---
title: "Multi-Curve and Collateral Discounting"
book: "Rates, Credit, XVA and Risk"
subject: quant
language: en
chapter: 2
exercises: 8
source: https://one-course.com/books/quant/6/en/chapter/2-multi-curve-and-collateral-discounting
---

# Chapter 2 — Multi-Curve and Collateral Discounting

Until the summer of 2007 a three-month interbank loan and a three-month loan against collateral cost almost the same: the spread between them was about ten basis points, and in euros under six. From August 2007 to May 2009 it stayed above fifty basis points, and after Lehman Brothers failed it was above a hundred for six months. Every swap pricer on the Street rested on one assumption that this broke: that a single curve both projects the floating rates a swap will pay and discounts every cash flow, so that a floating leg on the three-month rate is worth par. Once three-month and six-month interbank rates, and the overnight rate, each carried their own premium for bank credit and liquidity, there was no single curve left. Dealers moved in a few years to a framework with one curve per index for projection and one curve per collateral agreement for discounting, and in 2020 the clearing houses moved their discounting for the whole market in two weekends. This chapter builds that framework on chapter 1’s curve builder.

## 2.1 The 2007 break and the tenor basis

Before 2007 the textbook argument of One Quant Book 2, chapter 9, held: a floating leg paying the rate for each period, on a curve that both projects and discounts, is worth $P(t_0)-P(t_n)$. If a bank can borrow for six months at six-month Euribor and roll it, a six-month floating leg is worth par and three- and six-month legs are worth the same. After 2007 a lender for six months demanded more than two three-month loans in a row, because of the chance that the borrower fails in the second three months and the value of keeping one’s cash available. The difference is paid in a swap of two floating legs.

**Definition 2.1 (Tenor basis, tenor basis swap).**

A *tenor basis swap* exchanges two floating legs on indices of different tenors in the same currency, for example six-month Euribor against the overnight rate compounded, with a spread on one leg that makes it worth zero at inception. The *tenor basis* at a maturity is that par spread.

**Proposition 2.2 (Tenor basis from two swap curves).**

If an overnight-index swap and an interbank-index swap of the same maturity have identical fixed legs and are discounted on the same curve, the [tenor basis](#def-rc-multi-curve-and-collateral-discounting-basis) spread (paid on the overnight leg, on the fixed leg’s schedule) equals the difference of their par rates, $b_n = S^{\text{IRS}}_n - S^{\text{OIS}}_n$.

**Proof.** Each par swap equates its floating leg with $S_n A$, with the same annuity $A$. Receiving the interbank leg and paying the overnight leg plus $b$ is worth $S^{\text{IRS}}_nA - S^{\text{OIS}}_nA - bA$, zero for $b=S^{\text{IRS}}_n-S^{\text{OIS}}_n$. ∎

## 2.2 Discount curves and projection curves

**Definition 2.3 (Discount curve, projection curve, multi-curve framework).**

A *discount curve* gives the present value of a cash flow paid under a given collateral agreement; a *projection curve* of an index gives its forward fixings, as $F(t;T_1,T_2) = \bigl(P^{\text{proj}}(t,T_1)/P^{\text{proj}}(t,T_2)-1\bigr)/\delta$, without being used to discount anything. The *multi-curve framework* values a swap by projecting each floating fixing on its index’s projection curve and discounting every cash flow on the discount curve of the trade’s collateral.

The [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves)’s “discount factors” are only a device for storing forward fixings; in the language of One Quant Book 4, chapter 5, the forward of each fixing is its expectation under the forward measure of the *discount* curve, which is why the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) is calibrated with the [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) held fixed.

