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Explicit solutions to an optimal portfolio choice problem with stochastic income. (English) Zbl 1198.91188
Summary: This paper solves in closed form the optimal portfolio choice problem for an investor with negative exponential utility over terminal wealth facing imperfectly hedgeable stochastic income. The returns on income and the stock are imperfectly correlated, so the market is incomplete. We identify how an investor adjusts the portfolio of the stock and riskless asset via an intertemporal hedging demand, in reaction to the stochastic income. Under general assumptions on the process governing the income, the sign of the hedging demand is the opposite of the sign of the correlation between the income and stock. The optimal portfolio in the stock is long stock if the risk premium is positive and correlation negative, and short if these signs are reversed. Specializing in turn to normally distributed income and lognormally distributed income with or without mean-reversion, the effect of a number of parameters on the optimal portfolio in the stock can be studied. When the risk premium on the stock and correlation have opposite signs, the optimal portfolio decreases in magnitude with risk aversion, unhedgeable variance of income and time, and increases in magnitude with the magnitude of correlation under both the lognormal (without mean-reversion) and normal models for income. These relationships do not necessarily hold if the signs on the risk premium and correlation are the same, in particular the optimal portfolio is not necessarily monotone in risk aversion, violating well-known static results in models without income. When income follows a lognormal model with mean-reversion, more complicated behavior can occur. For instance, the optimal portfolio does not have to be monotonic in time, regardless of the signs of correlation and the risk premium.

##### MSC:
 91G10 Portfolio theory
Full Text:
##### References:
 [1] Bhattacharya, S., Project valuation with Mean-reverting cash flow streams, Journal of finance, 33, 1317-1331, (1978) [2] Bliss, R.; Panigirtzoglou, N., Option implied risk aversion estimates, Journal of finance, 59, 407-446, (2004) [3] Bodie, Z.; Merton, R.C.; Samuelson, W.F., Labor supply flexibility and portfolio choice in a life cycle model, Journal of economic dynamics and control, 16, 427-449, (1992) [4] Campbell, J.Y.; Viceira, L.M., Consumption and portfolio decisions when expected returns are time-varying, Quarterly journal of economics, 114, 433-495, (1999) · Zbl 0933.91021 [5] Campbell, J.Y.; Viceira, L.M., Strategic asset allocationportfolio choice for long term investors, (2002), Oxford University Press Oxford [6] Cairns, A.J.G., Blake, D., Dowd, K., 2003. Stochastic lifestyling: optimal dynamic asset allocation for defined-contribution pension plans. Preprint. · Zbl 1200.91297 [7] Chan, Y.L., Viceira, L.M., 2000. Asset allocation with endogenous labor income: the case of complete markets. Preprint. [8] Davis, M.H.A., 2000. Optimal Hedging with Basis Risk. Preprint, Imperial College, London. [9] Duffie, D., 1996. Dynamic Asset Pricing Theory, second ed. Princeton University Press, Princeton. · Zbl 1140.91041 [10] Duffie, D.; Jackson, M.O., Optimal hedging and equilibrium in a dynamic futures market, Journal of economic dynamics and control, 14, 21-33, (1990) · Zbl 0714.90003 [11] Duffie, D.; Zariphopoulou, T., Optimal investment with undiversifiable income risk, Mathematical finance, 3, 135-148, (1993) · Zbl 0884.90062 [12] Duffie, D.; Fleming, W.; Soner, H.M.; Zariphopolou, T., Hedging in incomplete markets with HARA utility, Journal of economic dynamics and control, 21, 753-782, (1997) · Zbl 0899.90026 [13] Eeckhoudt, L.; Gollier, C.; Schlesinger, H., Changes in background risk and risk taking behavior, Econometrica, 64, 683-689, (1996) · Zbl 0849.90002 [14] Elmendorf, D.W.; Kimball, M.S., Taxation of labor income and the demand for risky assets, International economic review, 41, 801-832, (2000) [15] Franke, G., Stapleton, R.C, Subrahmanyam, M.G., 2001. Standard risk aversion and the demand for risky assets in the presence of background