Assignment Module Credit Value Adjustment (CVA)

Assignment Module 8 2017 Credit Value Adjustment (CVA) Quantitative Risk Management MSc in Mathematical Finance (part-time) June 4, 2017

Contents 1 Introduction 4 2 A brief history of counterparty risk and CVA 5 3 Managing CVA and the role of the CVA desk 6 4 CVA mathematical background 7 4.1 Derivation of the CVA formula.................. 7 4.2 CVA formula under risk-neutral default probability...... 9 4.3 CVA as a spread.......................... 10 4.4 Incremental CVA......................... 10 5 CVA Capital Charge 11 6 CVA numerical example 11 6.1 Credit spreads and probability of default.............. 11 6.2 Numerical examples on the CVA formula and CVA as a spread.. 13 6.3 Numerical example on incremental CVA.............. 15 7 Conclusions 16 8 Appendix A 19 9 Appendix B 21 2

List of Figures 1 Illustration of the role of a CVA desk in a bank........... 6 2 Different shapes of credit spread curves with a 5Y spread of 220 bps. 12 3 Cumulative default probabilities for an upward-sloping, flat and inverted credit spread curve with a 5Y spread of 220 bps and an LDG of 60%............................. 12 4 FX forward expected exposure (EE) (as a percentage of notional) and default probability profile over time for an inverted credit spread curve................................. 14 5 FX forward expected exposure (EE) (as a percentage of notional) and default probability profile over time for an upward-sloping credit spread curve.......................... 14 6 Expected exposure profile of: (i) the original portfolio, (ii) the portfolio with the addition of a risk-increasing trade, and (iii) the portfolio with the addition of a risk-reducing trade........... 15 List of Tables 1 Three shapes of credit spread curves................ 11 2 Credit spread curves........................ 13 3 CVA, EPE, and approximate CVA as a spread of a 5Y and a 10Y FXFW................................ 13 4 CVA of the original portfolio, the portfolio with the addition of a risk-increasing trade, and the portfolio with the addition of a riskreducing trade............................ 15 3

1 Introduction Traditional pricing of derivatives considered, in general, solely the replacement value of a derivative, namely the present value of future cashflows [1]. For simple transactions, this problem was always considered to be relatively straightforward and was often simply a question of applying the correct risk-free discount rate. The pricing of derivatives was only difficult where the cashflows were themselves more complex, such as being non-linear, contingent or multidimensional, but for vanilla derivatives this was assumed to be quite trivial. An important consequence of the recent financial crisis is that the derivatives have become more expensive for the end users as more and more costs and risks are built into the pricing of derivatives [2]- [5]. Some of the embedded costs include: Credit Value Adjustment (CVA), i.e. the valuation of counterparty risk Funding Value Adjustment (FVA), i.e. the cost and benefit arising from the funding of the transaction Capital Value Adjustment (KVA), i.e. the cost of holding capital (typically regulatory) over the lifetime of the transaction Collateral Value Adjustment (ColVA), i.e. costs and benefits from the embedded optionality in the collateral agreement and any other non-standard collateral terms Initial Margin Value Adjustment (MVA), i.e. the cost of posting initial margin over the lifetime of the transaction The general term xva has been used to classify the variety of valuation adjustments that are included in the pricing of derivatives, such as the aforementioned types of derivative value adjustments [1]. In mathematical terms, this can be written as an add on to the replacement value of the derivative, i.e. V actual = V ideal + xva (1) where V actual denotes the adjusted value of the derivative that includes the various valuation adjustments (e.g. CVA, FVA, KVA, MVA etc.). In this paper, we will focus on CVA (Credit Value Adjustment), which is the adjustment made to the price of a derivative transaction as a result of the risky nature of the counterparty [6], [7]. CVA is essentially the expected loss on a trade that occurs from the default of the counterparty and along with the other xva components has become an effective price factor of a deal, just like the interest rate or the exchange rate. CVA could affect whether a trade is profitable or not once the valuation adjustment has been priced in and thus pricing counterparty credit risk through CVA could actually affect trading decisions and strategies [2]. 4

