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題名 確定提撥制退休金之評價:馬可夫調控跳躍過程模型下股價指數之實證
Valuation of a defined contribution pension plan: evidence from stock indices under Markov-Modulated jump diffusion model作者 張玉華
Chang, Yu Hua貢獻者 陳麗霞<br>林士貴
Chen, Li Shya<br>Lin, Shih Kuei
張玉華
Chang, Yu Hua關鍵詞 確定提撥制退休金
保證收益
馬可夫調控跳躍過程模型
EM演算法
Esscher轉換法
defined contribution
guarantee
Markov-Modulated jump diffusion model
expectation-maximization algorithm
Esscher transformation日期 2012 上傳時間 11-Jul-2013 16:36:19 (UTC+8) 摘要 退休金是退休人未來生活的依靠，確保在退休後能得到適足的退休給付，政府在退休金上實施保證收益制度，此制度為最低保證利率與投資報酬率連結。本文探討退休金給付標準為確定提撥制，當退休金的投資報酬率是根據其連結之股價指數的表現來計算時，股價指數報酬率的模型假設為馬可夫調控跳躍過程模型，考慮市場狀態與布朗運動項、跳躍項的跳躍頻率相關，即為Elliot et al. (2007) 的模型特例。使用1999年至2012年的道瓊工業指數與S&P 500指數的股價指數對數報酬率作為研究資料，採用EM演算法估計參數及SEM演算法估計參數共變異數矩陣。透過概似比檢定說明馬可夫調控跳躍過程模型比狀態轉換模型、跳躍風險下狀態轉換模型更適合描述股價指數報酬率變動情形，也驗證馬可夫調控跳躍過程模型具有描述報酬率不對稱、高狹峰及波動叢聚的特性。最後，假設最低保證利率為固定下，利用Esscher轉換法計算不同模型下型I保證之確定提撥制退休金的評價公式，從公式中可看出受雇人提領的退休金價值可分為政府補助與個人帳戶擁有之退休金兩部分。以執行敏感度分析探討估計參數對於馬可夫調控跳躍過程模型評價公式的影響，而型II保證之確定提撥制退休金的價值則以蒙地卡羅法模擬並探討其敏感度分析結果。
Pension plan make people a guarantee life in their retirement. In order to ensure the appropriate amount of pension plan, government guarantees associated with pension plan which ties minimum rate of return guarantees and underlying asset rate of return. In this paper, we discussed the pension plan with defined contribution (DC). When the return of asset is based on the stock indices, the return model was set on the assumption that markov-modulated jump diffusion model (MMJDM) could the Brownian motion term and jump rate be both related to market states. This model is the specific case of Elliot et al. (2007) offering. The sample observations is Dow-Jones industrial average and S&P 500 index from 1999 to 2012 by logarithm return of the stock indices. We estimated the parameters by the Expectation-Maximization (EM) algorithm and calculated the covariance matrix of the estimates by supplemented EM (SEM) algorithm. Through the likelihood ratio test (LRT), the data fitted the MMJDM better than other models. The empirical evidence indicated that the MMJDM could describe the asset return for asymmetric, leptokurtic, volatility clustering particularly. Finally, we derived different model`s valuation formula for DC pension plan with type-I guarantee by Esscher transformation under rate of return guarantees is constant. From the formula, the value of the pension plan could divide into two segment: government supplement and employees deposit made pension to their personal bank account. And then, we done sensitivity analysis through the MMJDM valuation formula. We used Monte Carlo simulations to evaluate the valuation of DC pension plan with type-II guarantee and discussed it from sensitivity analysis.參考文獻 [1] Black, F., Scholes, M., 1973.The pricing of options and corporate liabilities. Journal of Political Eeconomy 81, 3, 637-654.[2] Brennan, J.M., Schwartz, E.S., 1976. The pricing of equity-linked life insurance policies with an asset value guarantee. Journal of Financial Economics 3, 195-213.