Growth Calculations

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Interest on a bank deposit

[1] Yn = Y0(1+r)n

Can solve for any variable, as long as we know the other three.

Example of solving for the ending balance. Suppose that:

ending balance, Y4 = $1464.10

In a spreadsheet, such as Microsoft Excel, you use the formula +1000*(1+.10)^4

Next example--Solve for beginning balance. Same example as before, except we want to know how much to deposit now in order to have $1600 in four years.

1600 = Y0(1+0.10)4

Divide 1600 by (1.1)4 to get $1092.82

Next example--Solve for interest rate. Suppose you start with $1000 and end up with $1600 after four years. What was the interest rate?

1600 = 1000(1+r)4

Three steps:

  1. Divide both sides by 1000
    1.6 = (1+r)4
  2. Get rid of the exponent by taking both sides to the 1/n power
    (1.6)(1/4) = (1+r) = 1.125
  3. Solve for r and convert to percent: r = .125 = 12.5 percent

Final example--Solving for the number of years. Suppose you start with $1000, the interest rate is 10 percent, and you want to know how long it will take until you have $2000. Here, we use logs (you can use either regular logs or natural logs).

logYt = logY0 + t[log(1+r)]

Solving for t gives

t = (logYt - logY0)/log(1+r)

In our example

t = (log[2000] - log[1000])/log[1+.10] = 7.27 years

For problem set, note that:

In general, a ratio increases when the numerator grows faster than the denominator, and vice-versa.