2 A brief history of counterparty risk and CVA Counterparty risk first gained prominence in the late 1990s when the Asian crisis (1997) and the default of Russia (1998) highlighted some of the potential problems of major defaults in relation to derivative contracts [2]. It was actually the failure of Long-Term Capital Management (1998), a hedge fund (whose founders included R. Merton and M. Scholes) that was trading with many large banks which had the most impact, due to the fear of a knock-on impact that would lead to a cascade of defaults. Both the growth of the derivative markets and the default of some significant clients (e.g. Enron, WorldCom) led banks to better quantify and allocate such losses. Banks started to price counterparty risk into transactions, mainly focusing on the more risky trades and counterparties. Traders and salespeople were charged for this risk, which was often managed centrally by the CVA desk. CVA amounted to an expected loss and the CVA desk generally acted as an insurer of counterparty risk that built up a collective reserve to offset against counterparty defaults. The above approach to counterparty risk started to change in 2005 with changes introduced by the accounting standards (IAS 39, FAS 157) that required derivatives to be held at a fair value associated with the concept of exit price. This implied that CVA was a requirement that should be adjusted to the price of a derivative. Banks, however, made dramatic errors in their assessment of counterparty risk (e.g. monoline insurers), undertook regulatory arbitrage to limit their regulatory capital requirements, were selective about reporting CVA in financial statements and did not routinely hedge CVA risk, which contributed to the recent financial crisis. During the crisis, CVA losses increased dramatically [8], [9]. A loss-attribution exercise conducted by the UK Financial Service Authority (FSA) on the losses incurred on their market operations by large UK banks during the period 2007-2009 concluded that CVA losses were five times the amounts of actual default losses [10]. These occurred from the global deterioration of credit quality of most participants in the derivative markets, but they were highly concentrated on banks exposures to monoline insurers and Credit Derivative Product Companies (CDPC) that were providing credit protection on asset-backed securities and structured credit derivative instruments. CVA losses on this type of exposures were severe due to the fact that these exposures were generally un-collateralised and directional (all of these counterparties were net providers of credit protection). When the asset-backed securities and structured credit derivatives started to underperform (therefore increasing the banks exposures to the providers of credit protection), it became clear that the monoline insurers and CDPCs would have to make large payments in the future to compensate their clients, which led their credit spreads to widen. The fact that some large banks recognised billions in CVA losses in some instances led the Basel Committee to consider CVA risks as a potential source of financial instability against which capital should be held. This led to the introduction of the CVA Capital Charge [11] that will be further discussed in Section 5. One thing that became clear after the recent financial crisis was that no counterparty (triple-a entities, global investment banks, retail banks and sovereigns) could 5

ever be regarded as risk-free. CVA, which is defined as the price of counterparty risk, has gone from being a technical term that was rarely used to a buzzword constantly associated with derivatives [2], [12]. As a result, the pricing of counterparty risk into trades (via a CVA charge) has now become the rule. Whilst the largest investment banks had built trading desks, complex systems and models around CVA, all banks have now focused on expanding their capabilities in this respect, highlighting the importance of the CVA desk. 3 Managing CVA and the role of the CVA desk Banks have set up CVA desks that manage the derivative counterparty risk in the event of a counterparty default and the mark-to-market (MtM) volatility of CVA [2]. While the setup is different from bank to bank, the aim, in general, is for the counterparty risk of the originating trading or sales desk to be priced and managed. CVA desks charge and then underwrite the counterparty risk with the aim of minimising losses in the event of a default. They manage default and close-out processes and advise on restructuring of derivatives with distressed counterparties. An illustration of the role of a CVA desk in a bank can be seen in Fig. 1 [2]. Figure 1: Illustration of the role of a CVA desk in a bank. Certain transactions will be more significant than others from a CVA point of view, such as long-dated or un-collateralised derivatives for example. However, it should be noted that CVA is calculated on a portfolio basis and if a derivative increases the risk of the portfolio, it will incur a CVA charge, otherwise the CVA desk will have to rebate the originating desk. The hedging of counterparty credit risk and CVA has been facilitated by the growth of the credit derivatives market and in practice it is usually done on a dynamic basis with reference to credit spreads (mainly via CDS indices and/or singlename CDS) and other dynamic market variables (e.g. interest rates, FX, etc.) [2]. In the following section, we provide the mathematical background for the calculation of CVA, which will provide a better insight in the credit valuation adjustment on derivative pricing. 6

