Problem 13
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
Verified Answer
(a) \(\$1649\) after 10 years at 5\%; (b) Balance changes by \(\$165\) per 1\% change in rate.
1Step 1: Understand the function
The function \(g(r)\) represents the amount of money you'll have after investing \(\$1000\) for 10 years at an annual interest rate of \(r\%\) with continuous compounding. The balance depends on the interest rate \(r\).
2Step 2: Interpret statement (a)
The statement \(g(5) \approx 1649\) means that if you invest \(\\(1000\) at an annual interest rate of 5\%, compounded continuously for 10 years, you will end up with approximately \(\\)1649\).
3Step 3: Interpret statement (b)
The value \(g^{\prime}(5) \approx 165\) indicates the rate of change of the balance with respect to the interest rate when \(r = 5\%\). It means that for a small increase or decrease in the interest rate around 5\%, the balance \(B\) increases or decreases by approximately \(\$165\) per 1\% change in the interest rate.
4Step 4: Define the units of \(g^{\prime}(5)\)
The units of \(g^{\prime}(5)\) are "dollars per percent," which means that it represents how much the final balance changes for each 1\% change in the interest rate. In this context, it refers to \(\$165\) per 1\% change in \(r\).
Key Concepts
Interest RateRate of ChangeInvestment Balance
Interest Rate
Interest rates are essentially the price of borrowing money or the reward for saving it. When you invest money, the interest rate tells you how much extra money your investment will earn over a set period.
The interest rate can be expressed as a percentage. It is important to know how this rate affects the growth of your investment. In the context of continuous compounding, this interest is added to your investment balance at an infinitely small intervals, leading to exponential growth.
For example, when you see that an investment at a 5% annual interest rate compounded continuously for 10 years results in a return of around 1649 dollars for a 1000 dollar principal, it demonstrates how powerful compounding can be over time.
The interest rate can be expressed as a percentage. It is important to know how this rate affects the growth of your investment. In the context of continuous compounding, this interest is added to your investment balance at an infinitely small intervals, leading to exponential growth.
For example, when you see that an investment at a 5% annual interest rate compounded continuously for 10 years results in a return of around 1649 dollars for a 1000 dollar principal, it demonstrates how powerful compounding can be over time.
- Annual Rate: The yearly interest percentage added to the initial investment.
- Continuous Compounding: The process of constantly calculating and adding interest to an account balance.
Rate of Change
The rate of change in the balance of an investment gives insight into how sensitive your returns are to changes in the interest rate. This concept is mathematically represented by the derivative of the investment function or, in our case, the function called \( g'(r) \).
When you encounter \( g'(5) \approx 165 \), it suggests that at an interest rate of 5%, for each 1% increase or decrease in the rate, your balance changes by approximately 165 dollars.
This is not just useful for predicting future value but also in comparing different investment opportunities and their potential outcomes.
When you encounter \( g'(5) \approx 165 \), it suggests that at an interest rate of 5%, for each 1% increase or decrease in the rate, your balance changes by approximately 165 dollars.
- This information helps in understanding how vulnerable your investment is to fluctuations in rates.
- It essentially quantifies the responsiveness of your investment's final value to the interest rate changes.
This is not just useful for predicting future value but also in comparing different investment opportunities and their potential outcomes.
Investment Balance
Investment balance refers to the total amount in your investment account, including your initial principal and any interest that has been earned over time. The balance depends on several factors, such as the principal amount, the interest rate, the frequency of compounding, and the time period over which the interest is applied.
Continuous compounding results in the maximum possible balance because it assumes interest is being computed and added to the account balance at an infinite frequency.
If you initially invest 1000 dollars at a 5% continuous compounding rate for 10 years, your final balance is approximately 1649 dollars, demonstrating the significant growth achieved through continuous compounding.
Continuous compounding results in the maximum possible balance because it assumes interest is being computed and added to the account balance at an infinite frequency.
If you initially invest 1000 dollars at a 5% continuous compounding rate for 10 years, your final balance is approximately 1649 dollars, demonstrating the significant growth achieved through continuous compounding.
- Initial principal: The original sum of money placed in the account, in this case, 1000 dollars.
- Compound Frequency: Continuous compounding means interest is added constantly, maximizing returns.
- Time Duration: The length of time the money is invested or borrowed, impacting final returns greatly.
Other exercises in this chapter
Problem 13
A company's cost of producing \(q\) liters of a chemical is \(C(q)\) dollars; this quantity can be sold for \(R(q)\) dollars. Suppose \(C(2000)=5930\) and \(R(2
View solution Problem 13
IBM-Peru uses second derivatives to assess the relative success of various advertising campaigns. They assume that all campaigns produce some increase in sales.
View solution Problem 14
An industrial production process costs \(C(q)\) million dollars to produce \(q\) million units; these units then sell for \(R(q)\) million dollars. If \(C(2.1)=
View solution Problem 14
Let \(f(x)\) be the elevation in feet of the Mississippi River \(x\) miles from its source. What are the units of \(f^{\prime}(x)\) ? What can you say about the
View solution