---
title: "Credit and Debit Valuation Adjustments"
book: "Rates, Credit, XVA and Risk"
subject: quant
language: en
chapter: 18
exercises: 8
source: https://one-course.com/books/quant/6/en/chapter/18-credit-and-debit-valuation-adjustments
---

# Chapter 18 — Credit and Debit Valuation Adjustments

When the Basel Committee finalised its capital rules for counterparty risk in June 2011, it gave the reason in one sentence: during the financial crisis, roughly two-thirds of the losses attributed to [counterparty credit risk](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-ccr) came from changes in the [credit valuation adjustment](#def-rc-credit-and-debit-valuation-adjustments-cva), and only about one-third from actual defaults. Banks had lost money not because their counterparties failed, but because the market price of the chance that they might had risen, and that price was in the fair value of every uncollateralised derivative. This chapter prices the chance: the [credit valuation adjustment](#def-rc-credit-and-debit-valuation-adjustments-cva) that a bank deducts from the value of its derivatives for the counterparty’s default, the [debit valuation adjustment](#def-rc-credit-and-debit-valuation-adjustments-dva) it adds for its own, how both are computed from chapter 17’s exposures and chapter 13’s hazard curves, and how they are hedged, accounted for and treated in capital.

## 18.1 CVA as the price of counterparty default

**Definition 18.1 (Credit valuation adjustment).**

The *credit valuation adjustment* (CVA) of a [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) is the market value of the loss from the counterparty’s default: the value of the trades without counterparty risk minus their value with it. With the counterparty’s default time $\tau_C$, recovery $R_C$ and discounted exposure,

$$
\begin{align*}
\mathrm{CVA} &= (1-R_C)\,\E\bigl[\mathbf 1_{\tau_C\le T}\,D(0,\tau_C)\max(V_{\tau_C}-C_{\tau_C},0)\bigr]\\
&\approx (1-R_C)\sum_k\mathrm{DEE}(t_k)\bigl(Q_C(t_{k-1})-Q_C(t_k)\bigr),
\end{align*}
$$

where $\mathrm{DEE}(t) = \E[D(0,t)E_t]$ and the approximation assumes the exposure and the default independent.

The formula is a sum over default dates of three factors: the [loss given default](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-pd), the probability that default falls in each period (from the counterparty’s default-swap curve), and the discounted [expected exposure](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-profiles) then. It is the value of a contingent default swap whose notional is the future exposure.

**Example 18.2 (The netting set’s CVA).**

The counterparty of chapter 17’s [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) is quoted at 80, 95, 110, 140, 160 and 175 basis points at one to ten years (illustrative BBB), recovery 40%: its ten-year [probability of default](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-pd) is 26.4%. With the exposures discounted path by path along the Hull–White scenarios, the [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting)’s CVA is USD 891 342. Standalone, the swap’s CVA is 288 250 and the cross-currency swap’s 768 334: netting saves 16%. Over the ten-year default swap’s [risky annuity](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-risky) of 7.45, the CVA is 12.0 basis points a year on USD 100 million ([Figure 18.1](#fig-rc-credit-and-debit-valuation-adjustments-contrib)).

![The netting set’s CVA by year of the counterparty’s default: the product of the discounted expected exposure (rising, then flat) and the probability of default in the year (rising with the hazard curve, then discounted by survival). Years four to six contribute most. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-credit-and-debit-valuation-adjustments/fig-e69fd89b3b7e.svg)

***Figure 18.1.** The [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting)’s CVA by year of the counterparty’s default: the product of the discounted [expected exposure](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-profiles) (rising, then flat) and the [probability of default](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-pd) in the year (rising with the hazard curve, then discounted by survival). Years four to six contribute most. Data: the chapter’s tutorial.*

```python
def cva(dee: np.ndarray, times: np.ndarray, cpty: HazardCurve, recovery: float = 0.4,
        own: HazardCurve | None = None) -> float:
    """Unilateral CVA, or first-to-default CVA if the bank's own curve is given. EE at period midpoints."""
    dq, _ = _default_probs(cpty, times)
    mid = 0.5 * (dee[1:] + dee[:-1])
    w = np.ones_like(dq) if own is None else np.array([own.survival(float(t)) for t in times[1:]])
    return float((1.0 - recovery) * np.sum(mid * dq * w))
```

