11Credit and XVA

Answer to Question 11.1.

The hazard rate is about 3.33% a year, and the chance of default within five years is about 15.4%.

This is the credit triangle: spread = hazard $\times$ (1 $-$ recovery), so $h = 0.02/0.6$. Survival is $e^{-5h} = e^{-1/6} = 0.846$.

The logic is a fair exchange. Each year, the premium is the spread. The expected payout is the chance of default times the loss given default.

Answer to Question 11.2.

It is 3.92% in the first year, 3.77% in the second as seen from today, and 3.92% again given survival of the first.

$1 - e^{-0.04} = 0.0392$. From today, the second year needs survival and then default: $e^{-0.04} - e^{-0.08} = 0.0377$. A constant hazard has no memory, so the conditional figure repeats.

Answer to Question 11.3.

With no recovery it is worth 0.7788, a spread of exactly 3%. With 40% recovery it is worth 0.8292, a spread of about 1.75%.

No recovery: $e^{-(0.02 + 0.03) \times 5} = e^{-0.25}$. The spread is the hazard rate.

With recovery of face at maturity: $e^{-0.1}\,(e^{-0.15} + 0.4(1 - e^{-0.15})) = 0.9048 \times 0.9164$. The spread is $-\ln(0.9164)/5 = 1.746\%$, a little below $0.6 \times 3\% = 1.8\%$.

Answer to Question 11.4.

You pay $50,000 a quarter, and on default you receive $7 million. You are short credit: you gain when the name gets worse.

The annual premium is $0.02 \times \$10$ million $= \$200{,}000$. The payout is $(1 - 0.3) \times \$10$ million. Accrued premium up to the default date is settled as well.

Answer to Question 11.5.

The protection buyer pays the seller about 4.3 points up front. That is about $430,000 on $10 million.

Since 2009, North American contracts have traded with fixed coupons of 100 or 500 basis points, and the difference is settled up front. The buyer pays 100 a year for protection worth 200, so the gap is paid at the start.

The hazard is $h = 0.02/0.6 = 3.33\%$. The risky annuity is $(1 - e^{-(r + h)T})/(r + h)$ with $r + h = 6.33\%$, which is 4.29. So the up-front payment is $(0.02 - 0.01) \times 4.29 = 4.29\%$.

Answer to Question 11.6.

At 200, the up-front payment is 4.29 points at 40% recovery and 4.37 at 20%. At 2,000, it is 43.8 points and 51.1. So recovery moves the first by 0.1 point and the second by more than 7.

The formula is the same: up-front = (spread $-$ coupon) $\times$ risky annuity, with $h = s/(1 - R)$.

For a name at 200, the hazard moves from 3.33% to 2.5%, and the annuity moves only from 4.29 to 4.37. And the spread gap is small, so it hardly matters.

For a name at 2,000, the hazard moves from 33.3% to 25%. That moves the annuity from 2.30 to 2.69, and the gap being multiplied is 19 points.

Answer to Question 11.7.

Buy the bond and buy protection. That is a negative-basis trade, and it earns 40 basis points a year, or $40,000 on $10 million, for as long as it lasts.

Why the spreads must match. Hold a par floater with spread $s_b$ and protection at $s_c$, funded at the funding rate. If there is no default, you collect $s_b - s_c$. On a default, you are paid par less recovery by the protection, you deliver or sell the bond at recovery, and you repay par. So you hold no credit risk, and the running gain must be zero if the funding rate is the true financing rate: $s_b = s_c$. With hazard $h$ and recovery of par, both equal $h(1 - R)$.

What can go wrong.

• Your funding gets dearer, or the repo is pulled. The trade has no credit risk, but it has financing risk, and basis trades get cut when funding tightens.
• The bond is not par at default, so the payout does not match. A bond above par loses more than protection pays.
• The contract and the bond can differ on what counts as a default or on what can be delivered.
• The trade can run at a loss before it converges, and you can be forced out then.

Answer to Question 11.8.

Both default with probability 1.675%, and at least one with probability 8.325%. With probabilities of 5% and 1%, the correlation cannot go above about 0.438.

For default indicators, the covariance is $P(\text{both}) - pq$, and the correlation divides that by $\sqrt{p(1 - p)q(1 - q)}$. So

and at least one is $0.10 - 0.01675 = 0.08325$.