**Method 2.4 (Two-stage calibration).**

(1) Calibrate the [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) from overnight-index instruments ([Method 1.4](https://one-course.com/books/quant/6/en/chapter/1-curve-construction#met-rc-curve-construction-newton)). (2) Holding it fixed, calibrate the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) of each index from that index’s fixing, futures and swaps, each priced as $\sum_j F_j\delta_jP^{\text{disc}}(t_j)$ against $K\sum_i
\delta_iP^{\text{disc}}(t_i)$. (3) Check that every instrument reprices and that the forward basis (projected fixing minus the overnight forward for the same period) is positive and smooth.

```python
def calibrate_projection(spot: dt.date, disc, instruments: Sequence, kind: str = "monotone_convex",
                         tol: float = 1e-12, max_iter: int = 30) -> ZeroCurve:
    """Solve the projection curve's pillar zero rates with the discount curve held fixed."""
    times = [(i.maturity - spot).days / 365.0 for i in instruments]
    labels = [i.label or f"P{k}" for k, i in enumerate(instruments)]
    q = np.array([i.quote() for i in instruments])
    z = q.copy()

    def resid(zv):
        c = ZeroCurve(spot, times, list(zv), kind, labels)
        return np.array([i.model(c, disc) for i in instruments]) - q
    r = resid(z)
    for _ in range(max_iter):
        if np.max(np.abs(r)) < tol:
            break
        jac = np.empty((len(z), len(z)))
        for j in range(len(z)):
            e = z.copy()
            e[j] += 1e-7
            jac[:, j] = (resid(e) - r) / 1e-7
        z = z - np.linalg.solve(jac, r)
        r = resid(z)
    return ZeroCurve(spot, times, list(z), kind, labels)
```

***Listing 2.1.** Calibration of a projection curve on a fixed discount curve. code/firm/multicurve/firm_multicurve.py*

**Example 2.5 (A euro market).**

Take illustrative €STR swap rates from 1.95% at one year to 2.55% at ten and 2.60% at thirty, and six-month Euribor swap rates 20 basis points higher at one year, 15 at ten and 12 at thirty, with a six-month fixing of 2.13%. The [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) reprices all ten Euribor instruments; by [Proposition 2.2](#prop-rc-multi-curve-and-collateral-discounting-basisrates) the [tenor basis](#def-rc-multi-curve-and-collateral-discounting-basis) is 20, 15 and 12 basis points at one, ten and thirty years. The forward basis, fixing by fixing ([Figure 2.1](#fig-rc-multi-curve-and-collateral-discounting-forwards)), is 21.6 basis points on the first six-month period and 11.8 on the period starting in ten years: a par basis is an average of forward bases.

![Forward rates for six-month periods from the two euro curves of . The gap is the forward tenor basis: about 22 basis points on the first period, 12 after ten years, 9 at the long end. Data: the chapter’s illustrative curves and tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-multi-curve-and-collateral-discounting/fig-76e89f1a3290.svg)

***Figure 2.1.** Forward rates for six-month periods from the two euro curves of [Example 2.5](#ex-rc-multi-curve-and-collateral-discounting-euro). The gap is the forward [tenor basis](#def-rc-multi-curve-and-collateral-discounting-basis): about 22 basis points on the first period, 12 after ten years, 9 at the long end. Data: the chapter’s illustrative curves and tutorial.*

**Example 2.6 (One swap, two frameworks).**

A receiver of 3.50% against six-month Euribor for ten years on EUR 100 million, with the ten-year Euribor swap at 2.70%, is worth EUR 7 135 408 in the [multi-curve framework](#def-rc-multi-curve-and-collateral-discounting-curves) and EUR 7 075 108 on a single curve built from the Euribor swaps, which discounts at the higher Euribor rates. Both frameworks agree on the par rate, 2.70%, because each is calibrated to it; they disagree by EUR 60 300 on an off-market swap.

## 2.3 Collateral discounting and the credit support annex

Why discount at the overnight rate? Because that is the rate paid on the cash that secures the trade. Under a credit support annex (One Quant Book 2, chapter 10), a party whose derivative has positive value $V_t$ receives $V_t$ in cash from the other, and pays the agreed interest on it; the derivative is, in effect, funded by its own collateral.

**Definition 2.7 (Collateral rate, CSA discounting).**

The *collateral rate* $c_t$ of a trade is the rate the collateral taker pays on cash collateral under the trade’s collateral agreement, typically the overnight benchmark of the collateral currency. *CSA discounting* values each cash flow of a fully cash-collateralised trade on the curve of its collateral rate.

