狀態轉換漸進極值因子模型下擔保債權憑證之評價與避險 - 政大學術集成

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(1)國立政治大學商學院金融學系研究所碩士學位論文 Graduate Department of Money and Banking College of Commerce National Chengchi University Master Thesis. 狀態轉換漸近極值因子模型下擔保債權憑證之. 立. 治政評價與避險大. ‧ 國. 學. Pricing and Hedging of CDOs under a Regime Switching Asymptotic Single Factor Model. ‧. n. CKuan-Yu Lai hengchi. er. io. al. sit. y. Nat. 賴冠宇. i n U. v. 指導教授：江彌修博士 Advisor: Mi-Hsiu Chiang, Ph.D. 中華民國 104 年 6 月 June, 2015.

(2) 誌謝此篇論文得以順利完成，要感謝我的指導教授江彌修老師，謝謝您在我還對於未來迷惘的時候就答應讓我的指導教授，也謝謝您每學期都固定帶我們去酒足飯飽一番，更要謝謝您每個禮拜不斷督促我論文進度，指導我研究的方向，一切收穫，感激不盡。除了論文的指導外，老師平常對我人格上的循循善誘，使我對自己性格上的缺點以及如何去克服有更深刻的認識，老師不但是我的授業老師，更是一位重要的人生導師。另外，也要感謝博士班的信瑜、啟均以及明哲學長，. 政治大. 一路走來給我的幫助，特別是信瑜學長，感謝你每個星期撥空幫我檢查程式的錯. 立. 誤，以及協助我解決寫論文時所遇到的問題。. ‧ 國. 學. 最後，要感謝一路支持我念書的家人，讓我能無後顧之憂的完成我的學業。. ‧. 也要感謝一路陪伴我的宛璇，在那段暗無天日的程式語言裡，鼓勵我不要放棄。. sit. y. Nat. 政大六年轉眼即逝，帶不走的是回憶，帶走的是專業知識的成長，以及終. io. n. al. er. 身學習的態度，相信這些能力將陪伴我繼續走向人生的下一段旅程。再一次，謝. i n U. v. 謝所有一路上幫助過我的人，謹以本文獻給我深愛的朋友們。. Ch. engchi. 賴冠宇僅誌於政大金融所中華民國一百零四年六月 i.

(3) 摘要本篇論文使用了 LHP 的近似方法去評價擔保債權憑證，並推導出漸進極值因子模型，又稱單因子 copula 模型，單因子 copula 模型被廣泛運用在 CDO 之風險管理與一些風險因子模擬之應用，但由於 2008 年之金融海嘯造成市場標準模型 Gaussian copula model 會有評價上的誤差，所以為了能在市場不穩定時能更精確的求算出分券價差，我們必須找到一個更簡單且快速捕捉到市場不穩定性的模型。在這篇論文中，我們引用了 Anna Schloesser 在 2009 年所提出以 NIG copula model. 治政為基礎的兩個延伸，讓模型更穩健和且擁有良大好的性質去進行模擬， NIG 立 ‧ 國. 學. Regime-Switch 模型有兩大特色: (i)可以用一致的方法去評價不同到期日的分券，放寬了同一分券必須是相同到期日的假設，和(ii)有不同的相關係數狀態，對於金. ‧. 融風暴來說，狀態轉換可以有效地降低市場不穩定所帶來的評價誤差。本文也對. Nat. sit. y. 不同模型下的 CDO 進行風險分析與避險，分券的期望損失廣泛被信評公司視為一. n. al. er. io. 項審定信用評等重要的風險衡量指標，但是並無法真實反映出擔保債權憑證分券. Ch. i n U. v. 之間相對風險之大小，因此本文採用期望損失率的觀念，利用期望損失佔本金的. engchi. 比例來比較各分券之相對風險，且本文也求算出 CDO 之避險參數，讓投資人了解對合成行擔保債權憑證分券避險時所需之避險部位，分券持有人也可依據所要規避的風險類型，選擇市場上現有的信用違約交換指數或是單一資產之信用違約交換(single-name credit default swap)來進行避險。. ii.

(4) Abstract This paper presents the Large Homogeneous Portfolio (LHP) approach to the pricing of CDOs and we derive the one-factor copula model. It is popular that the one-factor copula models are very useful for risk management and measurement applications involving the generation of scenarios for the complete universe of risk factors. However, since the financial crisis in 2008 induces some errors in the valuation by Gaussian copula model, which is originally adopted by credit rating firms, it is. 治政大 which are first present by this paper we apply two extensions of the NIG copula model, 立. necessary to have a simple and fast model that can capture the market unstableness. In. Anna Schloesser (2009), since they make the model well defined and powerful for. ‧ 國. 學. scenario simulation. The NIG Regime-Switch copula model allows for two important. ‧. features: (i) tranches with different maturities modeled in a consistent way, and (ii). sit. y. Nat. different correlation regimes. The regime-switching component of the NIG copula. io. er. model is especially important in view of the financial crisis. This paper also targets on different models to conduct risk analysis and hedging strategy. The expected loss of. al. n. v i n tranches is widely used by creditC rating as one of the important indicators h eorganizations ngchi U for risk measurement. However, it can’t reflect the relative risk level between CDO’s. tranches. Therefore, our research adopts the concept of expected loss rate, which use the proportion of expected loss to total principal amount to compare the relative risk of each tranche. Moreover, when we want to hedge the spread risk of synthetic CDO tranches, the holders of tranches can choose the existing CDS index or the single-name CDS based on different risks types to hedge. The employment of the NIG Regime-Switch copula model not only has more precise estimation for the spread of tranches but also possess more stable hedge ratio to hedge.. iii.

(5) Table of Contents 誌謝 ....................................................................................................................................i 摘要 .................................................................................................................................. ii Abstract ............................................................................................................................ iii Table of Contents ..............................................................................................................iv List of Figures .................................................................................................................... v List of Tables ....................................................................................................................vi Chapter 1. Introduction .............................................................................................. 7. Chapter 2. Literature Review .................................................................................. 10. 3.2. 治政 Valuation of Synthetic CDOs........................................................................ 13 大 Constructing the立 Loss Distribution ............................................................... 15. 3.3. Asymptotic Single Factor Models ................................................................ 18. 3.4. Risk Measures and Hedging Parameters of Synthetic CDO ........................ 25. 3.1. 學. Empirical and Numerical Results......................................................... 29. ‧. Chapter 4. Methodology ........................................................................................... 13. ‧ 國. Chapter 3. Model Calibration ......................................................................................... 29. 4.2. Product Assumptions .................................................................................... 33. 4.3. Pricing Results .............................................................................................. 35. 4.4. Risk Characters of Synthetic CDO Tranches ............................................... 37. 4.5. Hedging Analysis .......................................................................................... 47. y. sit. n. al. er. io. Chapter 5. Nat. 4.1. Ch. engchi. i n U. v. Conclusions ............................................................................................. 55. References ....................................................................................................................... 57. iv.

(6) List of Figures Figure 2-1 Calibration with the 5-year data from iTraxx spread ..................................... 12 Figure 4-1 Leverages of expected loss of equity tranche at each time period using different models ............................................................................................ 44 Figure 4-2 Leverages of expected loss of senior tranche at each time period using different models ............................................................................................ 46. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. v. i n U. v.

(7) List of Tables Table 4-1 Calibration result of models (before financial crisis) ...................................... 30 Table 4-2 Calibration result of models (during financial crisis) ...................................... 31 Table 4-3: Assumptions of synthetic CDO ...................................................................... 34 Table 4-4: CDO Pricing (Unit: bp) .................................................................................. 35 Table 4-5: Expected Loss (Unit: million USD) ............................................................... 37 Table 4-6: Rate of Expected Loss (Unit: %).................................................................... 38. 治政大 Table 4-8: Risk Leverage of Double t ............................................................................. 41 立 Table 4-7: Risk Leverage of Gaussian ............................................................................. 39. Table 4-9: Risk Leverage of NIG .................................................................................... 42. ‧ 國. 學. Table 4-10: Risk Leverage of NIG-Regime .................................................................... 43. ‧. Table 4-11: Delta for tranche with 5-year maturity ......................................................... 48. sit. y. Nat. Table 4-12 Gamma for tranches with 5 year maturity ..................................................... 49. io. er. Table 4-13 Hedge of equity tranches ............................................................................... 51 Table 4-14 Hedge of mezzanine tranche A ...................................................................... 52. al. n. v i n C h B...................................................................... Table 4-15 Hedge of mezzanine tranche 52 engchi U Table 4-16 Hedge of senior tranche ................................................................................. 53. vi.

