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Optimal retirement planning under partial information. (English) Zbl 1437.91385

A retiree has an annuity which pays at a rate \(a\). She has further an amount \(x\) to invest. For the investment there is a riskless asset \(d B(t) = r B(t) \;d t\) and a risky asset \(d S(t) = S(t) \{ (r + \theta \sigma)\;d t + \sigma \;d W(t)\}\), where \(W\) is a standard Brownian motion. The remaining lifetime \(\tau\) of the retiree is independent of the Brownian motion and has a deterministic force of mortality \(\zeta(t)\). The market price of risk \(\theta\) (and thus the drift of the risky asset) is unknown and can take \(m\) different values \(\vartheta_1, \ldots, \vartheta_m\). Writing \(Y(t) = W(t) + \theta t\) the process \(Y(t) = \sigma^{-1} \{\log (S(t)/S(0)) - (r-\frac12 \sigma^2)t\}\) is observable. Bayes’ rule yields \(\mathbb{P}[\theta = \vartheta_k \mid \mathcal{F}_t^S] = \mathbb{P}[\theta = \vartheta_k \mid Y(t)]\), which are explicit expressions.
The retiree can decide on the investment strategy \((\pi(t))\) and on the consumption rate \((c(t))\), which need to be adapted. There is a minimal consumption \(\underline{c}\) that the retiree needs for living. Then the current wealth process fulfils \[ d X(t) = (r X(t) + a - c(t)) \;d t + \pi(t) \sigma \;d Y(t) \;.\] The goal is to maximise the expected utility of the consumption process \[ \sup_{(\pi,c) \in \mathcal{A}(x)}\mathbb{E}\Bigl[ \int_0^\tau e^{-\rho t}\frac{(c(t) - \underline{c})^{1-\gamma}}{1-\gamma}\;d t\Bigr]\;,\] where \(\gamma \in (0,\infty) \setminus\{1\}\) and \[\mathcal{A}(x) = \Bigl\{(\pi,c): X(0) = x, c(t) \ge \underline{c}, X(t) \ge \frac{\underline{c}-a}r, \int_0^\infty \pi(t)^2 \;d t < \infty\Bigr\}\;.\] The condition \(X(t) \ge (\underline{c}-a)/r\) is in order that there is enough wealth for the minimal consumption \(\underline{c}\).
The problem is solved by considering \(d \hat W(t) = d Y(t) - \mathbb{E}[\theta \mid Y(t)]\;d t\) and changing the measure, such that \((\hat W(t))\) becomes a standard Brownian motion. This makes the observation complete. The solution is then the solution to Merton’s problem with a correction term caused by the random life time. In the same way, logarithmic utility (\(\gamma = 1\)) is considered and it is shown that the strategy is just the limiting strategy as \(\gamma \to 1\). Some numerical examples and extensions to the model are discussed.

MSC:

