Problem 38 - Calculus and its applications

Question

Gusto Stick is a professional baseball player who has just become a free agent. His attorney begins negotiations with an interested team by asking for a contract that provides Gusto with an income stream given by \(R_{1}(t)=800,000+340,000 t,\) over \(10 \mathrm{yr}\), where \(t\) is in years. (Round all answers to the nearest \(\$ 100 .)\) a) What is the accumulated future value of the offer. assuming an interest rate of \(5 \%,\) compounded continuously? b) What is the accumulated present value of the offer, assuming an interest rate of \(5 \%,\) compounded continuously? c) The team counters by offering an income stream given by \(R_{2}(t)=600,000+210,000 t\). What is the accumulated present value of this counteroffer? d) Gusto comes back with a demand for an income stream given by \(R_{3}(t)=1,000,000+250,000 t\). What is the accumulated present value of this income stream? e) Gusto signs a contract for the income stream in part (d) but decides to live on \(\$ 500,000\) each year, investing the rest at \(5 \%,\) compounded continuously. What is the accumulated future value of the remaining income, assuming an interest rate of \(5 \%,\) compounded continuously?

Step-by-Step Solution

Answer

Future and present values require integrating income streams over the designated time period, adjusting for continuous compounding.

Step 1: Future Value of Initial Offer

The future value of an income stream can be calculated using the formula \(FV = P e^{rt}\), where \(P\) is the integral of the rate function over the time period. For \(R_1(t) = 800,000 + 340,000t\), the future value can be calculated as follows:\[FV = \int_{0}^{10} (800,000 + 340,000t) e^{0.05(10-t)} \, dt\]Evaluating this integral will give the future value of the income stream after 10 years.

Step 2: Present Value of Initial Offer

The present value of an income stream continuously compounded can be calculated by integrating the rate function, adjusted for the discount rate over the time period. For \(R_1(t)\), the present value \(PV\) is:\[PV = \int_{0}^{10} (800,000 + 340,000t) e^{-0.05t} \, dt\]Solving this integral will give the present value of the income stream at \(t=0\).

Step 3: Present Value of Counteroffer

For the counteroffer \(R_2(t) = 600,000 + 210,000t\), compute the present value using:\[PV = \int_{0}^{10} (600,000 + 210,000t) e^{-0.05t} \, dt\]Calculate this integral to find the present value of the counteroffer.

Step 4: Present Value of Gusto's Demand

For Gusto's demand \(R_3(t) = 1,000,000 + 250,000t\), the present value is given by:\[PV = \int_{0}^{10} (1,000,000 + 250,000t) e^{-0.05t} \, dt\]Solve this integral to find the present value of Gusto's new demand.

Step 5: Future Value of Remaining Income

Given Gusto lives on $500,000 and invests the remainder, calculate the invested amount as a function of \(t\): \(R_3(t) - 500,000 = 500,000 + 250,000t\).The future value is:\[FV = \int_{0}^{10} (500,000 + 250,000t) e^{0.05(10-t)} \, dt\]Evaluating this gives the future value of the invested income at the end of 10 years.

Key Concepts

Continuous Compounding

Continuous compounding is a powerful concept in calculus and financial mathematics that allows you to calculate the amount of money that will grow continuously over time without any breaks. When interest is compounded continuously, it means that the interest is added at every possible instant. This is different from simple or periodic compounding, which might happen annually or monthly.The mathematical expression for continuous compounding is given by the formula:\[ A = Pe^{rt} \]where:

• \(A\) is the accumulated amount including interest
• \(P\) is the principal amount (initial deposit or investment)
• \(r\) is the annual interest rate (as a decimal)
• \(t\) is the time in years
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828

Continuous compounding is particularly useful in high-frequency financial scenarios, such as when calculating the future value of an investment or accumulated savings over a long period.

Future Value Calculation

The future value calculation is all about determining what the amount of money in the future will be worth from a stream of income, taking into account how it grows with interest over time.To calculate the future value of an income stream, especially one that compounds continuously, we often use integral calculus to handle the complexity of such continuous growth.The general approach involves integrating the income stream function over the given time period and adjusting for the continuous compounding effect:\[ FV = \int_{0}^{T} R(t) e^{r(T-t)} \, dt \]In this formula:

• \(FV\) is the future value
• \(T\) is the total time period
• \(R(t)\) is the income stream as a function of time
• \(r\) is the continuous compounding rate

This method is essential for individuals and institutions alike aiming to predict and plan for financial future based on anticipated investments or continuous income flows.

Present Value Calculation

Present value calculation helps us determine the current worth of a stream of future income, or equivalently, how much those future cash flows are valued in today's dollars.This concept assumes that money now is more valuable than the same amount in the future due to its potential earning capacity. Hence, it is a crucial aspect of financial mathematics.When interest is compounded continuously, the present value \(PV\) of an income stream \(R(t)\) can be determined using the formula:\[ PV = \int_{0}^{T} R(t)e^{-rt} \, dt \]This integral computes the sum of each individual cash flow or income stream, adjusted for the fact that it is happening in the future.Understanding and calculating present value is vital for making informed financial decisions, such as evaluating investment opportunities or comparing various financial scenarios.

Integral Calculus

Integral calculus plays a pivotal role in financial mathematics, especially when dealing with continuous income streams and compounding interest rates. The power of integral calculus lies in its ability to find solutions to problems involving accumulation, area under curves, and general continuous change over time. This is exactly what is needed when we want to understand accumulating value or present value from an income that varies over time. Using integrals, we can model dynamic systems and evaluate accumulated growth or decay. For instance, when calculating the present or future value of an income stream, integral calculus allows us to account for continuous changes better than mere linear methods. In financial applications, integrals help in aspects like:

• Deriving total income or total worth over time
• Time-value of money concepts
• Risk management calculations

Thus, integral calculus is an indispensable tool in solving complex financial mathematics problems.

Financial Mathematics

Financial mathematics is a field of study focused on applying mathematical methods and models to solve problems in finance. It combines concepts from calculus, probability, statistics, and economic theories to guide optimal financial decision-making. Key applications of financial mathematics include:

• Valuation of securities and businesses
• Risk assessment and management
• Strategic financial planning and analysis

One significant area where financial mathematics is applied is through the use of compounding interest principles to assess investments, savings, loans, and income streams. It involves calculating the present and future values accurately to determine the viability of financial decisions and investments. By leveraging mathematical models, financial mathematics helps us make sound decisions about investments, understand market dynamics, and drive profitability and financial growth.