行政院國家科學委員會專題研究計畫 成果報告
固定比例投資組合保險策略在信用投資組合之應用及實證
分析
研究成果報告(精簡版)
計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 98-2410-H-004-061- 執 行 期 間 ： 98 年 08 月 01 日至 99 年 12 月 31 日 執 行 單 位 ： 國立政治大學財務管理學系 計 畫 主 持 人 ： 岳夢蘭 計畫參與人員： 博士班研究生-兼任助理人員：邱信瑜 博士班研究生-兼任助理人員：劉崇齡 報 告 附 件 ： 國外研究心得報告 處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，1 年後可公開查詢中 華 民 國 100 年 02 月 17 日
1
Introduction
Constant proportion portfolio insurance (CPPI) is an investment strategy widely em-ployed to create structured credit products with principal-protected features. How-ever, most of these portfolio insurance-related investments failed to provide the spec-ified percentage of principal redemption during the recent financial market turmoil. To what extent can the implementation of a CPPI mechanism attain the goal of prin-cipal protection? In this research, we investigate how the characteristics of credit risk exposures, along with the features of a CPPI strategy, affect the performance of a CPPI investment on credit index tranches.
Within a credit CPPI structure, the risky exposure is managed according to its mark-to-market portfolio value, such that returns are driven by spread premiums received through selling protection and movements in the market spread. Because the portfolio is dynamically rebalanced according to realized index spreads, an ana-lytical formula for a portfolio’s terminal value is not readily available. We therefore empirically study the performance of the credit CPPI strategy by examining whether a portfolio’s net asset value can always stay above the present value of the required floor level under various CPPI features.
The empirical results show that the performance of a credit CPPI portfolio depends heavily on the aggressiveness of the leverage strategy employed. The larger the mul-tiplier assigned to a CPPI strategy, the greater is the net asset value of a CPPI portfolio, as is the incidence of lower net asset values when the market performs badly, espe-cially since the financial market turmoil that began in June 2007. Moreover, for a given multiplier, a CPPI portfolio with a risky investment on an equity tranche will offer a more variable risk return profile than that with a risky investment on a mezzanine tranche of index portfolio. The iTraxx tranches themselves are already leveraged to the underlying reference portfolio, so investment in iTraxx tranches will introduce ad-ditional leverages into a credit CPPI portfolio. The equity tranche associated with a higher implicit leverage therefore will induce larger changes in the mark-to-market
portfolio value than those tranches associated with smaller implicit leverages. Finally, the empirical evidence shows that in general, a credit CPPI strategy performs poorly in highly volatile markets. In particular, if the value of the risky investment falls sharply before the portfolio can be readjusted, the CPPI mechanism may even fail to provide the principal with protection. The so-called gap risk related to a sharp decline in port-folio values that are too sudden to allow for a rebalancing is hard to manage for most principal-protected CPPI portfolios.
The remainder of the report is organized as follows: Section 2 briefly introduces the iTraxx tranches and provides valuation formulae for them. Section 3 offers an illustrative example to show how a quarterly-rebalanced CPPI portfolio adjusts its risky exposure on an iTraxx equity tranche over time. It then proposes investment rules for governing the dynamic switch of an investment mix between risk-free and risky credit index tranches of a credit CPPI portfolio. Section 4 presents an empirical analysis of the risk/return profile of credit CPPI portfolios. The final section offers some concluding remarks.
2
Valuation of Credit Index Tranches
A credit CPPI strategy is a dynamic asset allocation strategy because the amount in-vested in the credit portfolio is readjusted over time, depending on the actual per-formance of the credit portfolio. Increases or decreases of risky exposures on iTraxx tranches depend on spread movements of the underlying iTraxx indices. Therefore, to study the performance of a credit CPPI portfolio, we assess the mark-to-market value of risky exposures driven by changes in credit spread.