***Listing 18.1.** CVA from a discounted expected exposure profile and a hazard curve, unilateral or first-to-default. code/firm/cva/firm_cva.py*

## 18.2 DVA and the bilateral view

**Definition 18.3 (Debit valuation adjustment).**

The *debit valuation adjustment* (DVA) is the same quantity seen from the counterparty: the market value of the gain to the bank from its own default, when it would stop paying what it owes, computed from the discounted negative exposure and the bank’s own hazard curve and recovery.

**Definition 18.4 (Bilateral CVA).**

The *bilateral CVA* is the adjustment $\mathrm{CVA}-\mathrm{DVA}$ when each is computed on first-to-default: the counterparty’s default costs the bank only if the bank has not defaulted first (weight $Q_B(t)$ in the CVA sum), and symmetrically. It makes the two parties agree on the price: one’s CVA is the other’s DVA.

**Example 18.5 (Both sides).**

The bank itself is quoted at 40 to 90 basis points (illustrative single-A). Its [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) drifts in the counterparty’s favour (chapter 17), so its DVA of USD 1 055 437 exceeds its CVA. On first-to-default the CVA is 832 416 and the DVA 896 328: the bilateral adjustment is $-\text{USD}~63\,912$, a small net *benefit* to the bank. Under chapter 17’s zero-threshold CSA the CVA falls to USD 122 340 and the DVA to 70 490.

**Remark 18.6 (Can a bank monetise its own default?).**

DVA is real value to the bank’s creditors, not to its shareholders: it is realised only by defaulting. It cannot be hedged by buying protection on oneself; banks approximate the hedge by selling protection on similar banks, which adds correlation risk. Its growth when the bank’s credit worsens is why regulators remove it from capital.

**Definition 18.7 (Close-out amount).**

The *close-out amount* is the amount the surviving party determines under the master agreement for the terminated [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting): broadly the cost of replacing the transactions, which may include the replacement counterparty’s view of the survivor’s own credit (a risky close-out) or not (a risk-free close-out). The choice changes the CVA formula: with a risky close-out, the surviving party’s DVA survives the counterparty’s default.

## 18.3 Computing CVA

**Method 18.8 (CVA in a simulation engine).**

(1) Simulate exposures and deflators on common scenarios (chapter 17). (2) Compute the discounted expected positive and negative exposures of each [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting). (3) Bootstrap each counterparty’s hazard curve from its default-swap quotes, or from a proxy curve for a name without quotes (chapter 20). (4) Sum over the grid. (5) For [wrong-way risk](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-wwr), simulate the hazard rate with the market factors and average the pathwise loss.

**Example 18.9 (Wrong-way CVA).**

Tie the counterparty’s hazard to the euro:

$$
\lambda_t = \lambda^0_t\,\bigl(X_t/F_X(0,t)\bigr)^{-b},
$$

normalised so that the average survival curve is still the market’s, as in chapter 17. The cross-currency swap’s CVA is USD 766 532 without dependence ($b = 0$), 1 108 459 with $b = 2$ and 1 521 524 with $b = 5$: [wrong-way risk](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-wwr) doubles it ([Figure 18.2](#fig-rc-credit-and-debit-valuation-adjustments-wwr)). With very strong dependence the CVA falls back a little: the counterparty then defaults early on the paths where the euro falls first, before the exposure has grown.

![CVA of the cross-currency swap as its counterparty’s hazard depends more strongly on the euro, with the average survival curve held at the market’s. Wrong-way risk doubles the CVA; the curve turns down when defaults concentrate so early that the exposure is still small. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-credit-and-debit-valuation-adjustments/fig-60acf3c64f16.svg)

***Figure 18.2.** CVA of the cross-currency swap as its counterparty’s hazard depends more strongly on the euro, with the average survival curve held at the market’s. [Wrong-way risk](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-wwr) doubles the CVA; the curve turns down when defaults concentrate so early that the exposure is still small. Data: the chapter’s tutorial.*

## 18.4 Sensitivities and hedging

CVA moves with everything the exposure depends on (rates, FX, volatilities) and with the counterparty’s credit. A CVA desk hedges the first set with the underlying instruments and the second with default swaps on the counterparty, or on an index when the name does not trade.