91G05 Actuarial mathematics
91G10 Portfolio theory
60G44 Martingales with continuous parameter
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[1] N. Bäuerle and S. Grether, Extremal behavior of long-term investors with power utility, Int. J. Theor. Appl. Finance 20 (2017), no. 5, Article ID 1750029. · Zbl 1396.91674
[2] T. Björk, M. H. A. Davis and C. Landén, Optimal investment under partial information, Math. Methods Oper. Res. 71 (2010), no. 2, 371-399. · Zbl 1189.49053
[3] C. Blanchet-Scalliet, N. El Karoui, M. Jeanblanc and L. Martellini, Optimal investment decisions when time-horizon is uncertain, J. Math. Econom. 44 (2008), no. 11, 1100-1113. · Zbl 1153.91018
[4] Z. Bodie, J. B. Detemple, S. Otruba and S. Walter, Optimal consumption—portfolio choices and retirement planning, J. Econom. Dynam. Control 28 (2004), no. 6, 1115-1148. · Zbl 1179.91230
[5] S. Brendle, Portfolio selection under incomplete information, Stochastic Process. Appl. 116 (2006), no. 5, 701-723. · Zbl 1137.91012
[6] J. Y. Campbell and L. M. Viceira, Strategic Asset Allocation: Portfolio Choice for Long-Term Investors, Oxford University, Oxford, 2002. · Zbl 0933.91021
[7] A. Chen, P. Hieber and J. K. Klein, Tonuity: A novel individual-oriented retirement plan, Astin Bull. 49 (2019), no. 1, 5-30. · Zbl 1419.91352
[8] K. J. Choi and G. Shim, Disutility optimal retirement and portfolio selection, Math. Finance 16 (2006), no. 2, 443-467. · Zbl 1145.91343
[9] K. J. Choi, G. Shim and Y. H. Shin, Optimal portfolio, consumption-leisure and retirement choice problem with CES utility, Math. Finance 18 (2008), no. 3, 445-472. · Zbl 1141.91428
[10] J. F. Cocco and F. J. Gomes, Longevity risk, retirement savings, and financial innovation, J. Financial Econ. 103 (2012), no. 3, 507-529.
[11] J. C. Cox and C.-F. Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econom. Theory 49 (1989), no. 1, 33-83. · Zbl 0678.90011
[12] L. U. Delong and A. Chen, Asset allocation, sustainable withdrawal, longevity risk and non-exponential discounting, Insurance Math. Econom. 71 (2016), 342-352. · Zbl 1371.91154
[13] P. H. Dybvig and H. Liu, Lifetime consumption and investment: Retirement and constrained borrowing, J. Econom. Theory 145 (2010), no. 3, 885-907. · Zbl 1245.91044
[14] P. H. Dybvig and H. Liu, Verification theorems for models of optimal consumption and investment with retirement and constrained borrowing, Math. Oper. Res. 36 (2011), no. 4, 620-635. · Zbl 1238.93125
[15] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models. Estimation and Control, Appl. Math. (New York) 29, Springer, New York, 1995. · Zbl 0819.60045
[16] E. Farhi and S. Panageas, Saving and investing for early retirement: A theoretical analysis, J. Financial Econ. 83 (2007), no. 1, 87-121.
[17] G. Gennotte, Optimal portfolio choice under incomplete information, J. Finance 41 (1986), no. 3, 733-746.
[18] B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Philos. Trans. Roy. Soc. Lond. 115 (1825), 513-583.
[19] E. J. Gumbel, Statistics of Extremes, Columbia University, New York, 1958. · Zbl 0086.34401
[20] H. Hata and S.-J. Sheu, An optimal consumption and investment problem with partial information, Adv. Appl. Probab. 50 (2018), no. 1, 131-153. · Zbl 1434.91060
[21] H. He and N. D. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite-dimensional case, J. Econom. Theory 54 (1991), no. 2, 259-304. · Zbl 0736.90017
[22] H. He and N. D. Pearson, Consumption and portfolio policies with incomplete markets and short-sales constraints: The finite-dimensional case, Math. Finance 1 (1991), 1-10. · Zbl 0900.90142
[23] C.-F. Huang and H. Pagès, Optimal consumption and portfolio policies with an infinite horizon: Existence and convergence, Ann. Appl. Probab. 2 (1992), no. 1, 36-64. · Zbl 0749.60039
[24] I. Karatzas, J. P. Lehoczky and S. E. Shreve, Optimal portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. Control Optim. 25 (1987), no. 6, 1557-1586. · Zbl 0644.93066
[25] I. Karatzas, J. P. Lehoczky, S. E. Shreve and G.-L. Xu, Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim. 29 (1991), no. 3, 702-730. · Zbl 0733.93085
[26] I. Karatzas and X. Zhao, Bayesian adaptive portfolio optimization, Option Pricing, Interest Rates and Risk Management, Handb. Math. Finance, Cambridge University, Cambridge (2001), 632-669. · Zbl 1012.91022
[27] H.-S. Lee and Y. H. Shin, An optimal consumption, investment and voluntary retirement choice problem with disutility and subsistence consumption constraints: A dynamic programming approach, J. Math. Anal. Appl. 428 (2015), no. 2, 762-771. · Zbl 1314.91198
[28] Y. Lin and S. H. Cox, Securitization of mortality risks in life annuities, J. Risk Insurance 72 (2005), no. 2, 227-252.
[29] K. Lindensjö, Optimal investment and consumption under partial information, Math. Methods Oper. Res. 83 (2016), no. 1, 87-107. · Zbl 1414.91348
[30] M. Longo and A. Mainini, Learning and portfolio decisions for CRRA investors, Int. J. Theor. Appl. Finance 19 (2016), no. 3, Article ID 1650018. · Zbl 1337.91079
[31] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory 3 (1971), no. 4, 373-413. · Zbl 1011.91502
[32] M. A. Milevsky and T. S. Salisbury, Optimal retirement income tontines, Insurance Math. Econom. 64 (2015), 91-105. · Zbl 1348.91176
[33] C. Munk, Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences, J. Econom. Dynam. Control 32 (2008), no. 11, 3560-3589. · Zbl 1181.91299
[34] L. Pastor and P. Veronesi, Learning in financial markets, NBER Working Paper 14646 (2009).
[35] J. Piggott, E. A. Valdez and B. Detzel, The simple analytics of a pooled annuity fund, J. Risk Insurance 72 (2005), no. 3, 497-520.
[36] S. R. Pliska, A stochastic calculus model of continuous trading: Optimal portfolios, Math. Oper. Res. 11 (1986), no. 2, 370-382. · Zbl 1011.91503
[37] S. R. Pliska and J. Ye, Optimal life insurance purchase and consumption/investment under uncertain lifetime, J. Banking Finance 31 (2007), no. 5, 1307-1319.
[38] W. Putschögl and J. Sass, Optimal consumption and investment under partial information, Decis. Econ. Finance 31 (2008), no. 2, 137-170. · Zbl 1165.91410
[39] S. J. Richards, A handbook of parametric survival models for actuarial use, Scand. Actuar. J. (2012), no. 4, 233-257. · Zbl 1277.91148
[40] U. Rieder and N. Bäuerle, Portfolio optimization with unobservable Markov-modulated drift process, J. Appl. Probab. 42 (2005), no. 2, 362-378. · Zbl 1138.93428
[41] J. Sass and U. G. Haussmann, Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain, Finance Stoch. 8 (2004), no. 4, 553-577. · Zbl 1063.91040
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