---
title: "Interest-Rate Swaps"
book: "Markets II: Rates, FX and Credit"
subject: quant
language: en
chapter: 9
exercises: 8
source: https://one-course.com/books/quant/2/en/chapter/9-interest-rate-swaps
---

# Chapter 9 — Interest-Rate Swaps

A company has borrowed 200 million dollars for ten years at the overnight rate plus 1.5%, and its board wants to know what the interest will cost next year and every year after that. Its treasurer calls a bank, and the bank quotes a fixed rate of 4.05% for ten years: the company will pay that, the bank will pay the company the overnight rate, and the loan’s floating interest is turned into a fixed 5.55%. The bank’s trader, who has just taken on the other side of ten years of interest-rate risk, hedges it within the minute, in the interbank market or on an electronic platform, with an identical swap with another dealer or with Treasury futures. Interest-rate derivatives make up about four fifths of the notional of all over-the-counter derivatives, and the swap curve is a reference against which bank loans, corporate bonds and structured products are priced and hedged. This chapter prices a swap, builds the curve it is priced on, and measures its risk.

## 9.1 The swap and its legs

**Definition 9.1 (Interest-rate swap, fixed and floating legs).**

An *interest-rate swap* is an agreement to exchange, on a notional amount that is never itself exchanged, fixed-rate interest payments, the *fixed leg*, for floating-rate payments set by a benchmark rate, the *floating leg*, on a schedule of dates until maturity.

**Definition 9.2 (Payer and receiver swaps).**

In a *payer swap* one pays the fixed rate and receives floating; in a *receiver swap* one receives fixed and pays floating. A payer gains when rates rise: it is short duration, like a borrower who has fixed its rate, or like a short position in a bond.

![The company’s hedge. It keeps paying its lenders SOFR plus 1.50%; the swap pays it SOFR and charges 4.05%. The floating payments cancel and the company pays 5.55% fixed. Each year’s payments are netted and settled two business days after the period ends.](https://one-course.com/images/onecourse/chapters/quant-2/m2-interest-rate-swaps/fig-edfbfb5f3595.svg)

***Figure 9.1.** The company’s hedge. It keeps paying its lenders SOFR plus 1.50%; the swap pays it SOFR and charges 4.05%. The floating payments cancel and the company pays 5.55% fixed. Each year’s payments are netted and settled two business days after the period ends.*

**Example 9.3 (One year of the company’s swap).**

The swap starts on 29 September 2026. On 29 September 2027 the company pays $200\,000\,000 \times 4.05\% \times 365/360 = \text{USD}~8\,212\,500$ and receives the overnight rate compounded over those 365 days, whatever it turns out to be; on its loan it pays the same compounded rate plus 1.5%. The floating payments cancel, and its cost is fixed.

## 9.2 Overnight index swaps and conventions

**Definition 9.4 (Overnight index swap).**

An *overnight index swap* (OIS) is a swap whose [floating leg](#def-m2-interest-rate-swaps-swap) pays an [overnight benchmark rate](https://one-course.com/books/quant/2/en/chapter/1-central-banks-and-the-short-rate#def-m2-central-banks-and-the-short-rate-rfr) [compounded in arrears](https://one-course.com/books/quant/2/en/chapter/1-central-banks-and-the-short-rate#def-m2-central-banks-and-the-short-rate-arrears) over each period ([Definition 1.14](https://one-course.com/books/quant/2/en/chapter/1-central-banks-and-the-short-rate#def-m2-central-banks-and-the-short-rate-arrears)). Since the end of LIBOR it is the standard swap in the currencies whose interbank rate ceased; in euros it trades alongside swaps on Euribor, which is still published ([Box 8.1](https://one-course.com/books/quant/2/en/chapter/8-short-term-interest-rate-futures#dat-m2-short-term-interest-rate-futures-contracts)).

**As of September 2026 — Conventions of a dollar SOFR swap.**

Both legs pay annually, actual/360; the [floating leg](#def-m2-interest-rate-swaps-swap) pays SOFR [compounded in arrears](https://one-course.com/books/quant/2/en/chapter/1-central-banks-and-the-short-rate#def-m2-central-banks-and-the-short-rate-arrears), with payment two government-securities business days after the end of each period so that the last fixing is known; the swap starts accruing two business days after the trade date. Conventions differ by currency, and by the swap’s maturity (short swaps may pay once at maturity).