**Proposition 2.8 (The collateral rate discounts).**

If a trade is continuously and fully collateralised in cash paying $c_t$, and the hedges it needs are financed at the risk-free rate, its value is

$$
V_t = \E^{\mathbb Q}_t\Bigl[\exp\Bigl(-\int_t^T c_u\,du\Bigr)V_T\Bigr].
$$

**Partial proof.** The holder of the trade holds $V_t$ in collateral, owed back, on which it pays $c_t$; the hedging positions earn and cost $r_t$ and cancel. The account $V_t$ therefore grows at $c_t$: under $\mathbb Q$, $dV_t = c_tV_t\,dt +
dM_t$ with $M$ a martingale, whose solution is the formula. A full proof (Piterbarg, 2010) keeps track of the funding of the hedges. ∎

![A cash-collateralised derivative. The bank holds collateral equal to the trade’s value and pays the collateral rate on it; its position in the trade is funded by the collateral, so its value accrues at c_t and is discounted on the curve of c_t ().](https://one-course.com/images/onecourse/chapters/quant-6/rc-multi-curve-and-collateral-discounting/fig-fa9a5626ff4f.svg)

***Figure 2.2.** A cash-collateralised derivative. The bank holds collateral equal to the trade’s value and pays the [collateral rate](#def-rc-multi-curve-and-collateral-discounting-csa) on it; its position in the trade is funded by the collateral, so its value accrues at $c_t$ and is discounted on the curve of $c_t$ ([Proposition 2.8](#prop-rc-multi-curve-and-collateral-discounting-csa)).*

A cleared swap is the same, with the clearing house as counterparty: variation margin moves every day, and interest on it flows the other way.

**Definition 2.9 (Price alignment interest).**

*Price alignment interest* (PAI) is the interest a clearing house charges the receiver of cash variation margin and pays to the payer, at an overnight rate it chooses, so that a cleared swap is economically equivalent to a bilateral swap collateralised in cash at that rate. The clearing house discounts at the PAI rate.

## 2.4 Collateral choice and cross-currency collateral

Many collateral agreements let the poster choose among several currencies or securities, and substitute them later.

**Definition 2.10 (Collateral choice option).**

A *collateral choice option* is the right of the collateral poster to choose, and later change, which of several eligible collaterals it posts. Expressed in the trade’s currency, the poster chooses the collateral earning the highest rate, so the trade is discounted at $\max_i c^{(i)}_t$ over the eligible collaterals.

Collateral in a foreign currency earns that currency’s overnight rate; swapped into the trade currency at the cross-currency basis (One Quant Book 2, chapter 16), it earns the domestic overnight rate plus or minus the basis. With the basis for lending euros against dollars at $b(t)$, posting euros earns $r_{\text{SOFR}}-b$ in dollar terms: more than dollar cash when $b<0$.

**Example 2.11 (A dollar trade with a two-currency CSA).**

Take an illustrative basis of $-15$ basis points at the front rising linearly to $+5$ at ten years. Euros are the better collateral until 7.5 years, dollars after. A payment of USD 100 million in ten years is worth USD 68 598 292 under a dollar-only CSA and USD 68 213 510 when the payer may post either currency: the choice, used at every date, raises the effective discount rate by an average of $56.25/10 = 5.6$ basis points a year and costs the receiver USD 384 782. With volatile rates and basis the option is worth more than this intrinsic value.

![Instantaneous forward rates earned by two eligible collaterals, in dollars, and the effective discount rate when the poster may choose. The lines cross at 7.5 years (dashed), where the illustrative basis changes sign. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-multi-curve-and-collateral-discounting/fig-f2e65433aea1.svg)

***Figure 2.3.** [Instantaneous forward rates](https://one-course.com/books/quant/6/en/chapter/1-curve-construction#def-rc-curve-construction-pillar) earned by two eligible collaterals, in dollars, and the effective discount rate when the poster may choose. The lines cross at 7.5 years (dashed), where the illustrative basis changes sign. Data: the chapter’s tutorial.*

## 2.5 After the interbank rates: the 2020 discounting switches

When overnight benchmarks replaced EONIA and the dollar effective federal funds rate as the reference rates of their markets, the clearing houses changed the rate of their [price alignment interest](#def-rc-multi-curve-and-collateral-discounting-pai), and with it the [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) of every cleared swap in the currency.

