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Compound Interest and Exponential Equations

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Compound Interest and Exponential Equations Compound Interest and Exponential Equations Math 98 Among the many applications of exponential functions is compound interest, where, as opposed to simple interest, the interest rate is applied after a pred- ermined time has passed, to the initial investment plus the acquired interest to that point. In practice, this means that the following formula applies. Let P be the initial investment (the principal), and r the effective annual interest rate – that means that $1 will grow, to 1+ r after one year. Then, after t years (not 4 necessarily an integer: for example, t = 3 corresponds to 16 months) the invest- t ment will have grown to P (1 + r) (see below for the reason). We often write exponential functions in the form “abx”. In this case, a = P,b =1+ r, x = t. Example An initial investment of $2500 at 3.5% rate (r = 0.035) will grow, after 27 9 27 9 · 4 months (two years and three months or t = 12 = 4 ) to 2500 (1 + 0.035) = 9 2500 · 1.035 4 ≈ 2701.10 This exponential growth is markedly faster than the linear growth coming from simple interest, and is what applies to most any debt or investment you might incur into. A typical question we might ask is how long it will take for an investment to grow to a given value. This leads to an exponential equation, an equation where the unknown is in the exponent. Such an equation can be solved by taking logarithms of both sides, and using the properties of logarithms (see the book and the corresponding additional file). Example For the same situation in the previous example, how long will it take for the investment to grow to $5000 (to double)? We are asking for t such that t t 2500 · 1.035 = 5000, which is equivalent to 1.035 = 2. Taking logarithms of both sides we find t · log(1.035) = log(2) log 2 t = ≈ 20.149 log 1.035 or a bit more than 20 years. By the way, it doesn’t matter what base you use for your logarithmsin a formula like this (althoough you would default to decimal logs), since the change of base formula for logarithms implies that all bases will result in the same final result. 1 2 In general, the time needed for an investment of P dollars to grow to C dollars, at a compound yearly interest rate of r, can be found by solving the equation t t C P (1 + r) = C, or (1 + r) = P for the unknown t. The simplest way to find the solution is to take logarithms of both sides: t C log (1 + r) = log h i P C t · log (1 + r) = log P log C t = P log (1 +r) You will need a computer or a calculator to evaluate these logarithms and obtain a number for t, since logarithms and exponentials are not easily calculated “by hand”, except in extremely simple cases. x x Note In general, for a function of the form y = ab , b is the factor by which the initial value a (it corresponds to x = 0) is multiplied when the independent t value is x. Thus, in our interest examples, (1 + r) tells us by how much the initial investment P will be multiplied after t years. Why does compound interest grow exponentially Compound interest follows this logic: after a given amount of time, during which interest grows linearly (as in simple interest), the interest accrued is “merged" into the “principal” (the amount on which interest is computed), which is now larger, and so will gain more interest. The length of time for this to happen is the “compounding time”. It can be a year, as in the example above or a fraction of a year, even a day, or an even shorter amount of time. Since time is usually measured in years, and the interest rate usually refers to a year (but you may notice that some credit cards list a monthly rate, so as to make it less obvious how high the interest they charge actually is), if the compounding occurs n times a year (for example, monthly compound- ing means n = 12), after the first compounding period a capital of C will r have grown to C 1+ n . In the next period interest will be computed on this amount, so that at the end of the second period the investment will have r 2 grown to C 1+ n , and so on. After k periods, corresponding to a total time k r k of n = t years, so that k = nt, the investment will be worth C 1+ n = t r nt r n C 1+ n = C 1+ n . It is common to rewrite this formula in terms of the annual percentage rate (APR) which is the annual rate that would produce the same growth if the compounding happened yearly. That is a rate rˆ, such r n that 1+ n =1+ˆr. In the formulas in the first section, the interest rate is meant to be this APR..
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