**Example 18.10 (Bucketed CS01).**

Raising each of the counterparty’s six quoted spreads by one basis point, with the curve re-bootstrapped, changes the CVA by $-235$, $-165$, $-141$, $-22$, $+125$ and $+4\,693$ dollars ([Figure 18.3](#fig-rc-credit-and-debit-valuation-adjustments-cs01)): a parallel rise costs USD 4 254 per basis point. Almost all of it sits on the ten-year quote; raising a short quote with the long ones fixed lowers the later forward hazards, so short buckets are negative. The hedge is about USD 5.7 million of ten-year protection (the [CS01](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-cs01) over the default swap’s [risky annuity](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-risky) and a basis point).

![Bucketed CS01 of the netting set’s CVA. The ten-year quote carries almost all the sensitivity; shorter quotes have small negative sensitivities through the bootstrap. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-credit-and-debit-valuation-adjustments/fig-08852a180330.svg)

***Figure 18.3.** Bucketed [CS01](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-cs01) of the [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting)’s CVA. The ten-year quote carries almost all the sensitivity; shorter quotes have small negative sensitivities through the bootstrap. Data: the chapter’s tutorial.*

## 18.5 Accounting and regulation

**Definition 18.11 (XVA).**

*XVA* is the family of valuation adjustments that turn a derivative’s riskless value into the price a bank charges or reports: CVA and DVA for default, and the funding, margin and capital adjustments of chapter 19.

Accounting standards value derivatives at fair value, an exit price: IFRS 13, issued in May 2011, defines it with the assumptions market participants would use, including about risk. Counterparty credit is such a risk, so CVA is part of reported fair value, and so is the bank’s own credit, as DVA: the fair value of a liability reflects its non-performance risk, which includes the entity’s own credit risk (paragraph 42). JPMorgan’s third quarter of 2011, for instance, included a 1.9 billion dollar pretax DVA gain from the widening of its own credit spreads. Regulation takes a different view: CVA losses need capital (the charge the Basel Committee introduced after the crisis, chapter 19), and DVA, which rises when the bank weakens, is deducted from common equity.

**As of September 2026 — CVA and DVA in capital.**

The Basel Committee stated in June 2011 that about two-thirds of the crisis losses from counterparty risk were CVA losses. Basel III requires banks to derecognise from common equity tier 1 the unrealised gains and losses from changes in their own credit risk on fair valued liabilities; its final rule of July 2012 extended this to all DVA on derivatives, phased in from 20% in 2014 to a full deduction from 1 January 2018.

![DVA of the netting set as the bank’s own default-swap spreads widen in parallel: the worse the bank’s credit, the larger the accounting gain. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-credit-and-debit-valuation-adjustments/fig-693b996c9591.svg)

***Figure 18.4.** DVA of the [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) as the bank’s own default-swap spreads widen in parallel: the worse the bank’s credit, the larger the accounting gain. Data: the chapter’s tutorial.*

## 18.6 Tutorial: CVA of a netting set

**Goal.** Compute the CVA and DVA of chapter 17’s [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting), unilateral and first-to-default, with and without a CSA, their credit sensitivities, and the effect of [wrong-way risk](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-wwr). **End state:** the numbers of Examples [18.2](#ex-rc-credit-and-debit-valuation-adjustments-cva), [18.5](#ex-rc-credit-and-debit-valuation-adjustments-bcva), [18.9](#ex-rc-credit-and-debit-valuation-adjustments-wwr) and [18.10](#ex-rc-credit-and-debit-valuation-adjustments-cs01) and the four charts.

1. **Inputs** : chapter 17’s scenarios and values; `deflators(sc)` ; `bootstrap` for both credit curves.
2. **Adjustments** : `adjustments()` and `adjustments(csa=True)` .
3. **Risk** : `cs01()` ; `wrong_way()` .
4. **DVA** : `dva_shift()` ; `fig_rc_cva.py` writes the charts.