2.1
Calculation of Tranche Spread
A synthetic CDO tranche j can be specified by the attachment point C(j) and detach-ment point D(j). In turn, N(j), the size of tranche j is equal toD(j)−C(j)
investor receives a periodic spread premium from the issuer and makes a contingent default payment in the case of default events. Therefore, the value of a specific tranche, from the perspective of the investor, is the difference between the expected premiums that a tranche investor will receive and the expected losses he or she will suffer over the life of the contract.
Let t1 < . . . < tn = T be the set of spread payment dates, with T equal to the
maturity of the CDO tranche. Denote Li as the pool’s cumulative losses up to time ti; then L(ij), or the losses absorbed by tranche j up to time ti, can be calculated as
L(ij) =minhN(j), maxLi−C(j), 0
i .
Denote B(ti) = e−R0tirudu as the risk-free discount factor, where r
u is the risk-free interest rate; ∆iis the tenor between payment dates ti−1and tiin the unit of years; and
s(j) is the annualized fair spread for tranche j. The present value of the premium leg of tranche j is computed as the discount value of all expected spread fees received:
PV(premium-leg(j)) =s(j) n
∑
i=1 B(ti)∆ihN(j)−EL(j) i i .If there were no defaults by time ti, the premium would be paid on the notional of the tranche as s(j)N(j); otherwise, the premium is paid on the remaining amount N(j)−
L(ij)as s(j)N(j)−L(ij)
.
The present value of the default leg of tranche j is equal to the sum of discounted expected contingent payments that a tranche investor must pay if a default event has affected the tranche’s principal; that is:1
PV(default-leg(j)) = n
∑
i=1 B(ti) h EL(ij)−E L(i−j)1i.The fair annualized tranche spread s(j) is determined so that the premium leg and
1Although default payments are made immediately after default, for simplicity, we ignore the
default leg have equal values: s(j) = ∑ni=1B(ti) h EL(ij)−E L(i−j)1i ∑ni=1B(ti)∆ihN(j) −E L(ij)i .
This illustration shows that the CDO valuation problem may be reduced to the computation of tranche j’s expected cumulative losses up to time ti, EL(ij). Each tranche accordingly can be valued once the cumulative joint loss distribution is known. Thus, the key step in pricing a CDO tranche is the derivation of the joint loss distribu-tion of the underlying reference portfolio. Currently, the industry standard for pricing iTraxx index tranches is the one-factor Gaussian copula model proposed by Li (2000) and Laurent and Gregory (2005). Implementing the model involves the computation of a discretised version of the conditional loss distribution by means of a recursion formula, as well as integration over the common factor to obtain the unconditional loss distribution. We summarize the implementation of the Gaussian copula model in the Appendix. For a review of the recursive approach to constructing portfolio loss distribution under a Gaussian copula model, see Andersen, Sidenius, and Basu (2003) or Gregory and Laurent (2004). For a review of recent advances in pricing tranches of a CDO, see Wang, Rachev, and Fabozzi (2006). For the comparison of hedging perfor-mances among different CDO pricing models, see Finger (2009).
To rebalance a CPPI portfolio consisting of iTraxx index tranches dynamically, it is necessary to assess the changes in the values of these tranches over time. To determine the mark-to-market (MtM) of a given tranche j, we define a measure δV(j)(t) as the ratio of the expected present value change of tranche j to the notional amount of the underlying portfolio due to the change of 1 basis point in the underlying reference portfolio at time t. The MtM of tranche j is calculated as
MtM(j)(ti) =es(j)(ti) −es(j)(ti−1)×δV(j)(ti) ×
N(j)
where es(j)(ti) is the time ti realized spread for tranche j, N = ∑jN(j), and N is the total notional amount of the reference portfolio.
