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Math Help - A Model of Investment Gearing

  1. #1
    Newbie emterics90's Avatar
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    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 \pi_G is

    \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 L(1+r)^T is the value of your investment after T years and the second term 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,

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

    To see whether you should gear, find out whether \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.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by emterics90 View Post
    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 \pi_G is

    \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 L(1+r)^T is the value of your investment after T years and the second term 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,

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

    To see whether you should gear, find out whether \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.
    Would it not make more sense to redure R to an annual eqivalent rate R',
    then this is worth doing if R' < r.

    RonL
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  3. #3
    Newbie emterics90's Avatar
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    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 R = 500 \times 12.

    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)

    <br />
\pi_G = L(1+r)^T - RT<br />

    where L(1+r)^T is the price of the house after T years and RT is cost of the interest repayments. \pi_{G} is the profit from gearing. \pi_{NG} 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:

    <br />
\pi_{NG} = R(1+r) + R(1+r)^2 + R(1+r)^3 + ... + R(1+r)^T<br />
= R \int^{T}_{0} (1+r)^x dx<br />

    And so I should get a loan from the bank to buy property or shares if \pi_G - \pi_{NG} > 0 or

    <br />
L(1+r)^T - RT - R \int^{T}_{0} (1+r)^x dx > 0

    Further mathematical manipulation of the above gets the following rule:

    Borrow to invest if

    \frac{L}{R} > \frac{1}{\log(1+r)} + \frac{T}{(1+r)^{T}}

    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

    \frac{L}{R} > 15.866

    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 \frac{L}{R}=50>15.866, 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.
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