Optimizing Savings Growth with Continuous Compounding and Depletion

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In summary, the conversation discusses finding the maximum balance of a savings account after one year, given the initial balance of $1000, a continuous interest rate of 10% per annum, and continuous depletion at a rate of y^2 per million dollars per year. The differential equation dy/dt= y/10 - y^2/1000000 is used to solve for the balance after t years, with initial values of y=100000(e^(t/10))/(99+e^(t/10)). It is determined that the maximum balance occurs as t goes to infinity, with a value of 100000, and the balance reaches half of this maximum after t = ln(99) years.
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Alex Mamaev
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Homework Statement


Find the amount in a savings account after one year if the initial balance in the account was $1000, the interest is paid continuously into the account at a normal rate of 10% per annum (compounded continuously), and the account is being continuously depleted at the rate of y^2 per million dollars per year. the balance in the account after t years is y=y(t). How large can the account grow? How long will it take the account to grow to half of this maximum balance

Homework Equations


the differential equation which i think is correct is:
dy/dt= y/10 - y^2/1000000

The Attempt at a Solution


I solved the equation by separating and then doing partial fractions,
I got y= 100000Ce^(t/10)/(1+ke^(t/10))
with initial values this became 100000(e^(t/10))/(99+e^(t/10))
here is where i don't really know what to do. I took the derivative and that was always positive so I'm not sure how to find the maximum or if I've made an error.
Based on the initial differential equation, there should be an optimum value when y = 100000 but for some reason the function isn't defined there. Please help if possible really lost with this one, thanks.
 
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  • #2
No, you are not "really lost", you have just not realized a critical point: If the slope is always positive, then the function keeps increasing so the maximum occurs as to goes to infinity. It is easy to see that the limit, as t goes to infinity, is 100000. Now, what is t when the value is 50000?
 
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Omg thank you so much, i kept thinking it was an optimization problem, don't know why. I get it now:)
 

FAQ: Optimizing Savings Growth with Continuous Compounding and Depletion

What is a financial ODE problem?

A financial ODE problem is a mathematical problem that involves using Ordinary Differential Equations (ODEs) to model and solve financial situations or scenarios. These problems often involve variables such as interest rates, inflation, and stock prices, and are commonly used in finance and economics.

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ODEs allow for a more dynamic and accurate representation of financial systems and can account for changing variables over time. They also allow for the prediction of future outcomes based on current data and can be used to optimize financial decisions.

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Financial ODE problems are commonly used in areas such as investment analysis, risk management, and option pricing. They can also be applied to macroeconomic models, such as predicting the effects of policy changes on the economy.

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One of the main challenges of financial ODE problems is that they can be complex and require advanced mathematical knowledge to solve. They also rely heavily on accurate and up-to-date data, which can be difficult to obtain in some cases.

How can I learn more about financial ODE problems?

There are many resources available for learning about financial ODE problems, including textbooks, online courses, and academic papers. It is recommended to have a strong understanding of calculus and financial principles before diving into these problems.

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