4 CVA mathematical background 4.1 Derivation of the CVA formula Let V (t, T ) denote the risk-free value at time t of a netting set of derivatives with maximum maturity T. We are interested in finding an expression for the risky value Ṽ (t, T ) of the netting set that includes the counterparty credit risk. Let s assume that τ denotes the default time of the counterparty. There are two cases to consider [13], namely: Case 1: The counterparty has not defaulted before time T In this case, the risky position is equivalent to the risk-free position and the corresponding payoff can be written as I (τ > T ) V (t, T ) (2) where I (τ > T ) is the indicator function denoting the default of the counterparty, i.e. it is equal to 1 if the counterparty has not defaulted by time T and 0 otherwise. Case 2: The counterparty defaults before time T The payoff in this scenario consists of two terms, namely the value of the position that would be paid before the default time τ and the payoff at default. The cashflows up to default time can be written as whereas the default payoff can be written as I (τ T ) V (t, τ) (3) I (τ T ) ( RV (τ, T ) + + V (τ, T ) ) (4) where x + = max (0, x) and x = min (0, x). In the default scenario, if the MtM of the netting set V (τ, T ) is positive then the institution will receive a recovery fraction R of the risk-free value of the derivatives, whereas if it is negative it will still have to settled this amount. If we combine the two terms above, we get the value of the the risky position under the risk-neutral measure as [15] Ṽ (t, T ) = E Q I (τ > T ) V (t, T ) + I (τ T ) V (t, τ) + I (τ T ) ( RV (τ, T ) + + V (τ, T ) ) (5) Using the relationship x = x x +, the equation (5) can be re-written as I (τ > T ) V (t, T ) + Ṽ (t, T ) = E Q I (τ T ) V (t, τ) + I (τ T ) [ RV (τ, T ) + + V (τ, T ) V (τ, T ) +] = I (τ > T ) V (t, T ) + = E Q I (τ T ) V (t, τ) + I (τ T ) [ (R 1) V (τ, T ) + + V (τ, T ) ] (6) 7

Given that V (t, τ) + V (τ, T ) V (t, T ), the equation above can be written as I (τ > T ) V (t, T ) + Ṽ (t, T ) = E Q I (τ T ) V (t, T ) + I (τ T ) (R 1) V (τ, T ) + (7) The first two terms in the expectation can be combined given that I (τ > T ) V (t, T ) + I (τ T ) V (t, T ) V (t, T ) and therefore the equation (7) can be written as Ṽ (t, T ) = V (t, T ) E Q [ I (τ T ) (1 R) V (τ, T ) +] (8) The equation above defines the risky value Ṽ (t, T ) of the netting set of derivatives with respect to the risk-free value V (t, T ). The second term CV A (t, T ) = E Q [ I (τ T ) (1 R) V (τ, T ) +] (9) in equation (8) is known as CVA (Credit Value Adjustment) and is an adjustment to the risk-free value V (t, T ) of the netting set to account for counterparty credit risk. This can be written as Ṽ (t, T ) = V (t, T ) CV A (t, T ) (10) It should be noted that the notation V (s, T ) for the future MtM of the netting settled includes discounting. This was done for notation simplicity. Using more rigid notation, the equation (9) can be written as [ β (t) CV A (t, T ) = E Q I (τ T ) (1 R) V (τ, T )+ β (τ) were β (s) denotes the value of the money market account at time s. To derive the classic CVA formula, we can then write ] (11) CV A (t, T ) = ( 1 R ) E Q [ I (τ T ) V (τ, T ) +] (12) where R is the mean or expected recovery rate. We use V (u, T ) to denote V (u, T ) = V (u, T ) τ = u (13) The above statement requires the exposure at a future date, V (u, T ), knowing that the default of the counterparty has occurred at time τ = u. By ignoring wrong-way risk, we have V (u, T ) = V (u, T ). More details regarding CVA and wrong way risk can be found in [16], [17]. Since the expectation in the above equation is over all times before the final maturity T, we can integrate over all possible default times, as follows CV A (t, T ) = ( 1 R ) [ T ] E Q B (t, u) V (u, T ) + df (t, u) (14) t 8

