Maximizing Mortgage Borrowing Potential with Continuous Interest and Payments

[jaejoon89]

I am having trouble understanding conceptually the following DiffEq problem:

"Suppose you can afford no more than $500 per month of payment on a mortgage. The interest rate is 8% and the mortgage term is 20 yrs. If the interest is compounded continuously and payments are made continuously, what is the max. amount you can borrow and the total interest paid during the mortgage term?"

1) dA/dt = r*A
2) A(t)=A_0*e^rt

dA/dt is the rate of change of the value of the investment... would that just be the 500/month, and use that to find the original investment?

I'm not sure I understand... From the 2nd equation A'(t) = A_0*r*e^rt, but 500/month would be fixed so doesn't resemble A'(t)...

 

[chaoseverlasting]

What does it mean that "the Interest is compounded continuously"?

 

[bdforbes]

A(t) is the amount you owe at time t. It increases continuously due to interest, and decreases continuously due to repayments. I think you need the variable r to encompass both these effects.

 

[bdforbes]

I've been thinking about this problem I feel like it's not being asked properly... it seems like they want you to calculate the principle A_0 which would grow to $120 000 after 20 years of continuous compounding at 8%, ignoring repayments.

The fact that it says "continuous repayments" is also troubling. If we make payments continuously at a rate of $500 / month, but the amount continuously grows at a rate 0.08*A, the only way to ever reduce A to zero would be if the initial rate of growth is $500 / month. Otherwise A could only increase.

EDIT: I just read what I wrote and realized that I pretty much answered the question for you :P. Here's the trick:

$$\frac{dA}{dt}=r*A - 6000$$

Ie, the rate of change of the amount includes a fixed continuous rate of $6000 / year.

 

[jaejoon89]

Thanks.