risk. Preprint. [16] Franke, G., Peterson, S., Stapleton, R.C., 2002. Long term portfolio choice given uncertain personal savings. Preprint, EFA 2002. [17] Gollier, C.; Pratt, J.W., Weak proper risk aversion and the tempering effects of background risk, Econometrica, 64, 1109-1123, (1996) · Zbl 0856.90014 [18] He, H.; Pearson, N.D., Consumption and portfolio policies with incomplete markets and short Sale constraintsthe infinite dimensional case, Journal of economic theory, 54, 259-304, (1991) · Zbl 0736.90017 [19] Heaton, J.; Lucas, D., Portfolio choice and asset pricesthe importance of entrepreneurial risk, Journal of finance, 55, 1163-1198, (2000) [20] Henderson, V., Valuation of claims on nontraded assets using utility maximization, Mathematical finance, 12, 351-373, (2002) · Zbl 1049.91072 [21] Henderson, V.; Hobson, D., Substitute hedging, Risk, 15, 71-75, (2002) [22] Huang, C.; Litzenberger, R.H., Foundations for financial economics, (1988), Elsevier Science Publishing New York · Zbl 0677.90001 [23] Kahl, M.; Liu, J.; Longstaff, F.A., Paper millionaireshow valuable is stock to a stockholder who is restricted from selling it?, Journal of financial economics, 67, 385-410, (2003) [24] Karatzas, I.; Lehoczky, J.P.; Shreve, S.E.; Xu, G., Martingale and duality methods for utility maximization in an incomplete market, SIAM journal of control and optimization, 29, 702-720, (1991) · Zbl 0733.93085 [25] Keppo, J.S., Sullivan M.G., 2003. Modeling the optimal strategy in an incomplete market. Preprint, University of Michigan. [26] Kim, T.S.; Omberg, E., Dynamic nonmyopic portfolio behavior, Review of financial studies, 9, 141-161, (1996) [27] Kimball, M.S., Standard risk aversion, Econometrica, 61, 564-589, (1993) · Zbl 0771.90017 [28] Koo, H.K., 1995. Consumption and portfolio selection with labor income I: evaluation of human capital. Preprint, Washington University. [29] Koo, H., Consumption and portfolio selection with labor incomea continuous time approach, Mathematical finance, 8, 49-65, (1998) · Zbl 0911.90030 [30] Liu, J., 2001a. Portfolio selection in stochastic environments. Preprint, Anderson School of Management, UCLA. [31] Liu, J., 2001b. Dynamic portfolio choice and risk aversion. Preprint, Anderson School of Management, UCLA. [32] Merton, R.C., Lifetime portfolio selection under uncertaintythe continuous time case, The review of economics and statistics, 51, 247-257, (1969) [33] Merton, R.C., Optimum consumption and portfolio rules in a continuous time model, Journal of economic theory, 3, 373-413, (1971) · Zbl 1011.91502 [34] Merton, R.C., 1977. A reexamination of the capital asset pricing model. In: Friend, I., Bicksler, J. (Eds.), Risk and Return in Finance, Ballinger Press, Cambridge, MA. [35] Munk, C., Optimal consumption/investment policies with undiversifiable income risk and liquidity constraints, Journal of economic dynamics and control, 24, 1315-1343, (2000) · Zbl 0951.90052 [36] Rocha, K., Dias, M.A.G., Teixeira, J.P., 2003. The timing of development and the optimal production scale: a real option approach to oilfield E&P. Preprint, No. 981, Catholic University of Rio de Janeiro. [37] Svensson, L.E.O.; Werner, I.M., Non traded assets in incomplete markets, European economic review, 37, 1149-1168, (1993) [38] TeplĂˇ, L., Optimal hedging and valuation of nontraded assets, European finance review, 4, 231-251, (2000) · Zbl 1031.91056 [39] Viceira, L.M., Optimal portfolio choice for long-horizon investors with nontradeable labor income, Journal of finance, 56, 433-470, (2001) [40] Wachter, J.A., Portfolio and consumption decisions under Mean-reverting returnsan exact solution for complete markets, Journal of financial and quantitative analysis, 37, 63-91, (2002) [41] Weil, P., Nontraded assets and the CAPM, European economic review, 38, 913-922, (1994) [42] Zariphopoulou, T., Optimal investment and consumption models with non-linear stock dynamics, Mathematical methods of operations research, 50, 271-296, (1999) · Zbl 0961.91016 [43] Zariphopoulou, T., A solution approach to valuation with unhedgeable risks, Finance and stochastics, 5, 61-82, (2001) · Zbl 0977.93081
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