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 is
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 is the value of your investment after T years and the second term is the opportunity cost of your investment because you could have invested without borrowing.
Solving the integral, we get,
To see whether you should gear, find out whether . 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.
R is the yearly interest repayments. So for example suppose you buy a house and the bank lend you $100,000 so then L=100,000. The bank requires you to repay $500 a month, and so .
I'm a little uncertain about the second term, the integral.
Maybe it looks like this. If I buy a house and get L from the bank and have to pay back R per year then after T years I will have (assuming the houses appreciates by r per year)
where is the price of the house after T years and RT is cost of the interest repayments. is the profit from gearing. is the profit from not gearing.
If however instead of buying a house I use the money I would otherwise use for interest repayments for a house to buy, say, shares that go up at rate r per year then I get the following:
And so I should get a loan from the bank to buy property or shares if or
Further mathematical manipulation of the above gets the following rule:
Borrow to invest if
So for example suppose I wanted to be as wealthy as possible in 11 years time, so I make T=11. Returns on property or shares in the long run is about 9% so I will set r=0.09. Plugging these into the equation gives the following:
Borrow to invest if
Remember L is the loan amount and R is the yearly interest repayment. So suppose the bank offers to loan you $300,000 and asks for you to pay back $500 monthly (or $6000 yearly) then L=300,000 and R=6000 and , so you should accept this deal from the bank and borrow to invest.
Most people who buy houses tend to just buy it for emotional reasons, e.g. it looks nice, but I am trying to reduce here a rigid rule for whether or not taking out a mortgage is a good idea. Taking out a loan from the bank to invest needn't be for property investment. Investment in shares is also possible.