where B (t, u) is the risk-free discount factor and F (t, u) is the cumulative default probability of the counterparty. The discounted expected exposure (EE) calculated under the risk-neutral measure is given by EE d (u, T ) = E Q [ B (t, u) V (u, T ) +] Assuming that the default probabilities are deterministic, we have from (14) CV A (t, T ) = ( 1 R ) T EE d (u, T ) df (t, u) (15) The equation above can be computed via some integration scheme as t CV A (t, T ) ( 1 R ) m EE d (t, t i ) [F (t, t i ) F (t, t i 1 )] (16) i=1 where we have m periods given by [t 0 (= t), t 1,..., t m (= T )]. As long as m is reasonably large then this will be a good approximation. 4.2 CVA formula under risk-neutral default probability Assuming that the default probability is driven by a Poisson process, the default probability for a future period u is given by [18] F (u) = 1 exp ( hu) (17) where h defines the hazard rate of default, which is the conditional default probability in an infinitesimally small period. An approximate relationship 1 between the hazard rate and the credit spread is given by [18] h s (18) LGD where the assumed loss given default (LGD) is a percentage. From the equations (17) and (18) we get the following approximate expression for the risk-neutral default probability up to a given time u ( F (u) = 1 exp s ) LGD u (19) The above formula is a good approximation, although computing the implied default probability accurately requires solving numerically for the correct hazard rate assuming a certain underlying function form [18], [19]. Finally, the CVA from equation (16) can be written as CV A (t, T ) ( 1 R ) m i=1 [ ( EE d (t, t i ) exp s LGD t i 1 ) ( exp s )] LGD t i (20) 1 This assumes that the credit spread term structure is flat and that the CDS premiums are paid continuously. 9

4.3 CVA as a spread CVA is usually expressed as a stand-alone value, but sometimes it is useful to express it as a spread (per annum charge). Suppose that we approximate the (undiscounted) expected exposure term, EE (u, T ), as a fixed known amount as follows [20] EP E = 1 T EE (u, T ) du 1 m EE (t, t j ) (21) T t t m The approximation will be a good one if the relationship between EP E, default probability and discount factors is reasonably homogeneous through time [13]. Using this approach, the CVA is given by CV A (t, T ) = ( 1 R ) [ T ] E Q B (t, u) df (t, u) EP E (22) This is simply the value of a CDS protection on a notional that equals the EP E. Therefore, the following approximation provides the running CVA, which is expressed as a spread t j=1 CV A (t, T ) EP E Spread (23) Using risky annuity formulas, this can be converted to an upfront amount [13], [14]. 4.4 Incremental CVA To calculate the incremental CVA, we need to quantify the change before and after adding a new trade i to the portfolio CV A NS+i (t, T ) CV A NS (t, T ) = = ( 1 R ) m EE NS+i (t, t i ) [F (t, t i ) F (t, t i 1 )] i=1 ( 1 R ) m EE NS (t, t i ) [F (t, t i ) F (t, t i 1 )] = i=1 = ( 1 R ) m [ EE NS+i (t, t i ) EE NS (t, t i ) ] [F (t, t i ) F (t, t i 1 )] (24) i=1 We therefore simply need to use the incremental EE, i.e. EE NS+i (t, t i ) EE NS (t, t i ) in the standard CVA formula. Using equation (16), the incremental CVA in (24) under the risk-neutral default probability can be written as CV A NS+i (t, T ) CV A NS (t, T ) = = ( 1 R ) m [ EE NS+i (t, t i ) EE NS (t, t i ) ] [ exp ( s ) ( LGD t i 1 exp s )] LGD t i i=1 (25) 10

