Question

Assume you invest \(\$ 250\) at the end of each year for 10 years at an annual interest rate of \(r .\) The amount of money in your account after 10 years is given by \(A(r)=\frac{250\left((1+r)^{10}-1\right)}{r} .\) Assume your goal is to have \(\$ 3500\) in your account after 10 years. a. Show that there is an interest rate \(r\) in the interval \((0.01,0.10)-\) between \(1 \%\) and \(10 \%-\) that allows you to reach your financial goal. b. Use a calculator to estimate the interest rate required to reach your financial goal.

Short answer

Answer: The estimated interest rate required is approximately 4.84%.

Step-by-step solution

Evaluating A(r) at the interval's endpoints

First, let's evaluate the function A(r) at the end points of the given interval of interest, r=0.01 and r=0.10. At r=0.01: \(A(0.01) = \frac{250\left((1+0.01)^{10}-1\right)}{0.01}\) At r=0.10: \(A(0.10) = \frac{250\left((1+0.10)^{10}-1\right)}{0.10}\) Now, we need to check if these values are below and above our target amount (\(\$ 3500\)).

Checking if interval is valid

Let's check if \(A(0.01) < 3500 < A(0.10)\): A(0.01) = 2557.27, which is less than 3500 and A(0.10) = 3866.10, which is greater than 3500. Since \(A(0.01) < 3500 < A(0.10)\), it can be concluded that there exists an interest rate, r, in the interval (0.01, 0.10) or (1%, 10%) that will allow us to reach our financial goal of having \(\$ 3500\) in the account after 10 years.

Estimating the required interest rate using calculator

Now, let's use a calculator with numerical solving capability to estimate the interest rate, r, that will result in having \(\$ 3500\) in the account. We need to solve the equation: \(A(r) = 3500\) Using a calculator's solving function, we input the following equation: \(\frac{250\left((1+r)^{10}-1\right)}{r} = 3500\) The estimated interest rate comes out to be approximately r = 0.0484 or 4.84%. So, to reach the financial goal of having \(\$ 3500\) in the account after 10 years of investing \(\$ 250\) per year, the required interest rate is estimated to be approximately 4.84%.

Key concepts

Time Value of Money

Understanding the time value of money is crucial for anyone looking to make informed decisions in the realm of finance. Put simply, it refers to the idea that a dollar today holds more value than a dollar in the future because of its potential earning capacity. This principle underpins many financial concepts and calculations, such as interest rates and the growth of investments over time.

Let's consider an exercise where you're saving \(250 annually for 10 years, and you're aiming to achieve a future sum of \)3500. By taking into account the time value of money, we can calculate the interest rate needed to reach this target. The formula used in the exercise represents this principle exactly — it shows how consistent investments grow over time given a certain interest rate.

Future Value of Annuity

An annuity is a series of equal payments made at regular intervals. The future value of an annuity calculates how much these series of payments will be worth at a future date, based on compound interest. This is highly relevant for retirement planning, saving for college, or any scenario where you plan to make regular payments towards a financial goal.

In the given exercise, the annuity is the consistent \(250 year-end payments, and the future value is the amount you aim to have after 10 years (\)3500). The calculation requires compounding, which incorporates the principle that interest accumulates on both initial principal and accumulated interest from previous periods. Hence, compound interest magnifies the growth of your savings over time compared to simple interest.

Present Value

Present value is essentially the current worth of a future sum of money or flow of cash flows given a specific rate of return. The present value goes hand in hand with the concept of the time value of money, as it adjusts for the interest you would earn over time.

In this context, it would be the amount of money you need to invest today to reach your future financial goal, considering the expected rate of return. While the original exercise doesn't explicitly address the present value, it's the starting point for anyone looking to determine how much they need to start with to achieve a certain amount in the future, depending on their expected earnings from interest.

Financial Goal Setting

Setting financial goals is a fundamental step in planning your economic future. It involves understanding how much money you need to save and how long it will take you to save it, informed by calculations like those in our exercise.

Financial goal setting begins with defining a specific, measurable outcome — in this case, accumulating \(3500 over 10 years through annual investments of \)250. By using the future value formula of an annuity, you can identify the necessary interest rate to meet this goal as the exercise illustrates. The clarity provided by mathematical quantification helps in crafting a realistic and attainable financial plan.

Interest Rate Estimation

Interest rate estimation plays a key role when you are planning your savings or investment strategy. In our exercise, the interest rate determines how much the annual \(250 investments will grow over the 10-year period.

Finding the right interest rate involves solving for 'r' in the future value formula. Based on the calculation, if the estimated rate falls within a realistic and accessible range, the financial goal is attainable. The exercise indicates that by trial with a calculator or utilizing financial software, an estimated interest rate of approximately 4.84% would enable you to achieve your target of \)3500. Estimating interest rates with precision is important for personal financial planning and for understanding the broader economic environment.