According to The New York Times, “During the fourteen years...

According to The New York Times, “During the fourteen years [Mitt Romney] ran it, Bain Capital’s investments reportedly earned an annual rate of return of over 100 percent, potentially turning an initial investment of $\$ 1$ million into more than $\$ 14$ million by the time he left in 1998 "' Source: The New York Times. a. What rate of return, compounded annually, would turn $\$ 1$ million into $\$ 14$ million by 1998$?$ b. The actual rate of return of Bain Capital during the 14 years that Romney ran it was 113$\% .$ Source: The American. How much would $\$ 1$ million, compounded annually at this rate, be worth after 14 years?

According to The New York Times, “During the fourteen years [Mitt Romney] ran it, Bain Capital’s investments reportedly earned an annual rate of return of over 100 percent, potentially turning an initial investment of $\$ 1$ million into more than $\$ 14$ million by the time he left in 1998 "' Source: The New York Times.
a. What rate of return, compounded annually, would turn $\$ 1$ million into $\$ 14$ million by 1998$?$
b. The actual rate of return of Bain Capital during the 14 years that Romney ran it was 113$\% .$ Source: The American. How much would $\$ 1$ million, compounded annually at this rate, be worth after 14 years?

Key Concepts

Compound Interest

Compound interest refers to the process of earning interest on both the initial amount (principal) and the interest that has been accumulated over previous periods. In finance, this principle leads to exponential growth of investments, as each period’s interest is added to the principal, thereby generating more interest in subsequent periods.

Exponential Growth

Exponential growth is a pattern of data that shows a constant percentage increase per time period. In the context of finance, it describes how investments can grow over time when the returns are reinvested, making the growth rate dependent on the current value of the investment rather than just the original principal.

Compound Annual Growth Rate (CAGR)

The Compound Annual Growth Rate (CAGR) is a useful metric that describes the mean annual growth rate of an investment over a specified time period longer than one year. It is computed by finding the rate at which the initial investment would have to grow each year to reach the final investment value, assuming that the profits were reinvested at the end of each year.

Solving for Rate Using Logarithms

Solving for an unknown annual rate in a compound interest formula often requires the use of logarithms. By expressing the compound interest equation in logarithmic form, one can isolate the rate variable and solve it algebraically, which is particularly useful when finding the annual rate of return required to reach a certain investment goal over a given number of years.

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