## 9.3 Pricing, par rates and risk

**Definition 9.5 (Annuity and par swap rate).**

For fixed-leg payment dates $t_1, \dots, t_n$ with accrual fractions $\delta_i$ and discount factors $P(t_i)$, the *annuity* is $A =
\sum_i \delta_i P(t_i)$, the value of receiving 1 a year on the [fixed leg](#def-m2-interest-rate-swaps-swap)’s schedule. The *par swap rate* is the fixed rate at which the swap is worth zero at inception.

**Proposition 9.6 (Value of an OIS in a single curve).**

If overnight rates are discounted on the same curve that they project, the [floating leg](#def-m2-interest-rate-swaps-swap) of an OIS from $t_0$ to $t_n$ is worth $P(t_0) - P(t_n)$ per unit of notional, whatever its schedule. Hence the payer’s value is

$$
V \;=\; N\bigl[P(t_0) - P(t_n) - K\,A\bigr],
\qquad
K_{\text{par}} \;=\; \frac{P(t_0) - P(t_n)}{A},
$$

and $\partial V/\partial K = -N A$: the swap’s sensitivity to its own rate is its [annuity](#def-m2-interest-rate-swaps-par).

**Proof.** A period’s compounded overnight payment at $t_i$ is replicated by investing $P(t_{i-1})$ at $t_0$ in a deposit to $t_{i-1}$ and rolling 1 overnight from there, which yields $1/P(t_{i-1}, t_i)$ at $t_i$, of which 1 is returned and the rest is the payment; its value is $P(t_{i-1}) - P(t_i)$, and the sum telescopes. ∎

**Method 9.7 (Bootstrapping an OIS curve).**

Take [par swap rates](#def-m2-interest-rate-swaps-par) for increasing maturities. Solve for the discount factor at the first maturity so that its swap is worth zero; then, holding it, for the second, interpolating the discount factors of any intermediate payment dates between the pillars (here log-linearly in time); and so on. Check that every input swap reprices at par.

**Example 9.8 (A ten-year swap).**

With illustrative par rates of 3.90, 3.85, 3.83, 3.85, 3.92 and 4.05% at one, two, three, five, seven and ten years, the ten-year par rate is 4.05% by construction, the [annuity](#def-m2-interest-rate-swaps-par) is 8.228 and the swap’s DV01 on USD 100 million is USD 82 278 by $N A \times 10^{-4}$, and USD 82 262 when the curve is rebuilt after bumping each input by a basis point. All of it falls on the ten-year pillar: a par swap is hedged exactly by the swap that built its pillar.

![The curve bootstrapped from six illustrative par rates. Zero rates are smooth; forward rates are flat between pillars and jump at them, an artefact of interpolating the logarithm of discount factors linearly: every choice of interpolation is a choice of forward curve, and One Quant Book 6 treats better ones. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-2/m2-interest-rate-swaps/fig-07e2b61b8f64.svg)

***Figure 9.2.** The curve bootstrapped from six illustrative par rates. Zero rates are smooth; forward rates are flat between pillars and jump at them, an artefact of interpolating the logarithm of discount factors linearly: every choice of interpolation is a choice of forward curve, and One Quant Book 6 treats better ones. Data: the chapter’s tutorial.*

![Bucketed DV01 of two payer swaps of USD 100 million: the change in value when each input par rate rises by a basis point and the curve is rebuilt. The par ten-year loads only its own pillar. The five-year swap starting in five years is long ten-year rates and short five-year rates: it is a bet on the forward, and its hedge is a pair of par swaps. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-2/m2-interest-rate-swaps/fig-51c546016326.svg)

***Figure 9.3.** Bucketed DV01 of two [payer swaps](#def-m2-interest-rate-swaps-payer) of USD 100 million: the change in value when each input par rate rises by a basis point and the curve is rebuilt. The par ten-year loads only its own pillar. The five-year swap starting in five years is long ten-year rates and short five-year rates: it is a bet on the forward, and its hedge is a pair of par swaps. Data: the chapter’s tutorial.*

## 9.4 Swap spreads

**Definition 9.9 (Swap spread).**

The *swap spread* at a maturity is the [par swap rate](#def-m2-interest-rate-swaps-par) minus the yield of the government bond of the same maturity.