[3] Bo, L., Wang, Y., Yang, X., 2010. Markov-modulated jump-diffusion for currency option pricing. Insurance: Mathematics and Economics 46, 461-469.[4] Bodie, Z., Marcus, Alan J., Merton, Robert C., 1988. Defind benefit versus defined contribution pension plans: What are the real trade-offs? Pensions in the U.S. Economy. University of Chicago Press.[5] Charles, C., Fuh, C., D., Lin, S., K., 2013. A tale of two regimes: Theory and empirical evidence for a markov-modulated jump diffusion model of equity returns and derivative pricing implications. Working paper.[6] Cont, R., 2007. Volatility clustering in financial markets: empirical facts and agent-based models. Long Memory in Economics, 289-310.[7] Ding, Z., Granger, C., and Engle, R., 1993. A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 1, 83-106.[8] Duan, J.C., Popova, I., and Ritchken, P., 2002. Option pricing under regime switching. Quantitative Finance 2, 116-132.[9] Elliott, R.J., Chan, L., and Siu, T., 2005. Option pricing and Esscher transform under regime switching. Annals of Finance 1, 4, 423-432.[10] Elliott, R.J., Siu, T.K., Chan, L., 2007. Pricing options under a generalized markov-modulated jump-diffusion model. Stochastic Analysis and Application 25, 4, 821-843.[11] Elliott, R.J., Siu, T.K., 2012. Option pricing and filtering with hidden markov-modulated pure-jump processes. Applied Mathematical Finance, iFirst, 1-25.[12] Gerber, H., and Shiu, E., 1994. Option pricing by esscher transforms. Transactions of the Society of Actuaries 46, 99, 140.[13] Hamilton, J. D., 1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 , 357-384.[14] Haldrup, N. and Nielsen, M.O., 2006. A regime switching long memory model for electricity prices. Journal of Econometrics 135, 349–376[15] Hsu, Emma Y., Lin, S.K., Hung, M., Huang, T.H., 2013. Empirical analysis of stock indices under regime switching model with dependent jump sizes risk. Working paper.[16] Korn, R., Siu, T.K., Zhang, A., 2011. Asset allocation for a DC pension fund under regime switching environment. European Actuarial Journal 1, 361-377.[17] Kou, S., 2002. A jump-diffusion model for option pricing. Management Science, 1086-1101.[18] Lange, K. A, 1995. Gradient algorithm locally equivalent to the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 57, 2, 425-437.[19] Lin, S. K., Shyu, S. D., and Wang, S. Y., 2013. Option pricing under stock market cycles with jump risks: evidence from Dow Jones industrial average index and S&P 500 indix. Working paper. [20] Lin, S. K., Lin, C. S. and Chou, C. Y. 2013. A recursive formula for a participating contract embedding a surrender option under regime-switching model with jump risks: evidence from S&P 500 stock indix. Working paper.[21] Lin, S. K., Yang, Sharon S. and Lin, C. Y., 2013. Valuation of equity-indexed annuities under regime-switching jump model: evidence from stock indices. Working paper.[22] Lin, S. K., Liu, H., Lee, J. C., 2013. Option pricing under regime-switching jump model with dependent jump sizes: evidence from stock index options. Working paper.[23] Lin, S. K., and Wu, S. J., 2013. Estimating variance of parameter estimators by supplemented expectation maximization and gibbs sampling algorithm in regime-switching jump model. Working paper.