2.2
Data Description
The data used in this research consist of daily closing quotes (the midpoint quotes between quoted ask and bid spreads) for the iTraxx index spreads during the period from June 2004 to June 2008, which covers the entire history of the iTraxx index. For comparison, the iTraxx HiVol index, an equally weighted portfolio of the 30 entities with the highest CDS spreads from the iTraxx index, is also used. All index quotes are from Markit iTraxx CDS indices. The iTraxx indices typically trade with a five-year maturity and are issued by series.2
3
Empirical Results: Monthly Rebalanced
To analyze the capacity of a credit CPPI strategy as a capital protection method, in this section, we present the empirical risk return profiles of credit CPPI portfolios for different values of the key variables within a CPPI strategy. We use the algorithm pro-posed in the previous section to study the performance of a monthly rebalanced CPPI portfolio with risky exposures on CDO tranches. To keep things as simple as possible, we also assume that standard iTraxx index tranches (i.e., 0%−3%, 3%-6%,. . .) are the
underlying risky investment of the credit CPPI strategy. All parameter settings for the cases studied here are assumed to be the same as those in the illustrative example. To save space, the empirical results appear in figures; detailed values of the relevant variables in the portfolios being studied are available upon request.
A CPPI strategy maintains a highly leveraged exposure to maximize the return
2A new series of iTraxx indices is launched every six months when the composition of reference
entities is rebalanced based on a dealer liquidity poll, which occurs twice a year on the roll dates of 20 March or 20 September. A new version index will then be on-the-run for the next six months. Currently, the on-the-run iTraxx index is series 10. In this research, we use all the on-the-run series as the iTraxx indices.
from the risky investments while guaranteeing a sufficient amount is available to as-sure principal protection at maturity. To this end, the leveraged risky expoas-sure is designed to expand and contract in sync with the performance of the risky asset. To address the relationship between risky exposure and credit spread, Figure 1 plots the historical iTraxx index spread and risky exposures to three different iTraxx index tranches for a given multiplier m =4.
Figure 1 also shows that the leveraged exposure is adjusted on the basis of the credit spread in a negatively correlated way. The leveraged risky exposure grows in re-sponse to credit spread tightening, which indicates a favorable portfolio performance, and reduces in response to credit spread widening, or a poor portfolio performance.
For a CPPI method, the multiplier is the key parameter that determines the amount to be invested in the risky asset. It therefore plays a crucial role in balancing the risk versus return of a portfolio. To study the effect of a multiplier on the performance of a credit CPPI portfolio, Figure 2 plots the net asset value of a CPPI portfolio at different levels of multipliers for the risky investment to be iTraxx equity, mezzanine tranches, and iTraxx portfolio, respectively.
Figure 2 shows that the higher the multiplier, the larger is the net asset value of a CPPI portfolio, but also the higher is the incidence of the lower net asset value in the period of financial market turmoil that began in June 2007. For a given level of bond floor, the greater the multiplier is, the larger the amount invested in the risky asset is. Therefore, the probability of a larger change in the net asset value also increases. This reasoning explains why a CPPI portfolio with a greater multiplier has a higher net asset value and also is more likely to hit the floor.
4
Conclusion
In this research, we first describe how to implement a credit CPPI strategy to manage a principal-guaranteed portfolio, then examine how various features of a CPPI strategy
30−Jun−04 31−Mar−05 31−Jan−06 30−Nov−06 30−Sep−07 28−Jun−08 60 80 100 120 140 20 40 Credit Spread (bp)
iTraxx Equity Tranche, m=4
0.2 0.6 1 1.4 1.8 x 105 Time Risky Exposures *10 5 Risky Exposures Credit Spread
30−Jun−04 31−Mar−05 31−Jan−06 30−Nov−06 30−Sep−07 28−Jun−08
60 80 100 120 140 20 40 Credit Spread (bp)
iTraxx Mezzanine Tranche, m=4
2 4 6 8 0 x 104 Time Risky Exposures *10 4 Credit Spread Risky Exposures
30−Jun−04 31−Mar−05 31−Jan−06 30−Nov−06 30−Sep−07 28−Jun−08
60 80 100 120 140 20 40 Credit Spread (bp) iTraxx Portfolio, m=4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 104 Time Risky Exposures *10 4 Credit Spread Risky Exposures
30−Jun−04 −1.5 31−Mar−05 31−Jan−06 30−Nov−06 30−Sep−07 28−Jun−08 −1 −0.5 0 0.5 1x 10 6 Time
Net Asset Value (*10
6 )
iTraxx Equity Tranche
m=4 m=8 m=12
30−Jun−04 4 31−Mar−05 31−Jan−06 30−Nov−06 30−Sep−07 28−Jun−08
5 6 7 8 9 10x 10 4 Time
Net Asset Value (*10
4 )
iTraxx Mezzanine Tranche
m=4 m=8 m=12
30−Jun−04 2 31−Mar−05 31−Jan−06 30−Nov−06 30−Sep−07 28−Jun−08
3 4 5 6 7 8x 10 4 Time
Net Asset Value (*10
4 )
iTraxx Portfolio
m=12 m=8 m=4
will affect the net asset value of a credit CPPI portfolio.