For decades swap rates were above government yields, since the [floating leg](#def-m2-interest-rate-swaps-swap) was an interbank rate with bank credit risk in it. In September 2008, shortly after Lehman Brothers’ default, the thirty-year dollar [swap spread](#def-m2-interest-rate-swaps-spread) turned negative, and it has stayed negative since. An overnight-rate swap carries almost no credit risk in its [floating leg](#def-m2-interest-rate-swaps-swap), so a negative spread says something else: owning a Treasury ties up a balance sheet that must be financed in repo, while receiving fixed on a swap needs only margin, and long-dated demand for duration, from pension funds hedging their liabilities, meets dealers whose balance sheets are costly. The [swap spread](#def-m2-interest-rate-swaps-spread) is the price of that difference, and trading it, long Treasuries against paying fixed or the reverse, is one of the core relative-value trades of the rates market (One Quant Book 9).

## 9.5 Execution facilities and compression

**Definition 9.10 (Swap execution facility and trade compression).**

A *swap execution facility* (SEF) is a regulated trading platform for swaps in the United States; swaps that have been declared available to trade must be executed on one. *Trade compression* is the tearing up of offsetting swaps between several counterparties, replacing them by fewer swaps with the same net risk, so as to reduce gross notional, the number of trades and the capital and margin they consume.

The swap market was, until the rules that followed the 2008 crisis, largely a bilateral market between dealers and their clients, traded by telephone. It now has three tiers: central clearing for most standard swaps ([Chapter 10](https://one-course.com/books/quant/2/en/chapter/10-swap-clearing-and-the-clearing-house-basis#ch-m2-swap-clearing-and-the-ccp-basis)), mandatory execution of the most standard ones on electronic platforms, and reporting of every trade. Compression runs on top: because swaps are never closed out by selling them, only offset by new swaps, gross notional grows with every hedge unless it is periodically torn up.

![A compression cycle. Three dealers hold three identical swaps in a circle: each pays fixed on one and receives on another, so none has risk, yet all three hold gross notional, margin and capital against it. Compression tears the cycle up. Real cycles involve many participants and swaps that match only approximately; the service finds the largest set of tear-ups that leaves each participant’s risk within its tolerances.](https://one-course.com/images/onecourse/chapters/quant-2/m2-interest-rate-swaps/fig-f2da2af0164b.svg)

***Figure 9.4.** A compression cycle. Three dealers hold three identical swaps in a circle: each pays fixed on one and receives on another, so none has risk, yet all three hold gross notional, margin and capital against it. Compression tears the cycle up. Real cycles involve many participants and swaps that match only approximately; the service finds the largest set of tear-ups that leaves each participant’s risk within its tolerances.*

**As of September 2026 — The size of the market.**

Notional of over-the-counter derivatives outstanding: USD 846 trillion at end-June 2025 (the latest semiannual BIS survey), 16% more than a year earlier; interest-rate derivatives were 79% of it, about USD 668 trillion, with euro contracts larger than dollar contracts since 2022. In the United States, SEF registration was required from October 2013 and execution on a SEF of swaps declared available to trade from 15 February 2014.

## 9.6 Tutorial: a curve and a swap

**Goal.** Bootstrap an OIS curve from six par rates, reprice the inputs, and compute the bucketed DV01 of a par swap and a forward-starting swap. **End state:** Figures [9.2](#fig-m2-interest-rate-swaps-curve) and [9.3](#fig-m2-interest-rate-swaps-buckets) and the numbers of [Example 9.8](#ex-m2-interest-rate-swaps-ten).

