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If invested assets grow at rate r and you spend at constant rate s, both in continuous time, is there an analytical formula for the future value of a portfolio at time t? For example if you start with 100,000 dollars that would grow to 104,000 dollars in one year without withdrawals, but you actually withdraw $10,000 continuously during the year, how much will you have at year end? The problem could be discretized and then simulated, but I wonder if there is an analytical solution.

Fortranner
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    If you it grows to $104,000$ and you withdraw $10,000$ during the year then at the end of the year you have $94,000$. That´s easy. – callculus42 May 30 '19 at 09:16
  • That's true if you withdraw 10,000 at the end. I specified continuous spending. You earn less than 4,000 interest because you do not have 100,000 invested the whole time. – Fortranner May 30 '19 at 10:41
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    One possible approach: If $y(t)$ is the amount of money in your account at time $t$, $r$ is the continuous compounding rate, and $s$ is the continuous spending rate (in dollars per unit time), then you have $$\frac{dy}{dt} = ry- s.$$ You could solve this ODE (e.g. using an integrating factor) to find a formula for $y(t)$ (and use the initial condition like $y(0)=100000$ to find the value of the constant of integration). – Minus One-Twelfth Jun 01 '19 at 13:12
  • Thanks -1/12. Since spending has some positive dependence on wealth (although I ignored that in my question), the differential equation could be augmented to dy/dt = ry - s - c*y , where c measures the response of spending to wealth. – Fortranner Jun 03 '19 at 13:48

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