(8) Chapter 1. Introduction. On April 2, 2007, New Century Financial Corporation, the second-biggest subprime mortgage lender in the United States, and its related entities filed voluntary petitions for reorganization under the bankruptcy laws in the United States. New Century Financial Corporation listed liabilities of more than $100 million and announced that the employment of about 3,200 people, more than half the workforce, will be terminated. During the week of July 16, 2007, the Bear Stearns Company, one of the largest global investment banks and securities trading firms in the world,. 政治大. disclosed the two subprime hedge funds which had lost nearly all of their value amid a. 立. rapid decline in the market for subprime mortgages.. ‧ 國. 學. These just announced the beginning of the Subprime mortgage crisis. This crisis was started from several reasons: First, the rising of the floating subprime rate, which. ‧. makes the debtors insolvent and the bursting of the US housing bubble; Second, the. y. Nat. sit. subprime mortgage were rolled into Mortgage Backed Securities (known as MBSs) by. n. al. er. io. banks and were sold to brokerage firms. These securities were re-splited and packaged. i n U. v. as CDOs, it caused such kinds of the structure of credit derivatives were too. Ch. engchi. complicated to be aware of the risk; Third, the most important reason for the research in this article, because of these derivatives lacking of historical rating records, the mathematical pricing models became the only basis for rating; Finally the investors, such as investment banks, retirement funds, hedge funds, municipal funds and administrative funds, bought these credit derivatives according to the rate given by rating agencies. During 2006, about $100 billion subprime mortgages were packaged in nearly $ 375 billion CDOs and then were sold on the U.S. market. Many banks, mortgage lenders, real estate investment trusts (REITs), and hedge funds suffered significant losses as a result of mortgage payment defaults or mortgage asset 7.

(9) devaluation. As of April 30, 2008 financial institutions had recognized subprime-related losses or write-downs exceeding U.S. $280 billion. In this article, we hope to find a model more precise than one factor Gaussian copula model and double-t copula model to estimate the fair prices for CDOs. We first provide the one factor model with Normal Inverse Gaussian (NIG) distribution under LHP assumption. Different from normal distribution and student-t distribution, the NIG distribution includes more parameters to control the location, scale and shape of the distribution function and with more flexibility. And NIG distribution is. 治政 density and inverse distribution functions can still 大 be computed sufficiently fast. 立. convolution stable under certain conditions and the cumulative distribution function,. However, the missing term structure still is a disadvantage of the model, and we can. ‧ 國. 學. observe that correlations are especially high during the current sup-prime crisis. The. ‧. year before the crisis in July 2007 began, the correlation was in contrast very low. For. sit. y. Nat. this reason, we are going to generate an extension of the NIG factor copula model,. io. er. which allowing for different correlation regimes, and we also try to extend the method. al. of base correlations, which enables NIG model to be an extension of the. n. v i n We Ccompare results h e n gthecnumerical hi U. term-structure dimension.. of the one factor. Gaussian copula model, Double-t copula model, NIG copula model and NIG Regime-Switch model. At last, each tranche’s fair price from NIG Regime-Switch model is more precise. This paper also targets on different models to conduct risk analysis and hedging strategy. The expected loss of tranches is widely used by credit rating organizations as one of the important indicators for risk measurement. However, it can’t reflect the relative risk level between CDO’s tranches. Therefore, our research adopts the concept of expected loss rate, which use the proportion of expected loss to total 8.

(10) principal amount to compare the relative risk of each tranche. Moreover, we also utilize the leverage of expected loss to analyze the changes of each tranche’s default risk at different point of time. When considering the spread risk, our paper also computes delta and gamma according to different models and we also use CDS index to conduct delta hedging analysis. Nevertheless, although we can reach to complete hedging when using CDS index to conduct delta hedging, high hedging cost will make mezzanine tranches and senior tranches have negative net income after hedging. Therefore, we will encounter difficulties when we utilize it practically. As a result, we. 治政 volatile credit spread of a specific target because the大 spread sensitivity of tranches 立 suggest that the holders of tranches can sell a single asset CDS to avoid the risk of. toward a single CDS is smaller than its own credit spread. Therefore, delta usually. ‧ 國. 學. smaller than 1 so it may involve lower hedging cost for holders of mezzanine tranches. ‧. as well as senior tranches if they adopt single asset CDS as their hedging strategy.. sit. y. Nat. This article is organized as follows: In Chapter 1 we introduced some. io. er. background for CDOs. We then make some brief introductions for several CDOs. al. pricing models from early research in Chapter 2. In Chapter 3, we use the assumptions. n. v i n C h function underUdifferent models and we also of LHP to derive the loss distribution engchi. define the risk indicators and hedge parameters. In Chapter 4, we compare numerical results of pricing synthetic CDO tranches of the iTraxx with four different models and we conduct risk analysis and calculation of hedge parameters toward different models. Finally, we give a brief conclusion in last chapter.. 9.

(11) Chapter 2. Literature Review. The calculation of loss distributions of the portfolio of reference instruments over different time horizons is the central problem of pricing synthetic CDOs. The factor copula approach for modeling correlated defaults has become very popular. Making the additional simplifying assumption of a large homogeneous portfolio (LHP), i.e. assuming it is possible to approximate the real reference credit portfolio with a portfolio consisting of a large number of equally weighted identical instruments (having the same term structure of default probabilities, recovery rates, and. 政治大. correlations to the common factor), we get a closed-form analytic synthetic CDO. 立. pricing formula. This LHP limit approximation employing the Law of Large Numbers. ‧ 國. 學. was first proposed by Vasicek.. In the case of the one factor Gaussian copula all integrals in the pricing formulas. ‧. can be computed analytically. Due to its simplicity, this model has become market. y. Nat. sit. standard. However, there is a fundamental problem: if we calculate the correlations. n. al. er. io. that are implied by the market prices of tranches of the same CDO using the LHP. i n U. v. approach, we do not get the same correlation over the whole structure but observe a. Ch. engchi. correlation smile. The main explanation of this phenomenon is the lack of tail dependence of the Gaussian copula. After that, many authors proposed other different approaches (i.e. use other copulas with more tail dependence), like the Student t-copula in O’kane and Schloegl (2003) and the double t-copula in Hull and White (2004). Burtschell et al. (2005) compared Gaussian factor copula model with several copula models, such as the stochastic correlation extension to Gaussian copula, Student t-copula, double t-copula, Clayton copula and Marshall-Olkin copula. It showed that the results of Student-t copula and Clayton copula models were similar to the Gaussian factor copula model. The Marshall-Olkin copula leads to a dramatic 10.

(12) fattening of the tail of the loss distributions. The results for double t-copula and stochastic correlation copula were closer to market quotes than others. The random factor loading introduced in Andersen and Sidenius (2005) leads to a correlation smile to stochastic correlation copula. Unfortunately, the integrals in the synthetic CDO pricing formula in the LHP model that is based on the double t copula cannot be computed analytically. The major problem is the instability of the Student t distribution under convolution. The calculation of the default thresholds requires a numerical root search procedure. 治政 Thus, finding a different heavy tailed distribution that大 is similar to the Student t but 立 involving numerical integration that increases the computation time dramatically.. stable under convolution would help to decrease the computation time tremendously.. ‧ 國. 學. In our opinion, the Normal Inverse Gaussian (NIG) distribution is an appropriate. ‧. distribution to solve the problem. The family of NIG distributions is a special case of. sit. y. Nat. the generalized hyperbolic distributions (Barndorff-Nielsen). Due to their specific. io. al. er. characteristics, NIG distributions are very interesting for applications in finance - they. n. are a generally flexible four parameter distribution family that can produce fat tails and skewness, the class is. v i n Ch convolution under certain e n gstable chi U. conditions and the. cumulative distribution function, density and inverse distribution functions can still be computed sufficiently fast. The distribution has been employed, e.g., for stochastic volatility modeling (Barndorff-Nielsen). After the correlation smile was improved by using other distributions, the missing term structure still is a disadvantage of the model. Both, the Gaussian and the NIG models (as well as all analogue models with different distribution assumptions) just average the correlations and other model parameters over the complete lifetime of a CDO tranche. Thus, applying the model to the long-dated tranches is not consistent 11.

(13) with the short-dated ones. The practitioners tried to fix this problem by extending the method of base correlations. In contrast to the Gaussian model, the NIG model allows for an extension into the term-structure dimension. This extension is not only helpful for the pricing of CDO tranches with different maturities, but also important for defining a consistent simulation framework for risk management applications. From observing the trend of iTraxx spread, Anna Schloesser (2009) detects that correlations are especially high during the current sup-prime crisis. The year before the crisis in July 2007 began, the correlation was in contrast very low. For this reason,. 治政 regimes is expected to better reflect the reality than大 the model with the constant 立. an extension of the NIG factor copula model allowing for different correlation. correlation.. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 2-1 Calibration with the 5-year data from iTraxx spread It detect the second state during the market turbulences after then downgrade of Ford and General Motors in May, 2005, and during the sub-prime crisis starting in July, 2007, with a short break in September-October,. 2007.(Source:. Anna,. Schloesser. regime-switching credit portfolio modeling through the crisis). 12. (2009).. Crash-NIG. copula. model:.

