Problem 18 - 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) \(g(2) \approx 1221.\) (b) \(g^{\prime}(2) \approx 122 .\) What are the units of \(g^{\prime}(2) ?\)

Step-by-Step Solution

Answer

(a) With a 2% interest, $1000 grows to $1221 in 10 years. (b) A 1% rate increase raises the balance by $122. Units: dollars/percentage point.

Step 1: Understand the function and context

The problem is about continuous compound interest, where the balance after a certain time is given by the formula: \[ B = P e^{rt} \]where \( P \) is the principal amount, \( e \) is the base of the natural logarithm (approximately 2.718), \( r \) is the annual interest rate, and \( t \) is the time in years. In this scenario, \( P = 1000 \) and \( t = 10 \). The function \( g(r) \) represents the balance \( B \) as a function of the rate \( r \% \).

Step 2: Interpret statement (a)

The statement \( g(2) \approx 1221 \) indicates that when the interest rate is 2%, the balance after 10 years is approximately \(1221. This means that with an annual interest rate of 2%, the investment grows from \)1000 to approximately $1221 over 10 years.

Step 3: Interpret statement (b)

The statement \( g'(2) \approx 122 \) provides the derivative of the function \( g \) at \( r = 2 \). This indicates how the balance \( B \) changes with respect to a small change in the interest rate around 2%. Specifically, it implies that for a 1% increase in the interest rate, the balance \( B \) would increase by approximately $122.

Step 4: Determine units for \( g'(2) \)

The derivative \( g'(r) \) represents the rate of change of the balance \( B \) in dollars with respect to the interest rate \( r \) in percentage points. Therefore, the units for \( g'(2) \) are dollars per percentage point.

Key Concepts

Compound Interest Formula

The continuous compound interest formula is a cornerstone in financial calculations. It is given by the equation:

• B = P e^rt

In this formula, B represents the balance after interest accrues, P is the principal amount originally invested, e is the base of the natural logarithm (roughly 2.718), r is the annual interest rate expressed as a decimal, and t is the time in years that the money is invested or borrowed for.

The continuous compounding aspect means that interest is calculated and added to the principal balance at every possible moment. This differs from simple or discrete compounding, where interest addition occurs at fixed intervals, like monthly or yearly. Because of this constant addition, continuous compounding typically results in higher accumulated interest compared to discrete methods with the same nominal rate.

In the given problem, with a principal P of $1000, an interest rate r of 2%, and a time t of 10 years, the balance B becomes $1221 using continuous compounding.

Derivatives in Finance

Derivatives are a powerful tool in finance, providing insight into how sensitive certain financial metrics are with respect to changes in other variables. In this context, when referring to continuous compound interest, the derivative of the balance function with respect to the interest rate tells us how much the balance will change if the interest rate changes.

Mathematically, if the balance is represented by a function g(r), then the derivative g'(r) indicates the rate of change of the balance per unit change in the interest rate.

• g(2) ≈ 1221 says that at an interest rate of 2%, the balance is about $1221.
• g'(2) ≈ 122 signifies that for each 1% increase in the interest rate when it is around 2%, the balance increases by roughly $122.

This derivative gives a snapshot of the potential growth in balance due to small rate changes, making it highly valuable for decision-making and interest rate analysis.

Interest Rate Analysis

Interest rate analysis involves examining changes to interest rates and understanding their impact on financial variables. When it comes to investments like savings accounts or bonds, understanding the implications of interest rate fluctuations is essential.

In the exercise context, analyzing the interest rate involves looking at both the computed balance at a fixed rate and the derivative to see how the balance changes around that rate. For instance, when g(2) ≈ 1221, it tells us the balance amount at a specific interest rate of 2%. Meanwhile, g'(2) ≈ 122 helps forecast how much extra balance we might earn if the rate were slightly higher.

Units play an important role too. Here, the units for g'(2) are dollars per percentage point. This means that any small variation in the interest rate translates directly into a calculated change in the balance, facilitating easy comparisons and forecasts of potential earnings under different scenarios.