John Roberts has 42,180.53 in a brokerage account, and he...

John Roberts has $\$ 42,180.53$ in a brokerage account, and he plans to contribute an additional $\$ 5,000$ to the account at the end of every year. The brokerage account has an expected annual return of 12 percent. If John's goal is to accumulate $\$ 250,000$ in the account, how many years will it take for John to reach his goal?

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Key Concepts

Compound Interest

Compound interest is the process where interest earned on an investment is reinvested, so that in future periods, interest is calculated on both the initial principal and the accumulated interest from previous periods. This accelerates the growth of the investment over time and is a fundamental concept in finance and investment planning.

Annuity

An annuity refers to a series of equal payments made at regular intervals, such as annual contributions to a brokerage account. The future value of these contributions can be calculated using the annuity formula, which accounts for the compounded growth of each payment over time.

Future Value Calculation

Future value calculation involves determining the amount an investment made today will grow to over a period of time at a given rate of return. This includes separate calculations for an initial lump sum and for the series of periodic contributions, which are then combined to estimate the total accumulated value.

Solving for Time in Exponential Equations

When determining how many periods it takes to reach a specific financial goal, one must solve equations in which the unknown appears in the exponent. This typically requires the use of logarithms or iterative methods, as the growth due to compound interest creates an exponential equation.

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