Using the standardized iTraxx tranches as the underlying risky investment in a CPPI structure, we find that the performance of a credit CPPI portfolio is very de-pendent on the aggressiveness of the leverage strategy employed. The higher the multiplier assigned to a CPPI strategy, the larger is the net asset value of a CPPI port-folio. The incidence of lower net asset values also increases when the market per-forms badly. Moreover, because iTraxx tranches themselves can be viewed as lever-aged exposures to the underlying reference portfolio, a credit CPPI portfolio can seek additional leverage through investment in iTraxx tranches. For a given multiplier, a portfolio with risky investment in an equity tranche will offer a more variable risk return profile than that with risky investment in a mezzanine tranche or index port-folio. The result suggests that managers of credit CPPI portfolios should be aware of the additional leverages that might be introduced by investment in underlying CDO tranches that involve highly complex techniques for valuation and hedging. The em-pirical evidence also shows that in general a CPPI strategy performs poorly in highly volatile markets. In particular, if the value of the risky investment falls abruptly before the portfolio can be readjusted, the CPPI mechanism may fail to provide the princi-pal protection. The so-called gap risk, related to a sharp decline in portfolio values that are too sudden to allow for a rebalancing, becomes larger when the degree of leverage increases. To manage the gap risk, instead of keeping a constant multiplier all the time, a dynamic multiplier might provoke a quicker adjustment to risky expo-sures in response to a sudden sharp decline in the portfolio value, which would be an interesting issue for further investigation.
References
[1] L. Andersen, J. Sidenius, and S. Basu. All Your Hedges in One Basket. Risk, November:67–72, 2003.
[2] C. Finger. Testing Hedges Under the Standard Tranched Credit Pricing Model.
RiskMetrics Journal, 9(1):1–29, 2009.
[3] J. Gregory and J. P. Laurent. In the Core of Correlation. Risk, October:87–91, 2004. [4] J. P. Laurent and J. Gregory. Basket Default Swaps, CDOs and Factor Copulas.
Journal of Risk, 7:103–122, 2005.
[5] D. X. Li. On Default Correlation: A Copula Function Approach. Journal of Fixed Income, 9:43–54, 2000.
[6] D. Wang, S. T. Rachev, and F. J. Fabozzi. Pricing Tranches of a CDO and a CDS Index: Recent Advances and Future Research. Working Paper, 2006.
