# Tail risk in pension funds: an analysis using ARCH models and bilinear processes

- 361 Downloads
- 3 Citations

## Abstract

Pension funding rules and practice contain implicit smoothing and counter-cyclical mechanisms. We set up a stylized model to investigate whether this may give rise to tail risk, in the form of large but rare losses, when pension liabilities are imperfectly but optimally hedged by pension fund assets. We find that pension losses follow a nonlinear dynamic process, and we derive a complete description of the stochastic properties of this process using Markov chain and bilinear stochastic process theory. The resulting pension dynamics resembles that of a modified ARCH model, which suggests that bursts in volatility may occur, and tail risk may be present. Simulations confirm that pension losses exhibit skewness, leptokurtosis and heavy tails, specially when cash flow smoothing is pronounced. Regulators and investors should be aware of the total amount of smoothing in pension funds as this may contribute to extreme losses, which may adversely affect the security of employee benefits as well as the valuation of firms with corporate pension plans.

## Keywords

Pensions Risk Heavy-tailed distribution LARCH process## JEL Classification

G23 G28 G32## References

- Adams J, Smith DJ (2009) Mind the gap: using derivatives overlays to hedge pension duration. Financ Anal J 65(4):60–67CrossRefGoogle Scholar
- ASB (2007a) Actuarial standard of practice no. 4. Measuring pension obligations and determining pension plan costs or contributions. Actuarial Standards Board, WashingtonGoogle Scholar
- ASB (2007b) Actuarial standard of practice no. 27. Selection of economic assumptions for measuring pension obligations. Actuarial Standards Board, WashingtonGoogle Scholar
- ASB (2009) Actuarial standard of practice no. 44. Selection and use of asset valuation methods for pension valuations. Actuarial Standards Board, WashingtonGoogle Scholar
- Bergstresser D, Desai MA, Rauh JD (2006) Earnings manipulation, pension assumptions, and managerial investment decisions. Q J Econ 121:157–195CrossRefGoogle Scholar
- Berkelaar A, Kouwenberg R (2003) Retirement saving with contribution payments and labor income as a benchmark for investments. J Econ Dyn Control 27:1069–1097CrossRefGoogle Scholar
- Blake D (2006a) Pension Economics. Wiley, ChichesterGoogle Scholar
- Blake D (2006b) Pension Finance. John Wiley, ChichesterGoogle Scholar
- Bollerslev T (1986) Generalized autoregressive conditional heteroscedasticity. J Econom 31:307–327CrossRefGoogle Scholar
- Bougerol P, Picard N (1992a) Strict stationarity of generalized autoregressive processes. Ann Probab 20(4):1714–1730CrossRefGoogle Scholar
- Bougerol P, Picard N (1992b) Stationarity of GARCH processes and of some nonnegative time series. J Econom 52:115–127CrossRefGoogle Scholar
- Broeders D, Chen A (2010) Pension regulation and the market value of pension liabilities: a contingent claims analysis using Parisian options. J Bank Financ 34(6):1201–1214CrossRefGoogle Scholar
- Campbell JL, Dhaliwal DS, Schwartz WC Jr (2010) Equity valuation effects of the Pension Protection Act of 2006. Contemp Acc Res 27(2):469–536CrossRefGoogle Scholar
- Campbell JL, Dhaliwal DS, Schwartz WC Jr (2012) Financing constraints and the cost of capital: evidence from the funding of corporate pension plans. Rev Financ Stud 25(3):868–912CrossRefGoogle Scholar
- Citigroup (2010) Citigroup pension liability index—revised methodology. http://www.soa.org/Files/Sections/pen-cpli-methodology.pdf. Accessed 10 July 2012
- Committee on Retirement Systems Research (2001) Survey of asset valuation methods for defined benefit pension plans. Pension Forum 13(1):1–49. Society of Actuaries, SchaumburgGoogle Scholar
- Davidson J (1994) Stochastic limit theory. Oxford University Press, OxfordGoogle Scholar
- Dickson DCM, Hardy MR, Waters HR (2009) Actuarial mathematics for life contingent risks. Cambridge University Press, CambridgeGoogle Scholar
- Dowd K (2002) Measuring market risk. Wiley, ChichesterGoogle Scholar
- Duffie D, Richardson HR (1991) Mean-variance hedging in continuous time. Ann Appl Probab 1:1–15CrossRefGoogle Scholar
- EIOPA (2012) Solvency II. European Insurance and Occupational Pensions Authority (EIOPA), Frankfurt am Main. https://eiopa.europa.eu/activities/insurance/solvency-ii/. Accessed 12 July 2012
- Fan J, Yao Q (2003) Nonlinear time series: nonparametric and parametric methods. Springer, BerlinGoogle Scholar
- FASB (1985) Statement of financial accounting standards no. 87: employers’ accounting for pensions. Financial Accounting Standards Board (FASB), StamfordGoogle Scholar
- FASB (2006) Statement of financial accounting standards no. 158: employers’ accounting for defined benefit pensions and other postretirement plans. Financial Accounting Standards Board (FASB), StamfordGoogle Scholar
- Feller W (1971) An introduction to probability theory and its applications, vol. 2, 2nd ed. Wiley, LondonGoogle Scholar
- Franzoni F (2009) Underinvestment vs. overinvestment: evidence from price reactions to pension contributions. J Financ Econ 92:491–518CrossRefGoogle Scholar
- Franzoni F, Marín JM (2006) Pension plan funding and stock market efficiency. J Financ 61(2):921–956CrossRefGoogle Scholar
