1478:
1024:
981:
1473:{\displaystyle {\begin{aligned}\nu &=\left(\rho -(1-\gamma )\left({\frac {(\mu -r)^{2}}{2\sigma ^{2}\gamma }}+r\right)\right)/\gamma \\&=\rho /\gamma -(1-\gamma )\left({\frac {(\mu -r)^{2}}{2\sigma ^{2}\gamma ^{2}}}+{\frac {r}{\gamma }}\right)\\&=\rho /\gamma -(1-\gamma )(\pi (W,t)^{2}\sigma ^{2}/2+r/\gamma )\\&=\rho /\gamma -(1-\gamma )((\mu -r)\pi (W,t)/2\gamma +r/\gamma ).\end{aligned}}}
714:
1521:
represents portfolios having the stock/bond proportion derived by Merton in the absence of transaction costs. As long as the point which represents the current portfolio is near the Merton line, i.e. between the upper and the lower boundary, no action needs to be taken. When the portfolio crosses
1536:
Morton and Pliska considered trading costs that are proportional to the wealth of the investor for logarithmic utility. Although this cost structure seems unrepresentative of real life transaction costs, it can be used to find approximate solutions in cases with additional assets, for example
976:{\displaystyle c(W,t)={\begin{cases}\nu \left(1+(\nu \epsilon -1)e^{-\nu (T-t)}\right)^{-1}W&{\textrm {if}}\;T<\infty \;{\textrm {and}}\;\nu \neq 0\\(T-t+\epsilon )^{-1}W&{\textrm {if}}\;T<\infty \;{\textrm {and}}\;\nu =0\\\nu W&{\textrm {if}}\;T=\infty \end{cases}}}
1574:
change over time. An interest rate model could be added and would lead to a portfolio containing bonds of different maturities. Some authors have added a stochastic volatility model of stock market returns.
695:
569:
465:
1029:
1961:
1522:
above the upper or below the lower boundary, one should rebalance the portfolio to bring it back to that boundary. In 1994 Shreve and Soner provided an analysis of the problem via the
1016:
1572:
1577:
Bankruptcy can be incorporated. This problem was solved by
Karatzas, Lehoczky, Sethi and Shreve in 1986. Many models incorporating bankruptcy are collected in Sethi (1997).
254:
310:
1992:
1657:
Karatzas, Ioannis; Lehoczky, John P.; Sethi, Suresh P.; Shreve, Steven E. (May 1986). "Explicit
Solution of a General Consumption/Investment Problem".
1523:
1968:
708:
do not appear on the right-hand side; a constant fraction of wealth is invested in stocks, no matter what the age or prosperity of the investor.
45:
in 1969 both for finite lifetimes and for the infinite case. Research has continued to extend and generalize the model to include factors like
37:. An investor must choose how much to consume and must allocate their wealth between stocks and a risk-free asset so as to maximize expected
2097:
2056:
1877:
1791:
1659:
629:
2023:
2006:
Karatzas, I.; Lehoczky, J. P.; Sethi, S. P.; Shreve, S. E. (1985). "Explicit solution of a general consumption/investment problem".
1509:
where the solution is known. For a graphical representation, the amount invested in each of the two assets can be plotted on the
2138:
316:
499:
1517:-axes; three diagonal lines through the origin can be drawn: the upper boundary, the Merton line and the lower boundary. The
508:
324:
34:
20:
623:
problem, a closed-form solution exists. The optimal consumption and stock allocation depend on wealth and time as follows:
2123:
1643:
Sethi, S.P. and Taksar, M.I., “A Note on Merton's 'Optimum
Consumption and Portfolio Rules in a Continuous-Time Model,”
1533:
the problem was addressed by
Eastman and Hastings in 1988. A numerical solution method was provided by Schroder in 1995.
2133:
2128:
1506:
312:
is a constant which expresses the investor's risk aversion: the higher the gamma, the more reluctance to own stocks.
1835:
1690:
607:
are known and constant, in this (1969) version of the model, although Merton allowed them to change in his
1488:
Many variations of the problem have been explored, but most do not lead to a simple closed-form solution.
989:
1537:
individual stocks, where it becomes difficult or intractable to give exact solutions for the problem.
264:
1545:
744:
1542:
The assumption of constant investment opportunities can be relaxed. This requires a model for how
1934:
Morton, A. J.; Pliska, S. R. (1995). "Optimal
Portfolio Management with Fixed Transaction Costs".
