Problem 14 - Applied Calculus

Question

Your company needs \(\$ 500,000\) in two years' time for renovations and can earn \(9 \%\) interest on investments. (a) What is the present value of the renovations? (b) If your company deposits money continuously at a constant rate throughout the two-year period, at what rate should the money be deposited so that you have the \(\$ 500,000\) when you need it?

Step-by-Step Solution

Answer

(a) \$420,873.21; (b) continuous rate: \$228,153.19 per year.

Step 1: Formula for Present Value

To find the present value (PV) of an amount we need in the future, we use the formula for present value: \( PV = \frac{FV}{(1+r)^n} \) where FV is the future value, \(r\) is the annual interest rate, and \(n\) is the number of years.

Step 2: Calculate Present Value

Substitute \(FV = 500,000\), \(r = 0.09\) and \(n = 2\) into the formula: \( PV = \frac{500,000}{(1+0.09)^2} \). Calculate the denominator: \((1+0.09)^2 = 1.1881\). Now divide: \( PV = \frac{500,000}{1.1881} \approx 420,873.21 \). The present value is approximately \$420,873.21.

Step 3: Formula for Continuous Compound Interest

For continuous compounding, the amount \(A\) in the future is given by \(A = Pe^{rt}\), where \(P\) is the principal (initial investment), \(r\) is the interest rate, and \(t\) is time in years. To find how much must be deposited continuously to reach \$500,000, we rearrange the formula to solve for the continuous payment rate.

Step 4: Setup and Solve for Continuous Deposit Rate

We use the formula for continuous payment \( FV = \frac{R}{r}(e^{rt} - 1) \). We want \(FV = 500,000\), \(r = 0.09\), and time \(t = 2\). Rearrange the formula to solve for \(R\): \( R = \frac{FV \cdot r}{e^{rt} - 1} \). Substitute in the values: \( R = \frac{500,000 \times 0.09}{e^{0.18} - 1} \). Calculate \(e^{0.18} \approx 1.19722\), thus \( R = \frac{45,000}{0.19722} \approx 228,153.19 \). The continuous deposit rate is approximately \$228,153.19 over the two years.

Key Concepts

Future Value

When we talk about future value, we're referring to the amount of money an investment will grow to over a period of time at a specific interest rate. It gives us a clear picture of what our current investments will yield in the future. The basic idea is simple: money grows over time, thanks to interest, resulting in a larger sum down the road.

To calculate the future value, we typically use the formula:

• \( FV = PV imes (1 + r)^n \)

where:

• \( FV \) is the future value
• \( PV \) is the present value or the initial amount of money invested
• \( r \) is the annual interest rate expressed as a decimal
• \( n \) is the number of years

This formula assumes that the interest compounds annually. But keep in mind, the more frequently interest is compounded, the more the initial investment will grow over time.

Understanding future value can be very useful for setting financial goals, planning for future expenses, or evaluating investment opportunities.

Continuous Compounding

Continuous compounding considers a scenario where interest is added to the principal an infinite number of times per period. In other words, it compounds every moment, letting your investment grow more rapidly than traditional compounding methods, which might compound annually or quarterly.

The mathematical representation of future value under continuous compounding is given by the formula:

• \( A = Pe^{rt} \)

where:

• \( A \) is the amount of money accumulated after \( t \) years, including interest.
• \( P \) is the principal amount (or initial investment).
• \( r \) is the annual interest rate expressed as a decimal.
• \( t \) is the time the money is invested for in years.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Continuous compounding becomes more significant in situations where interest rates are high, or the investment period is long, as even small differences in compounding frequency can have a large impact over time.

It's an essential concept in finance, offering insights into how investments can maximize growth through constant interest application.

Interest Rate

An interest rate is a percentage that expresses the price of borrowing money or the reward for investing it. It heavily influences how much your money can earn over time or how expensive a loan might be. Interest rates can be fixed, meaning they remain constant, or variable, meaning they may fluctuate based on market conditions.

In many investment or savings scenarios, the rate of interest plays a crucial role, determining how fast your money grows. For an investment earning compound interest, as we've seen, the snowballing effect of applying interest on earned interest becomes quite powerful.

When you encounter an interest rate in formulas, like those for present value or future value, it's usually expressed as a decimal rather than a percentage — for example, 9% becomes 0.09. Paying close attention to the interest rate is vital when you're planning financial goals. Changes in the rate can significantly alter the outcome of an investment or savings plan.

In financial contexts, even small differences in the interest rate can compound over time, leading to significantly different financial outcomes. This is why choosing the right rate for your investment or loan is critical in achieving your financial objectives.