- Giraitis L, Leipus R, Robinson PM, Surgailis D (2004) LARCH, leverage and long memory. J Financ Econom 2(2):177–210Google Scholar
- Giraitis L, Robinson PM, Surgailis D (2000) A model for long memory conditional heteroscedasticity. Ann Appl Probab 10(3):1002–1024Google Scholar
- Girola JA (2011) The HQM yield curve: basic concepts. US Department of the Treasury. http://www.treasury.gov/resource-center/economic-policy/corp-bond-yield/Documents/ycp_oct2011.pdf. Accessed 10 July 2012
- Grant CT, Grant GH, Ortega WR (2007) FASB’s quick fix for pension accounting is only first step. Financ Anal J 63:21–35Google Scholar
- Hann RN, Heflin F, Subramanayam KR (2007) Fair-value pension accounting. J Account Econ 44(3):328–358Google Scholar
- Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, CambridgeGoogle Scholar
- Jiang X (2011) The smoothing of pension expenses: a panel analysis. Rev Quant Financ Acc 37:451–476Google Scholar
- Josa-Fombellida R, Rincón-Zapatero JP (2006) Optimal investment decisions with a liability: the case of defined benefit pension plans. Insur Math Econ 39:81–98Google Scholar
- Kesten H (1973) Random difference equations and renewal theory for products of random matrices. Acta Math 131:207–248Google Scholar
- Kristensen D (2009) On stationarity and ergodicity of the bilinear model with applications to GARCH models. J Time Ser Anal 30(1):125–144Google Scholar
- Leibowitz ML, Henrikssan RD (1988) Portfolio optimization within a surplus framework. Financ Anal J 44(2):43–47Google Scholar
- Love DA, Smith PA, Wilcox DW (2011) The effect of regulation on optimal corporate pension risk. J Financ Econ 101:18–35Google Scholar
- McGill DM, Brown KN, Haley JJ, Schieber SJ (2004) Fundamentals of private pensions, 8th ed. Oxford University Press, OxfordGoogle Scholar
- Mercer (2012) Mercer pension discount yield curve and index rates in the US. http://www.mercer.com/articles/1213490. Accessed on 10 July 2012
- Mitra S, Hossain M (2009) Value-relevance of pension transition adjustments and other comprehensive income components in the adoption year of SFAS No. 158. Rev Quant Financ Acc 33:279–301Google Scholar
- Novy-Marx R, Rauh J (2011) Public pension promises: how big are they and what are they worth? J Financ 66(4):1211–1249Google Scholar
- OECD (2007) Protecting pensions. Policy analysis and examples from OECD countries. Private pensions series no. 8, Organization for Economic Cooperation and Development. OECD Publishing, ParisGoogle Scholar
- Owadally MI, Haberman S (1999) Pension fund dynamics and gains/losses due to random rates of investment return. N Am Actuar J 3(3):105–117Google Scholar
- Owadally MI, Haberman S (2004a) Efficient gain and loss amortization and optimal funding in pension plans. N Amer Actuar J 8(1):21–36Google Scholar
- Owadally MI, Haberman S (2004b) The treatment of assets in pension funding. ASTIN Bull 34(2):425–433Google Scholar
- Petersen MA (1994) Cash flow variability and firm’s pension choice. A role for operating leverage. J Financ Econ 36:361–383Google Scholar
- Pham DT (1985) Bilinear Markovian representation and bilinear models. Stoch Proc Appl 20:295–306Google Scholar
- Pham DT (1986) The mixing property of bilinear and generalised random coefficient autoregressive models. Stoch Proc Appl 23:291–300Google Scholar
- Pham DT (1993) Bilinear time series models. In: Tong H (ed) Dimension estimation and models. World Scientific, Singapore, pp. 191–223Google Scholar
- Picconi M (2006) The perils of pensions: does pension accounting lead investors and analysts astray? Acc Rev 81:925–955Google Scholar
- Rauh JD (2006) Investment and financing constraints: evidence from the funding of corporate pension plans. J Financ 61:33–71Google Scholar
- Rauh JD (2009) Risk shifting versus risk management: investment policy in corporate pension plans. Rev Financ Stud 22(7):2687–2733Google Scholar
- Rouge R, El Karoui N (2000) Pricing via utility maximization and entropy. Math Financ 10:259–276Google Scholar
- Rudolf M, Ziemba WT (2004) Intertemporal surplus management. J Econ Dyn Control 28(5):975–990Google Scholar
- Sharpe WF, Tint LG (1990) Liabilities—a new approach. J Portfolio Manage 16:5–10Google Scholar
- Shivdasani A, Stefanescu I (2010) How do pensions affect corporate capital structure decisions? Rev Financ Stud 23(3):1287–1323Google Scholar
- Sundaresan S, Zapatero F (1997) Valuation, optimal asset allocation and retirement incentives of pension plans. Rev Financ Stud 10:631–660Google Scholar
- Tapia W (2008) Comparing aggregate investment returns in privately managed pension funds: an initial assessment. OECD working papers on insurance and private pensions no. 21, Organization for Economic Cooperation and Development. OECD Publishing, ParisGoogle Scholar
- Tong H (1990) Non-linear time series. A dynamical system approach. Clarendon Press, OxfordGoogle Scholar
- Towers Watson (2011) Global pension asset study. Towers Watson, New YorkGoogle Scholar
- Yermo J, Severinson C (2010) The impact of the financial crisis on defined benefit plans and the need for counter-cyclical funding regulations. OECD working papers on finance, insurance and private pensions no. 3, Organization for Economic Cooperation and Development. OECD Publishing, ParisGoogle Scholar