Question

Maria currently has \(\$ 10,000\) in her retirement fund. She wants to see how much money she will have in her fund for several different years in the future, assuming that her portfolio has a steady annual growth rate of \(10 \% .\) What function \(f(n)\) would model the amount she should have in her portfolio in \(n\) years? (A) \(f(n)=10,000^{n}\) (B) \(f(n)=10,000 \times 0.1^{n}\) (C) \(f(n)=10,000 \times 1.1^{n}\) (D) \(f(n)=10,000 \times 1.11^{n}\)

Short answer

The correct function is (C) \( f(n) = 10,000 \times 1.1^n \).

Step-by-step solution

Understand the Growth Model

The problem involves modeling the growth of Maria's retirement fund over time with a constant annual growth rate. Since the portfolio grows at a rate of 10% per year, this is a case of exponential growth. The function to model such growth is in the form \( A = P(1 + r)^n \), where \( P \) is the principal amount, \( r \) is the growth rate, and \( n \) is the number of years.

Identify the Principal and Growth Rate

In this problem, the principal amount \( P \) is Maria’s current retirement fund, which is \$10,000. The growth rate \( r \) is 10%, or 0.10 when expressed as a decimal.

Construct the Function

Substitute the known values into the exponential growth formula. Using \( P = 10,000 \) and \( r = 0.10 \), the formula becomes: \[ f(n) = 10,000(1 + 0.10)^n = 10,000(1.1)^n \].

Match with Given Options

Compare the function \( f(n) = 10,000(1.1)^n \) derived in the previous step with the provided options. The correct function, which matches our derived equation, is option (C) \( f(n) = 10,000 \times 1.1^n \).

Key concepts

Understanding Retirement Funds

A retirement fund is a vital financial tool for planning your future financial needs post-employment. By setting aside money in a retirement fund, you ensure that you will have a source of income when you no longer work. Retirement funds often involve investing in various financial assets that grow over time. There are many types of retirement funds, such as 401(k)s, IRAs, and pensions. The core aim of a retirement fund is to accumulate enough money that will sustain your lifestyle after retirement. Starting early is beneficial as it takes advantage of compound interest—earning interest on interest over the years. Managing your retirement fund well involves knowing your risk tolerance, how much you can contribute regularly, and understanding the power of exponential growth through your investments.

Annual Growth Rate and Its Importance

The annual growth rate is a measure that reflects how much a particular quantity increases or decreases over a year. In financial contexts, it often signifies the percentage increase in investments like stocks or retirement funds. A steady annual growth rate is crucial for long-term financial planning. For Maria's retirement fund, an annual growth rate of 10% means that each year, her funds are expected to increase by 10% on top of the existing balance. This consistency allows for more reliable future financial projections. Understanding how the growth rate affects your investments helps in setting realistic expectations for your savings goals. It is essential to remember that higher growth rates often come with increased risk, so it's vital to balance growth rate aspirations with your risk capacity.

Exponential Functions in Financial Growth

Exponential functions are mathematical expressions that describe a quantity that grows or decays at a consistent relative rate. In the context of financial growth, an exponential function helps in modeling how investment portfolios grow over time due to compound interest.Maria's situation is modeled with an exponential function: \[ f(n) = 10,000 \times (1.1)^n \]Here, 10,000 represents the initial investment, while \((1.1)^n\) denotes the growth factor increasing by 10% each year. This setup is typical in scenarios involving compounded returns, as it accurately mirrors real-world financial growth dynamics.Exponential growth illustrates that investment returns grow larger over time, not just because of the added returns but due to earning on top those returns as well. Therefore, understanding how exponential functions work is critical in predicting how initial investments like retirement funds will mature over time.

The Role of Financial Mathematics

Financial mathematics involves applying mathematical methods to solve problems in finance, including strategic planning, assessing the value of investments, and managing risk. It equips you with tools to analyze and predict the behavior of financial markets or personal investments. In our exercise about Maria's retirement fund, financial mathematics allows us to predict the future value of her fund based on the current amount, growth rate, and time. The concept of exponential growth applied here is a cornerstone of financial mathematics, simplifying complex financial predictions into understandable formats. By mastering these principles, you can make better financial decisions, such as deciding the right time to invest, understanding the expected returns, and managing your portfolio strategically to optimize growth. Whether you're working on retirement planning or any financial endeavor, a solid grip on financial mathematics is invaluable.