[24] Lindest, S., 2004. Relative guarantees. The geneva papers on risk and insurance theory 29, 187-209.[25] Mandelbrot, B., 1963. The variation of certain speculative prices. The Journal of Business 36, 4, 394-419.[26] Margrabe, W., 1978. The value of an option to exchange one asset for another. Journal of Finance 33, 177-186.[27] Merton, R.C., 1976. Option pricing when underlying stock returns are discontinuous* 1. Journal of Financial Economics 3, 1-2, 125-144.[28] Merton, R.C., 1983. On the role of Social Security as a means for efficient risk sharing in an economy where human capital is not tradeable. Financial aspects of the United States pension system, edited by Zvi Bodie and John Shoven. Chicago：University of Chicago Press.[29] Pennacchi, G.G, 1999. The value of guarantees on pension fund returns. Journal of Risk and Insurance 66, 219-237.[30] Person, S.A., Aase, K.K., 1997. Valuation of the minimum guaranteed return embedded in life insurance products. Journal of Risk and Insurance 64, 599-617.[31] Ramezani, C.A., Zeng, Y., 1999. Maximum likelihood estimation of asymmetric jump-diffusion processes: application to security prices. Working Paper.[32] Schaller, H., and Norden, S. V., 1997. Regime switching in stock market returns. Applied Financial Economics 7, 177-191.[33] Schwert, G. W., 1989. Business cycles, financial crises, and stock volatility. Carnegie Rochester Conference Series on Public Policy 31, 83-126.[34] Siu, T.K., Yang, H., Lau, J.W., 2008. Pricing currency options under two-factor markov-modulated stochastic volatility models. Insurance: Mathematics and Economics 43, 295-302.[35] Yang, S.S., Yueh, M.L., Tang, C.H., 2008. Valuation of the interest rate guarantee embedded in defined contribution pension plans. Insurance: Mathematics and Economics 42, 920-934. 描述 碩士
國立政治大學
統計研究所
100354007
101資料來源 http://thesis.lib.nccu.edu.tw/record/#G0100354007 資料類型 thesis dc.contributor.advisor 陳麗霞<br>林士貴 zh_TW dc.contributor.advisor Chen, Li Shya<br>Lin, Shih Kuei en_US dc.contributor.author (Authors) 張玉華 zh_TW dc.contributor.author (Authors) Chang, Yu Hua en_US dc.creator (作者) 張玉華 zh_TW dc.creator (作者) Chang, Yu Hua en_US dc.date (日期) 2012 en_US dc.date.accessioned 11-Jul-2013 16:36:19 (UTC+8) - dc.date.available 11-Jul-2013 16:36:19 (UTC+8) - dc.date.issued (上傳時間) 11-Jul-2013 16:36:19 (UTC+8) - dc.identifier (Other Identifiers) G0100354007 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/58782 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 統計研究所 zh_TW dc.description (描述) 100354007 zh_TW dc.description (描述) 101 zh_TW dc.description.abstract (摘要) 退休金是退休人未來生活的依靠，確保在退休後能得到適足的退休給付，政府在退休金上實施保證收益制度，此制度為最低保證利率與投資報酬率連結。本文探討退休金給付標準為確定提撥制，當退休金的投資報酬率是根據其連結之股價指數的表現來計算時，股價指數報酬率的模型假設為馬可夫調控跳躍過程模型，考慮市場狀態與布朗運動項、跳躍項的跳躍頻率相關，即為Elliot et al. (2007) 的模型特例。使用1999年至2012年的道瓊工業指數與S&P 500指數的股價指數對數報酬率作為研究資料，採用EM演算法估計參數及SEM演算法估計參數共變異數矩陣。透過概似比檢定說明馬可夫調控跳躍過程模型比狀態轉換模型、跳躍風險下狀態轉換模型更適合描述股價指數報酬率變動情形，也驗證馬可夫調控跳躍過程模型具有描述報酬率不對稱、高狹峰及波動叢聚的特性。最後，假設最低保證利率為固定下，利用Esscher轉換法計算不同模型下型I保證之確定提撥制退休金的評價公式，從公式中可看出受雇人提領的退休金價值可分為政府補助與個人帳戶擁有之退休金兩部分。