5 CVA Capital Charge A CVA capital charge was recently introduced with the aim to supplement the existing Basel II requirements which capitalise only potential losses due to default and credit migrations [2]. The motivation for this was that in the recent financial crisis only one-third of the counterparty risk-related losses were due to defaults and the rest were MtM-based [21]. As a result, Basel III has increased the counterparty risk capital requirements via a new additional component related to CVA rather than imposing a more conservative application of the existing rules (e.g. via a higher multiplier) [11]. It was noted that CVA accounting volatility represents a substantial risk and as such it needs to be capitalised. This risk is represented by the VaR, since CVA can be seen as a market risk for the trading book of a bank. Banks have been trying to optimise their capital position over the past few years and hedging CVA (thus reducing the relevant CVA capital requirements) often leads to profit-and-loss (P&L) volatility. There is actually a trade-off where reducing the CVA capital charge requirements could lead to real losses [23] and this is something that needs to be managed and optimised. The details of the CVA capital charge calculation (advanced and standardised method) are outside the scope of this paper and the reader is referred to [2], [11], [22] for more details on the regulatory CVA capital charge calculation. 6 CVA numerical example In this section, we provide a numerical example of the CVA calculation and discuss the different components used in the calculation of CVA. 6.1 Credit spreads and probability of default We consider three different credit spread curves, i.e. an upward-sloping curve, a flat curve and an inverted curve. The curves considered in this example can be found in Table 1 and their plot in Fig. 2. Table 1: Three shapes of credit spread curves Time Inverted curve (bps) Flat curve (bps) Upward-sloping curve (bps) 1Y 350 220 150 3Y 300 220 175 5Y 220 220 220 7Y 170 220 240 10Y 120 220 250 11

Figure 2: Different shapes of credit spread curves with a 5Y spread of 220 bps. In Fig. 3, the cumulative probability of default is plotted for the three credit spread curves presented in Table 1 using equation (19). All three curves have a five-year spread of 220 bps and an LGD of 60%. It can be seeing in Fig. 3 that the three curves agree on the five-year cumulative default probability, however, the shape of the curve before and after the five-year point is different for each curve. For an upward-sloping curve, a default is less likely in the early years and more likely in the later years, whereas the opposite can be seen for the inverted credit spread curve [2]. Thus, in order to calculate the risk-neutral default probabilities properly, both the level and the shape of the credit spread curve are important. Figure 3: Cumulative default probabilities for an upward-sloping, flat and inverted credit spread curve with a 5Y spread of 220 bps and an LDG of 60%. 12

6.2 Numerical examples on the CVA formula and CVA as a spread We consider an FX forward (FXFW) that matures in five (5Y) and ten (10Y) years and calculate the CVA and approximate CVA as a spread for the upward-sloping and inverted credit spread curves in Table 2 using the equations (20) and (23). The LGD is assumed to be 60%. The CVA results can be found in Table 3 and the profiles used in the CVA calculations can be found in Appendix A. Table 2: Credit spread curves Time Inverted credit curve (bps) Upward-sloping credit curve (bps) 1Y 800 400 3Y 700 500 5Y 600 600 7Y 500 700 10Y 400 800 Table 3: CVA, EPE, and approximate CVA as a spread of a 5Y and a 10Y FXFW 5Y FXFW 10Y FXFW 5Y FXFW 10Y FXFW Spread upward-sloping upward-sloping inverted inverted CVA -1.51% -3.73% -1.28% -1.87% EPE 6.17% 8.58% 6.17% 8.58% CVA approx (bps) -37.00-68.65-37.00-34.32 It can be seen from Table 3 that the CVA for the long-dated FXFW (10Y) is higher compared to the short-dated FXFW (5Y), which is expected based on equation (20) and the EE profile seen in Fig. 4. This is because the CVA calculation for the 10Y FXFW includes the additional positive terms post Year 5. As the EE profiles and credit spreads up to Year 5 are exactly the same between the two FXFWs, it is expected that the longer-dated FXFW will have a higher CVA due to the increasing expected exposure profile of the FXFW. Moreover, it can be seen from Table 3 that the CVA spread (CVA approx (bps)) for the 10Y FXFW is higher for the upward-sloping credit spread curve compared to the inverted credit spread curve. This is due to the fact that the 10Y credit spread used in the approximate CVA as spread formula is higher in the upward-sloping curve (800 bps) compared to the inverted credit spread curve (400 bps), as seen in Table 2. This is expected and can be explained based on equation (23). In addition, the 5Y FXFW CVA approx (bps) is the same between the two credit spread curves in Table 3, as both FXFWs have the same EPE and 5Y credit spread value (600 bps). 13

