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  • Jennifer Alonso-García (a1) and Pierre Devolder (a2)


The notional defined contribution pension scheme combines pay-as-you-go financing and a defined contribution pension formula. The return on contributions is based on an index set by law, such as the growth rate of GDP, average wages or contribution payments. The volatility of this return compromises the system's pension adequacy and therefore guarantees may be needed. Here, we provide a minimum return guarantee to the pension contributions. The price is calculated in a utility indifference framework. We obtain a closed-form solution for a general dependence structure with exponential preferences and in presence of stochastic short interest rates.


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Amin, K.I. and Jarrow, R.A. (1992) Pricing options on risky assets in a stochastic interest rate economy1. Mathematical Finance, 2, 217237.
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9, 203228.
Björk, T. (2004) Arbitrage Theory in Continuous Time. Oxford University Press.
Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637654.
Börsch-Supan, A. (2006) What are NDC systems? What do they bring to reform strategies? In Pension Reform: Issues and Prospects for Notional Defined Contribution (NDC) Schemes (eds. Holzmann, R. and Palmer, E.), pp. 3557. Washington, D.C.: The World Bank, chap. 3.
Boulier, J.-F., Huang, S. and Taillard, G. (2001) Optimal management under stochastic interest rates: The case of a protected defined contribution pension fund. Insurance: Mathematics and Economics, 28, 173189.
Boyle, P.P. and Schwartz, E.S. (1977) Equilibrium prices of guarantees under equity-linked contracts. Journal of Risk and Insurance, 639–660.
Brennan, M.J. and Schwartz, E.S. (1976) The pricing of equity-linked life insurance policies with an asset value guarantee. Journal of Financial Economics, 3, 195213.
Brigo, D. and Mercurio, F. (2007) Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit 2nd ed. Berlin Heidelberg: Springer-Verlag.
Brunello, G. (2009) The effects of cohort size on European earnings. Journal of Population Economics, 23, 273290.
Campbell, J.Y. and Viceira, L.M. (2002) Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. Oxford University Press.
Cochrane, J.H. and Saá-Requejo, J. (2000) Beyond arbitrage: Good-deal asset price bounds in incomplete markets. The Journal of Political Economy, 108, 79119.
Cvitanic, J., Pham, H. and Touzi, N. (1999) A closed-form solution to the problem of super-replication under transaction costs. Finance and Stochastics, 3, 3554.
De Menil, G., Murtin, F. and Sheshinski, E. (2006) Planning for the optimal mix of paygo tax and funded savings. Journal of Pension Economics and Finance, 5, 125.
Denuit, M. (1999) The exponential premium calculation principle revisited. ASTIN Bulletin, 29, 215226.
Devolder, P. and Melis, R. (2015) Optimal mix between pay as you go and funding for pension liabilities in a stochastic framework. ASTIN Bulletin, 45, 551575.
Estrella, A. and Mishkin, F.S. (1998) Predicting US recessions: Financial variables as leading indicators. Review of Economics and Statistics, 80, 4561.
Floroiu, O. and Pelsser, A. (2013) Closed-form solutions for options in incomplete markets. Netspar Discussion Paper, no. 02/2013-004.
Follmer, H. and Schweizer, M. (1991) Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis (eds. Davis, M.H.A. and Elliott, R.J.), pp. 389414. Amsterdam: Gordon and Breach Science Publishers.
Frittelli, M. (2000) The minimal entropy martingale measure and the valuation problem in incomplete markets. Mathematical Finance, 10, 3952.
Heath, D., Jarrow, R., Morton, A., (1992) Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica, 60, 77105.
Henderson, V. (2002) Valuation of claims on nontraded assets using utility maximization. Mathematical Finance, 12, 351373.
Hodges, S. and Neuberger, A. (1989) Optimal replication of contingent claims under transaction costs. Review of Futures Markets, 8, 222239.
Holzmann, R. and Palmer, E. (2006) The status of NDC discussion: Introduction and overview. In Pension Reform: Issues and Prospects for Non-Financial Defined Contribution (NDC) Schemes (eds. Holzmann, R. and Palmer, E.), pp. 116. Washington, D.C.: The World Bank, chap. 1.
Knell, M. (2010) The optimal mix between funded and unfunded pension systems when people care about relative consumption. Economica, 77, 710733.
Korn, R. and Kraft, H. (2002) A stochastic control approach to portfolio problems with stochastic interest rates. SIAM Journal on Control and Optimization, 40, 12501269.
Matsen, E. and Thøgersen, Ø. (2004) Designing social security - A portfolio choice approach. European Economic Review, 48, 883904.
Merton, R.C. (1973) Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4, 141183.
Miao, J. and Wang, N. (2007) Investment, consumption, and hedging under incomplete markets. Journal of Financial Economics, 86, 608642.
Miyahara, Y. (1996) Canonical martingale measures of incomplete assets markets. In Probability Theory and Mathematical Statistics (Tokyo, 1995), pp. 343352.
Musiela, M. and Zariphopoulou, T. (2004) An example of indifference prices under exponential preferences. Finance and Stochastics, 8, 229239.
Næs, R., Skjeltorp, J.A. and Ødegaard, B.A. (2011) Stock market liquidity and the business cycle. The Journal of Finance, 66, 139176.
OECD (2013) Private pensions and public pension reserves. In Pensions at a Glance 2013: OECD and G20 Indicators (ed. OECD), pp. 187204. OECD Publishing, chap. 8.
Palmer, E. (2006) What's NDC? In Pension Reform: Issues and Prospects for Non-Financial Defined Contribution (NDC) Schemes (eds. Holzmann, R. and Palmer, E.), pp. 1735. Washington, D.C.: The World Bank, chap. 2.
Pennacchi, G.G. (1999) The value of guarantees on pension fund returns. Journal of Risk and Insurance, 66, 219237.
Persson, M. (2002) Five fallacies in the social security debate. In Social Security Reform in Advanced Countries: Evaluating Pension Finance (eds. Ihori, T. and Tachibanaki, T.), pp. 3951. London: Routledge Press.
Poterba, J.M. (2001) Demographic structure and asset returns. Review of Economics and Statistics, 83, 565584.
Pratt, J.W. (1964) Risk aversion in the small and in the large. Econometrica: Journal of the Econometric Society, 122–136.
Queisser, M. (1995) Chile and beyond: The second-generation pension reforms in Latin America. International Social Security Review, 48, 2339.
Rouge, R. and El Karoui, N. (2000) Pricing via utility maximization and entropy. Mathematical Finance, 10, 259276.
Samuelson, P. (1958) An exact consumption loan model of interest with or without the social contrivance of money. Journal of Political Economy, 66, 467482.
Severinson, C. and Stewart, F. (2012) Review of the swedish national pension funds. OECD Working Papers on Finance, Insurance and Private Pensions, no. 17.
Valdés-Prieto, S. (2000) The financial stability of notional account pensions. Scandinavian Journal of Economics, 102, 395417.
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177188.
Young, V.R. (2004) Pricing in an incomplete market with an affine term structure. Mathematical Finance, 14, 359381.
Zariphopoulou, T. (2001) A solution approach to valuation with unhedgeable risks. Finance and Stochastics, 5, 6182.



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