1. **[Annuity](#def-m2-interest-rate-swaps-par), par rate, value** ([Proposition 9.6](#prop-m2-interest-rate-swaps-value)). `def annuity (curve: Curve, dates: list [dt.date]) -> float : """Sum of accrual fraction times discount factor over the fixed leg (actual/360).""" return sum ((b - a).days / 360.0 * curve.df(b) for a, b in zip (dates, dates[1 :], strict=False )) def par_rate (curve: Curve, dates: list [dt.date]) -> float : return (curve.df(dates[0 ]) - curve.df(dates[-1 ])) / annuity(curve, dates) def swap_pv (curve: Curve, dates: list [dt.date], fixed: float , notional: float , payer: bool = True ) -> float : """Value to the payer of fixed (receiver of floating), or the reverse.""" floating = curve.df(dates[0 ]) - curve.df(dates[-1 ]) v = notional * (floating - fixed * annuity(curve, dates)) return v if payer else -v` **Listing 9.1.** Annuity, par rate and value of an OIS in a single curve. code/firm/curve/firm_curve.py
2. **Bootstrap**: one pillar at a time, by bisection on the logarithm of the discount factor. `def bootstrap (spot: dt.date, tenors: list [int ], rates: list [float ]) -> Curve: """Pillars at the swaps' maturities; each pillar's discount factor solved so that its swap reprices at par, earlier pillars fixed (bisection on the log discount factor).""" curve = Curve(spot, [], []) for n, r in zip (tenors, rates, strict=True ): end = add_years(spot, n) curve.times.append(curve.t(end)) curve.dfs.append(1.0 ) lo, hi = -5.0 , 1.0 # bounds on log(df): long ends, negative rates dates = schedule(spot, n) for _ in range (200 ): mid = 0.5 * (lo + hi) curve.dfs[-1 ] = math.exp(mid) if par_rate(curve, dates) > r: # df too low -> par rate too high -> raise df lo = mid else : hi = mid curve.dfs[-1 ] = math.exp(0.5 * (lo + hi)) return curve` **Listing 9.2.** Bootstrapping discount factors so that each input swap is at par. code/firm/curve/firm_curve.py Every input swap should reprice to zero within a thousandth of a dollar.
3. **Bucketed DV01**: bump each input, rebuild, reprice. `def bucket_dv01 (spot: dt.date, tenors: list [int ], rates: list [float ], dates: list [dt.date], fixed: float , notional: float , payer: bool = True , bump: float = 1e-4 ) -> list [float ]: """Change in value for a one-basis-point rise of each par input in turn (rebuild the curve).""" base = swap_pv(bootstrap(spot, tenors, rates), dates, fixed, notional, payer) out = [] for k in range (len (rates)): bumped = [r + (bump if i == k else 0.0 ) for i, r in enumerate (rates)] out.append(swap_pv(bootstrap(spot, tenors, bumped), dates, fixed, notional, payer) - base) return out` **Listing 9.3.** Bucketed DV01 by bump and rebuild. code/firm/curve/firm_curve.py
4. **Run** `swaps_demo.ten_year()` and `swaps_demo.forward_5y5y()` ; `fig_swaps.py` writes the charts.

**What to change next.** Replace log-linear interpolation of discount factors by linear interpolation of zero rates and watch the forward curve change while every input still reprices; then price a swap whose fixed rate is 1% above par and read its buckets.

## 9.7 Build: the OIS curve and swap pricer

**Purpose.** Every rates position of the miniature firm, and every discounting of a future cash flow, reads this curve. It is the first curve of the pricing library (One Quant Book 5 and 6 extend it to several curves, to turns and to meeting dates, [Section 8.6](https://one-course.com/books/quant/2/en/chapter/8-short-term-interest-rate-futures#bld-m2-short-term-interest-rate-futures-meetings)).

**Interface.** `Curve(spot, times, dfs)` with `df(date)`, `zero(date)`; `schedule(start, years)`; `annuity`; `par_rate`; `swap_pv(curve, dates, fixed, notional, payer)`; `bootstrap(spot, tenors, rates)`; `bucket_dv01(…)`.

**Rules.** Annual periods, actual/360 accruals, no business-day adjustment in this version; log-linear interpolation of discount factors, flat forward beyond the last pillar; one curve for projection and discounting.

**Acceptance tests.** `code/firm/curve/tests/`: every input reprices at par; a flat input curve gives flat par rates at intermediate maturities; a payer gains when rates rise; a par swap loads only its own pillar; a 5y5y forward is short the five-year pillar and long the ten-year; leap days.

**Stretch.** Business-day calendars and payment delays (the conventions of [Box 9.1](#dat-m2-interest-rate-swaps-conventions)); futures and meeting dates at the front; a separate discount curve ([Chapter 10](https://one-course.com/books/quant/2/en/chapter/10-swap-clearing-and-the-clearing-house-basis#ch-m2-swap-clearing-and-the-ccp-basis)).