以執行敏感度分析探討估計參數對於馬可夫調控跳躍過程模型評價公式的影響，而型II保證之確定提撥制退休金的價值則以蒙地卡羅法模擬並探討其敏感度分析結果。 zh_TW dc.description.abstract (摘要) Pension plan make people a guarantee life in their retirement. In order to ensure the appropriate amount of pension plan, government guarantees associated with pension plan which ties minimum rate of return guarantees and underlying asset rate of return. In this paper, we discussed the pension plan with defined contribution (DC). When the return of asset is based on the stock indices, the return model was set on the assumption that markov-modulated jump diffusion model (MMJDM) could the Brownian motion term and jump rate be both related to market states. This model is the specific case of Elliot et al. (2007) offering. The sample observations is Dow-Jones industrial average and S&P 500 index from 1999 to 2012 by logarithm return of the stock indices. We estimated the parameters by the Expectation-Maximization (EM) algorithm and calculated the covariance matrix of the estimates by supplemented EM (SEM) algorithm. Through the likelihood ratio test (LRT), the data fitted the MMJDM better than other models. The empirical evidence indicated that the MMJDM could describe the asset return for asymmetric, leptokurtic, volatility clustering particularly. Finally, we derived different model`s valuation formula for DC pension plan with type-I guarantee by Esscher transformation under rate of return guarantees is constant. From the formula, the value of the pension plan could divide into two segment: government supplement and employees deposit made pension to their personal bank account. And then, we done sensitivity analysis through the MMJDM valuation formula. We used Monte Carlo simulations to evaluate the valuation of DC pension plan with type-II guarantee and discussed it from sensitivity analysis. en_US dc.description.tableofcontents 第一章 緒論 1第二章 文獻回顧 62.1 退休金 62.2 財務模型 82.2.1 BS模型與狀態轉換模型 82.2.2 卜瓦松跳躍模型 92.2.3 馬可夫調控跳躍模型 112.2.4 其他模型 13第三章 契約、財務模型與估計檢定 143.1 退休金契約 143.2 財務模型 163.2.1 狀態轉換模型 163.2.2 跳躍風險下狀態轉換模型 173.2.3 馬可夫調控跳躍過程模型 183.3 馬可夫調控跳躍過程模型之估計與檢定 19第四章 確定提撥制下退休金之定價 224.1 Esscher轉換 224.1.1 狀態轉換模型下Esscher 轉換 224.1.2 跳躍風險下狀態轉換模型下Esscher 轉換 244.1.3 馬可夫調控跳躍過程模型下Esscher 轉換 264.2 確定提撥制退休金定價 284.2.1 狀態轉換模型下確定提撥制退休金定價 284.2.2 跳躍風險下狀態轉換模型確定提撥制退休金定價 304.2.3 馬可夫調控跳躍過程模型下確定提撥制退休金定價 32第五章 實證分析 345.1 實證分析 345.1.1 模型參數估計與檢定 345.1.2 偏態與峰態 395.1.3 狀態與跳躍動態分析 405.1.4 波動叢聚 435.2 敏感度分析 45第六章 結論 53參考文獻 56附錄A：EM演算法估計模型參數之過程 59附錄B：狀態轉換模型確定提撥制退休金評價公式 61附錄C：跳躍風險下狀態轉換模型確定提撥制退休金評價公式 66附錄D：馬可夫調控跳躍過程模型確定提撥制退休金評價公式 73附錄E：馬可夫調控跳躍過程模型動差公式 80附錄F：馬可夫調控跳躍過程模型自我相關函數公式 82表目錄表 1：1999年至2012年道瓊工業指數報酬率之統計 3表 2：道瓊工業指數及S&P 500指數在四種模型中之參數估計與檢定結果 35表 3：日報酬資料與各模型平均數、變異數、偏態系數與峰態系數之估計 39表 4：型I保證之轉移機率敏感度分析 46表 5：型I保證之布朗運動項標準差敏感度分析 47表 6：型I保證之跳躍幅度平均數與標準差敏感度分析 47表 7：型I保證之跳躍頻率敏感度分析 48表 8：型II保證之轉移機率敏感度分析 49表 9：型II保證之布朗運動項標準差敏感度分析 50表 10：型II保證之跳躍幅度平均數與標準差敏感度分析 51表 11：型II保證之跳躍頻率敏感度分析 52表 12：1999年至2012年道瓊工業指數報酬率之統計 55表 13：1999年至2012年S&P 500指數報酬率之統計 55圖目錄圖 1：道瓊工業指數動態圖 3圖 2：道瓊工業指數報酬率動態圖 4圖 3：道瓊工業指數之指數、報酬率、狀態及跳躍機率動態圖 41圖 4：S&P 500指數之指數、報酬率、狀態及跳躍機率動態圖 42圖 5：道瓊工業指數報酬率動態、資料與狀態轉換跳躍相關模型自我相關函數 44圖 