Question

Geometric Mean The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. To find the geometric mean of n values (all of which are positive), first multiply the values, then find the nth root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of 5.154%, 2.730%, 0.488%, 0.319%, 0.313%, and 0.268%. Identify the single percentage growth rate that is the same as the five consecutive growth rates by computing the geometric mean of 1.05154, 1.02730, 1.00488, 1.00319, 1.00313, and 1.00268.

Short answer

The growth rate is 1.5290%.

Step-by-step solution

Given information

The observations for growth rates are 1.05154, 1.02730, 1.00488, 1.00319, 1.00313, and 1.00268, corresponding the interest rates of 5.154%, 2.730%, 0.488%, 0.319%, 0.313%, and 0.268%, respectively.

Geometric mean formula

For n observations, the geometric mean is the nth root of the product of all positive observations.

Mathematically, it can be written as

$\overline{x}=\sqrt[n]{\prod x}$for x as the observations.

Substitute the values to obtain thegeometric mean of the growth rates.

$$\begin{gathered} \overline{x}=\sqrt[6]{1.05154\times 1.02730\times 1.00488\times 1.00319\times 1.00313\times 1.00268} \\ =\sqrt[6]{1.095318} \\ =1.01529 \end{gathered}$$

The associated growth rate will be 1.529%.

Thus, the single percentage growth is 1.5290%