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Chapter 7: Problem 42

Find the present value of \(\$ 500\) due in the future under each of the following conditions: a. 12 percent nominal rate, semiannual compounding, discounted back 5 years. b. 12 percent nominal rate, quarterly compounding, discounted back 5 years. c. 12 percent nominal rate, monthly compounding, discounted back 1 year.

Short Answer

Expert verified
a) $280.37, b) $274.54, c) $443.62.

Step by step solution

01

Understand the Present Value Formula

The present value is calculated using the formula: \( PV = \frac{FV}{(1 + r/n)^{nt}} \), where \( FV \) is the future value, \( r \) is the nominal interest rate, \( n \) is the number of compounding periods per year, and \( t \) is the time in years.
02

Calculate Present Value for Part (a)

For part (a), the nominal rate is 12%, compounded semiannually, for 5 years. Here, \( r = 0.12 \), \( n = 2 \), and \( t = 5 \). Substitute these into the formula: \[ PV = \frac{500}{(1 + \frac{0.12}{2})^{2 \times 5}} \] This gives: \[ PV = \frac{500}{(1 + 0.06)^{10}} = \frac{500}{1.06^{10}} \approx 280.37 \]
03

Calculate Present Value for Part (b)

For part (b), the nominal rate is 12%, compounded quarterly, for 5 years. Here, \( r = 0.12 \), \( n = 4 \), and \( t = 5 \). Substitute these into the formula: \[ PV = \frac{500}{(1 + \frac{0.12}{4})^{4 \times 5}} \] This gives: \[ PV = \frac{500}{(1 + 0.03)^{20}} = \frac{500}{1.03^{20}} \approx 274.54 \]
04

Calculate Present Value for Part (c)

For part (c), the nominal rate is 12%, compounded monthly, for 1 year. Here, \( r = 0.12 \), \( n = 12 \), and \( t = 1 \). Substitute these into the formula: \[ PV = \frac{500}{(1 + \frac{0.12}{12})^{12 \times 1}} \] This gives: \[ PV = \frac{500}{(1 + 0.01)^{12}} = \frac{500}{1.01^{12}} \approx 443.62 \]

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Key Concepts

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

Compounding Periods
Compounding periods refer to the frequency with which interest is applied to the principal balance of an investment or loan within a year.
This frequency can significantly impact the growth of money over time. For example, interest might be compounded annually, semiannually, quarterly, monthly, or even daily.
  • **Annually:** Interest is compounded once a year.
  • **Semiannually:** Compounding occurs twice a year.
  • **Quarterly:** Interest is applied four times a year.
  • **Monthly:** Compounding occurs twelve times a year.
  • **Daily:** Interest is compounded every day.
The more frequently interest is compounded, the more often the principal amount earns interest. This results in potentially higher returns because each compounding period builds on the previously accrued interest.
In the given exercise, understanding the impact of compounding periods is crucial because the calculation changes when interest is compounded semiannually, quarterly, or monthly, affecting the final present value significantly.
Nominal Interest Rate
The nominal interest rate is the stated interest rate on a loan or investment without adjusting for inflation or compounding effects.
It is essentially the percentage that is advertised or agreed upon when the financial contract is established.
The nominal interest rate is a key figure in financial transactions, but it doesn鈥檛 always represent the actual cost of borrowing or the real yield from an investment.
That's because it doesn鈥檛 account for how often interest is compounded. When interest is compounded more frequently, the effective interest rate will be higher than the nominal rate.
In practical terms:
  • When a loan is disclosed with a 12% nominal interest rate, this rate doesn't take into account how many compounding periods are involved.
  • As demonstrated in the exercise, the same 12% rate results in different outcomes depending on whether interest is compounded semiannually, quarterly, or monthly.
Thus, to fully understand the cost or return of a financial product, it's crucial to consider both the nominal rate and the compounding frequency.
Time Value of Money
The time value of money (TVM) is a fundamental financial concept that describes the idea that money available today is worth more than the same amount in the future.
This principle is based on the potential earning capacity of money; if you have cash now, you can invest it and earn interest over time.
This concept is key in explaining why interest is paid or earned because it accounts for opportunities lost when committing money to a future date. The Present Value (PV) calculation is a practical application of the time value of money concept.
It helps in determining what an amount of money in the future is worth today, considering a specific interest rate and compounding periods.
  • It allows businesses and investors to evaluate the profitability of projects by converting future cash flows into present terms.
  • In the exercise, calculating the present value involves understanding how different compounding intervals can affect the value of money over time.
Hence, managing the time value of money effectively is essential for making informed financial decisions and maximizing investment returns.

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Most popular questions from this chapter

As the manager of Oaks Mall Jewelry, you want to sell on credit, giving customers 3 months in which to pay. However, you will have to borrow from the bank to carry the accounts receivable. The bank will charge a nominal 15 percent, but with monthly compounding. You want to quote a nominal rate to your customers (all of whom are expected to pay on time) that will exactly cover your financing costs. What nominal annual rate should you quote to your credit customers?

If you deposit money today into an account that pays 6.5 percent interest, how long will it take for you to double your money?

Your broker offers to sell you a note for \(\$ 13,250\) that will pay \(\$ 2,345.05\) per year for 10 years. If you buy the note, what interest rate (to the closest percent) will you be earning?

The prize in last week's Florida lottery was estimated to be worth \(\$ 35\) million. If you were lucky enough to win, the state will pay you \(\$ 1.75\) million per year over the next 20 years. Assume that the first installment is received immediately. a. If interest rates are 8 percent, what is the present value of the prize? b. If interest rates are 8 percent, what is the future value after 20 years? c. How would your answers change if the payments were received at the end of each year?

What is the future value of a 5 -year ordinary annuity that promises to pay you \(\$ 300\) each year? The rate of interest is 7 percent.
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