Sources and further reading

- Bank for International Settlements, *OTC derivatives statistics at end-June 2025* , December 2025.
- S. Klingler and S. Sundaresan, “An explanation of negative swap spreads: demand for duration from underfunded pension plans”, *Journal of Finance* 74, 2019.
- Clarus Financial Technology, “SOFR swap nuances”; ARRC, *An Updated User’s Guide to SOFR* , 2021.
- CFTC, final rule on core principles and other requirements for swap execution facilities, 2013.

## 9.8 Exercises

**Exercise 9.1 ★.**

A [payer swap](#def-m2-interest-rate-swaps-payer) of USD 100 million at 4.05% has an annual period of 365 days. What fixed payment is due at its end? What does the payer receive?

**Solution of Exercise 9.1.**

$100\,000\,000 \times 0.0405 \times 365/360 = \text{USD}~4\,106\,250$. It receives SOFR compounded over the same 365 days, actual/360, on USD 100 million, paid two business days after the period ends.

**Exercise 9.2 ★.**

A pension fund receives fixed on a thirty-year swap. Is it long or short duration? Does it gain or lose if rates rise?

**Solution of Exercise 9.2.**

Receiving fixed is long duration, like owning a thirty-year bond financed at the overnight rate: it loses when rates rise, as its liabilities (which it is hedging) fall in value too.

**Exercise 9.3 ★.**

The ten-year swap rate is 4.05% and the ten-year Treasury yields 4.20%. Give the [swap spread](#def-m2-interest-rate-swaps-spread) and say what it would have meant before 2008.

**Solution of Exercise 9.3.**

$4.05 - 4.20 = -15$ basis points. Before 2008, when [floating legs](#def-m2-interest-rate-swaps-swap) paid an interbank rate with bank credit risk, a negative spread would have meant that the market priced banks’ term borrowing below the government’s, which did not happen; [swap spreads](#def-m2-interest-rate-swaps-spread) were positive.

**Exercise 9.4 ★★.**

The ten-year swap’s DV01 is USD 82 278 from the [annuity](#def-m2-interest-rate-swaps-par) and USD 82 262 from bumping and rebuilding the curve. Why do they differ, and which would you hedge with?

**Solution of Exercise 9.4.**

$N A \times 10^{-4}$ is the sensitivity to a change in the swap’s own fixed rate holding the curve’s discount factors; bumping the par input and rebuilding also changes the discount factors that value the [fixed leg](#def-m2-interest-rate-swaps-swap), so the two differ by a second-order term (USD 16 on 82 000). Hedge with the rebuilt, bucketed number: it is what the hedging instruments’ prices actually do.

**Exercise 9.5 ★★.**

The 5y5y forward swap of [Figure 9.3](#fig-m2-interest-rate-swaps-buckets) has a par rate of 4.295%. Explain its bucketed DV01 and give the par swaps that hedge it.

**Solution of Exercise 9.5.**

A payer 5y5y gains USD 82 651 per basis point on the ten-year input and loses USD 45 394 on the five-year: it is long the ten-year rate and short the five-year, since it is the ten-year swap minus the five-year swap. Hedge: receive fixed on a ten-year par swap with the same ten-year DV01 (USD 100.5 million, since a ten-year par swap of USD 100 million has 82 262 on that pillar) and pay fixed on a five-year par swap with 45 394 (USD 100.3 million, at 45 278 per 100 million), leaving only the small residuals at the other pillars.

**Exercise 9.6 ★★.**

Dealer A pays fixed to B, B pays fixed to C, C pays fixed to A, each USD 100 million of the same ten-year swap. What is the gross notional, the net risk of each dealer, and the result of compression?

**Solution of Exercise 9.6.**

Gross notional USD 300 million; each dealer pays fixed on one swap and receives on another of identical terms, so its net risk is zero. Compression tears up all three swaps: gross notional falls to zero, and so do the margin and capital the three trades consumed, with no change in anyone’s risk.

**Exercise 9.7 ★★★.**

*Coding.* With `firm_curve`, bootstrap the curve of [Example 9.8](#ex-m2-interest-rate-swaps-ten) and report the four-year and eight-year par rates and the ten-year discount factor.

**Solution of Exercise 9.7.**

Four-year par rate 3.8425%, eight-year 3.9743%, ten-year discount factor 0.666773. The intermediate par rates depend on the interpolation; the inputs do not.

**Exercise 9.8 ★★★.**

*Find the flaw.* “The thirty-year [swap spread](#def-m2-interest-rate-swaps-spread) is negative, so the market thinks banks are safer than the US government.” Correct it.