6：S&P 500指數報酬率動態、資料與狀態轉換跳躍相關模型自我相關函數44 zh_TW dc.format.extent 1161790 bytes - dc.format.mimetype application/pdf - dc.language.iso en_US - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0100354007 en_US dc.subject (關鍵詞) 確定提撥制退休金 zh_TW dc.subject (關鍵詞) 保證收益 zh_TW dc.subject (關鍵詞) 馬可夫調控跳躍過程模型 zh_TW dc.subject (關鍵詞) EM演算法 zh_TW dc.subject (關鍵詞) Esscher轉換法 zh_TW dc.subject (關鍵詞) defined contribution en_US dc.subject (關鍵詞) guarantee en_US dc.subject (關鍵詞) Markov-Modulated jump diffusion model en_US dc.subject (關鍵詞) expectation-maximization algorithm en_US dc.subject (關鍵詞) Esscher transformation en_US dc.title (題名) 確定提撥制退休金之評價:馬可夫調控跳躍過程模型下股價指數之實證 zh_TW dc.title (題名) Valuation of a defined contribution pension plan: evidence from stock indices under Markov-Modulated jump diffusion model en_US dc.type (資料類型) thesis en dc.relation.reference (參考文獻) [1] Black, F., Scholes, M., 1973.The pricing of options and corporate liabilities. Journal of Political Eeconomy 81, 3, 637-654.[2] Brennan, J.M., Schwartz, E.S., 1976. The pricing of equity-linked life insurance policies with an asset value guarantee. Journal of Financial Economics 3, 195-213.[3] Bo, L., Wang, Y., Yang, X., 2010. Markov-modulated jump-diffusion for currency option pricing. Insurance: Mathematics and Economics 46, 461-469.[4] Bodie, Z., Marcus, Alan J., Merton, Robert C., 1988. Defind benefit versus defined contribution pension plans: What are the real trade-offs? Pensions in the U.S. Economy. University of Chicago Press.[5] Charles, C., Fuh, C., D., Lin, S., K., 2013. A tale of two regimes: Theory and empirical evidence for a markov-modulated jump diffusion model of equity returns and derivative pricing implications. Working paper.[6] Cont, R., 2007. Volatility clustering in financial markets: empirical facts and agent-based models. Long Memory in Economics, 289-310.[7] Ding, Z., Granger, C., and Engle, R., 1993. A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 1, 83-106.[8] Duan, J.C., Popova, I., and Ritchken, P., 2002. Option pricing under regime switching. Quantitative Finance 2, 116-132.[9] Elliott, R.J., Chan, L., and Siu, T., 2005. Option pricing and Esscher transform under regime switching. Annals of Finance 1, 4, 423-432.[10] Elliott, R.J., Siu, T.K., Chan, L., 2007. Pricing options under a generalized markov-modulated jump-diffusion model. Stochastic Analysis and Application 25, 4, 821-843.[11] Elliott, R.J., Siu, T.K., 2012. Option pricing and filtering with hidden markov-modulated pure-jump processes. Applied Mathematical Finance, iFirst, 1-25.[12] Gerber, H., and Shiu, E., 1994. Option pricing by esscher transforms. Transactions of the Society of Actuaries 46, 99, 140.[13] Hamilton, J. D., 1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 , 357-384.[14] Haldrup, N. and Nielsen, M.O., 2006. A regime switching long memory model for electricity prices. Journal of Econometrics 135, 349–376[15] Hsu, Emma Y., Lin, S.K., Hung, M., Huang, T.H., 2013. Empirical analysis of stock indices under regime switching model with dependent jump sizes risk. Working paper.