1894:
1857:
1816:
1761:
1623:
608:
121:
1738:
2093:
2052:
2019:
1986:
1615:
1598:(1 August 1969). "Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case".
295:
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2011:
1943:
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1686:
1668:
1607:
1595:
272:
46:
42:
1783:
620:
1875:
Eastham, Jerome F.; Hastings, Kevin J. (1988). "Optimal
Impulse Control of Portfolios".
1947:
492:
2117:
1757:
1709:
1765:
1505:
the problem was solved by Davis and Norman in 1990. It is one of the few cases of
2010:. Lecture Notes in Control and Information Sciences. Vol. 78. p. 209.
1914:"Optimal Portfolio Selection with Fixed Transaction Costs: Numerical Solutions"
2048:
1913:
1852:
1619:
1812:
275:(which applies both to consumption and to the terminal wealth, or bequest,
1890:
1804:
1672:
1718:
2015:
1898:
1861:
1820:
1627:
38:
30:
700:
This expression is commonly referred to as Merton's fraction. Because
1739:"Labor supply flexibility and portfolio choice in a life cycle model"
1691:"Optimum consumption and portfolio rules in a continuous-time model"
1611:
2089:
592:
is unrestricted (that is borrowing or shorting stocks is allowed).
482:) are the expected return and volatility of the stock market and
690:{\displaystyle \pi (W,t)={\frac {\mu -r}{\sigma ^{2}\gamma }}.}
81:(which may include the present value of wage income). At time
2084:. Stochastic Modelling and Applied Probability. Vol. 39.
94:, and what fraction of wealth to invest in a stock portfolio,
1836:"Optimal Investment and Consumption with Transaction Costs"
969:
564:{\displaystyle u(x)={\frac {x^{1-\gamma }}{1-\gamma }}.}
460:{\displaystyle dW_{t}=\,dt+W_{t}\pi _{t}\sigma \,dB_{t}}
595:
Investment opportunities are assumed constant, that is
1548:
1027:
992:
717:
632:
511:
327:
298:
124:
85:
he must choose what amount of his wealth to consume,
2041:Optimal Consumption and Investment with Bankruptcy
1566:
1492:Flexible retirement age can be taken into account.
1472:
1010:
975:
689:
563:
459:
304:
248:
125:
1495:A utility function other than CRRA can be used.
19:For Merton’s one-period portfolio problem, see
8:
1784:"Portfolio Selection with Transaction Costs"
288:parameterizes the desired level of bequest,
103:(the remaining fraction 1 −
57:The investor lives from time 0 to time
41:. The problem was formulated and solved by
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1206:
1187:
1156:
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1028:
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991:
950:
949:
921:
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888:
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828:
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654:
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510:
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140:
123:
74:. He starts with a known initial wealth
1746:Journal of Economic Dynamics and Control
112:being invested in the risk-free asset).
1587:
495:, i.e. the stochastic term of the SDE.
1991:: CS1 maint: archived copy as title (
1984:
1600:The Review of Economics and Statistics
1498:Transaction costs can be introduced.
292:is the subjective discount rate, and
7:
1834:Shreve, S. E.; Soner, H. M. (1994).
1639:
1637:
1011:{\displaystyle 0\leq \epsilon \ll 1}
315:The wealth evolves according to the
1948:10.1111/j.1467-9965.1995.tb00071.x
1878:Mathematics of Operations Research
1792:Mathematics of Operations Research
1660:Mathematics of Operations Research
963:
916:
841:
16:Problem in continuous-time finance
14:
1840:The Annals of Applied Probability
1524:Hamilton–Jacobi–Bellman equation
574:Consumption cannot be negative:
317:stochastic differential equation
29:is a problem in continuous-time
2082:Methods of Mathematical Finance
2008:Stochastic Differential Systems
500:constant relative risk aversion
498:The utility function is of the
1567:{\displaystyle r,\mu ,\sigma }
1503:proportional transaction costs
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1432:
1420:
1414:
1402:
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35:intertemporal portfolio choice
21:Mutual fund separation theorem
1:
619:Somewhat surprisingly for an
1923:. Michigan State University.
1758:10.1016/0165-1889(92)90044-F
1710:10.1016/0022-0531(71)90038-X
1526:and its viscosity solutions.
1737:; Samuelson, W. F. (1992).