**Solution of Exercise 9.8.**

An overnight-rate swap has almost no credit risk in its [floating leg](#def-m2-interest-rate-swaps-swap), and its counterparty risk is removed by clearing: its rate is not a bank’s borrowing cost. A negative spread says that receiving fixed on a swap is cheaper to hold than owning a Treasury, which must be financed on a costly balance sheet, and that long-dated demand for receiving (pension funds) exceeds what dealers will absorb. It is a price of balance sheet and of demand, not of credit.

## 9.9 Problem: The Corporate Hedge

**Problem 9.1.**

Weekend problem — fixing a ten-year loan

A company has borrowed USD 200 million for ten years at compounded SOFR plus 1.50%. On 25 September 2026 it enters a ten-year [payer swap](#def-m2-interest-rate-swaps-payer) starting on 29 September at the par rate of the curve of [Example 9.8](#ex-m2-interest-rate-swaps-ten).

**Part I — The swap.**

1. Give the fixed rate and the company’s all-in fixed cost.
2. Give the swap’s DV01 for the company, with its sign.
3. What is the swap worth to the company on the trade date?
4. Which pillar carries its risk?
5. Why does the swap start on 29 September and not on the trade date?

**Part II — Scenarios.**

6. All par rates rise by 50 basis points. What is the swap worth?
7. All fall by 50 basis points.
8. Why are the two not equal and opposite?
9. The curve steepens: $-20$ basis points at one year rising linearly to $+15$ at ten. Give the shocks at each pillar and the swap’s value.
10. Why is the steepener worth almost exactly 15 times the DV01?

**Part III — The accounts.**

11. After the 50-basis-point fall the swap shows a loss. Has the company lost money?
12. What does hedge accounting try to achieve, and why do treasurers care?
13. The bank quoted 4.07% instead of the 4.05% mid. What did the company pay for the hedge, in dollars today?
14. What credit risk does each side take, and how is it managed?
15. What would a five-year swap have left unhedged?

**Part IV — Judgement.**

16. Why might a company prefer a swap to issuing a fixed-rate bond?
17. How does the bank hedge the swap it has just written?
18. Why is the company’s hedge a [payer swap](#def-m2-interest-rate-swaps-payer) and a pension fund’s a [receiver swap](#def-m2-interest-rate-swaps-payer) ?
19. State the *named result* : the fixed rate, the DV01 of the hedge, and its value after the steepening.
20. In one sentence: what does a swap exchange?

**Solution of Problem 9.1.**

**1.** 4.05%; all-in $4.05 + 1.50 = 5.55\%$. **2.** $+\text{USD}~164\,474$ per basis point: the payer gains when rates rise. **3.** Zero at the mid-market par rate. **4.** The ten-year pillar only. **5.** Dollar swaps start two business days after the trade: 25 September 2026 was a Friday. **6.** $+\text{USD}~8.02$ million. **7.** $-\text{USD}~8.44$ million. **8.** A [payer swap](#def-m2-interest-rate-swaps-payer) is short a fixed-rate bond and long a floating one; it is short convexity, so it gains less on a rise than it loses on an equal fall. **9.** Shocks $-20.0$, $-16.1$, $-12.2$, $-4.4$, $+3.3$ and $+15.0$ basis points at one, two, three, five, seven and ten years; value $+\text{USD}~2.46$ million. **10.** Its risk sits entirely on the ten-year pillar, which moved 15 basis points: $15 \times 164\,474 = 2.47$ million, less a small convexity term. **11.** No: the swap’s loss is offset by the lower value of the fixed-rate debt it has synthetically created; its interest cost is still 5.55%. The loss is the value of having fixed at 4.05% when the market would now fix lower. **12.** To book the swap’s changes in value against the hedged item’s, so that reported earnings do not swing with rates when the economic position is hedged; treasurers care because volatile earnings are costly to explain. **13.** Two basis points for ten years on its DV01: about $2 \times 164\,474 =
\text{USD}~328\,947$ of present value. **14.** Each is exposed to the other’s default when the swap has value to it; managed by collateral under a credit support annex, by central clearing for dealers, and for corporate clients by credit lines and charges in the price. **15.** The interest of years six to ten, which would float again. **16.** It keeps the bank loan’s flexibility and pricing, can be adjusted or unwound separately, and fixes the rate without a bond issue’s documentation and investor base. **17.** By receiving fixed in the interdealer market or on a SEF, or with Treasury futures and bonds, in the same DV01 and pillar, then managing the residual curve risk in its book. **18.** The company has floating debt and wants to fix what it pays: it pays fixed. A pension fund has long fixed liabilities and wants assets that gain when rates fall: it receives fixed. **19.** Named result: *the corporate hedge* fixes at 4.05% (5.55% all in), with a DV01 of USD 164 474 per basis point, and is worth USD 2.46 million after the steepening. **20.** A fixed rate for a floating one, on a notional that never changes hands.