With unequal probabilities, the joint default cannot go above the smaller one, 1%. So the highest possible correlation is

Answer to Question 11.9.

Independent: about 2.94% a year, close to the sum of the five spreads of 0.6% each. Perfectly correlated: 0.6%, the same as one name. A second-to-default goes the other way, and rises with correlation.

The chance of at least one default is $1 - 0.99^5 = 4.90\%$, and with recovery of 40% the expected loss is $0.6 \times 4.90\% = 2.94\%$. With perfect correlation they all go together with probability 1%, so the loss is $0.6\%$.

The second-to-default needs two defaults. Independent, that is $1 - 0.99^5 - 5 \times 0.01 \times 0.99^4 = 0.098\%$, which is rare. As correlation rises, defaults come in clusters. Under a Gaussian copula, at a correlation of 0.6, the chance of a first default falls from 4.90% to about 3.7%, a spread of about 2.2%, and the chance of a second rises from 0.098% to about 0.9%, a spread of about 0.55%.

Answer to Question 11.10.

$\mathrm{EE}(t) = 0.399 \times 0.01 \times 100 \sqrt{t}\,(5 - t)$. It peaks at $t = 5/3$ years, one third of the life, at about 1.72 on 100 of notional.

At time $t$, the rate has moved by a normal with standard deviation $\sigma\sqrt{t}$, and the swap’s value is that move times the remaining annuity, $100(5 - t)$. The expected positive part of a centred normal is $\sigma_X/\sqrt{2\pi} = 0.399\,\sigma_X$.

Setting the derivative of $\sqrt{t}\,(5 - t)$ to zero gives $(5 - t)/(2\sqrt{t}) = \sqrt{t}$, so $t = 5/3$. The value is $0.399 \times 1 \times 1.291 \times 3.333 = 1.72$.

There are two forces. The rate has more time to move, which grows like $\sqrt{t}$. And fewer cash flows are left, which shrinks linearly.

Answer to Question 11.11.

The CVA is about 0.0714 on 100 of notional, which is about 7 cents per 100.

$\mathrm{CVA} = (1 - R)\int_0^5 \mathrm{EE}(t)\, h\,\mathrm{d}t$, and the integral of the profile is

So $\mathrm{CVA} = 0.6 \times 0.02 \times 5.947 = 0.0714$.

Since $(1 - R)h$ is the spread $s = 1.2\%$, and the average exposure is $5.947/5 = 1.189$, this is $s \times \overline{\mathrm{EE}} \times T = 0.012 \times 1.189 \times 5$. That is the rule you should say aloud: you are paying a CDS premium on your average exposure.

Answer to Question 11.12.

With no netting it is 0.798. With netting it is $0.399\sqrt{2 + 2\rho}$: 0, 0.399, 0.564, 0.691 and 0.798 for those five correlations.

With no netting, each trade contributes $0.399$ on its own. With netting, the exposure is on the sum, which is normal with variance $2 + 2\rho$. The expected positive part is 0.399 times its standard deviation.

Netting never makes exposure worse, because $\max(a + b, 0) \le \max(a, 0) + \max(b, 0)$. At a correlation of 1 it gives no gain. At $-1$ the two trades cancel.

Answer to Question 11.13.

It is about $3.99 million with no collateral and about $0.79 million with daily margin. It is not zero because the market moves in the 10 days between the last margin call and the close-out.

With no collateral, $0.399 \times \$10$ million $\times \sqrt{1}$. With margin, only the move during the margin period of risk counts: $0.399 \times 10 \times \sqrt{10/252} = 0.795$.

So the ratio is $\sqrt{10/252} \approx 0.2$. Halving the period saves only 29%.

Also, the margin period is often longer than one might think: disputes, a failure to post, and then replacing the hedge in a stressed market. And exposure over the period tends to come with stress.

Answer to Question 11.14.

Without weighting, CVA is 0.0714 and DVA is 0.0357, so the bilateral adjustment is 0.0357 against you. With first-to-default weighting at $e^{-(h_c + h_o)t}$, they are 0.0670 and 0.0335, so 0.0335.

An at-the-money swap has a symmetric profile, so the negative exposure has the same integral, 5.947. So $\mathrm{CVA} = 0.6 \times 0.02 \times 5.947$, and $\mathrm{DVA} = 0.6 \times 0.01 \times 5.947$. The weighting counts only the first of the two defaults, since after that the contract has stopped.

