6 million after 100; 284.4 million after 160. **9.** At least GBP 50 billion against about 12 billion of daily volume in those maturities, several days of the market’s whole turnover in hours; the price falls until other buyers appear, which raises yields and calls for more sales. **10.** LDI funds also held interest-rate and inflation swaps in which they received fixed; rising rates made those swaps lose value, and variation margin had to be posted in cash daily. **11.** 351 basis points. **12.** It sells the [gilts](#def-m2-european-and-japanese-government-bonds-ldi) it holds as collateral to recover its loan, adding to the supply that caused the loss. **13.** 39.9% of the holding, GBP 399 million. **14.** A 100-basis-point rise costs 19.2% of the holding: a cushion of about a fifth, less than half of what 250 requires. **15.** The same cushion protects a short-duration fund against much larger yield moves than a long one: resilience is a yield move, cushion divided by duration, not a percentage. **16.** Because the long end had stopped functioning: without a buyer the spiral would have forced sales by banks of their collateral and disrupted funding markets. The operation was sized and timed to the dysfunction, not to the inflation outlook. **17.** To buy time for the funds to raise capital and reduce leverage, without making purchases a monetary-policy tool at a time of tightening. **18.** Its liabilities fell with long yields, by more than its non-LDI assets: its funding position was, if anything, better; what it lacked was cash, fast. **19.** Named result: *the exhaustion yield* of the fund is a rise of 351 basis points; after the September move of 160 its cushion is 43% of the starting value, and restoring leverage would require selling GBP 284 million of [gilts](#def-m2-european-and-japanese-government-bonds-ldi) into a falling market. **20.** Leverage that had to be topped up with cash faster than cash could be raised, in a market that could not absorb the sales.

## 7.9 Interview questions

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

Why do Italian and German ten-year bonds, in the same currency, trade at different yields? What is in the spread?

**Solution of Interview question 7.1.**

Italy borrows in a currency it does not issue, so it can default in euros; Germany is the benchmark of the safest credit and the most liquid market. The spread pays for Italy’s credit risk, the liquidity difference, and [redenomination risk](#def-m2-european-and-japanese-government-bonds-spread), the chance of repayment in a new, weaker currency; in a crisis, also for the flight to [Bunds](#def-m2-european-and-japanese-government-bonds-bund), which lowers the benchmark.

*What the interviewer is looking for: credit without a printing press, liquidity, redenomination.*

**Interview question 7.2 ★ trader, bank.**

What happened in the [gilt](#def-m2-european-and-japanese-government-bonds-ldi) market in September 2022?

**Solution of Interview question 7.2.**

After the government’s fiscal announcement of 23 September, long [gilt](#def-m2-european-and-japanese-government-bonds-ldi) yields rose faster than any time since 2000, 160 basis points in a few days. Pension-fund LDI strategies, leveraged through repo and swaps, faced collateral calls they could meet fastest by selling [gilts](#def-m2-european-and-japanese-government-bonds-ldi), which pushed yields higher: a spiral. On 28 September the Bank of England bought long [gilts](#def-m2-european-and-japanese-government-bonds-ldi) temporarily, 19.3 billion pounds by 14 October, and regulators then required LDI funds to withstand a 250-basis-point move.

*What the interviewer is looking for: the trigger, the leverage mechanism, and the targeted backstop.*

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

How does [yield-curve control](#def-m2-european-and-japanese-government-bonds-ycc) work, and what are the risks of exiting it?

**Solution of Interview question 7.3.**

The central bank targets a long yield and buys whatever is offered there; it gives up control of its balance sheet. Exiting risks a jump in yields as the buyer of last resort withdraws, losses for holders (and for the central bank on its holdings), and spillovers to other markets whose investors had moved abroad for yield. Exits are therefore done in steps, loosening the band before ending the target.

*What the interviewer is looking for: price versus quantity control and the credibility problem.*

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

A spread widened by 100 basis points. How do you tell how much was the issuer and how much was the benchmark, and why does it matter for a hedge?

**Solution of Interview question 7.4.**

Decompose the change: issuer’s yield change minus benchmark’s. In a crisis the benchmark often falls while the issuer rises. A hedge that shorts the benchmark against a long in the issuer loses on the benchmark’s rally; a trader positioned for the spread must size both legs by DV01 and know which leg is doing the moving.

*What the interviewer is looking for: the decomposition and its hedging consequence.*

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

What is the difference between the ECB’s OMT and TPI?

**Solution of Interview question 7.5.**

OMT (2012): purchases without ex ante limit, conditional on an adjustment or precautionary programme with the European Stability Mechanism. TPI (2022): purchases against unwarranted, disorderly market dynamics that threaten the transmission of policy, conditional on four criteria of fiscal and macroeconomic soundness, without a programme. The first is a lender of last resort with conditions; the second a transmission tool with eligibility.

*What the interviewer is looking for: the conditionality of each.*

**Interview question 7.6 ★★★ researcher, bank.**

You are asked to stress-test a leveraged bond fund. How do you choose the size of the shock, and what else must the test capture?

**Solution of Interview question 7.6.**

Choose the shock from history at the relevant speed and horizon (how far yields can move in the time it takes to raise capital), from the extremes of comparable markets, and from reverse stress: the move that exhausts capital. The test must capture full repricing rather than duration, liquidity (how much can be sold, at what impact, in that time), margin and collateral calls on derivatives, the counterparty’s reaction (haircut increases), and the feedback when many funds act alike.

*What the interviewer is looking for: reverse stress, liquidity and feedback, not only a larger number.*