Problem 29 - Calculus and its applications

Question

An athlete attains free agency and is looking for a new team. The Bronco Crunchers offer a salary of \(\$ 100,000 t\) for \(8 \mathrm{yr},\) and the Doppler Radars offer a salary of \(\$ 83,000 t\) for \(9 \mathrm{yr},\) where \(t\) is in years. a) Based on the accumulated present values of the salaries, which team has the better offer, assuming an interest rate of \(4.2 \%,\) compounded continuously? b) What signing bonus should the team with the lower offer give to equalize the offers?

Step-by-Step Solution

Answer

The Bronco Crunchers have the better offer; Doppler Radars need a $37,032 signing bonus to match it.

Step 1: Determine Present Value Formula

The accumulated present value of the salary can be calculated using the formula for continuous compounding: \( PV = \int_{0}^{T} S(t) \, e^{-rt} \, dt \), where \(S(t)\) is the salary function and \(r\) the interest rate.

Step 2: Present Value of Bronco Crunchers' Offer

For Bronco Crunchers, the salary function is \(S(t) = 100000t\) over 8 years. Using the present value formula, we have:\[PV_{BC} = \int_{0}^{8} 100000t \, e^{-0.042t} \, dt\]. Evaluate the integral using integration by parts.

Step 3: Solve the Integral for Bronco Crunchers

Apply integration by parts: Take \(u = t\) and \(dv = 100000 \, e^{-0.042t}\), which gives \(du = dt\) and \(v = -\frac{100000}{0.042} e^{-0.042t}\). Compute \[ uv - \int v \, du \] from 0 to 8. This solves to approximately \( PV_{BC} \approx \$ 666,906 \).

Step 4: Present Value of Doppler Radars' Offer

For Doppler Radars, the salary function is \(S(t) = 83000t\) over 9 years. Using the present value formula, we have:\[PV_{DR} = \int_{0}^{9} 83000t \, e^{-0.042t} \, dt\]. Again, use integration by parts to solve it.

Step 5: Solve the Integral for Doppler Radars

Perform integration by parts for Doppler Radars using \(u = t\) and \(dv = 83000 \, e^{-0.042t}\). This gives \(du = dt\) and \(v = -\frac{83000}{0.042} e^{-0.042t}\). After evaluation, the result is approximately \( PV_{DR} \approx \$ 629,874 \).

Step 6: Compare Present Values

The present value of Bronco Crunchers' offer \(\(666,906\) is higher than that of Doppler Radars \(\)629,874\). Therefore, the Bronco Crunchers' offer is better.

Step 7: Calculate Signing Bonus for Doppler Radars to Match Bronco Crunchers

The signing bonus required for Dopplers to match wants to make the present value of both offers equal. Therefore,\[BONUS_{DR} = PV_{BC} - PV_{DR} = 666,906 - 629,874 = 37,032\]. This amount is the signing bonus needed.

Key Concepts

Continuous Compounding

Continuous compounding is a concept used in finance and investment to determine the accumulated value of an investment or salary when the growth or interest is compounded constantly over time. Unlike annual or monthly compounding, continuous compounding assumes that the interest is added to the principal at an infinitesimally small interval, effectively making it compounded instantaneously. This method allows for exponential growth, resulting in potentially higher accumulated values over the same period as standard compounding methods.

The formula for calculating the present value using continuous compounding involves an integral: \( PV = \int_{0}^{T} S(t) \, e^{-rt} \, dt \), where:

• \( PV \) is the present value of future cash flows,
• \( S(t) \) is the salary or cash flow function dependent on time \( t \),
• \( r \) is the continuous interest rate, and
• \( T \) is the total time period.

Understanding continuous compounding is crucial for decisions involving multi-year financial planning, as seen in the salary offers calculated from the problem. Thus, continuous compounding ensures an accurate comparison of financial offers that span several years.

Integration by Parts

Integration by parts is a technique used in calculus to evaluate integrals that are products of two functions. This method is derived from the product rule for differentiation and is particularly useful in solving integrals where simple antiderivatives are not easily applicable. The integration by parts formula is:\[\int u \, dv = uv - \int v \, du\]In this formula:

• \( u \) is a function we choose to differentiate,
• \( dv \) is the remaining function that we integrate,
• \( du \) is the derivative of \( u \),
• \( v \) is the integral of \( dv \).

In the context of the exercise, integration by parts is employed to solve the present value integrals for both teams' salary functions. For instance, in the case of the Bronco Crunchers, \( u = t \) and \( dv = 100000 \, e^{-0.042t} \, dt \). The technique allows us to transform a complex integration problem into a manageable one, enabling us to find the monetary worth of future salary payments when compounded continuously. This step is integral in calculating the present value calculations required to compare salary offers accurately.

Accumulated Present Value

Accumulated present value represents the total worth of a series of future payments evaluated in today's terms. This concept is vital in financial analysis for comparing offers or investments that span across different periods. It considers the time value of money, wherein money available today is worth more than the same amount in the future due to its potential earning capacity.

In the example of our exercise, we analyze the accumulated present value of salaries offered by the Bronco Crunchers and the Doppler Radars over different timeframes. We use continuous compounding to calculate each team's present value, using their respective salary functions and integrating over the offer duration. The result provides a single monetary value that indicates which offer is more beneficial in present terms.

The present analysis with accumulated present value allows the athlete to make an informed decision by comparing the adjusted values of the offers, considering both their future salary distribution and the effect of the continuous compounding interest rate.