If your own credit worsens, DVA grows, and you book a profit. A firm in trouble posts a gain for being in trouble.

Desks are wary of this for good reasons:

• You can only realise it by defaulting, or by selling protection on yourself, which you cannot do.
• You can hedge it only with a proxy, such as a basket of peers, and that adds risk.
• Capital rules take it out of regulatory capital, so it does not support the balance sheet.

Accounting fair value still includes it, as an exit price would reflect your own credit.

Answer to Question 11.15.

The funding cost at 100 basis points is about 0.0595 on 100 of notional. When the spread is purely credit, at 60 basis points, the funding benefit is 0.0357, which is exactly your DVA. So counting both double-counts your own credit.

FVA. When the client’s swap is in the money to you, the hedge is out of the money, and you post collateral on it. You borrow to fund that collateral, and the client posts nothing. So you pay the funding spread on the positive exposure: $0.01 \times 5.947 = 0.0595$.

When the client’s swap is out of the money, you receive collateral on the hedge, and that saves you borrowing. So you gain the spread on the negative exposure, which is the funding benefit.

The overlap. With funding spread $s_o = h_o(1 - R) = 0.6\%$, the benefit is $s_o \times 5.947 = 0.0357$. And $\mathrm{DVA} = (1 - R)h_o \times 5.947$, the same number. It is one thing seen twice: you may never repay that debt, which is why your borrowing costs more and also why your liability is worth less.

So a coherent price takes the cost side from funding and the benefit side from either DVA or funding, not both.

The debate is whether FVA belongs in a price at all, as a pure no-arbitrage argument would say funding should not change value. Banks book it because they pay it. If the funding spread is zero, FVA is zero, as it should be.

Answer to Question 11.16.

The expected exposure given default is 1.99 at $\rho = 0$, 4.25 at $0.3$, 6.22 at $0.5$, and 0.62 at $-0.3$. So $\rho = 0.3$ more than doubles the CVA, and $\rho = 0.5$ triples it.

Exposure is large when $Z$ is low, and with positive $\rho$ default is also more likely when $Z$ is low. So you lose most when the counterparty is most likely to go: that is wrong-way risk.

Given default, the factor is shifted. The default threshold for 2% is $-2.05$, and the mean of a normal below that is $-\varphi(c)/\Phi(c) = -2.42$. So $Z$ has a conditional mean of $-2.42\rho$, and $E[\max(-5Z, 0) \mid \text{default}]$ is an integral:

At $\rho = 0$ it is $5\varphi(0) = 1.99$, the plain number.

Wrong-way examples. Buying protection on a bank from a bank in the same system. Holding a dollar receivable from a borrower in a weak currency, where the currency falls as it fails. Lending against the counterparty’s own shares.

Right-way example. An airline that has bought fuel forward from you. You are exposed when fuel falls, and falling fuel helps the airline. Always check the direction: an oil producer that has bought oil forward from you is wrong way, as falling oil raises your exposure and weakens the producer.

Answer to Question 11.17.

The table gives the expected loss as a share of each tranche.

Equity loses value as correlation falls and gains as it rises. The senior tranche moves the other way. What stays fixed is the pool’s expected loss, $0.02 \times 0.6 = 1.2\%$, since a sum of expectations ignores correlation.

Given the market factor $M$, a large pool’s loss is deterministic:

The tranche’s loss is $E[\min(\max(L - a, 0), b - a)]/(b - a)$, by quadrature over $M$.

There are two limits. As $\rho \to 0$, the pool loses exactly 1.2%, so the first tranche has lost $1.2/3 = 40\%$, and the rest nothing. As $\rho \to 1$, all default together with probability 2% and the loss is 60%, so every tranche up to 60% has lost 2%.

So equity is long correlation, and senior is short correlation. The middle tranche moves one way and then the other: it rises here, but must go back toward 2% as $\rho \to 1$.

Answer to Question 11.18.

Equity is worth 24.15, debt 75.85, the risk-neutral default probability is 18.7%, and the spread is about 232 basis points. At three months, the spread is about 71 basis points, and at five years about 210.

Equity is a call on the assets. So $d_1 = 1.1376$, $d_2 = 0.8876$, and $E = 100\,\Phi(d_1) - 80 e^{-0.03}\Phi(d_2) = 24.15$. Debt is what is left: $100 - 24.15 = 75.85$. A second route is debt as a riskless bond less a put: $77.64 - 1.78 = 75.85$.