**What to change next.** Replace the counterparty curve by a flat 175 basis points and compare; compute the CVA with undiscounted EE times the zero-coupon bond, and measure the error of ignoring the correlation of rates and exposure.

## 18.7 Build: the CVA engine

**Purpose.** The firm’s CVA and DVA: fair-value adjustments of every [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting), their credit sensitivities for the hedging desk, and the inputs of chapter 20’s pricing.

**Interface.** `deflators(sc)`; `discounted_profiles(E, D)`; `cva(dee, times, cpty, recovery, own=None)`; `dva`; `cva_pathwise(E, D, times, hazards)`; `cs01_buckets`; `running_spread`.

**Rules.** Exposure at period midpoints; independence unless a hazard is simulated; recovery 40% by default; curves from `firm_cdscurve`.

**Acceptance tests.** `code/firm/cva/tests/`: flat exposure and hazard give the closed form; the pathwise engine with a deterministic hazard equals the profile engine; first-to-default CVA is below unilateral; bucketed [CS01](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-cs01) add up to the parallel [CS01](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-cs01).

**Stretch.** Stochastic recovery; risky close-out; CVA Greeks by adjoint algorithmic differentiation; incremental CVA (chapter 20).

Sources and further reading

- Basel Committee on Banking Supervision, press release of 1 June 2011 on the capital treatment of bilateral counterparty credit risk.
- Basel Committee on Banking Supervision, *Basel III* , revised June 2011, paragraph 75; press release of 25 July 2012 on valuation adjustments to derivative liabilities.
- IFRS Foundation, IFRS 13 *Fair Value Measurement* , 2011.

## 18.8 Exercises

**Exercise 18.1 ★.**

A [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) has a flat discounted EE of USD 2 million for five years; the counterparty’s hazard is a flat 2% and its recovery 40%. Give the CVA.

**Solution of Exercise 18.1.**

$0.6\times2\,000\,000\times(1-e^{-0.02\times5}) = \text{USD}~114\,195$.

**Exercise 18.2 ★.**

Estimate the chapter’s CVA as EPE times the ten-year spread times the [risky annuity](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-risky), and compare.

**Solution of Exercise 18.2.**

$6.44\text{ million}\times0.0175\times7.45 = \text{USD}~840\,420$, against the exact 891 342: the approximation uses an undiscounted, time-averaged exposure and the ten-year spread for every date, and it works because the spread already prices $\lambda(1-R)$.

**Exercise 18.3 ★.**

Why is the bilateral adjustment of [Example 18.5](#ex-rc-credit-and-debit-valuation-adjustments-bcva) negative, and what does the counterparty see?

**Solution of Exercise 18.3.**

The [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) drifts in the counterparty’s favour, so the bank expects to owe more than it is owed: its DVA exceeds its CVA even though the counterparty is riskier. The counterparty sees the mirror image: its CVA on the bank is USD 896 328 on first-to-default, its DVA 832 416, a net cost of 63 912 that is the bank’s net benefit.

**Exercise 18.4 ★★.**

Why is the [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting)’s CVA less than the sum of the standalone CVAs?

**Solution of Exercise 18.4.**

CVA is linear in the [expected positive exposure](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-profiles), and the positive part of a sum is at most the sum of the positive parts: netting lets the two trades’ values offset on each path.

**Exercise 18.5 ★★.**

How much ten-year protection hedges the CVA’s credit sensitivity, and what does the hedge leave open?

**Solution of Exercise 18.5.**

$4\,254/(7.45\times10^{-4}) \approx \text{USD}~5.7$ million of ten-year protection. It leaves the changes of the exposure with rates and FX (hedged separately), [cross-gamma](https://one-course.com/books/quant/6/en/chapter/3-rates-risk#def-rc-rates-risk-crossgamma) between credit and exposure, [wrong-way risk](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-wwr), the basis between the counterparty’s curve and any proxy, and jump-to-default: protection sized on [CS01](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-cs01) does not pay the actual loss at default.

