Problem 9 - 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) Approximately \$420,076.79 is needed today. (b) Deposit \$228,104.94 continuously per year.

Step 1: Understanding the Present Value Formula

To determine the present value needed to secure a future amount, we use the present value formula: \( PV = \frac{FV}{(1 + r)^n} \), where \( PV \) is the present value, \( FV \) is the future value, \( r \) is the interest rate, and \( n \) is the number of years. Here, \( FV = 500,000 \), \( r = 0.09 \), and \( n = 2 \).

Step 2: Calculating Present Value

Substitute the given values into the formula: \( PV = \frac{500,000}{(1 + 0.09)^2} = \frac{500,000}{1.1881} \approx 420,076.79 \). This means you would need approximately \$420,076.79 today.

Step 3: Determining the Continuously Deposited Rate

For continuous deposits, we use the formula \( FV = \frac{d}{r} (e^{r \cdot t} - 1) \), where \( d \) is the continuous deposit rate, \( r \) is the annual interest rate, \( t \) is time in years, and \( FV \) is the future value. We solve for \( d \) knowing \( FV = 500,000 \), \( r = 0.09 \), and \( t = 2 \).

Step 4: Solving for the Continuous Deposit Rate

Rearrange the continuous formula to solve for \( d \): \( d = \frac{FV \cdot r}{e^{r \cdot t} - 1} = \frac{500,000 \times 0.09}{e^{0.18} - 1} \approx \frac{45,000}{0.197217} \approx 228,104.94 \). This means you must deposit approximately \$228,104.94 continuously each year.

Key Concepts

Interest Rate

Interest rates are pivotal in understanding financial calculations. They represent the cost of borrowing money or the return on investment for savings and investments. In this context, an interest rate of 9% per annum (0.09 in decimal form) is used to determine how much money your savings will earn over time. The interest rate affects how the present and future values are calculated. As the interest compounds over each period (in our example annually), this rate greatly impacts the overall return on investment. When solving for present value or determining continuous deposit rates, identifying the correct interest rate is crucial. Remember:

• A higher interest rate increases the future value of savings or investments.
• It conversely decreases the present value needed to achieve a specific future financial goal.
• Understanding how to manipulate interest rates can optimize your investment strategies.

Continuous Deposit

Continuous deposits refer to a scenario where money is consistently added to an account over a period, rather than all at once. This concept is essential when trying to accumulate a future sum of money, particularly when you do not have the lump sum to invest immediately.In the context of the exercise, solving for continuous deposits means finding the rate at which money needs to be continuously added to the investment over a two-year period to reach a future value of $500,000.The formula used is \[FV = \frac{d}{r} (e^{r \cdot t} - 1)\]where:

• \(d\) = deposit rate
• \(r\) = annual interest rate (9% in this example)
• \(t\) = time in years (2 years)
• \(e\) = the base of the natural logarithm

Rearranging to solve for the deposit rate, \(d\), allows investors to determine how much they need to continuously deposit each year to meet their future financial requirements.

Future Value

Future value is the amount of money expected or desired after a specific period, given a certain interest rate and initial investment. Understanding future value helps in planning financial goals and investments, allowing you to predict how much you will have or need in the future. In this example, the future value is the $500,000 your company plans to have in two years for renovations. The future value can be calculated in different contexts:

• Single lump-sum investment over time
• Continuous deposits, as illustrated in the solution

This concept helps investors measure the growth potential of an investment and ensures that future financial needs are met effectively. Future value is impacted by:

• The interest rate (higher rates yield higher future values)
• Time duration (longer investment periods increase future values)

Investment Formula

The investment formula integrates several elements to calculate both present and future values related to investments. A key formula derived for calculating the present value is: \[PV = \frac{FV}{(1 + r)^n}\]where:

• \(PV\) = present value
• \(FV\) = future value
• \(r\) = interest rate
• \(n\) = number of years

This helps determine how much needs to be invested today for a future payoff, like the \(420,076.79 calculation, meaning that much needs to be invested now for \)500,000 in two years at 9% interest.For continuous deposits, we use:\[d = \frac{FV \cdot r}{e^{r \cdot t} - 1}\]These formulas combine to show how present investments and continuous savings can grow over time. Understanding these formulas and their variables empowers investors to make informed decisions and plan effectively to meet future financial goals.