Default is when assets end below 80, so the risk-neutral probability is $\Phi(-d_2) = 18.7\%$.

The spread is $-\ln(75.85/80) - 0.03 = 2.32\%$.

Term structure. At one month, the spread is about 2 basis points, and as maturity goes to zero, it goes to zero. It peaks near two years at about 248, then falls: 210 at five years, 162 at ten.

The short end is the failure. A company can only reach default by a continuous diffusion, and over a short time it can’t get there. So the model gives almost nothing for short-term credit, while real short spreads are not close to zero. Models add jumps, or uncertainty about the asset level or the barrier, to fix it.

Also, asset value and volatility are not observed. Practitioners back them out of equity and equity volatility, using $\sigma_E E = \Phi(d_1)\sigma_A V$.

Answer to Question 11.19.

The triangle gives 1.67%, 2.92% and 4.58%. The bootstrap gives 1.67%, 2.95% and 4.81%.

Triangle. Each spread gives an average hazard: $0.01/0.6$, $0.015/0.6$, $0.02/0.6$. Forward pieces follow from total hazard: $(3 \times 2.5\% - 1.67\%)/2 = 2.92\%$ and $(5 \times 3.33\% - 3 \times 2.5\%)/2 = 4.58\%$.

Bootstrap. Solve in order. Take the first hazard so that the one-year contract prices to zero. Then, holding it, take the second so the three-year contract prices to zero, then the third. Premium leg: $\sum_i \tfrac14 D_i S_i$ plus half a quarter of accrued on default. Protection leg: $(1 - R)\sum_i D_i (S_{i-1} - S_i)$.

Why higher. A CDS spread is a weighted average of the forward spreads, and the weights are the risky discount factors $D_iS_i$, not equal. Early periods weigh more, because they are less discounted and more likely to be reached. So the triangle’s equal time weights underestimate how high the last piece has to be to lift the average to 200.

The inverted curve. At 2,000 and 800, the first year has a hazard of about 33%. Total hazard over five years by the triangle is $5 \times 13.3\% = 66.7\%$, so the next four years average about 8.3%. The market says: this name is in immediate danger, and if it gets through that, it will be much safer. A distressed name’s long-dated contract is close to a claim on the same near-term event.

Answer to Question 11.20.

The CVA is about $110,800. Its sensitivity to the spread is about $515 per basis point, so buy about $1.1 million of protection. After the rate move, CVA is $182,500. After the spread move it is $160,600. After both it is $265,400. So the two together add $33,100 beyond the sum.

Value. With $h = 0.02/0.6 = 3.33\%$ and $\mathrm{EE}(t) = 0.399\sqrt{t}\,(5 - t)$ per 100,

Without the survival weighting, it would be $0.02 \times 5.947 = 0.119$.

Hedge. Revalue with the spread at 201: CVA rises by $515. A $1 million 5-year CDS gains its risky annuity times one basis point: $(1 - e^{-5h})/h \times 10^{-4} \times \$1$ million $= 4.61 \times \$100 = \$461$. So you need $515/461 = \$1.12$ million of protection, which is roughly 1% of the swap’s notional. It is small because exposure is small against notional.

Rate move. If the rate has moved so that the swap is in your favour by 50 basis points, exposure gets a mean $m = 0.005$ and

That gives $182,500. A 50-basis-point move against the swap would give $64,100, so the sensitivity is not symmetric, and the delta at the money is about $1,180 per basis point.

Both. Adding the separate moves gives $110{,}800 + 71{,}700 + 49{,}700 = \$232{,}200$, and the true value is $265,400. The extra $33,100 is cross-gamma between rates and credit. When you are in the money the CS01 is $855, up from $515. So your spread hedge needs to be $1.86 million, and it was short by $0.74 million for exactly the move that hurts.

So you must rebalance the credit hedge when rates move, and the rates hedge when spreads move. And a desk that hedges each risk separately still pays the cross term. With wrong-way risk, the cross term gets larger, as spread and exposure then move together.

No traded CDS. Hedge with a proxy, such as a sector and rating index or a related name. You remove the systematic spread risk, and you keep the idiosyncratic risk. That is a real residual, and you hold reserves or capital against it. You state plainly that the jump to default is not hedged.