Interest and discrete growth

When money is invested at a bank, or elsewhere, with a constant interest rate, the result is exponential growth of the money. Although most real investments involve complications such as varying interest rates, daily or monthly interest calculations, stock prices, or limited time periods, for present purposes let us consider a simplified hypothetical investment with a constant interest rate, with interest calculated after each year.

Suppose the interest rate is 5% per annum. Then the amount \(M(0)\) of money invested will receive \(0.05 \, M(0)\) interest after a year, giving a total of \(1.05 \, M(0)\). After \(t\) years, the amount of money will be multiplied by 1.05, \(t\) times, for a resulting total of \(M(t) = (1.05)^t \, M(0)\).

In general, let the interest rate be \(r\) per annum. (Above, \(r = 5\% = 0.05\).) An initial amount \(M(0)\) will be multiplied by \((1+r)\) each year; after \(t\) years the total amount \(M(t)\) is

\[ M(t) = (1+r)^t \, M(0). \]

As in our previous examples, we are measuring a quantity that varies over time. However, the processes considered in previous examples were approximately 7 continuous. The quantities were assumed to grow or decay continuously, and \(t\) could be any real number. With bank accounts, however, in practice interest is calculated at regular intervals, not continuously.

Processes which do not occur continuously over time, but rather in regular steps, are called discrete processes. Radioactive decay is a continuous process (at least to a good approximation); the regular addition of interest to a bank account is a discrete process.

Whereas continuous processes are described by differential equations such as \(\dfrac{dx}{dt} = kx\), discrete processes are described by difference equations such as

\[ x(t+1) - x(t) = k \, x(t). \]

Each time \(x\) changes, it changes by \(kx\).

With an interest rate of \(r\) per annum, the change in the amount of money \(M\) in the account each year is given by

\[ M(t+1) - M(t) = r \, M(t). \] We can write this equation as \(\Delta M = rM\); the interest rate \(r\) can be considered as a discrete growth rate.

In the Links forward section, we will discuss further the relationship between discrete and continuous processes.

Exercise 8

A bank account contains $100 and accumulates interest at a rate of \(r\) per annum.

  1. Write a formula for the amount \(M\) in the account after \(t\) years.
  2. Assuming that this formula is valid for all real numbers \(t\), when would the amount of money in the account double?
  3. Assuming again that the formula is valid for all real numbers \(t\), find \(\dfrac{dM}{dt}\) and hence find the continuous growth rate of \(M\).

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