**Definition 2.12 (Discounting switch).**

A *discounting switch* is a change of the [collateral rate](#def-rc-multi-curve-and-collateral-discounting-csa) of a population of trades, and hence of their [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves), on a set date, with the resulting changes in value compensated in cash and, where required, the changes in risk compensated with basis swaps.

**As of September 2026 — The 2020 switches.**

EONIA was computed as €STR plus 8.5 basis points from 1 October 2019 until it was discontinued on 3 January 2022. LCH moved discounting and price alignment on all euro swaps from EONIA to €STR flat on 27 July 2020, compensating each account in cash for the change in net present value, with projected cash flows held fixed (a “constant-forward” method). It moved dollar swaps from federal funds to SOFR over the weekend of 16–19 October 2020 with cash for the value change and federal-funds/SOFR basis swaps for the risk change: more than one million contracts and USD 120 trillion of notional; clients who did not want the basis swaps had them auctioned, USD 24 billion net notional, to 18 dealers at close to zero cost.

Because EONIA was fixed at €STR plus 8.5 basis points, the euro switch was a deterministic shift of the [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves): every discount factor rose by $e^{0.00085\,t}$. An in-the-money swap, whose future cash flows are net receipts, gained value; an out-of-the-money swap lost; the compensation reversed each change.

```python
def switch_value(years: int, fixed: float, notional: float, payer: bool = False) -> dict[str, float]:
    """Value of a cleared swap before (EONIA discounting) and after (ESTR flat), by the clearing
    house's constant-forward method: the cash flows projected under the EONIA regime are held
    fixed and only the discount factors change. Compensation = before - after, paid by the gainer."""
    estr, eonia, _ = curves_2020()
    ins = [Fixing(SWITCH, add_months(SWITCH, 6), FIX6M_2020, "6M")] + [
        IborSwap(SWITCH, n, r, 6, f"{n}Y") for n, r in zip(YEARS, IRS_2020, strict=True)]
    proj = calibrate_projection(SWITCH, eonia, ins)
    before = ibor_swap_pv(proj, eonia, SWITCH, years, fixed, notional, payer=payer)
    after = ibor_swap_pv(proj, estr, SWITCH, years, fixed, notional, payer=payer)
    return {"before": before, "after": after, "change": after - before, "compensation": before - after}
```

***Listing 2.2.** The clearing house’s constant-forward valuation of the switch. code/rates-credit-risk/02-multi-curve-and-collateral-discounting/python/rc_multicurve.py*

![Gain at the euro discounting switch of a receiver of 1.50% against six-month Euribor on EUR 100 million, by remaining maturity, on illustrative curves of July 2020 (euro swap rates between -0.50\% and +0.08\%). The clearing house’s cash compensation took each gain back. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-multi-curve-and-collateral-discounting/fig-434deccae7a2.svg)

***Figure 2.4.** Gain at the euro [discounting switch](#def-rc-multi-curve-and-collateral-discounting-switch) of a receiver of 1.50% against six-month Euribor on EUR 100 million, by remaining maturity, on illustrative curves of July 2020 (euro swap rates between $-0.50\%$ and $+0.08\%$). The clearing house’s cash compensation took each gain back. Data: the chapter’s tutorial.*

## 2.6 Tutorial: a euro market on two curves

**Goal.** Build an €STR [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) and a six-month Euribor [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves), read the [tenor basis](#def-rc-multi-curve-and-collateral-discounting-basis), value a swap in both frameworks, and value the 2020 [discounting switch](#def-rc-multi-curve-and-collateral-discounting-switch). **End state:** Figures [2.1](#fig-rc-multi-curve-and-collateral-discounting-forwards) and [2.4](#fig-rc-multi-curve-and-collateral-discounting-switch) and the numbers of [Example 2.6](#ex-rc-multi-curve-and-collateral-discounting-compare).