**Exercise 18.6 ★★.**

Why does the CSA cut the CVA by 86% but the DVA by 93%?

**Solution of Exercise 18.6.**

Under the CSA both adjustments are driven by the value’s moves over the margin period of risk, which are symmetric; without it, the [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting)’s drift in the counterparty’s favour made the negative exposure large. The CSA removes more of the larger side.

**Exercise 18.7 ★★★.**

*Coding.* Compute the wrong-way CVA for $b = 0$, 2 and 5, and explain the turn of the curve at large $b$.

**Solution of Exercise 18.7.**

USD 766 532, 1 108 459 and 1 521 524. With a very strong dependence, paths where the euro falls early default almost surely in the first years, when the exposure is still small, and the average survival must be kept, so the defaults are taken from the later, larger exposures: the CVA turns down after $b\approx7$.

**Exercise 18.8 ★★★.**

*Find the flaw.* “Our DVA gain this quarter proves that our derivatives business is more valuable than the market thought.”

**Solution of Exercise 18.8.**

A DVA gain says the bank’s own credit worsened: the market now thinks it likelier to default on what it owes. It is value to creditors realised only by defaulting, not a sign of a better business; that is why it is deducted from capital and usually excluded from traders’ P&L.

## 18.9 Problem: The Gain from Getting Riskier

**Problem 18.1.**

Weekend problem — own credit and the income statement

In a quarter of stress the bank’s own default-swap spreads widen by 50 basis points at every maturity while nothing else moves. Consider chapter 17’s [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) and the bank’s curve of [Example 18.5](#ex-rc-credit-and-debit-valuation-adjustments-bcva).

**Part I — The gain.**

1. Give the unilateral DVA before and after.
2. Give the gain, and say where it appears in the accounts.
3. Why does the DVA rise when the bank’s credit worsens?
4. Scale the gain to a derivatives book 500 times larger.
5. Who is worse off in the same quarter?

**Part II — Capital.**

6. What does Basel III do with the gain?
7. When was the deduction fully phased in?
8. Why do regulators treat DVA differently from CVA?
9. What would happen to the bank’s common equity tier 1 ratio if the gain were counted?
10. How would investors read the gain?

**Part III — Hedging it.**

11. Why can the bank not buy protection on itself?
12. What proxy hedge could it use, and what risk does it add?
13. Should the bank hedge DVA at all?
14. How does a CSA change the DVA and the gain?
15. What does the counterparty see in its CVA?

**Part IV — Judgement.**

16. Is DVA value? For whom?
17. How should a trader’s P&L treat DVA?
18. Why is the bilateral adjustment the one both parties can agree on?
19. State the *named result* : the DVA gain from a 50-basis-point widening on the [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) .
20. In one sentence: why is DVA deducted from capital?

**Solution of Problem 18.1.**

**1.** USD 1 055 437 before, 1 513 008 after. **2.** USD 457 571, a gain in the fair value of derivatives through profit and loss. **3.** The liabilities (the negative exposure) are now less likely to be paid in full, so their fair value falls: the bank gains as a debtor. **4.** About USD 229 million. **5.** The bank’s counterparties: their CVA on the bank rises by the same amount. **6.** It deducts the DVA from common equity tier 1: the gain does not count as capital. **7.** From 1 January 2018, after a phase-in from 20% in 2014. **8.** CVA is a loss that a counterparty’s default would realise; DVA is a gain realised only by the bank’s own default, when capital is needed and absent. **9.** The ratio would rise as the bank weakened, the opposite of what capital should do. **10.** As a non-recurring item to strip out: most analysts report results before own-credit adjustments. **11.** A default swap on oneself would pay only when the bank has defaulted and cannot collect; no one would sell it at a useful price. **12.** Selling protection on peer banks, which gains when bank spreads tighten together; it adds the peers’ [jump-to-default risk](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-jtd) and the basis between the bank’s and the peers’ spreads. **13.** Most banks do not: the hedge adds real risk to protect an accounting number that capital already excludes. **14.** Under the zero-threshold CSA the DVA is USD 70 490, so the gain from the same widening is about 93% smaller. **15.** Its CVA on the bank rises by the same USD 457 571, a loss in its accounts. **16.** Value to the bank’s creditors (they bear the shortfall), not to its shareholders. **17.** Report it separately and exclude it from the desk’s performance, or charge it to a central function; traders should not be rewarded for the bank’s credit deteriorating. **18.** Each party’s [bilateral CVA](#def-rc-credit-and-debit-valuation-adjustments-bcva) is the other’s bilateral DVA, so a price that includes both is symmetric; a unilateral price counts the same default twice or not at all. **19.** Named result: *the gain from getting riskier*: a 50-basis-point widening raises the [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting)’s DVA from USD 1 055 437 to 1 513 008, a gain of USD 457 571. **20.** Because it grows exactly when the bank’s ability to absorb losses shrinks.