Figure 4: FX forward expected exposure (EE) (as a percentage of notional) and default probability profile over time for an inverted credit spread curve. Finally, it can be clearly seen from Fig. 4, Fig. 5 and equation (20) why the CVA for the 10Y FXFW in Table 3 is higher under an upward-sloping credit spread curve compared to an inverted credit spread curve. This can be explained by the increasing nature of the EE profile of the FXFW and the decreasing default probability profile over time of the inverted credit spread curve. As the expected exposure of the FXFW increases, the default probability of the counterparty decreases, which as a result leads to lower CVA. Figure 5: FX forward expected exposure (EE) (as a percentage of notional) and default probability profile over time for an upward-sloping credit spread curve. 14

6.3 Numerical example on incremental CVA Finally, we study the incremental CVA when adding to a portfolio of nettable derivatives: (i) a risk-increasing trade, and (ii) a risk-reducing trade. For the calculation of CVA, we consider an LGD of 60% and the upward-sloping credit spread curve (and respective default probability) given in Appendix A. The expected exposure profiles of the original portfolio, the portfolio with the addition of a riskincreasing trade, and the portfolio with the addition of a risk-reducing trade are plotted in Fig. 6 and the respective values in GBP are given in Appendix B. Figure 6: Expected exposure profile of: (i) the original portfolio, (ii) the portfolio with the addition of a risk-increasing trade, and (iii) the portfolio with the addition of a risk-reducing trade. The CVA of the three portfolios is given in Table 4. As expected the addition of a risk-increasing trade to the portfolio leads to an increase in CVA, whereas the opposite is observed for the addition of a risk-reducing trade. This is in line with equation (25) and in practical terms it means that the CVA losses would be higher when the risk-increasing trade is included in the portfolio and this needs to be reflected in the price of the derivative. In this scenario, the CVA desk would charge the originating trading/sales desk with the incremental CVA. Table 4: CVA of the original portfolio, the portfolio with the addition of a riskincreasing trade, and the portfolio with the addition of a risk-reducing trade. Original Portfolio with a Portfolio with a portfolio risk-increasing trade risk-reducing trade CVA -54m -73m -48m Incremental CVA N/A -19m 6m 15

7 Conclusions In this paper, we discussed how the pricing of derivatives has changed post the recent financial crisis to include derivative value adjustments known as xva components that reflect the additional costs and risks embedded in derivatives. We focused on the credit value adjustment (CVA) and discussed its history and motivation, how it is calculated and embedded in the price of derivatives, how it is managed by banks, and its importance as viewed by the regulators with the introduction of the Basel III CVA Capital Charge that requires banks to hold extra capital for counterparty credit risk. Finally, we provided the mathematical background of CVA along with numerical examples and discussed how the various components impact the CVA calculation. It was noted that the credit spread curve and expected exposure profile can have a significant effect on the CVA of a derivative, and that, given that CVA is calculated on a portfolio basis, a risk-reducing trade will decrease the CVA of the portfolio, whereas the opposite can be observed for a risk-increasing trade. Finally, it was discussed that there is a trade-off between P&L and CVA, which is an area of increasing interest for the banks as their are trying to optimise their capital usage to maximise their profits. 16

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8 Appendix A Time Upward-sloping spread Default Probability 5Y FXFW (years) (bps) EE 0 350 0% 0% 0.25 362 1.5% 2% 0.5 375 1.58% 2.83% 0.75 387 1.65% 3.46% 1 400 1.73% 4% 1.25 413 1.8% 4.47% 1.5 425 1.84% 4.9% 1.75 438 1.91% 5.29% 2 450 1.94% 5.66% 2.25 463 2.01% 6% 2.5 475 2.02% 6.32% 2.75 488 2.09% 6.63% 3 500 2.08% 6.93% 3.25 512 2.1% 7.21% 3.5 525 2.16% 7.48% 3.75 537 2.13% 7.75% 4 549 2.14% 8% 4.25 562 2.19% 8.25% 4.5 574 2.14% 8.49% 4.75 587 2.19% 8.72% 5 600 2.18% 8.94% 5.25 613 2.17% 0% 5.5 626 2.15% 0% 5.75 639 2.13% 0% 6 652 2.11% 0% 6.25 665 2.08% 0% 6.5 677 2% 0% 6.75 689 1.96% 0% 7 700 1.87% 0% 7.25 711 1.84% 0% 7.5 721 1.75% 0% 7.75 730 1.66% 0% 8 739 1.62% 0% 8.25 748 1.58% 0% 8.5 756 1.49% 0% 8.75 764 1.45% 0% 9 771 1.36% 0% 9.25 779 1.37% 0% 9.5 786 1.28% 0% 9.75 793 1.24% 0% 10 800 1.21% 0%

