{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nConsider a market with a riskless asset with interest rate $r$ and a stock.\nThe stock price follows the [[geometric Brownian motion|D48DR46B97]] $dS_t = \\mu S_t \\, dt + \\sigma S_t \\, dW_t$, where $\\mu \u003e r$, $\\sigma \u003e 0$ and $W$ is a Wiener process.\nAn investor with initial wealth $x \u003e 0$ chooses, at each time $t$, the fraction $\\pi_t$ of wealth held in the stock and a consumption rate $c_t \\ge 0$.\nBoth are adapted to the information generated by $W$.\nThe wealth then evolves by\n$$dX_t = \\left( r X_t + \\pi_t (\\mu - r) X_t - c_t \\right) dt + \\pi_t \\sigma X_t \\, dW_t, \\qquad X_0 = x.$$\nLet $\\rho \u003e 0$ be a discount rate and let $U(c) = \\dfrac{c^{1 - \\gamma}}{1 - \\gamma}$ with $\\gamma \u003e 0$ and $\\gamma \\ne 1$.\nThe problem is to find the strategies $(\\pi, c)$ with $X_t \\ge 0$ for all $t$ that maximise the expected discounted utility of consumption\n$$E \\int_0^\\infty e^{-\\rho t} U(c_t) \\, dt.$$\n\n## Remarks\n\nThe problem is solved, and it is a standard example of [[stochastic optimal control|B24T6DQCDR]].\nWrite $\\theta = (\\mu - r) / \\sigma$ and assume $\\nu = \\dfrac{\\rho - (1 - \\gamma)\\left( r + \\theta^2 / (2 \\gamma) \\right)}{\\gamma} \u003e 0$.\nThe value function is $V(x) = \\nu^{-\\gamma} U(x)$.\nThe optimal strategy holds the constant fraction $\\pi^* = \\dfrac{\\mu - r}{\\gamma \\sigma^2}$ in the stock and consumes at the rate $c_t = \\nu X_t$.\nThese follow from the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] $\\rho V = \\max_{\\pi, c} \\left\\{ U(c) + V'(x) \\left( r x + \\pi (\\mu - r) x - c \\right) + \\frac{1}{2} \\sigma^2 \\pi^2 x^2 V''(x) \\right\\}$ with the guess $V = A U$ for a constant $A \u003e 0$.\nMerton also treated a finite horizon with utility of final wealth, and the logarithmic utility $U = \\ln$, which is the limit case $\\gamma = 1$ and gives $\\pi^* = (\\mu - r) / \\sigma^2$ and $c_t = \\rho X_t$.\n\n## Explanations\n\n### L1 human\n\nHow much of your savings should you keep in risky stocks, and how fast should you spend?\nMerton's answer, for this model, is to keep a fixed share of your wealth in stocks and to spend a fixed share of it per unit time.\nThe share in stocks grows with the excess return $\\mu - r$ and falls with the risk $\\sigma^2$ and with your aversion to risk $\\gamma$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "B24T6DQCDR"
    ],
    "history": {
      "first_stated": "1969",
      "posed_by": "Robert C. Merton"
    },
    "kind": "problem",
    "msc": [
      "91G10",
      "93E20",
      "49L20"
    ],
    "name": "Merton's portfolio problem",
    "same_as": {
      "wikidata": "Q3467003",
      "wikipedia": "Merton's portfolio problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Merton%27s_portfolio_problem"
      }
    ],
    "tag": "DZKBF576W5",
    "uses": [
      "B24T6DQCDR",
      "D48DR46B97",
      "SJP9HW82VX"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/DZKBF576W5/",
  "links": [
    {
      "field": "about",
      "target": "B24T6DQCDR"
    },
    {
      "field": "uses",
      "target": "B24T6DQCDR"
    },
    {
      "field": "uses",
      "target": "D48DR46B97"
    },
    {
      "field": "uses",
      "target": "SJP9HW82VX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DZKBF576W5.md",
  "msc": [
    "91G10",
    "93E20",
    "49L20"
  ],
  "name": "Merton's portfolio problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DZK/DZKBF576W5.md",
  "status": null,
  "tag": "DZKBF576W5",
  "uri": "m2://DZKBF576W5",
  "url": "https://tamnd.github.io/math2/wiki/mertons-portfolio-problem/",
  "writing": {
    "tag": "DZKBF576W5",
    "kind": "problem",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}