(14) Chapter 3 3.1. Methodology. Valuation of Synthetic CDOs. First, we consider a synthetic CDO with a reference portfolio consisting of credit default swaps only. A protection seller of a synthetic CDO tranche receives from the protection buyer spread payments on the outstanding notional at regular payment dates (usually quarterly). If the total loss of the reference credit portfolio exceeds the notional of the subordinated tranches, the protection seller has to make compensation payments for these losses to the protection buyer.. 政治大 (with 0 ≤ K1 < K2 ≤ 1, with 立K1 represents tranches start taking loss and K2 represents. Basically, the pricing of a synthetic CDO tranche that takes losses from K1 to K2. ‧ 國. 學. tranches end taking loss) of the reference portfolio works in the same way as the pricing of a credit default swap. Let’s assume that. ‧. 0  t0  ...  tn1. (3.1). Nat. sit. n. al. er. io. synthetic CDO.. y. denote the spread payment dates, and T (with 𝑡𝑛−1 < 𝑡𝑛 = T ) is the maturity of the. i n U. v. Then, the value of the premium leg of the tranche is the present value of all expected spread payments:. Ch. engchi. . n. . Premium Leg    ti  spread  1  EL K1 , K2   ti 1   B  t0 , ti  i 1. (3.2). where ∆𝑡𝑖 = 𝑡𝑖 − 𝑡𝑖−1 , 𝐵(𝑡0 , 𝑡𝑖 ) is the discount factor and 𝐸𝐿(𝐾1 ,𝐾2 ) (𝑡𝑖 ) is the expected percentage loss of the (𝐾1 − 𝐾2 ) CDO tranche. The value of the protection leg can be calculated according to: tn. Default Leg  B  t0 , s  dEL K1 , K2   s  t0. .   EL n. i 1. K1 , K 2 .  ti   EL K , K   ti 1    B  t0 , ti . 13. 1. 2. (3.3).

(15) At issuance of the CDO tranche the tranche spread is determined so that the values of premium leg and protection leg are equal:.  s. n.  ti   EL K , K   ti 1    B  t0 , ti  n i1  ti  1  EL K ,K   ti1    B t0 , ti . i 1.  EL. K1 , K 2 . 1. 1. 2. (3.4). 2. Equations (3.2)-(3.4) show that in order to price a CDO tranche it is necessary to know the distribution function of the tranche loss or of the overall portfolio loss. Given the portfolio loss 𝐿𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 (𝑡), the corresponding percentage tranche loss is calculated as. min  L 治 t  , K   K  政  大 L t   K  K 立 R portfolio. R  K1 , K2 . 2. 2. . 1. (3.5). 1. ‧ 國. 學. Assume, the discrete distribution of the aggregate loss of the reference portfolio after applying recoveries up to time t is known. There are m possible values that it can. ‧. take:. y. Nat. ,k LRportfolio  t   LRportfolio  t  with risk neutral probability F R t , k . sit. + (min(𝐿𝑅 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 (t),𝐾2 )−𝐾1 ). er. io. al. (3.6). k 1,, m. n. Then the(𝐾1 − 𝐾2 )CDO tranche suffers a percentage loss of. Ch. engchi. i n U. v. 𝐾2 −𝐾1. with probability 𝐹 𝑅 (𝑡, 𝑘) and the expected loss of the tranche up to time t can be easily calculated:. .  min LR  portfolio  t  , K2   K1 R EL K1 , K2   t   E Q   K 2  K1 . . . .    . m 1 ,k  min  LRportfolio  t , K2   K1  K 2  K1 k 1. . .  F R t, k. . (3.7). Now we consider the case when a continuous portfolio loss distribution function 𝐹 𝑅 (𝑡, 𝑘), accounting for portfolio loss after the recovery is applied, is known, the percentage expected loss of the (𝐾1 − 𝐾2 ) CDO tranche can be computed as: 14.

(16) 1. 1 (min  x, K 2   K1 )dF R  t , x  t    K 2  K1 K1. R  K1 , K2 . EL. (3.8). The expected tranche loss can be written as. ELR K1 , K2   t  . 1 1  1  R x K dF t , x x  K 2  dF R  t , x         1   K 2  K1  K1 K2 . (3.9). Thus, the central problem in the pricing of a CDO tranche is to derive the loss distribution of the reference portfolio. In the next sections we present the factor copula model of correlated defaults as well as semi-analytical and analytical approximation methods to compute the portfolio loss distribution and the expected loss of a tranche.. ‧ 國. 學. 3.2. 立. 政治大. Constructing the Loss Distribution. ‧. Introducing the assumption of the large homogeneous portfolio allows to derive. Nat. n. al. er. io. that makes the valuation of synthetic CDOs very fast.. sit. y. analytical formulas for the portfolio loss distribution and the expected tranche loss,. i n U. v. Definition (Large Homogeneous Portfolio (LHP)). The large homogeneous. Ch. engchi. portfolio is a portfolio consisting of a sufficiently large number of issuers having the same characteristics: i.. the same portfolio weights. ii.. the same default probability 𝑄(𝑡). iii.. the same recovery 𝑅. iv.. the same correlation to the market factor a. The analytic expression for the distribution function of the overall portfolio loss. of a large homogeneous portfolio, under the assumption of a factor copula for the dependence structure, is the central result for this class of analytic models for basket 15.

(17) credit derivatives. In this section, we want to derive a generalized result that can be used in different assumptions of distributions.. Theorem. Consider an infinitely large homogeneous portfolio with the asset returns following a one factor copula model. Ai  t   aM  t   1  a 2 X i  t . (3.10). Where 𝐹𝑀 (𝑡, 𝑥) is the distribution function of 𝑀(𝑡), 𝐹𝑋 (𝑡, 𝑥) is the distribution. 政治大. function of 𝑋𝑖 (𝑡) and 𝐹𝐴 (𝑡, 𝑥) is the distribution function of 𝐴𝑖 (𝑡). 𝑀(𝑡) and 𝑋𝑖 (𝑡). 立. are independent.. ‧ 國. 學. Further, we assume for simplicity that all portfolio assets have zero recovery. Then the distribution of the portfolio loss is given by. ‧. (3.11). sit. y. Nat.  C  t   1  a 2 F 1  t , x   X  F  t , x   1  FM  t ,   a  . n. al. er. io. With x ∈ [0,1] the percentage portfolio loss and 𝐶(𝑡) = 𝐹𝐴−1 (𝑡, 𝑄(𝑡)), where 𝑄(𝑡). i n U. v. is the risk-neutral default probability of each issuer in the portfolio.. Ch. engchi. 16.

(18) Now, we return to the large homogeneous portfolio with non-zero recovery 𝑅. For x between zero and one denoting the fraction of defaulted assets in the portfolio, only (1 − 𝑅)𝑥 represents the portfolio loss. Proposition. Then the loss of the senior tranche with attachment point K is equal 1  R  x  K . . (3.12). Thus, the expected loss of the senior tranche between K and 1 is given by 1. .  1  R  x  K  dF t , x   0. 1.  1  R  x  K  dF t , x  . K 1 R. 政  治 K   1 R   大  x  1  R  dF  t , x  1. 立. . (3.13). K 1 R. ‧ 國. 學. So the total loss of the equity tranche of K will occur only when assets of the total 𝐾 1−𝑅. have defaulted. Only afterwards the senior tranche from K to 1 will. ‧. amount of. start suffering loss.. y. Nat. er. io. sit. Now we consider the calculation of expected loss of a mezzanine tranche between 𝐾1 and 𝐾2 under the assumption of a non-zero recovery rate R of portfolio assets:. al. n. R  K1 , K2 . EL. t . Ch. engchi. i n U. v.  1  1 1    1  R  x  K1  dF  t , x    1  R  x  K 2  dF  t , x     K 2  K1  K1 K2   1 R 1 R .  1  1 K1  K2  1 R      x dF  t , x     x  dF  t , x        K 2  K1  K1  1  R  1 R  K2    1 R 1 R   EL K1. K2   1 R ,1 R   . t . (3.14). This is a general result, independent from the distribution assumption.. 17.

(19) 3.3. Asymptotic Single Factor Models. Definition (One factor Gaussian copula). Consider a portfolio of m credit instruments. The standardized asset return up to time t of the i th issuer in the portfolio, 𝐴𝑖 (𝑡) , is assumed to be of the form:. Ai  t   ai M  t   1  ai2 X i  t . (3.15). where 𝑀(𝑡) and 𝑋𝑖 (𝑡), i = 1,…, m are independent standard normally distributed random variables. Under this copula model the variable 𝐴𝑖 (𝑡) is mapped to default time 𝜏𝑖 of the i th issuer using a percentile-to-percentile transformation, i.e. the issuer i defaults before time t when. 立. Φ  Ai  t    Qi  t . (3.16). ‧ 國. 學. or equivalently. 政治大. ‧. Ai  t   Φ1  Qi  t    Ci  t . (3.17). Qi  t   Q  i  t . n. al. (3.18). er. io. sit. y. Nat. where 𝑄𝑖 (𝑡) denotes the probability of the issuer i to default before time t. i n U. v. The risk-neutral probabilities are implied from the observable market prices of credit. Ch. engchi. default instruments (e.g. bonds or CDS).. The factor M can be interpreted as the systematic common market factor and 𝑋𝑖 as the idiosyncratic factors. Correlation between the asset returns of the issuers i and j equals 𝑎𝑖 𝑎𝑗 . Conditionally on M, the asset returns of the different issuers are independent. According to (3.15), the i th issuer defaults up to time t if. X i t  . Ci  t   ai M  t  1  ai2. (3.19). Then the probability that the i th issuer defaults up to time t, conditional on the factor 18.