Time Inverted spread Default Probability 10Y FXFW (years) (bps) EE 0 850 0% 0% 0.25 838 3.43% 2% 0.5 825 3.21% 2.83% 0.75 813 3.02% 3.46% 1 800 2.82% 4% 1.25 787 2.64% 4.47% 1.5 775 2.49% 4.9% 1.75 762 2.31% 5.29% 2 750 2.19% 5.66% 2.25 737 2.03% 6% 2.5 725 1.93% 6.32% 2.75 712 1.77% 6.63% 3 700 1.69% 6.93% 3.25 688 1.58% 7.21% 3.5 675 1.44% 7.48% 3.75 663 1.38% 7.75% 4 651 1.28% 8% 4.25 638 1.15% 8.25% 4.5 626 1.11% 8.49% 4.75 613 0.98% 8.72% 5 600 0.9% 8.94% 5.25 587 0.82% 9.17% 5.5 574 0.75% 9.38% 5.75 561 0.67% 9.59% 6 548 0.6% 9.8% 6.25 535 0.53% 10% 6.5 523 0.53% 10.2% 6.75 511 0.47% 10.39% 7 500 0.47% 10.58% 7.25 489 0.42% 10.77% 7.5 479 0.43% 10.95% 7.75 470 0.46% 11.14% 8 461 0.41% 11.31% 8.25 452 0.37% 11.49% 8.5 444 0.4% 11.66% 8.75 436 0.36% 11.83% 9 429 0.4% 12% 9.25 421 0.29% 12.17% 9.5 414 0.34% 12.33% 9.75 407 0.3% 12.49% 10 400 0.27% 12.65% 20

9 Appendix B Time Original Portfolio with a risk- Portfolio with a risk- (years) portfolio EE increasing trade EE reducing trade EE 0 0 0 0 0.25 88,844,297 98,844,297 82,846,530 0.5 111,700,425 125,842,561 102,455,269 0.75 134,178,628 151,499,136 123,146,176 1 149,405,061 169,405,061 137,530,084 1.25 158,526,866 180,887,546 145,033,906 1.5 162,688,806 187,183,704 148,546,711 1.75 166,287,700 192,745,213 151,402,912 2 169,463,084 197,747,355 153,833,651 2.25 172,330,285 202,330,285 156,249,311 2.5 177,340,525 208,963,301 160,381,423 2.75 182,297,679 215,463,927 164,796,110 3 181,125,940 215,766,956 163,418,007 3.25 181,275,206 217,330,718 163,963,587 3.5 178,469,480 215,886,054 160,849,441 3.75 178,050,540 216,780,373 160,487,976 4 181,599,710 221,599,710 164,130,410 4.25 182,070,104 223,301,160 165,029,130 4.5 180,374,988 222,801,395 163,561,814 4.75 170,547,149 214,136,139 153,813,075 5 165,782,517 210,503,877 149,415,847 5.25 113,749,309 159,575,066 97,735,044 5.5 112,995,429 159,899,586 97,345,284 5.75 108,844,806 156,803,121 93,604,681 6 107,005,266 155,995,061 92,321,912 6.25 96,647,062 146,647,062 82,581,569 6.5 96,179,430 147,169,625 82,938,159 6.75 88,987,569 140,949,093 76,416,107 7 84,065,990 136,981,016 72,404,064 7.25 77,641,815 131,493,463 66,996,138 7.5 71,553,106 126,325,361 61,868,734 7.75 66,134,997 121,812,641 57,367,373 8 61,968,242 118,536,785 54,135,148 8.25 55,587,704 113,033,331 48,761,622 8.5 48,100,166 106,409,685 42,286,815 8.75 40,551,118 99,711,916 35,715,734 9 34,121,656 94,121,656 30,186,288 9.25 26,722,842 87,550,467 23,791,674 9.5 17,965,105 79,609,245 16,012,873 9.75 9,488,658 71,938,638 8,521,915 10 0 63,245,553 0