Problem 17 - Applied Calculus

Question

The amount of money, $$\$ B,$$ in a bank account earning interest at a continuous rate, \(r\), depends on the amount deposited, $$\$ P,$$and the time, \(t\), it has been in the bank, where $$B=P e^{r t}$$ Find \(\partial B / \partial t, \partial B / \partial r\) and \(\partial B / \partial P\) and interpret each in financial terms.

Step-by-Step Solution

Answer

\( \frac{\partial B}{\partial t} = P r e^{rt} \), \( \frac{\partial B}{\partial r} = P t e^{rt} \), \( \frac{\partial B}{\partial P} = e^{rt} \). These derivatives show sensitivity to time, rate, and principal.

Step 1: Identify the Formula

The given formula for the amount of money in the bank account is \( B = P e^{rt} \). This represents the balance \( B \) as a function which depends on the principal \( P \), the continuous interest rate \( r \), and the time \( t \).

Step 2: Differentiate with Respect to Time, t

To find \( \frac{\partial B}{\partial t} \), we differentiate the formula \( B = P e^{rt} \) with respect to \( t \). The derivative of \( e^{rt} \) with respect to \( t \) using the chain rule is \( r e^{rt} \). Thus, \( \frac{\partial B}{\partial t} = P r e^{rt} \).

Step 3: Interpretation of \( \partial B / \partial t \)

\( \frac{\partial B}{\partial t} = P r e^{rt} \) represents the rate of change of the balance \( B \) with respect to time \( t \). This shows how the balance increases as time goes on, owing to continuous compounding at the rate \( r \).

Step 4: Differentiate with Respect to Rate, r

To find \( \frac{\partial B}{\partial r} \), differentiate \( B = P e^{rt} \) with respect to \( r \). Using the chain rule, the derivative is \( P t e^{rt} \). Therefore, \( \frac{\partial B}{\partial r} = P t e^{rt} \).

Step 5: Interpretation of \( \partial B / \partial r \)

\( \frac{\partial B}{\partial r} = P t e^{rt} \) indicates how the balance \( B \) changes with respect to the interest rate \( r \). This shows the sensitivity of the balance to changes in the interest rate over time.

Step 6: Differentiate with Respect to Principal, P

To find \( \frac{\partial B}{\partial P} \), differentiate \( B = P e^{rt} \) with respect to \( P \). The derivative is \( e^{rt} \). Thus, \( \frac{\partial B}{\partial P} = e^{rt} \).

Step 7: Interpretation of \( \partial B / \partial P \)

\( \frac{\partial B}{\partial P} = e^{rt} \) represents the rate of change of the balance \( B \) with respect to the initial deposit \( P \). This shows how the balance grows proportionally with an increase in the principal.

Key Concepts

Continuous Compounding

When you deposit money in a bank with continuous compounding, your interest is compounded constantly. This creates a snowball effect where the balance grows more quickly compared to standard compounding periods, like yearly or quarterly.
Continuous compounding has a unique formula: \(B = Pe^{rt}\) , where:

• \(B\) represents the balance in your account.
• \(P\) is the principal or initial amount deposited.
• \(r\) is the interest rate.
• \(t\) is the time period.

The beauty of continuous compounding lies in its efficiency. It allows your money to grow faster, maximizing your interest earnings. What's fascinating is how it handles derivatives—partial derivatives, to be precise. This is how mathematicians and economists assess sensitivity with respect to time, rate, and principal.
By computing \(\frac{\partial B}{\partial t} = Pr e^{rt}\), we find how quickly the balance grows over time. This shows how continuous compounding accelerates growth as time progresses.

Interest Rate Sensitivity

Interest rate sensitivity describes how your bank balance responds to changes in the interest rate. If a small increase in the interest rate leads to a large increase in your balance, your account is highly sensitive to the rate.
In mathematical terms, this is determined by the partial derivative \(\frac{\partial B}{\partial r} = P t e^{rt}\). This equation captures the impact of varying interest rates on your total balance over time.
It's noteworthy that:

• Higher interest rates lead to a larger impact, growing the balance.
• The sensitivity depends on both the initial principal and the time the money has been compounding.

Understanding this can be crucial for investment strategies, as it helps investors assess the risk and potential growth of their finances.

Principal Dependency

The principal dependency concept focuses on how the initial deposit affects the total balance over time. The more you deposit, the greater the potential for the balance to grow. This dependency is quantified by the derivative \(\frac{\partial B}{\partial P} = e^{rt}\).
This formula illustrates that:

• Regardless of the initial amount, the growth factor \(e^{rt}\) remains constant over time.
• It highlights that every additional dollar deposited grows at the rate associated with continuous compounding.

Therefore, increasing the principal not only adds to the immediate balance but also enhances the compounding effect. This is why even small incremental deposits can lead to significant growth over a long time horizon.