(20) 𝑀(𝑡), is.  C t   a M t   i  pi (t | M )  Φ  i 2   1  a i  . (3.20). Under the assumption of LHP, the default thresholds 𝐶(𝑡) of all issuers are the same as well and the default probability of all issuers in the portfolio conditional on M is given by.  C  t   aM  t   p(t | M )  Φ   1  a2  . (3.21). 治政 the asset returns following a one factor Gaussian copula大 model is given by 立. By the theorem, the loss distribution of an infinitely large homogeneous portfolio with. ‧ 國. 學.  1  a 2 Φ1  x   C  t    F  t , x   Φ    a  . ‧. with x ∈ [0,1] the percentage portfolio loss.. (3.22). sit. y. Nat. io. al. er. Definition (Double-t copula). Consider a portfolio of m credit instruments. The. n. standardized asset return up to time t of the i th issuer in the portfolio, 𝐴𝑖 (𝑡) , is assumed to be of the form:. Ch. engchi. i n U. v. Ai  t   ai M  t   1  ai2 X i  t . (3.23). where 𝑀(𝑡) and 𝑋𝑖 (𝑡), i = 1,…, m are Student t distributed random variables. One natural extension of the LHP approach is to use a distributional assumption that produces heavy tail. The double t one factor model proposed by Hull and White assumes Student t distributions for the common market factor M as well as for the individual factors 𝑋𝑖 . Then the loss distribution 𝐹∞ in (13) becomes:.  1  a 2 T 1  x, v   C  t   X F  t , x, vX , vM   T  , vM    a   19. (3.24).

(21) where T denotes the Student t distribution function with degrees of freedom. In general, the degrees of freedom 𝑣𝑋 and 𝑣𝑀 can be different. Unfortunately, it is not possible to solve the integral in (5) analytically using this loss distribution and one has to use some numerical integration method. The asset returns Ai do not follow necessarily Student-t distributions since the Student-t distribution is not stable under convolution. The distribution function 𝐻𝑖 of 𝐴𝑖 must be computed numerically. Afterwards, it is possible to calculate the default thresholds 𝐶𝑖 by 𝐻𝑖−1 (𝑞𝑖 ). This procedure is quite time consuming and it makes the. 政治大. double t model too slow for Monte Carlo based risk management applications.. 立. ‧ 國. 學. Definition (One factor NIG copula model). Consider a homogeneous portfolio of m. ‧. credit instruments. The standardized asset return up to time t of the i th issuer in the. n. al. y. (3.25). er. io. Ai  t   aM  t   1  a 2 X i  t . sit. Nat. portfolio, 𝐴𝑖 (𝑡) , is assumed to be of the form:. with independent random variables. Ch. e n g c hi. i n U. v. 3  M  t ?~ NI G   ,  ,  2 , 2     . (3.26).  1  a2 1  a2 1  a 2  2 1  a 2  3  X i  t ?~ NI G  , , ,  2 2   a a a  a   . (3.27). 2. where γ = √𝛼 2 − 𝛽 2 . Under this copula model the variable 𝐴𝑖 (𝑡) is mapped to default time 𝑡𝑖 of the i th issuer using a percentile-to-percentile transformation. The issuer i defaults before time t if. 20.

(22) F. 1 NI G  a.  A t   Q  t . (3.28). i. or equivalently Ai  t   F 1. 1 NI G  a. Q t   C t . (3.29). where 𝑄(𝑡) denotes the risk-neutral probability of the instruments to default before time t . By the theorem, considering an infinitely large homogeneous portfolio with the asset returns following the one factor NIG copula model. Then the distribution of the. 政治大 F Q t  1 a F. portfolio loss before recovery is given by. 立.   . ‧ 國. 1 1 NI G  a. 2. 1.  1 a 2 NI G  a . 學.   F  t , x   1  FNI G1    . a.    .  x       . (3.30). ‧. with x ∈ [0,1] the percentage portfolio loss and 𝑄(𝑡) the risk-neutral default. y. Nat. n. er. io. al. sit. probability of each issuer in the portfolio.. Ch. engchi. i n U. v. The correlation parameters were assumed to be constant over time so far. It is no problem for a pricing application since the correlation parameter can be updated every day by a new calibration. However, it is not realistic to assume the correlations being constant for a simulation application. Especially the financial crisis has shown an extreme correlation shift. For this reason, we want to integrate the possibility of different correlation regimes to the NIG model. First, we consider two regimes: the first is the regime of an usual correlation and the second of a high correlation.. 21.

(23) Definition (NIG-Regime Switch copula model). The asset return of the i th issuer, 𝐴𝑖 (𝑡), is assumed to be of the form:. dAi  t   adM  t   1  a 2 dX i  t . (3.31). where 𝑀(𝑡) , 𝑋𝑖 (𝑡) , i = 1,…,m are independent processes with the following distributions: 2 3   2  2   ,  ,  Λ dt , Λ dt   t t 2 2    . dM  t  ~. (3.32). 2 2  1  a2 1  Λt2 ai2 1  a 2  2 1  Λt a j 1  a 2  3  1  a2 dX i  t ?~ NI G , , dt , dt   a a 1  ai2 a 1  a 2j a  2  2  (3.33). 立. 政治大. Λ 𝑡 is a Markov process with state space{1, 𝜆}, an initial distribution π = {𝜋1 , 𝜋2 }. ‧ 國. 學. and a (2 × 2) transition function{𝑃(ℎ)}ℎ≥0. The distribution of the increment of the. ‧. asset return is d𝐴𝑖𝑗 (𝑡) = 𝑁(1) (𝑡), i.e., 𝑎. (3.34). er. io. sit. y. Nat. 1 1 1  2 1 3  dAi  t  ~ NI G  ,  ,  dt , dt  a 2 a 2  a a. al. n. v i n Proposition. Consider an asset C return h eofnthegi cth hissuer, i U𝐴𝑖 (𝑡), as defined in Definition. Assume, the process Λ 𝑡 was in state 𝑟 ∈ {1,2} at the time 0. Let 𝑇 𝑟 (𝑡) ≔ (𝑇1𝑟 (𝑡), 𝑇2𝑟 (𝑡))′ be a stochastic process giving the duration of the stay in state i starting from the state r at time t = 0: t. Ti r  t   1state i at time s ds. (3.35). 0. Then the distributions of 𝑀(𝑡) and 𝑋𝑖 (𝑡), the cumulated returns on[0, 𝑡], conditional on the realization of 𝑇 𝑟 (𝑡), are NIG with the following parameters:. 22.

(24)   2 3  M  t  | T r  t  ~ NI G  ,  ,  T1r  t    2T2r  t   2 , T1r  t    2T2r  t   2      X i t  | T r t  ~. (3.36).  1  a2 t  a 2 T1r  t    2T2r  t   1  a 2  2 t  a 2 T1r  t    2T2r  t   1  a 2  3  1  a2   , , ,  a a 1  a2 a 1  a2 a 2  2  . (3.37) The distribution of 𝐴𝑖 (𝑡) is as before. 1 1 1  2 1  3  Ai  t  ~ NI G  ,  ,  t, t 2 2  a a a  a   . (3.38). 治政大 of the factor 𝑀(𝑡) are: Proposition. The moments of the unconditional distribution 立 ‧ 國. 學. E  M t   0. V  M  t    E T1r  t    2T2r  t  . ‧.   1   S  M t   2 E  T1r  t    2T2r  t      3. io. sit. y. Nat. n. al. er. 2  2 1     K  M  t    3  3 1  4    E  r  4 2 r       T1  t    T2  t    . Ch. engchi. i n U. v. The moments of the unconditional distribution of the factor 𝑋𝑖𝑗 (𝑡) are:. E  M t   0 V  M t  . S  M t  . t  a 2 E T1r  t    2T2r  t   1  a 2j. 3 a. 2.   1   E  t  a 2 T r t   2T r t   1   2    . 2  a 2 2  1      K  M  t    3  3 1  4    E     t  a 2 T r  t    2T r  t     4    1 2    . 23.