[16] Korn, R., Siu, T.K., Zhang, A., 2011. Asset allocation for a DC pension fund under regime switching environment. European Actuarial Journal 1, 361-377.[17] Kou, S., 2002. A jump-diffusion model for option pricing. Management Science, 1086-1101.[18] Lange, K. A, 1995. Gradient algorithm locally equivalent to the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 57, 2, 425-437.[19] Lin, S. K., Shyu, S. D., and Wang, S. Y., 2013. Option pricing under stock market cycles with jump risks: evidence from Dow Jones industrial average index and S&P 500 indix. Working paper. [20] Lin, S. K., Lin, C. S. and Chou, C. Y. 2013. A recursive formula for a participating contract embedding a surrender option under regime-switching model with jump risks: evidence from S&P 500 stock indix. Working paper.[21] Lin, S. K., Yang, Sharon S. and Lin, C. Y., 2013. Valuation of equity-indexed annuities under regime-switching jump model: evidence from stock indices. Working paper.[22] Lin, S. K., Liu, H., Lee, J. C., 2013. Option pricing under regime-switching jump model with dependent jump sizes: evidence from stock index options. Working paper.[23] Lin, S. K., and Wu, S. J., 2013. Estimating variance of parameter estimators by supplemented expectation maximization and gibbs sampling algorithm in regime-switching jump model. Working paper.[24] Lindest, S., 2004. Relative guarantees. The geneva papers on risk and insurance theory 29, 187-209.[25] Mandelbrot, B., 1963. The variation of certain speculative prices. The Journal of Business 36, 4, 394-419.[26] Margrabe, W., 1978. The value of an option to exchange one asset for another. Journal of Finance 33, 177-186.[27] Merton, R.C., 1976. Option pricing when underlying stock returns are discontinuous* 1. Journal of Financial Economics 3, 1-2, 125-144.[28] Merton, R.C., 1983. On the role of Social Security as a means for efficient risk sharing in an economy where human capital is not tradeable. Financial aspects of the United States pension system, edited by Zvi Bodie and John Shoven. Chicago：University of Chicago Press.[29] Pennacchi, G.G, 1999. The value of guarantees on pension fund returns. Journal of Risk and Insurance 66, 219-237.[30] Person, S.A., Aase, K.K., 1997. Valuation of the minimum guaranteed return embedded in life insurance products. Journal of Risk and Insurance 64, 599-617.[31] Ramezani, C.A., Zeng, Y., 1999. Maximum likelihood estimation of asymmetric jump-diffusion processes: application to security prices. Working Paper.[32] Schaller, H., and Norden, S. V., 1997. Regime switching in stock market returns. Applied Financial Economics 7, 177-191.[33] Schwert, G. W., 1989. Business cycles, financial crises, and stock volatility. Carnegie Rochester Conference Series on Public Policy 31, 83-126.[34] Siu, T.K., Yang, H., Lau, J.W., 2008. Pricing currency options under two-factor markov-modulated stochastic volatility models. Insurance: Mathematics and Economics 43, 295-302.[35] Yang, S.S., Yueh, M.L., Tang, C.H., 2008. Valuation of the interest rate guarantee embedded in defined contribution pension plans. Insurance: Mathematics and Economics 42, 920-934. zh_TW