1. **[Discount curve](#def-rc-multi-curve-and-collateral-discounting-curves)** : `discount()` , nine €STR swaps with chapter 1’s builder, monotone convex.
2. **[Projection curve](#def-rc-multi-curve-and-collateral-discounting-curves)** : `projection()` calibrates the fixing and nine Euribor swaps with the [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) held ( [Listing 2.1](#lst-rc-multi-curve-and-collateral-discounting-proj) ); check `basis_table()` returns 20, 19, 18, 17, 16, 15, 14, 13 and 12 basis points.
3. **Two frameworks** : `discounting_comparison()` .
4. **The switch** : `switch_value(20, 0.015, 5e8)` ; `fig_rc_multicurve.py` writes the charts.

**What to change next.** Make the basis flat at 15 basis points and check that the forward basis becomes flat too; value the switch with the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) recalibrated on the €STR [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) instead of held fixed, and compare with the clearing house’s method.

## 2.7 Build: discount and projection curves

**Purpose.** Every rates instrument of the book is priced with a [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) for its index and a [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) for its collateral.

**Interface.** `IborSwap`, `Fixing`; `calibrate_projection(spot, disc, instruments, kind)`; `ibor_swap_pv(proj, disc, …)`; `tenor_basis`; `ShiftedCurve(base, shift)`; `CollateralChoiceCurve(base, spreads)`; all curves expose `df(date)`.

**Rules.** The [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) is calibrated first and never changed by the projection calibration; fixed legs annual ACT/360, floating legs on the index’s tenor; collateral choice as the pointwise maximum of eligible rates; `firm_curvebuild` and Book 2’s `firm_curve` imported, not edited.

**Acceptance tests.** `code/firm/multicurve/tests/`: the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) reprices its swaps on its [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves); the [tenor basis](#def-rc-multi-curve-and-collateral-discounting-basis) equals the par-rate difference; with no basis, projection and discount forwards agree; payer and receiver values are opposite; a collateral choice never raises the value of a receivable and reduces to the single curve with one collateral.

**Stretch.** Several indices (one, three, six, twelve months) from basis swaps; a cross-currency [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) from cross-currency basis swaps; the collateral choice with volatile basis by simulation.

Sources and further reading

- V. Piterbarg, “Funding beyond discounting: collateral agreements and derivatives pricing”, *Risk* , February 2010; “Cooking with collateral”, *Risk* , 2012.
- A. De Socio, “The interbank market after the financial turmoil”, Banca d’Italia, Temi di Discussione 819, 2011.
- LCH, *SwapClear transition to €STR discounting* ; *SOFR discounting: plan for the SwapClear compensation process* , 2019–2020.
- M. Henrard, *Interest Rate Modelling in the Multi-Curve Framework* , Palgrave Macmillan, 2014.

## 2.8 Exercises

**Exercise 2.1 ★.**

The ten-year swap against six-month Euribor is at 2.70% and the ten-year €STR swap at 2.55%, with identical annual fixed legs. What is the ten-year [tenor basis](#def-rc-multi-curve-and-collateral-discounting-basis), and on which leg is it paid?

**Solution of Exercise 2.1.**

$2.70-2.55 = 15$ basis points ([Proposition 2.2](#prop-rc-multi-curve-and-collateral-discounting-basisrates)), added to the overnight leg: the party receiving six-month Euribor pays €STR compounded plus 15 basis points.

**Exercise 2.2 ★.**

Before 2007 the three-month unsecured-secured spread was about ten basis points; afterwards above fifty. Explain in two sentences why that ended the single-curve framework.

**Solution of Exercise 2.2.**

The single-curve argument needs a six-month loan to cost the same as two three-month loans or 180 overnight loans; a persistent spread of fifty basis points between unsecured and secured term lending made each tenor a different price. A floating leg on one index was no longer worth par on a curve built from another, and discounting at a bank’s term rate was no longer the cost of a collateralised position.