(25) Remark. Approximation of 𝑀(𝑡) and 𝑋𝑖 (𝑡) with.   2 3  M  t  ~ NI G  ,  ,   h1r  t    2 h2r  t   2 ,  h1r  t    2 h2r  t   2     . (3.39).  1  a2 t  a 2  h1r  t    2 h2r  t   1  a 2  2 t  a 2  h1r  t    2h2r t   1  a 2  3  1  a2   , , ,  a a 1  a2 a 2 1  a2 a 2   . X i t  ~. (3.40) fits the first two moments of the exact distributions. The third and the fourth moments. 政治大 the special case of a non-skewed distributions, i.e. 𝛽 = 0, the skewness is zero for the 立. of the approximate distribution are not higher than those of the exact distribution. In. ‧ 國. 學. approximate and the exact distributions.. Now all distributions necessary to describe the portfolio loss distribution are. ‧. available. The approximate loss distribution of an infinitely large homogeneous. sit. al. er. io. by. y. Nat. portfolio with the asset returns following a NIG-Regime Switch copula model is given. v. n.   2 3  FLHP  t , x   1  FNI G  lt1  x  ; ,  ,   h1r  t    2 h2r  t   2 ,  h1r  t    2 h2r  t   2  (3.41)    . Ch. engchi. i n U. with x ∈ [0,1] denoting the percentage portfolio loss. The function 𝑙𝑡 (𝑀(𝑡)) is the portfolio loss conditional on the realization of the systematic factor 𝑀(𝑡) and is given by:. lt  M  t    J.  1  R  w F j. j 1. NI G.  C (t)  aM (t) 1  a 2 t  a 2  h1r  t    2 h2r  t   1  a 2  2 t  a 2  h1r  t    2 h2r  t   1  a 2  3  1  a2   ; , , , 2  a a 1  a2 a 2 1  a2 a 2  1  a  . (3.42) The default thresholds are computed as. 24.

(26)  1 1 1  2 1  3  C  t   FNI1G  Q  t  ;  ,  ,  t, t a a a 2 a 2  . 3.4. Risk. Measures. and. Hedging. Parameters. (3.43). of. Synthetic CDO The risks of CDO tranches are mainly from the uncertainties of loss. If the risks of CDO are measured by the basis of its loss distribution function, then it can better reflect the loss uncertainties from investors of tranches. The followings are the. 政治大. definitions of CDO’s risk measurements.. 立. (1) Expected loss as the risk indicator. ‧ 國. 學. We can retrieve the expected loss of tranches from the loss distribution of targeted portfolio directly. However, since expected loss represents the absolute loss amount,. ‧. we can’t use it as the indicator for measuring relative risks. We can adopt the risk. y. Nat. sit. indicators that are based on expected loss, such as the expected loss rate and the. n. al. er. io. leverage of expected loss. Expected loss rate is the expected loss of tranche to total. i n U. v. nominal principal by definition. Leverage of expected loss is the expected loss rate of. Ch. engchi. tranches to expected loss rate of targeted portfolio ,which is expressed as the following:. Expected loss rate of tranches =. ELi D C.  .  E Lti  ELi    Leverage of expected loss =  /   D  C   sizeof reference portfolio . 𝐸𝐿𝑖 is the expected loss at 𝑡𝑖 , and 𝐸(𝐿𝑡𝑖 ) is the expected loss of the target portfolio. 25.

(27) Leverage of tranches’ expected loss is an indicator for relative risk. If the leverage equals to 20, it represents that every $1 principal from tranches takes 15 times of risks than $1 principal from targeted portfolio. (2) Sensitivity of tranches in regarding of the change in credit spread Before defining the sensitivity of tranches in regarding of the change in credit spread, we have to define the mark-to-market value of tranches. From the viewpoint of tranches’ investors, mark-to-market value of tranche is the present value of premium minus the present value of default, which can be shown as: MTM  Tranche PV  Premium Leg PV  Default Leg PV. (3.44) 政治大 Assume that we don’t consider the accruals of tranches. Since the credit spread of 立. ‧ 國. 學. each tranche, s, are designated in the beginning, the change in credit swap spread of targeted portfolio during the contract period will affect the default probability.. ‧. Therefore, it will further influence the expected loss of tranches, which are default leg. sit. y. Nat. (DL) and premium leg (PL).. io. al. er. Sensitivity of tranches in regarding of the change in credit spread represents the. n. influence from the simultaneous movements of targeted portfolio’s credit spread, which can be shown as:. Ch. engchi. i n U. v. Tranche PV  Tranche PV  s0  1bp   Tranche PV  s0  s0. (3.45). 𝑠0 is the targeted credit spread before change occurs. (3) Hedging Parameters of Synthetic CDO 𝑆 𝑡𝑟 is the credit spread of CDO’s tranche in the beginning and it’s also the credit spread that makes the contract value equals to 0. As the spread of targeted credit swap changes, market value of each tranche will be affected in different level. The purpose of hedging in tranches is that we want to make sure tranche’s market value after. 26.

(28) hedging won’t be affected by the change in targeted credit swap spread. We can utilize the sensitivity we mentioned in the previous section to achieve the hedging parameters of synthetic CDO, Delta and Gamma, and the followings are the definitions of hedging parameters of tranches. Most common hedging parameters for synthetic CDO include tranche Delta and tranche Gamma. Tranche Delta is the changing amount in tranche market value over the changing amount in targeted market value when spread differs. Furthermore, tranche delta can also be used in targeted portfolios as well as single-name CDS.. 治政大 As a result, we have to have to sell the CDS index, which equals to buy protection. 立. From investor’s point of view, when we want to hedge the spread risk of tranches, we. pay the amount of credit spread as the cost of hedging.. ‧ 國. 學. Tranche Delta to CDS Index.  PercentagechangeinTrancheValue  Percentagechangein IndexValue. ‧. . y. Nat. sit. n. al. er. io. 1 Tranche PV 1 Tranche PV   1bp  /  1bp  s s  Tranche Notional   Index Notional .  Tranche PV 01 /  Index PV 01. Ch. Tranche Delta to Single-name CDS . engchi. i n U. v.  PercentagechangeinTrancheValue  PercentagechangeinCDSValue. 1 Tranche PV 1 CDS PV   1bp  /  1bp  s s  Tranche Notional   CDS Notional . Delta of tranche in regarding of targeted asset portfolio (ex. CDS index) represents the percentage change in tranche market value over the percentage change in targeted market value when spread differs. Besides, we usually define PV01 of tranches as “the changing amount in every dollar of principal of tranches when the 27.

(29) credit spread of targeted portfolio change 1bp in average” in practical use. Therefore, the definition of delta can also represent by the PV01 of tranches and targeted portfolio. Delta of tranche in regarding of single-name CDS represents the percentage change in tranche market value over the percentage change in CDS market value when CDS spread (s) differs. We are going to take the delta of tranche in regarding of targeted asset portfolio as example in the following part. We will prove that when we buy the tranches and sell the CDS index according to the hedge ratio can avoid credit. 治政大1 bp, the change in the When credit spreads of targeted portfolio change 立. spread risk completely. I.. market value of tranche:. ‧ 國. 學. Change in the market value of hedging position:. ‧. II.. Tranche Notional   Tranche PV 01. y. Nat. Tranche Notional   Tranche Delta    Index PV 01. sit.  Tranche Notional   Tranche PV 01. al. er. io.  Tranche Notional   Tranche PV 01 /  Index PV 01   Index PV 01. n. v i n C hvalue after hedgingUis 0. (I)+(II)，then the change in tranche engchi. Furthermore, we are going to define the Gamma of synthetic CDO’s tranche. Gamma represents the changing amount of tranche’s Delta when the credit spread of targeted portfolio changes 1 bp in average. As the following: TrancheGamma . Tranche Delta 1bp  s. (3.46). Therefore, tranche’s Gamma represents sensitivity of tranche’s Delta to the average changes in credit spread and this can be a reference for investors to adjust its hedging percentage. 28.

(30) Chapter 4 4.1. Empirical and Numerical Results. Model Calibration. In this section we want to investigate the fitting ability of the one factor NIG-Regime copula model. We compare them with those of the one factor Gaussian, double-t models and one factor NIG copula model. To do so, we use the market quotes from the 31, January 2007 and 28, September 2007, of the 5th series of the tranches iTraxx Europe with 5 years maturity. Recall that the reference portfolio of this index consists of equally weighted credit default swaps of 125 European firms.. 政治大. However, all models under comparison employ the LHP assumption, i.e. that the. 立. reference portfolio contains infinitely many firms having the same characteristics.. ‧ 國. 學. Since the corresponding 5 year iTraxx index is trading at 23 bp and 36 bp for January. ‧. 31 and September 28 respectively, we assume that all reference firms in the portfolio have a CDS spread of 36 bp. Further, the default intensity of the large homogeneous. y. Nat. er. io. sit. portfolio is estimated from the CDS spread. The constant recovery rate is assumed to be 40%. The standard tranches have attachment/detachment points at 3%, 6%, 9%,. n. al. Ch. i n U. v. 12% and 22%. The investors of the tranches receive quarterly spread payments on the. engchi. outstanding notional and compensate for losses when these hit the tranche they are invested in. The investor of the equity tranche receives an up-front fee that is quoted in the market and an annual spread of 500 bp quarterly. Gaussian and double-t factor copulas have only one parameter, the correlation. The versions of the NIG factor copula we consider have one parameter α(β=0) or two parameters α and β besides the correlation parameter. At last, the versions of the NIG-Regime factor model we consider have two parameters α and β besides the. 29.

