Problem 10 A city government has approved t... [FREE SOLUTION]

Chapter 2: Problem 10

A city government has approved the construction of an $\$ 800$ million sports arena. Up to $\$ 480$ million will be raised by selling bonds that pay simple interest at a rate of $6 \%$ annually. The remaining amount (up to $\$ 640$ million) will be obtained by borrowing money from an insurance company at a simple interest rate of $5 \%$. Determine whether the arena can be financed so that the annual interest is $\$ 42$ million.

Short Answer

Expert verified

Yes, the financing plan works with $200 million in bonds and $600 million from the insurance.

Step by step solution

Understanding the Problem

The city needs to raise a total of $800 million. The interest payments on this amount should not exceed $42 million annually. There are two possible ways to finance the arena: by selling bonds at 6% interest and borrowing from an insurance company at 5% interest.

Assign Variables

Let $ x $ be the amount (in millions) raised by selling bonds with 6% interest, and $ y $ be the amount (in millions) borrowed from the insurance company with 5% interest. We know that $ x + y = 800 $.

Set Up Interest Equations

The total interest paid in a year will be the sum of the interest from the bonds and the insurance. This gives us the equation $ 0.06x + 0.05y = 42 $ since the interest cannot exceed $42 million.

Solve the System of Equations

We have the system: $ x + y = 800 $ and $ 0.06x + 0.05y = 42 $. Substitute the first equation into the second: $ 0.06x + 0.05(800 - x) = 42 $. Simplify and solve for $ x $.

Calculate $ x $

Distribute and simplify the equation: $ 0.06x + 40 - 0.05x = 42 $ leads to $ 0.01x + 40 = 42 $. Solving for $ x $ gives $ x = 200 $.

Find $ y $

Using $ x = 200 $, substitute back into $ x + y = 800 $ to find $ y $: $ 200 + y = 800 $, which gives $ y = 600 $.

Check Conditions

The total financing consists of $200 million from bonds and $600 million from the insurance. Check the interest: $ 0.06(200) + 0.05(600) = 12 + 30 = 42 $. The conditions are satisfied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bonds

Bonds are investments where an investor lends money to a borrower for a fixed period. In return, the borrower pays interest periodically and returns the principal at the bond's maturity. Bonds are often issued by governments, municipalities, and corporations as a way to raise capital for specific projects, like the construction of a sports arena.

Bonds usually offer a fixed interest rate, providing predictability in returns.
They are considered less risky than stocks since they typically have priority in repayment over equities if the issuer goes bankrupt.
Bonds can be traded on significant exchanges, offering liquidity to investors.

In the city government scenario, bonds raised at 6% interest allow for a substantial part of the financing. The lower risk of this funding method makes it an attractive option to secure up to $480 million for the project.

Insurance

Insurance companies often extend loans to different organizations as part of their investment strategy, providing a stable return on their large pool of funds. Borrowing from an insurance company is a common financing strategy because they can offer favorable interest terms.

Insurance companies can lend large sums, which is advantageous for massive projects.
Loans from these institutions usually come with fixed interest rates, allowing predictable interest calculations.
This method of borrowing can be integrated into a broader financing strategy, like combining with bonds to ensure a balanced flow of capital.

For the sports arena project, borrowing $600 million from an insurance company at a 5% interest rate complements the bond issuance, ensuring that the city's financing requirements are met efficiently.

System of Equations

A system of equations is a set of equations with the same variables, which you solve simultaneously to find a solution that satisfies all equations at once. It is a vital tool in financial decisions, allowing for the modeling of complex scenarios where multiple conditions must be addressed together.

Each equation represents a specific condition or constraint of the problem.
For the sports arena project, two primary equations were set up:
- The sum of funds raised through bonds and borrowing: $ x + y = 800 $ million.
- The sum of simple interest for the bonds and the insurance loan: $ 0.06x + 0.05y = 42 $ million.
The solution reveals how much can be borrowed in each method to meet the yearly interest constraint.

By solving these equations, we find how to distribute the $800 million total project cost effectively.

Interest Rate

The interest rate represents the cost of borrowing money or the return on investment for lending it. In simple interest calculations, the formula is straightforward: \[ \text{Simple Interest} = \text{Principal} \times \text{Interest Rate} \times \text{Time} \]For the arena project, the interest rate is crucial both in determining the project's feasibility and its financial impact.

A higher interest rate means higher returns for lenders but increased costs for borrowers.
In this context, 6% interest on bonds and 5% on insurance borrowing affect the total annual interest payment.
The importance of careful interest rate management comes into play as the city must keep the total interest payments under $42 million.

Understanding how interest rates work helps in making informed decisions that align with long-term financial objectives, ensuring that the project remains within budget constraints.

91影视

Short Answer

Step by step solution

Understanding the Problem

Assign Variables

Set Up Interest Equations

Solve the System of Equations

Calculate \( x \)

Find \( y \)

Check Conditions

Key Concepts

Bonds

Insurance

System of Equations

Interest Rate

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