**Exercise 2.3 ★.**

A euro swap is collateralised in euro cash paying €STR. Which curve discounts it, which projects its six-month fixings, and what changes if the CSA pays €STR plus 10 basis points?

**Solution of Exercise 2.3.**

The €STR curve discounts; the six-month Euribor [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) projects. With €STR plus 10 basis points the [collateral rate](#def-rc-multi-curve-and-collateral-discounting-csa) is higher, so the [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) is the €STR curve shifted by 10 basis points (discount factors times $e^{-0.001t}$); the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves), recalibrated on it, moves slightly, and an off-market trade changes value.

**Exercise 2.4 ★★.**

In [Example 2.11](#ex-rc-multi-curve-and-collateral-discounting-choice), compute the integral of the effective spread over ten years and the resulting discount factor ratio, and check the USD 384 782.

**Solution of Exercise 2.4.**

The effective spread is $\max(0,-b(t)) = (15-2t)$ basis points for $t<7.5$ and zero after: $\int_0^{7.5}(15-2t)\,dt = 112.5-56.25 = 56.25$ basis-point years. The ratio of discount factors is $e^{-0.005625}=0.99439$, and $68\,598\,292\times(1-0.99439) = \text{USD}~384\,782$.

**Exercise 2.5 ★★.**

Explain the EUR 60 300 of [Example 2.6](#ex-rc-multi-curve-and-collateral-discounting-compare): which way does the single curve err, and why are both frameworks right about the par rate?

**Solution of Exercise 2.5.**

The single curve discounts at Euribor rates, about 15 basis points above the €STR rates at which the collateralised swap is really funded, so it undervalues the receiver’s future net receipts, by EUR 60 300. Both frameworks reprice the par ten-year swap by construction, so they agree at the par rate and differ only for off-market trades, by an amount that grows with the distance from par.

**Exercise 2.6 ★★.**

At the euro switch, a client is *paying* 1.50% on EUR 100 million for ten years. Using [Figure 2.4](#fig-rc-multi-curve-and-collateral-discounting-switch), give its value change and who pays the compensation to whom.

**Solution of Exercise 2.6.**

A payer of 1.50% when the par rate is far lower has a liability: its value is $-\text{EUR}~16.70$ million, and lower discount rates make the liability larger, a loss of EUR 73 738, the mirror of the ten-year bar. The clearing house compensates it: the client receives EUR 73 738, paid out of the receivers’ gains.

**Exercise 2.7 ★★★.**

*Coding.* With `forward_table()`, give the forward basis on the six-month period starting in ten years, compare with the ten-year par basis of 15 basis points, and explain the difference.

**Solution of Exercise 2.7.**

11.76 basis points, against a ten-year par basis of 15. The par basis is an annuity-weighted average of the forward bases over ten years; the forward basis starts at 21.6 and declines, so the ten-year average exceeds the forward at ten years. Hedging a forward-starting basis position with par basis swaps needs the whole term structure.

**Exercise 2.8 ★★★.**

*Find the flaw.* “Our CSA lets the counterparty post euros or dollars; today dollars are cheaper to post, so the option is out of the money and we can value the trade on the dollar curve.” Correct it.

**Solution of Exercise 2.8.**

The poster can switch collateral at any time over the trade’s life, so the option is on the whole path of the basis, not today’s value: even with dollars cheapest today, future dates where euros are better lower the value of a receivable (the effective rate is the maximum at each date), and volatility of the basis adds time value. Valuing on the dollar curve ignores both.

## 2.9 Problem: The Switch Weekend

**Problem 2.1.**

Weekend problem — a pension fund’s swap at the euro discounting switch

On Friday 24 July 2020 a pension fund receives 1.50% annually against six-month Euribor on EUR 500 million, cleared at LCH, with twenty years left; it was traded in 2010. On the illustrative July 2020 curves of the chapter (€STR swaps from $-0.50\%$ to $-0.05\%$, Euribor swaps 12 to 17 basis points higher), the twenty-year Euribor par rate is 0.08%. On Monday 27 July the clearing house discounts at €STR flat instead of EONIA ($=$ €STR $+$ 8.5 basis points).