(31) Table 4-1 Calibration result of models (before financial crisis) Market. Gauss. t(3). NIG(1). NIG(2). NIG-Regime. 0-3%. 19.13%. 19.22%. 19.39%. 19.13%. 19.13%. 19.13%. 3-6%. 91.96 bp. 183.73 bp. 89.89 bp. 91.78 bp. 6-9%. 37.17 bp. 64.06 bp. 64.82 bp 治 91.42 bp 政大. 36.35 bp. 36.91 bp. 9-12%. 24.04 bp. 12-22%. 15.04 bp. Correlation. -. alpha. -. Nat. beta. -. -. 立. 26.02 bp. 22.69 bp. 20.43 bp. 20.96 bp. 21.24 bp. 5.76 bp. 21.42 bp. 19.75 bp. 19.19 bp. 18.05 bp. 25.96%. 26.00%. 18.53%. 19.17%. 18.64. -. 0.7748. 0.6627. 0.7134. 0.1822. 0.1571. 10.21 bp. 6.25 bp. -. 0. n. Absolute error. -. 138.92 bp. Ch. y. sit. io. al. er. -. ‧. ‧ 國. 36.23 bp. 學. 32.40 bp. engchi. 65.64 bp. 30. i n U. v. 10.01 bp.

(32) Market. Gauss. t(3). NIG(1). NIG(2). NIG-Regime. 0-3%. 10.34%. 10.39%. 10.49%. 10.29%. 10.35%. 10.31%. 3-6%. 42 bp. 207.23 bp. 17.35 bp. 32.24 bp. 6-9%. 12 bp. 114.99 bp. 11.80 bp. 15.30 bp. 9-12%. 5.6 bp. 71.66 bp. 38.04 bp. 20.43 bp. 9.90 bp. 10.27 bp. 12-22%. 2 bp. 32.03 bp. 31.51 bp. 8.2 bp. 0.11 bp. 1.35 bp. Correlation. -. 50.19%. 43.53%. 39.48%. ‧. 37.81%. 36.74%. alpha. -. Nat. -. 0.1517. y. Table 4-2 Calibration result of models (during financial crisis). 0.4887. 0.5039. beta. -. -. -. 0. -0.3777. -0.2713. 31.36 bp. 21.38 bp. n. Absolute error. -. 369.80 bp. sit. io. al. er. -. 學. ‧ 國. 立. 82.64 bp 治 30.36 bp 政大 50.01 bp 23.04 bp. iv n C h156.10 bp e n g c h i U48.19 bp. 31.

(33) correlation parameter and transition matrix is calibrated by iTraxx index. We minimize the sum of the absolute errors over all tranches to estimate these parameters. We can observe the estimating result before financial crisis from Table 4-1. Gaussian copula has larger errors when comparing tranches spread with market price, except equity tranche generates smaller errors. Furthermore, we also can see that senior tranches will be underestimated since Gaussian copula is a thin-tail distribution. As a result, it will underestimate the probability of loss in extreme situations and this will let senior tranches to be underestimated. The fat-tail characteristic raised by Hull and White’s Double-t copula indeed improved the problems of underestimation in. 治政大 spread also decrease to senior tranches. Moreover, the absolute error of tranche 立 65.64bp. However, the spread of equity tranche raises because the characteristic of. ‧ 國. 學. Double-t copula result in the probability increase in both not-default and default. ‧. circumstances.. sit. y. Nat. The results of the NIG copulas are similar to the results of double-t copulas. The. io. er. additional free parameter in the NIG copula makes it more flexible so the second tranche can be fitted exactly as well. Surprising is that one more free parameter 𝛽. al. n. v i n C hin this example. U doesn’t improve the fitting results e n g c h i The NIG models overprice the most senior tranche and underprice the third tranche as well as the fourth tranche similar to the double-t model. NIG-Regime copula and NIG copula have the same result when comparing the estimations of each tranche; however, since we consider an additional factor of market status change, NIG-Regime has the smallest absolute error among other four models. Table 4-2 states the estimation result after the financial crisis. We find that the implied correlation coefficients are higher than the value before the financial crisis in all models. This proves that the default correlation between assets will increase during 32.

(34) the financial crisis. Moreover, we also can see that the absolute errors all raise up between each tranche to their market price. For example, the absolute error of Gaussian copula grows from138.92 bp to 369.8bp. However, since NIG-Regime considers the factor of changes in market status, the absolute error only increases from 6.25 bp to 21.38 bp. Thus, if factor copula model can consider the transition in status, we can narrow the error in valuation when market is relatively unstable.. 4.2. Product Assumptions. 政治大 5-year maturity and targeted 立 at a portfolio with 100 Credit Default Swap (CDS), Assume that there is a synthetic Collateralized Debt Obligation (CDO) with. ‧ 國. 學. totaled 0.1 billion USD. Moreover, 4 tranches are entitled with different claiming priority (Table 4.3). The size of each tranche represents the proportion of issue. ‧. amount to targeted portfolio’s principal as well as the maximum percentage of default. Nat. sit. y. loss. Credit enhancement indicates the cumulated loss percentage of the most. n. al. er. io. subordinate tranches start to take position before other tranches. As a result, the most. i n U. v. subordinate tranches serve as a protection for more senior ones. Besides, if we assume. Ch. engchi. that it is a post-paid type, which pays interests every 3 months, then equity tranche will takes the first 3% of total cumulated default loss, mezzanine tranche (A) takes 3% to 6%, mezzanine tranche (B) takes 6% to 10% and senior tranche will burden the total cumulated default loss that is larger than 10%. Generally speaking, the credit ratings for senior tranches mostly are AAA/Aaa, mezzanine tranches lay between A and BBB/Baa and no ratings for equity tranches.. 33.

(35) Table 4-3: Assumptions of synthetic CDO Type of Tranche Issue Amount Tranche Size Credit Enhancement Credit Rating. Mezzanine. Mezzanine. Tranche (A). Tranche (B). 3 million USD. 3 million USD. 4 million USD. 90 million USD. 3%. 3%. 4%. 90%. N/A. 3%. 6%. 10%. Unrated. BBB/Baa. A. AAA/Aaa. Equity Tranche. Senior Tranche. 政治大. Furthermore, if the CDSs in our targeted portfolio are homogeneous and the. 立. default density among our contract period is a constant, then the default density can be. ‧ 國. 學. retrieved from the credit spread of a targeted CDS. The assumptions of the targeted CDS are shown in the followings:. ‧. (1) Issue Amount: 1 million USD. y. Nat. io. sit. (2) Maturity (T): 5 year. n. al. er. (3) Recovery rate (R): 40% (4) Credit spread: 60bp. (5) Default density (λ): 0.01. Ch. engchi. i n U. v. (6) Risk free rate: Fixed at 5% during the contract period (7) Pair-wise correlation among targeted CDS: 30%. Therefore, the loading factor for systematic risk of the random variable, 𝑥𝑖 , in the single-factor model is √0.3. 34.

(36) 4.3. Pricing Results Table 4-4: CDO Pricing (Unit: bp) Gaussian. Double-t. NIG(1). NIG-Regime. Equity. 1509. 1938. 1823. 1648. Mezzanine. 468. 341. 279. 226. 199. 121. 101. 100. 7. 13.1. 13.4. 14. Tranche (A) Mezzanine Tranche (B) Senior. 政治大. The reason why that tranche spread is not alike between different models is. 立. because diverse assumptions for default structure will produce different type of loss. ‧ 國. 學. distribution function. Under the assumptions of double-t copula, when the systematic risk factor has Student-t distribution, the characteristic of fat-tail makes the. ‧. probability of both default and survive increase. Moreover, this also increases the. y. Nat. io. sit. default correlation between credit assets which lower the credit spread of equity. n. al. er. tranches and the senior tranches possess higher credit spread due to rising. Ch. i n U. v. uncertainties in macroeconomic factors. When unsystematic risk factor is a Student-t. engchi. distribution, then the default probability for a single asset will increase, which makes equity tranches have higher credit spread and senior tranches have a lower one. This effect is similar to the decrease in the default correlation among credit assets. Through the pricing result shown in Table 4-4, in regarding of equity tranches, when nonsystematic risk factor is a Student-t distribution, it will influence more comparing to the circumstances that systematic risk factor is a Student-t distribution. As a result, the credit spread of equity tranche that is under double-t copula assumptions will larger than the one under Gaussian-copula assumptions. In regarding of senior tranches, when systematic risk factor is a Student-t distribution, it will influence more 35.

