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Chapter 4: Problem 81

The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. (a) Find the present value of \(\$ 10,000\) if interest is paid at a rate of \(9 \%\) per year, compounded semiannually, for 3 years. (b) Find the present value of \(\$ 100,000\) if interest is paid at a rate of \(8 \%\) per year, compounded monthly, for 5 years.

Short Answer

Expert verified
(a) $7679.70; (b) $67297.66.

Step by step solution

01

Understand Present Value Formula

The present value (PV) is calculated using the formula: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]where \(FV\) is the future value, \(r\) is the annual interest rate as a decimal, \(n\) is the number of times interest is compounded per year, and \(t\) is the number of years.
02

Calculate Present Value for Part (a)

For \(\$10,000\) with a \(9\%\) interest rate compounded semiannually:- Set \(FV = 10,000\), \(r = 0.09\), \(n = 2\), \(t = 3\).Plug these into the formula:\[ PV = \frac{10,000}{(1 + \frac{0.09}{2})^{2 \times 3}} \]Calculate the denominator:\[ (1 + \frac{0.09}{2})^6 = 1.045^6 \approx 1.303 \]Finally, calculate \(PV\):\[ PV = \frac{10,000}{1.303} \approx 7679.70 \]
03

Calculate Present Value for Part (b)

For \(\$100,000\) with an \(8\%\) interest rate compounded monthly:- Set \(FV = 100,000\), \(r = 0.08\), \(n = 12\), \(t = 5\).Plug these into the formula:\[ PV = \frac{100,000}{(1 + \frac{0.08}{12})^{12 \times 5}} \]Calculate the denominator:\[ (1 + \frac{0.08}{12})^{60} \approx 1.488 \]Finally, calculate \(PV\):\[ PV = \frac{100,000}{1.488} \approx 67297.66 \]

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Key Concepts

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

Compound Interest
Compound interest is a powerful concept in financial mathematics that calculates interest on not only the initial principal but also on the accumulated interest from previous periods. This means the amount of money you earn or owe can grow faster than with simple interest, which is calculated only on the principal amount.
The formula for compound interest helps us see how this works over time:
  • The interest rate is divided over the number of compounding periods per year.
  • The amount is compounded several times a year, depending on the terms.
  • The longer the money is left to compound, the greater the final amount will be.
This concept is important for any financial decision that involves interest, whether it be investing in a bank account or taking out a loan. Understanding compound interest can help make more informed decisions about saving and investing money for the future.
Future Value
Future value (FV) represents the amount of money an investment will grow to over a period of time when it has earned interest. It allows you to understand what today's money will be worth in the future if you invest it wisely. This is crucial for financial planning and setting realistic investment goals.
The future value formula is:\[ FV = PV imes (1 + rac{r}{n})^{nt} \]
  • Where \(PV\) is the present value, \(r\) is the annual interest rate, \(n\) is the number of compounding periods per year, and \(t\) is the time in years.
  • More compounding periods (e.g., monthly vs. annually) result in a higher future value.
  • The future value is not just about the principal but includes the effects of compounded interest over time.
By understanding and calculating future value, individuals can better plan for future expenses, like retirement or major purchases, ensuring they will have adequate funds when the time comes.
Interest Rate
The interest rate is a percentage that reflects the cost of borrowing money or the gain from investing it. It's a key factor in calculating present and future value as it directly affects how rapidly money grows.
Interest rates can be of different types:
  • Fixed Interest Rate: Remains constant throughout the period.
  • Variable Interest Rate: Can change based on market conditions.
High-interest rates can rapidly increase the amount owed on a loan if not managed well, but they can also be advantageous for investments as they can lead to greater returns. Always check how interest is compounded (e.g., annually, semiannually, monthly) to understand the true cost or gain over time.
Financial Mathematics
Financial mathematics involves using mathematical models and formulas to solve problems related to money management. It forms the crux of making informed decisions in finance, from personal budgeting to corporate finance strategy.
Key elements include:
  • Time Value of Money: Understanding that money now is worth more than the same sum in the future.
  • Risk Management: Using models to assess potential risks and returns.
  • Valuation: Calculating present and future values to assess the worth of investments.
It covers various topics like annuities, loans, investment appraisal, and derivatives. Financial mathematics is critical for anyone looking to make sound investments and manage their finances wisely. Such calculations enable better judgement for achieving financial goals, avoiding debt, and building wealth.

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Most popular questions from this chapter

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