Problem 14 - Applied Calculus

Question

Investing \(\$ 1000\) at an annual interest rate of \(r \%,\) compounded continuously, for 10 years gives you a balance of \(\$ B,\) where \(B=g(r) .\) Give a financial interpretation of the statements: (a) \(g(5) \approx 1649\) (b) \(g^{\prime}(5) \approx 165 .\) What are the units of \(g^{\prime}(5) ?\)

Step-by-Step Solution

Answer

(a) $1,649 is the balance at 5% interest after 10 years. (b) $165 is the balance increase per 1% rise in interest rate. Units are dollars per percentage point.

Step 1: Understanding Continuous Compounding Formula

Continuous compounding is based on the formula for compound interest, which is given by \( B = Pe^{rt} \), where \( P \) is the principal amount, \( r \) is the annual interest rate (expressed as a decimal), and \( t \) is the time in years. Here, given \( P = 1000 \), \( t = 10 \), so the balance function is \( B = 1000e^{0.01rt} \) or \( B = 1000e^{0.1r} \) for \( g(r) = B \).

Step 2: Interpreting (a) g(5) ≈ 1649

The interpretation of \( g(5) \approx 1649 \) is that when the annual interest rate \( r = 5\% \), the balance of the investment after 10 years is approximately \( \\( 1649 \). This indicates how much the original \( \\) 1000 \) investment has grown due to continuous compounding at a 5% interest rate over 10 years.

Step 3: Interpreting (b) g'(5) ≈ 165

The derivative \( g'(r) \) gives us the rate at which the balance \( B \) changes with respect to changes in the interest rate \( r \). Thus, \( g'(5) \approx 165 \) indicates that for each 1% increase in the interest rate from 5%, the balance \( B \) will increase by approximately \( \$ 165 \) after 10 years.

Step 4: Determining the Units of g'(r)

The derivative \( g'(r) \) has units of dollars per percentage point. This is because \( g'(r) \) represents the change in the balance \( B \) in dollars relative to a change in the interest rate \( r \), which is expressed in percentage points.

Key Concepts

Interest Rate

Interest rate is a critical aspect of investing and borrowing. It represents the cost of borrowing money or the gain on an investment over a period of time. When discussing continuous compounding, the interest rate, denoted as \( r \), is expressed as a percentage.

Continuous compounding means the interest is added to the principal balance of an account at every possible instant.

This leads to exponential growth of the investment, a topic we'll cover soon.

For instance, with a principal of \( \$1000 \) invested at an interest rate of \( 5\% \), compounded continuously, the formula \( B = 1000e^{0.1r} \) helps determine how the balance grows.

Here, continuous compounding will yield a higher final amount than simpler compounding methods like annually or monthly. This is because compounds happen more frequently in time.

Derivative

In calculus, the derivative is a measure of how a function changes as its input changes. It gives the slope of the function at any point. In the context of the interest rate, the derivative of the balance function, \( g'(r) \), is especially important.

The specific notation \( g'(r) \) indicates how much the balance \( B \) changes as the interest rate \( r \) changes.

This tells us the sensitivity of the balance to changes in the interest rate.

For example, if \( g'(5) \approx 165 \), it means that a small increase of 1% in the interest rate from \( 5\% \) leads to an approximate increase of \( \$165 \) in the total balance after 10 years. Such information helps investors understand the impact of interest rate fluctuations on their investment.

Exponential Growth

Exponential growth occurs when the growth rate of a quantity is proportional to its current value. With continuous compounding, an investment's growth follows this exponential pattern, described by the formula \( B = Pe^{rt} \).

The expression \( e^{rt} \) is where the magic of exponential growth lies.

• \( P \) is the initial principal amount.
• \( r \) is the rate of interest.
• \( t \) is the time the investment is held.

This kind of growth implies that amounts grow faster as the time periods increase without any breaks.

That's why continuous compounding usually results in higher returns over the same period compared to other compounding methods. Investors can clearly see their principal amount increase substantially over a long period due to compounding continuously.

Financial Interpretation

Financial interpretation is the process of understanding the real-world implications of mathematical expressions and values in finance.

Let's break down the financial interpretation of our exercise:

• **\( g(5) \approx 1649 \):** This means that an investment of \( \\(1000 \) at a \( 5\% \) interest rate, compounded continuously, will grow to approximately \( \\)1649 \) in 10 years. It's a practical illustration of how much your money can grow under the specified conditions.
• **\( g'(5) \approx 165 \):** This derivative tells us that an increase of 1% in the interest rate results in the investment balance increasing by \( \$165 \). The units, dollars per percentage point, express the magnitude of the balance's sensitivity to interest rate changes.

This helps investors make informed decisions as they understand how sensitive their investments are to interest rate fluctuations, translating mathematical results to actionable financial insights.