1507:stochastic singular control
249:{\displaystyle \max E\left}
2155:
1698:Journal of Economic Theory
1645:Journal of Economic Theory
27:Merton's portfolio problem
18:
2049:10.1007/978-1-4615-6257-3
1782:; Norman, A. R. (1990).
491:is the increment of the
474:is the risk-free rate, (
2139:Intertemporal economics
2108:Continuous Time Finance
1853:10.1214/aoap/1177004966
1531:fixed transaction costs
305:{\displaystyle \gamma }
61:; their wealth at time
1568:
1474:
1012:
977:
691:
583: ≥ 0, while
565:
461:
306:
250:
2039:Sethi, S. P. (1997).
1912:Schroder, M. (1995).
1891:10.1287/moor.13.4.588
1805:10.1287/moor.15.4.676
1673:10.1287/moor.11.2.261
1569:
1475:
1013:
978:
692:
566:
462:
307:
251:
1936:Mathematical Finance
1647:, 46, 1988, 395-401.
1546:
1025:
990:
715:
630:
509:
325:
296:
265:expectation operator
122:
2124:Financial economics
2110:, Blackwell (1990).
2076:Karatzas, Ioannis;
150:
2134:Portfolio theories
2129:Stochastic control
2016:10.1007/BFb0041165
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1008:
973:
968:
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609:intertemporal CAPM
561:
457:
302:
246:
136:
33:and in particular
2099:978-0-387-94839-3
2078:Shreve, Steven E.
2058:978-1-4613-7871-6
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115:The objective is
53:Problem statement
47:transaction costs
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1967:. Archived from
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273:utility function
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49:and bankruptcy.
43:Robert C. Merton
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2070:Further reading
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1962:"Archived copy"
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1780:Davis, M. H. A.
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1612:10.2307/1926560
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1529:When there are
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621:optimal control
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2090:10.1007/b98840
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2024:
1998:
1953:
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1904:
1867:
1826:
1771:
1725:
1704:(4): 373–413.
1678:
1667:(2): 261–294.
1649:
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1606:(3): 247–257.
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2106:Merton R.C.:
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2025:3-540-16228-3
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1988:
1974:on 2014-11-08
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1687:Merton, R. C.
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1596:Merton, R. C.
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1969:the original
1956:
1939:
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1876:
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1733:Bodie, Z.;
1519:Merton line
271:is a known
65:is denoted
2118:Categories
1978:2014-10-28
1942:(4): 337.
1885:(4): 588.
1846:(3): 609.
1799:(4): 676.
1582:References
1484:Extensions
1620:0034-6535
1562:σ
1556:μ
1458:γ
1444:γ
1418:π
1409:−
1406:μ
1394:γ
1391:−
1382:−
1379:γ
1371:ρ
1355:γ
1327:σ
1301:π
1292:γ
1289:−
1280:−
1277:γ
1269:ρ
1249:γ
1229:γ
1219:σ
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1194:μ
1177:γ
1174:−
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1154:ρ
1141:γ
1114:γ
1105:σ
1083:−
1080:μ
1063:γ
1060:−
1051:−
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1033:ν
1003:≪
1000:ϵ
997:≤
964:∞
942:ν
929:ν
917:∞
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882:ϵ
873:−
857:≠
854:ν
842:∞
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798:−
789:ν
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772:−
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670:σ
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634:π
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538:−
441:σ
432:π
395:−
373:−
370:μ
358:π
300:γ
215:ρ
212:−
202:γ
198:ϵ
160:ρ
157:−
138:∫
2080:(1998).
1987:cite web
1766:16699153
1689:(1971).
615:Solution
611:(1973).
1899:3689945
1862:2245058
1821:3689770
1628:1926560
603:,
599:,
478:,
263:is the
39:utility
31:finance
2096:
2055:
2022:
1897:
1860:
1819:
1764:
1626:
1618:
1513:- and
986:where
605:σ
601:μ
585:π
480:σ
476:μ
470:where
290:ρ
286:ε
259:where
105:π
96:π
1972:(PDF)
1965:(PDF)
1917:(PDF)
1895:JSTOR
1858:JSTOR
1817:JSTOR
1787:(PDF)
1762:S2CID
1742:(PDF)
1694:(PDF)
1624:JSTOR
2094:ISBN
2053:ISBN
2020:ISBN
1993:link
1616:ISSN
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1018:and
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2086:doi
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