**Part I — The position.**

1. Is the swap an asset or a liability of the fund, and why?
2. Give its value on Friday under EONIA discounting.
3. Which cash flows of the swap are known, which are projected?
4. What does the fund receive on its variation margin, before and after the switch?
5. Why does the discount rate of a cleared swap equal the PAI rate?

**Part II — The switch.**

6. By how much does every discount factor to $t$ years change?
7. Give the swap’s value on Monday by the clearing house’s constant-forward method.
8. Give the value change. Who gains?
9. Who pays the cash compensation, how much, and to whom?
10. Why hold the projected cash flows fixed rather than recalibrate the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) on the new [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) ?

**Part III — The risk.**

11. After the switch, how much does the swap lose if the €STR [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) rises by one basis point, projections held?
12. Before the switch the same number described EONIA risk. What risk has changed, and why did the euro switch need no basis-swap compensation while the dollar switch did?
13. Would a payer of 1.50% have gained or lost at the switch?
14. Why is the thirty-year receiver’s gain more than twice the fifteen-year’s ( [Figure 2.4](#fig-rc-multi-curve-and-collateral-discounting-switch) )?
15. Does the switch change the fund’s economic position? Its accounts?

**Part IV — Judgement.**

16. Why did the market move all cleared swaps on one weekend rather than each trade at its own pace?
17. What would a bilateral trade under a CSA paying EONIA have needed?
18. Could a dealer have profited from knowing the switch date? Why not much?
19. State the *named result* : the fund’s value change and compensation, and its discount sensitivity after the switch.
20. In one sentence: what decides the [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) of a derivative?

**Solution of Problem 2.1.**

**1.** An asset: it receives 1.50% while the par rate is 0.08%, so its future net receipts are large. **2.** EUR 145 577 447. **3.** The fixed payments and the current Euribor period are known; later Euribor fixings are projected from the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves). **4.** It has received variation margin equal to the swap’s value and pays [price alignment interest](#def-rc-multi-curve-and-collateral-discounting-pai) on it at EONIA before the switch, at €STR (8.5 basis points less) after; with negative rates the “payment” is a receipt. **5.** The fund’s position is financed by the variation margin at the PAI rate, so its value accrues at that rate ([Proposition 2.8](#prop-rc-multi-curve-and-collateral-discounting-csa)). **6.** Each discount factor is multiplied by $e^{0.00085\,t}$: 1.0171 at twenty years. **7.** EUR 146 774 873. **8.** $+\text{EUR}~1\,197\,426$: the fund gains, as a lower discount rate raises the value of net receipts. **9.** The fund pays EUR 1 197 426 in cash to the clearing house, which passes it to the members whose positions lost. **10.** The market quotes of Euribor swaps did not move over the weekend; the method isolates the pure discounting effect, is simple to replicate by each member, and matches the working group’s view that forwards do not depend on the CSA. **11.** EUR 141 575 per basis point. **12.** It was EONIA discounting risk and is now €STR discounting risk. EONIA was €STR plus a fixed 8.5 basis points, so the two risks were the same and nothing needed hedging; federal funds and SOFR move independently, so the dollar switch changed members’ risk, which basis swaps restored. **13.** Lost: its liability grew as discount factors rose. **14.** The gain is roughly value times duration times 8.5 basis points, and both the value of an in-the-money swap and its duration grow with maturity; the thirty-year gain is 4.1 times the fifteen-year. **15.** No: the compensation returns it to where it was. Its accounts may show a gain and an equal cash payment, and a slightly different discount risk. **16.** Mixed discounting within one netting pool would break the equivalence between members’ positions and the clearing house’s margin, and a single date lets compensation net across all accounts. **17.** A bilateral amendment of the CSA to €STR (plus or minus a spread), negotiated trade by trade, with its own compensation payment. **18.** Little: EONIA was mechanically €STR plus 8.5 basis points, so the switch was a known deterministic transfer, compensated in cash; there was no unknown to trade. **19.** Named result: *the switch weekend* raised the value of the fund’s swap by EUR 1 197 426, which it paid back as cash compensation; after the switch its discount sensitivity is EUR 141 575 per basis point of €STR. **20.** The rate paid on the collateral that secures it.