## 9.10 Interview questions

**Interview question 9.1 ★ trader, bank.**

What is an [interest-rate swap](#def-m2-interest-rate-swaps-swap), and why would a company enter one?

**Solution of Interview question 9.1.**

An exchange of fixed-rate interest for floating-rate interest on a notional amount, over a schedule. A company with floating-rate debt pays fixed to lock its cost; one with fixed debt may receive fixed to benefit from falling rates; an investor uses it to add or remove duration without buying or selling bonds.

*What the interviewer is looking for: the exchange of cash flows and a concrete use.*

**Interview question 9.2 ★ researcher, bank.**

How do you price the [floating leg](#def-m2-interest-rate-swaps-swap) of an [overnight index swap](#def-m2-interest-rate-swaps-ois)?

**Solution of Interview question 9.2.**

Compounded overnight interest over a period is replicated by rolling a deposit, so in a single-curve world each period’s payment is worth $P(t_{i-1}) -
P(t_i)$, and the whole leg $P(t_0) - P(t_n)$. With separate discount and projection curves, project the compounded rate from the forward curve and discount on the collateral curve.

*What the interviewer is looking for: the telescoping argument, and awareness of multi-curve pricing.*

**Interview question 9.3 ★★ researcher, developer.**

Walk me through bootstrapping a discount curve from swap rates. What choices do you make, and how do you test the result?

**Solution of Interview question 9.3.**

Choose instruments (deposits or futures at the front, swaps beyond), pillars at their maturities, and an interpolation (of log discount factors, zero rates or forwards); solve for each pillar in turn so that its instrument reprices, with a root finder, or all at once. Test: every input reprices to within a tolerance; forwards have no spurious oscillation; bumping one input changes the curve locally; the curve is stable to the order of inputs; published curves are reproduced.

*What the interviewer is looking for: the choices (instruments, interpolation) and a test list.*

**Interview question 9.4 ★★ trader, researcher.**

Why have thirty-year US [swap spreads](#def-m2-interest-rate-swaps-spread) been negative since 2008?

**Solution of Interview question 9.4.**

Owning a Treasury ties up balance sheet and must be financed in repo; receiving fixed on a swap needs only margin. Long-dated duration demand, notably from pension funds, is met more cheaply in swaps, and dealers’ balance sheets are too costly to arbitrage the gap away. The [floating leg](#def-m2-interest-rate-swaps-swap), once LIBOR and now SOFR or an overnight rate, carries little credit risk, so nothing pushes swap rates above government yields.

*What the interviewer is looking for: balance sheet and demand, not credit.*

**Interview question 9.5 ★★ trader.**

You are receiving fixed on a 5y5y forward swap. What is your risk, and how do you hedge it with par swaps?

**Solution of Interview question 9.5.**

Receiving a 5y5y is long the ten-year rate’s decline and short the five-year’s: it is the ten-year receiver minus the five-year receiver. Hedge by paying fixed on a ten-year par swap and receiving on a five-year par swap in the corresponding DV01s; the residual is exposure to the shape between the pillars, which depends on interpolation.

*What the interviewer is looking for: the decomposition into two par swaps.*

**Interview question 9.6 ★★★ developer, researcher.**

Your bucketed DV01s do not sum to the parallel DV01. List the possible causes.

**Solution of Interview question 9.6.**

Non-linearity: the parallel bump is not the sum of the local bumps for a convex position (second-order terms); interpolation that couples pillars; bumping par rates in one run and zero rates in another; different curve rebuild settings or tolerances between runs; instruments outside the pillar set (extrapolation); a discount curve bumped in one computation and not in the other; numerical noise from a loose root-finder tolerance.

*What the interviewer is looking for: second-order effects versus implementation inconsistencies.*