國外短期研究進修心得報告 政大財管岳夢蘭 此次利用暑假至日本東京一橋大學（Hitotsubashi University）做二個星期的 短期研究進修，可謂獲益良多。一橋大學在接到我短期研究的申請案後，便在短 時間內回覆我可以提供予我使用的資源。他們在 faculty 與交換學者使用的空間 中，提供我一張辦公桌、一台雷射印表機以及網路連線。該系所提供的研究設備 對一個僅做兩個星期短期進修的研究人員而言，實在非常慷慨，也讓我在東京該 校的研究期間中，可以不受干擾地思考從事我的學術研究。 抵達該校的第一天，系秘書 Emiko 便帶我認識該系及學校的環境。由於仍 值暑假期間，大樓內並沒有大量學生，只有利用暑假期間仍到校做研究的教授 們，因此顯得非常安靜。Emiko 帶我至 Professor Miura 的辦公室和他打招呼，告 知我以抵達。並約好下星期一中午的一個 welcome lunch，要讓我認識系上其它 的 faculty。 我在一橋大學短期進修的兩個星期中，Professor Miura 提供了許多研究資源 上的協助。除了讓我使用該校圖書館的資源外，也花時間帶我參觀並使用該校所 購置的財務資訊系統 Bloomberg。 Professor Miura 以在財金系當了十多年的系主任，前兩年才卸去系主任一 職，可謂相當資深。雖然 Professor Miura 的研究專長較偏重統計，不過他對於財 務工程領域相當熟悉，也常參加國際性的財務方面研討會。Professor Miura 和我 分享了他最近的幾篇 working papers 以及目前的研究領域。其中一篇關於 quantile exotic option 訂價問題的研究，Professor Miura 說此商品在實務上並沒有交易， 但數學問題卻相當有趣。我在思考是否有機會利用此商品建構出可供實際交易的 結構性商品，這部分的研究想法是此次短期進修的一項收穫。Professor Miura 的 另一個計劃是關於 hedge fund 報酬的研究。他目前的研究方向是將 hedge fund 視 為一項資產，並分析加入此項新資產後，對投資人資產配置問題的影響。這項研 究和我之前所指導的碩一生的碩士論文有些共通點。兩者相異的是，我們的研究 由於經費缺乏，使用的是 public available 的 free data – hedge fund index。而 Professor Miura 則是使用系上經費購置的 hedge fund data base，因此可以較清楚 地分析個別避險基金的報酬特性。
該系目前的系主任 Professor Ohashi 曾有一篇文章發表在 Journal of Financial Economics 上。他目前的研究方向之一是 security design，這個 topic 也是 subprime crisis 之後較熱門的一個研究領域。另外和我研究相關的是 Professor Ohashi 和其 指導的博士生的一篇關於 commodity spread 的論文。此論文的第一部份是理論模 型的建構，目前已經完全。而他們現在進行的是第二部分的實證工作。由於他們 使用的是我也非常熟悉的 Health, Jarrow and Morton 的模型架構，故我們也就模 型的實證估計議題，分享了彼此的經驗。
Professor Nakagawa 是一橋大學財金系的老師們中，和我目前研究領域最接 近的一位老師。他目前也在做和信用風險相關的研究。其中又以傳染效果 （contagion effect）為主要探討議題。由於 Professor Natagawa 有很強的數理背 景，因此他的研究中，統計和數學的語言非常多。我和他討論了一個信用衍生性 商品的訂價問題，由於我們兩個人所 follow 的文獻極為相似，因此在這個基礎 上，我們開始了共同的研究對話。 在兩個星期的短期進修中，一橋大學財金系也為我安排了一場 Seminar，讓 我有機會和系上老師們分享我的研究成果，也讓系上老師們更了解我的研究方 向，增加未來可能的合作機會。 整體而言，在暑假接近尾聲的開學前夕，能有機會至東京一橋大學短期進修 兩個星期，是一個難得的寶貴經驗。系上老師們的誠心接待，也讓我一個人在東 京的兩個星期的進修期間，感到非常舒適充實。也希望明年暑假前，能夠將合作 的研究完成，投稿至相關領域的期刊。
國科會補助計畫衍生研發成果推廣資料表
日期:2011/02/17國科會補助計畫
計畫名稱: 固定比例投資組合保險策略在信用投資組合之應用及實證分析 計畫主持人: 岳夢蘭 計畫編號: 98-2410-H-004-061- 學門領域: 財務無研發成果推廣資料
98 年度專題研究計畫研究成果彙整表
計畫主持人：岳夢蘭 計畫編號：98-2410-H-004-061- 計畫名稱：固定比例投資組合保險策略在信用投資組合之應用及實證分析 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 1 1 100% 博士生 2 2 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 1 1 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國外 參與計畫人力 （外國籍） 專任助理 0 0 100% 人次其他成果