## 2.10 Interview questions

**Interview question 2.1 ★ researcher, bank.**

Why do we discount collateralised swaps at the overnight rate and not at the index they pay?

**Solution of Interview question 2.1.**

Because the trade is funded by its collateral: the holder of a positive-value trade holds cash collateral equal to its value and pays the overnight rate on it, so the value accrues at that rate. The index only determines the cash flows, which are projected on its own curve.

*What the interviewer is looking for: discounting tied to the [collateral rate](#def-rc-multi-curve-and-collateral-discounting-csa), projection to the index.*

**Interview question 2.2 ★★ researcher, developer.**

Walk me through building a six-month Euribor curve after 2008. What is calibrated first, and why?

**Solution of Interview question 2.2.**

First the overnight-index (€STR) [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) from OIS; then, holding it fixed, the six-month Euribor [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) from the six-month fixing, the forward rate agreements or futures, and swaps against six-month Euribor, each priced with Euribor forwards discounted on €STR. The [discount curve](#def-rc-multi-curve-and-collateral-discounting-curves) comes first because every projection instrument’s price depends on it, not the reverse.

*What the interviewer is looking for: the order and the reason.*

**Interview question 2.3 ★★ trader.**

What is a [tenor basis swap](#def-rc-multi-curve-and-collateral-discounting-basis), and why is the six-month basis positive?

**Solution of Interview question 2.3.**

Two floating legs of different tenors (six-month Euribor against €STR or three-month Euribor) with a spread on one to make it fair. It is positive because a six-month unsecured loan carries more credit and liquidity premium than a rolled sequence of shorter or secured loans.

*What the interviewer is looking for: credit and liquidity premium by tenor.*

**Interview question 2.4 ★★ bank, risk.**

A CSA allows collateral in three currencies. How does that change the value of the trades under it, and which way?

**Solution of Interview question 2.4.**

The poster will post whichever collateral is best for it at each date, the one earning the highest rate once converted to the trade currency, so trades are discounted at the pointwise maximum of the [collateral rates](#def-rc-multi-curve-and-collateral-discounting-csa): receivables are worth less and payables less negative. With volatile rates and basis the option also has time value; it is often valued by simulation.

*What the interviewer is looking for: the maximum rule and the sign of the effect.*

**Interview question 2.5 ★★★ researcher, risk.**

A clearing house changes its discounting rate. What happens to the value and the risk of a swap book, and how would you compensate members fairly?

**Solution of Interview question 2.5.**

Values change by the difference of the present values of the cash flows under the two [discount curves](#def-rc-multi-curve-and-collateral-discounting-curves) (in-the-money receivers of fixed gain when rates fall); discount risk changes from one index to the other. Fair compensation: cash equal to the value change, computed with forwards held constant, netted per account; and, where the two rates are not tied, basis swaps that restore the old risk profile, with an auction for members who do not want them.

*What the interviewer is looking for: value and risk compensated separately.*

**Interview question 2.6 ★★★ developer.**

Your pricer uses the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves) to discount by mistake. Which trades show the error, and how large is it?

**Solution of Interview question 2.6.**

Par swaps on the [projection curve](#def-rc-multi-curve-and-collateral-discounting-curves)’s own index still reprice (both curves are calibrated to them), so the error hides on off-market and seasoned trades, where the discount rate is too high by the basis: an error of roughly value times duration times the basis, 15 basis points here, EUR 60 300 on a ten-year swap 80 basis points in the money on EUR 100 million. A test with an off-market trade finds it.

*What the interviewer is looking for: par trades hide the bug; off-market trades expose it.*