## 18.10 Interview questions

**Interview question 18.1 ★ trader, risk.**

Write the CVA formula and explain each term.

**Solution of Interview question 18.1.**

$\mathrm{CVA} = (1-R)\sum_k\mathrm{DEE}(t_k)\,\mathrm{PD}(t_{k-1},t_k)$: [loss given default](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-pd), discounted [expected positive exposure](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-profiles) of the [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) at each date, and the risk-neutral probability of the counterparty defaulting in each period from its default-swap curve; independence assumed.

*What the interviewer is looking for: the three terms and the independence assumption.*

**Interview question 18.2 ★★ researcher.**

What is DVA, and why is it controversial?

**Solution of Interview question 18.2.**

The counterparty’s CVA on the bank: the value of the bank’s option to default on what it owes. Controversial because it rises when the bank’s credit worsens, can be realised only by default, cannot be hedged directly, and flatters earnings in bad times; regulators deduct it from capital.

*What the interviewer is looking for: the definition and the three objections.*

**Interview question 18.3 ★★ trader.**

How would you hedge the CVA of a portfolio of uncollateralised swaps with a BBB corporate?

**Solution of Interview question 18.3.**

Hedge the exposure drivers (rates delta and vega) with swaps and swaptions, the credit with default swaps on the name or a proxy index sized on [CS01](https://one-course.com/books/quant/6/en/chapter/13-reduced-form-credit#def-rc-reduced-form-credit-cs01), and manage jump-to-default with single-name protection where it trades; seek a CSA or break clauses; monitor [cross-gamma](https://one-course.com/books/quant/6/en/chapter/3-rates-risk#def-rc-rates-risk-crossgamma).

*What the interviewer is looking for: market and credit hedges and their limits.*

**Interview question 18.4 ★★ developer, researcher.**

How do you include [wrong-way risk](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-wwr) in a CVA engine?

**Solution of Interview question 18.4.**

Simulate the counterparty’s hazard with the market factors (a stochastic intensity correlated with them, or a function of them) and compute pathwise default probabilities; or shift exposures conditional on default; or use stress scenarios. Keep the average survival curve calibrated to the market.

*What the interviewer is looking for: joint simulation and calibration of the marginal.*

**Interview question 18.5 ★★★ risk, bank.**

Why did CVA cause more losses than defaults in the crisis?

**Solution of Interview question 18.5.**

Most counterparties did not default, but their spreads widened sharply and exposures grew as rates and FX moved: the CVA, marked to market, rose on every uncollateralised [netting set](https://one-course.com/books/quant/6/en/chapter/17-counterparty-exposure#def-rc-counterparty-exposure-netting) at once, while defaults hit only a few.

*What the interviewer is looking for: mark-to-market of credit versus realised defaults.*

**Interview question 18.6 ★★★ researcher.**

Risk-free versus risky close-out: what is the difference and why does it matter?

**Solution of Interview question 18.6.**

Risk-free close-out values the terminated trades without the survivor’s credit; risky close-out lets the replacement price include it. With a risky close-out, the survivor’s DVA remains in the [close-out amount](#def-rc-credit-and-debit-valuation-adjustments-closeout), which changes the bilateral formulas and the recovery of the defaulted party’s estate.

*What the interviewer is looking for: the two conventions and their effect on DVA.*