A Model of Investment Gearing

I have made up by myself a mathematical model of gearing that is supposed to help me decide whether I should gear or not. I am still young and not a finance PhD, so I would like someone to see whether my model is correct or whether I have made some serious errors.

Gearing means borrowing to invest. If I borrow to invest, then my profit from gearing $\displaystyle \pi_G$ is

$\displaystyle \pi_G = L(1+r)^T - R \int^{T}_{0} (1+r)^x dx$

where L is the amount you loan from the bank, r is the annual rate of growth of your investment, T is the duration of your loan, and R is the *yearly* interest repayment.

The first term $\displaystyle L(1+r)^T$ is the value of your investment after T years and the second term $\displaystyle R \int^{T}_{0} (1+r)^x dx$ is the opportunity cost of your investment because you could have invested without borrowing.

Solving the integral, we get,

$\displaystyle \pi_G = L(1+r)^T - \frac{R(1+r)^T}{log(1+r)}$

To see whether you should gear, find out whether $\displaystyle \pi_G > 0$. If it is, borrow to invest. Otherwise, don't.

By looking at the formula we can see that gearing is a better option if L (the loan amount) is high and R (the interest repayments) is low.