(37) comparing to the circumstances that nonsystematic risk factor is a Student-t distribution; therefore, the credit spread of senior tranche that is under double-t copula assumptions will larger than the one under Gaussian-copula assumptions. For both mezzanine tranches, the pricing results under double-t copula assumptions are smaller than those under Gaussian copula. For fat-tailed distributions, Normal Inverse Gaussian and Normal Inverse Gaussian with Regime Switch, both systematic risk factors as well as nonsystematic risk factors have the same characteristics that Student-t distribution possesses.. 治政大 which unsystematic risk factors are NIG or NIG-Regime distribution, the influence, 立. However, for equity tranches, when systematic risk factors and unsystematic risk. factors have larger impacts over systematic risk factors, will be smaller comparing to. ‧ 國. 學. the distributions under Student-t, so the credit spread of equity tranches will smaller. ‧. than double-t copula. For senior tranches, the influence, which systematic risk factors. sit. y. Nat. have larger impacts over unsystematic risk factors, will be larger comparing to the. io. double-t copula.. er. distributions under Student-t, so the credit spread of senior tranches will larger than. al. n. v i n Since equity tranches willCend credit spreads that are higher than h eupnhaving gchi U. expected under the assumptions of double-t copula, if we use the fat-tailed distributions, such as NIG and NIG-Regime, we can not only keep the characteristics of fat-tailed distribution but also fix the problems that equity tranches will result in higher credit spread. Additionally, from Table 4-4 we can also observe that the fixing amount of credit spread from the NIG-Regime will be better comparing with NIG.. 36.

(38) 4.4. Risk Characters of Synthetic CDO Tranches Table 4-5: Expected Loss (Unit: million USD) Portfolio. Equity. Mezzanine. Mezzanine. Tranche (A). Tranche (B). Senior. Gaussian. 2.9435. 1.5263. 0.6529. 0.4028. 0.3616. Double-t. 3.2102. 1.8682. 0.4981. 0.2385. 0.6054. NIG(1). 2.9874. 1.8002. 0.4120. 0.2034. 0.5718. NIG-Regime. 2.9082. 1.6350. 0.3420. 0.2054. 0.7258. 政治大 be measured by the size of each tranche’s relative risks. Take Table 4-5 for example, 立 Since each tranche has different issue amount, the expected loss amount can not. under Gaussian copula, the expected loss amount at the maturity day for mezzanine. ‧ 國. 學. tranche (B) and senior tranche are 0.4028 million and 0.3616 million, respectively.. ‧. Although the amount doesn’t has significant difference, the issue amount for these. sit. y. Nat. two are 4 million dollars and 90 million dollars, respectively. Similarly, under the. io. er. NIG copula, the expected loss amount at the maturity day for mezzanine tranche (B). al. and senior tranche are 0.2034 million and 0.5718 million, respectively. Although the. n. v i n C ish larger than the mezzanine expected loss for senior tranche tranche (B), we can engchi U. observe that the risk of senior tranche is much smaller than mezzanine tranche (B) when we consider their nominal principal. To sum up, although the expected loss of tranches are widely considered as one of the important risk indicators for credit rating firms, it actually can not reveal the relative risks among CDO tranches. Therefore, our paper adopts the concept of rate of expected loss. We use the proportion of expected loss to principal in order to compare each tranche’s relative risks.. 37.

(39) Table 4-6: Rate of Expected Loss (Unit: %) Portfolio. Equity. Mezzanine. Mezzanine. Tranche (A). Tranche (B). Senior. Gaussian. 2.94. 50.88. 21.76. 10.07. 0.40. Double-t. 3.21. 64.48. 16.60. 5.96. 0.67. NIG(1). 2.99. 58.01. 13.73. 5.09. 0.64. NIG-Regime. 2.91. 54.5. 11.4. 5.14. 0.81. According to Table 4-6, in Gaussian copula, the investors of equity tranches will have 50.88% expected principal loss, mezzanine tranche (A) and mezzanine tranche. 政治大 Compared to the expected立 loss rate of 2.94% from credit portfolio, it reveals the. (B) will have the expected loss rate equals to 21.76% and 10.07%, respectively.. ‧ 國. 學. characteristic that CDO tranche possess high return as well as high risks. Since senior tranche has up to 10% protection from subordination level, it only has 0.4% expected. ‧. loss rate at the maturity and the risk of senior tranche is even lower than the risk of. sit. y. Nat. credit portfolio itself.. n. al. er. io. Under double-t copula、NIG copula and NIG-Regime copula, the expected loss. i n U. v. rate will be higher than the result using Gaussian copula assumptions. As for the. Ch. engchi. expected loss rate of mezzanine tranches, it will be smaller than the result from Gaussian copula. This conclusion shows that three types of fat-tailed assumptions of factor copula model, the risks from equity tranches and senior tranches are larger than those from Gaussian copula. As a result, the reasonable credit spreads of equity tranches and senior tranches are higher when we utilize double-t copula、NIG copula and NIG-Regime copula. Based on the same analyzing logic, we can see that mezzanine tranches have relatively lower risk, which also indicates lower credit spread under the double-t copula、NIG copula and NIG-Regime copula.. 38.

(40) Besides the expected loss rate of tranches, if we use the expected loss rate of credit portfolio as basis, we can further retrieve the risk leverage of tranche’s expected loss. We take expected loss rate of credit portfolio in each period as a unit and we calculate the multiples of expected loss rate relative to credit portfolio’s expected loss rate in every period as another indicator for measuring relative risks among different tranches. Our paper lists the tranche’s risk leverage of expected loss in each period in the following Table 4-7 to Table 4-10. Table 4-7: Risk Leverage of Gaussian Time. Equity. 0.25. 30.382. 0.50. 29.029. 2.859. 0.760. 0.75. 28.156. 3.295. 1.067. 0.015. 1.00. 26.625. 4.886. 1.163. 0.018. 1.25. 26.101. 4.531. 1.402. 1.50. 25.336. 4.948. 1.587. 23.721. 5.561. 1.916. 5.942. 1.918. 6.238. 0.001 0.014. y. sit. er. al. 22.547. Senior. ‧. 23.186. n. 2.25. io. 2.00. Tranche 治 (A) Tranche (B) 政大 2.286 0.472. Nat. 1.75. Mezzanine. 學. ‧ 國. 立. Mezzanine. 0.028 0.031 0.050 0.055. 2.184. v. 0.055. 2.311. 0.062. i n U. 2.50. C 21.892 h. 2.75. 21.276. 6.622. 2.443. 0.073. 3.00. 20.884. 6.637. 2.550. 0.080. 3.25. 20.447. 6.734. 2.747. 0.083. 3.50. 20.033. 6.966. 2.751. 0.090. 3.75. 19.357. 7.091. 2.902. 0.101. 4.00. 19.103. 7.199. 3.023. 0.100. 4.25. 18.725. 7.259. 3.075. 0.105. 4.50. 18.097. 7.306. 3.310. 0.114. 4.75. 17.700. 7.346. 3.355. 0.127. 5.00. 17.285. 7.393. 3.421. 0.136. e 6.490 ngchi. 39.

(41) Table 4-7 indicates that the risk leverage of equity tranches is up to 17.285 at maturity under Gaussian copula. Comparing to others risk leverage range from 0.136 to 7.393, it fully shows that equity tranches possess high risk. As time goes by, the risk leverage of equity tranches display a decreasing trend, which means that equity tranches face higher risks in the early period of contract. However, risk leverages of mezzanine tranches are increasing as time passes, which demonstrates that mezzanine tranches face higher risks in the later period of contract. As for the risk leverages of senior tranches, there aren’t significant changes during contract period, and this is also. 治政大tranches show the decreasing Under double-t copula, the risk leverages of equity 立. viewed as the characteristic of low return as well as low risk of senior tranches.. trend as time passes and risk leverage of others are decreasing as well, which. ‧ 國. 學. corresponds to our results under Gaussian copula assumptions. Furthermore, we can. ‧. see that equity tranches and senior tranches have higher risks under double-t copula. sit. y. Nat. than Gaussian copula, which are 20.087 and 0.21, respectively. In the early stage of. io. er. contract, equity tranches own relatively higher risks, which are resulted from the first. al. 3% of cumulated default amount in credit portfolio that took by equity tranches first.. n. v i n Cwill In other words, equity tranches most of the loss in contract’s early h econsume ngchi U periods, and when the cumulated default amount is larger than 3% of total credit. portfolio, then mezzanine tranches will start suffering loss. Therefore, mezzanine tranches often take the loss position from the medium to later periods of contract.. 40.

(42) Table 4-8: Risk Leverage of Double t Time. Equity. Mezzanine. Mezzanine. Senior. Tranche (A). Tranche (B). 0.25. 28.655. 0.790. 0.634. 0.084. 0.50. 28.076. 1.017. 0.683. 0.101. 0.75. 27.867. 1.126. 0.703. 0.112. 1.00. 27.250. 1.288. 0.763. 0.127. 1.25. 26.416. 1.513. 0.811. 0.132. 1.50. 25.866. 1.763. 0.888. 0.138. 1.75. 25.530. 1.988. 0.978. 0.142. 2.00. 24.723. 2.227. 1.086. 0.146. 2.25. 24.676. 2.50. 24.083. 2.75. 23.770. 2.908. 1.222. 3.00. 23.509. 3.288. 1.273. 0.168. 3.25. 23.391. 3.500. 1.355. 0.172. 3.50. 22.511. 3.797. 1.465. 3.75. 22.383. 4.051. 1.484. 21.512. 4.290. 1.596. 21.207. 4.551. 1.657. al. 4.817. Ch. 4.75. 20.760. 5.00. 20.087. e n5.018 gchi 5.172. 0.165. 0.177. sit. y. 0.182. er. 21.095. 0.158. ‧. n. 4.50. io. 4.25. Nat. 4.00. 0.150. 學. ‧ 國. 立. 1.100 政2.490治大 2.796 1.207. iv n U1.843 1.784. 1.857. 0.186 0.190 0.196 0.202 0.210. Under NIG copula and NIG-Regime copula, we can also see that the risk leverages of equity tranches will decrease as time goes by; however, risk leverages of others will increase. In regarding of equity tranches, risk leverages have the largest value under double-t copula at the maturity. For senior tranches, NIG-Regime assumptions make them have the largest risk leverages, which also corresponds to the pricing result that equity tranches possess the largest credit spread under double-t copula and senior tranches exhibit the largest credit spread under NIG-Regime. 41.

(43) However, this is the exact characteristic of high risk as well as high return.. Table 4-9: Risk Leverage of NIG Mezzanine. Mezzanine. Senior. Tranche (A). Tranche (B). 0.25. 28.446. 1.380. 0.827. 0.101. 0.50. 27.964. 1.460. 0.980. 0.110. 0.75. 26.848. 1.789. 0.987. 0.122. 1.00. 26.323. 1.932. 0.999. 0.129. 1.25. 25.886. 1.948. 1.038. 0.134. 1.50. 25.344. 0.140. 1.75. 24.572. 治 1.064 政2.061 2.275 1.168 大. 2.00. 24.333. 2.514. 1.219. 0.165. 2.25. 24.003. 2.538. 1.230. 0.174. 2.50. 23.890. 2.736. 1.259. 0.175. 2.75. 23.570. 2.823. 1.265. 0.179. 3.00. 22.873. 2.966. 1.278. ‧. 22.570. 3.157. 1.294. y. Equity. 0.183. 22.125. 3.233. 1.348. sit. Time. 0.196. al. 3.630. 1.420. er. 0.181. 0.197. 4.25. iv 21.200C 3.811 1.442 n hengchi U 20.958 3.823 1.588. 4.50. 20.265. 4.206. 1.596. 0.200. 4.75. 19.791. 4.505. 1.605. 0.203. 5.00. 19.399. 4.597. 1.702. 0.213. 4.00. 21.849. n. 3.75. io. 3.50. Nat. 3.25. 學. ‧ 國. 立. 0.153. 42. 0.197 0.198.

(44) Table 4-10: Risk Leverage of NIG-Regime Mezzanine. Mezzanine. Tranche (A). Tranche (B). 26.135. 0.626. 0.391. 0.181. 0.50. 24.370. 0.672. 0.521. 0.184. 0.75. 23.969. 0.727. 0.557. 0.192. 1.00. 23.465. 0.791. 0.659. 0.202. 1.25. 22.979. 0.892. 0.672. 0.218. 1.50. 22.834. 1.014. 0.678. 0.219. 1.75. 22.479. 1.390. 0.739. 0.234. 2.00. 22.377. 1.504. 0.973. 0.235. 2.25. 22.340. 2.50. 22.288. 2.75. 22.266. 1.801. 1.013. 3.00. 22.163. 2.011. 1.041. 0.251. 3.25. 22.152. 2.056. 1.083. 0.251. 3.50. 22.064. 2.358. 1.170. 3.75. 21.954. 2.658. 1.204. 21.634. 3.048. 1.317. 20.833. 3.374. 1.338. 19.733. al. 3.531. 4.75. 19.111. 5.00. 18.740. e n3.797 gchi. Ch. 3.920. 0.246 0.250. ‧. n. 4.50. io. 4.25. Nat. 4.00. 1.516 iv n U 1.647 1.766. senior. 0.242. 學. ‧ 國. 立. 政 1.511治大0.981 1.677 0.986. 0.258 0.267. y. 0.25. sit. Equity. er. Time. 0.268 0.268 0.274 0.275 0.277. When assets in the credit portfolio encounter default scenario, equity tranches will suffer first. From Table 4-7 to Table 4-10, the reason why risk leverages of equity tranches will decrease by time is because the realized loss makes it has lower uncertainties of loss occurrences than other tranches. For mezzanine tranche and senior tranche, since equity tranches can only take limited amount of loss, its principal and subordination level decrease as loss occur, which cause other tranches has higher loss uncertainties; as a result, the risk leverage will be an increasing trend by time. 43.

(45) Table 4-7 to Table 4-10 provides hedging information of each tranche for investors (or the sell side of credit protection). For investors of equity tranches, they mainly need to hedge the risk that asset may default too early. If targeted asset defaults in the early stage of the contract, the holder of equity tranches may suffer from losing part of their principal until they receive enough credit spreads. However, if targeted asset defaults in the later stage of contract, then the credit spreads which we received before may compensate principal’s partial loss of equity tranches. As a result, we are able to extend our conclusion to that investors of senior tranches mainly hedge targeted asset’s default risk in the later period of contract.. 立. 政治大. ‧ 國. 學. Figure 4-1 Leverages of expected loss of equity tranche at each time period using. ‧. different models. n. er. io. sit. y. Nat. al. Ch. engchi. 44. i n U. v.

(46) Furthermore, we are going to visualize the leverages of equity tranches and senior tranches in Figure 4-1. We can observe that the leverage calculated from Gaussian copula is the largest among these three models, which means that it has the highest risk. As a result, Gaussian copula requires higher cost when hedging. However, we can find that the leverages of Double-t, NIG and NIG-Regime copula will possess a slower decreasing speed than Gaussian copula does, which make Gaussian copula has the smallest leverage, or the least risky characteristics in the end of contract.. 治政大 leverage of expected loss, among different models. NIG-Regime copula has the largest 立 In Figure 4-2, we illustrate leverages of senior tranches at each time period. NIG copula as well as Double-t copula follow and Gaussian copula has the smallest. ‧ 國. 學. value since fat-tail characteristic will let the probabilities of not-default as well as. ‧. default increase. This will make senior tranches possess higher leverage of expected. sit. y. Nat. loss in the models that possess fat-tail distribution. Here also corresponds to our. io. al. er. previous valuation result which states that fat-tail distribution will make senior. n. tranches have higher credit spread due to the macroeconomic uncertainties raise.. Ch. engchi. 45. i n U. v.

(47) Figure 4-2 Leverages of expected loss of senior tranche at each time period using different models. 立. 政治大. ‧. ‧ 國. 學 y. Nat. sit. To sum up, when defaults happen, the realized loss of reference portfolio in. n. al. er. io. regarding of relative risk in each tranche has different level of influence. Although. i n U. v. realized loss makes equity tranches suffer from loss, it lowers the loss uncertainties in. Ch. engchi. equity tranches. However, in the mezzanine tranche, when realized loss erodes its subordination level, realized loss will increase the probability of loss occurring and this will raise the relative risk. Mezzanine tranche will lower its loss uncertainties until realized loss happens in mezzanine tranche.. 46.

(48) 4.5. Hedging Analysis. When we want to hedge the synthetic CDO, the holders of tranches can choose existing CDS index or single-name CDS to hedge based on the risk type they want to mitigate. If we use CDS index as our hedging solution, it means that we want to avoid the risk that result from the simultaneously change in both macroeconomic environment and asset credit spread. When we use single-name CDS as our hedging method, we want to mitigate the risk of default and change in targeted credit spread. As a result, we can see that there are obvious differences in hedging cost as well as. 政治大 hedging parameters and we 立will also analyze the hedging cost of each tranche in hedging effects of these two hedging methods. In the following, we will calculate the. ‧ 國. 學. regarding of four factor copula, including Gaussian copula, Double-t copula, NIG copula and NIG-Regime copula. Furthermore, we will also compare the different. ‧. effects in different hedging strategies.. sit. y. Nat. n. al. er. io. (1) Hedging with CDS index. i n U. v. In recently years, there are many standardized CDS index products in the market,. Ch. engchi. such as DJ CDX NA and DJ iTraxx Europe. When synthetic CDO portfolio targets at these CDS index, investors can use those index to hedge directly. Furthermore, they can also apply the models in Chapter 3 and the definitions of hedging parameters to calculate the tranche delta in synthetic CDO.. 47.