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Chapter 2: Problem 24

Present value for various compounding periods Find the present value of \(\$ 500\) due in the future under each of these 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. d. Why do the differences in the PVs occur?

Short Answer

Expert verified
PVs for (a), (b), and (c) are approximately $282.57, $276.84, and $443.27 respectively. Differences are due to compounding frequency.

Step by step solution

01

Understanding the Formula

The present value (PV) in all situations can be calculated using the formula: \[ PV = \frac{FV}{(1 + r/n)^{n \times t}} \] 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 number of years.
02

Calculate Present Value for Semiannual Compounding

For part (a), we are given a nominal rate of 12% with semiannual compounding over 5 years. This means \(r = 0.12\), \(n = 2\), and \(t = 5\). Substitute these into the formula: \[ PV = \frac{500}{(1 + 0.12/2)^{2 \times 5}} = \frac{500}{(1.06)^{10}} \approx 282.57 \]
03

Calculate Present Value for Quarterly Compounding

For part (b), use a nominal rate of 12% with quarterly compounding over 5 years. Here, \(r = 0.12\), \(n = 4\), and \(t = 5\). Substitute into the formula: \[ PV = \frac{500}{(1 + 0.12/4)^{4 \times 5}} = \frac{500}{(1.03)^{20}} \approx 276.84 \]
04

Calculate Present Value for Monthly Compounding

For part (c), use a nominal rate of 12% with monthly compounding over 1 year. Here, \(r = 0.12\), \(n = 12\), and \(t = 1\). Substitute into the formula: \[ PV = \frac{500}{(1 + 0.12/12)^{12 \times 1}} = \frac{500}{(1.01)^{12}} \approx 443.27 \]
05

Explain PV Differences

The differences in present values are due to the frequency of compounding. More frequent compounding results in a higher effective interest rate, thus lowering the present value for the same nominal rate and period.

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Key Concepts

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

Compounding Periods
Compounding periods are crucial when calculating the present value of a future sum of money. It refers to how often the interest is applied to the principal within a year. The more frequently compounding occurs, the less time each compounding period has to accumulate interest before the next period's interest is added. Understanding compounding periods is important because the more frequent the compounding, the greater the total amount of interest accrued over time will be.

Different schedules like annually, semiannually, quarterly, or monthly can significantly impact how much interest is earned. In our exercise, the present value calculations for semiannual, quarterly, and monthly compounding show varying results because each approach changes the effective interest rate applied to the invested amount.
Nominal Interest Rate
The nominal interest rate is the stated interest rate of an investment, not accounting for compounding within the year. Hence, it does not tell the whole story about how much you'll effectively earn or owe. In our exercise, a 12% nominal rate means that is the interest on the principal before taking into consideration any effect from frequent compounding periods.

When financial products are described, they often use this nominal rate because it is straightforward to communicate, although it doesn鈥檛 reveal the true rate paid or received over a particular period once compounding is factored in. It's important to look beyond the nominal rate to understand the true cost or benefit of an investment, which is where the compounding effect and the effective interest rate come into play.
Future Value
Future value (FV) is an important concept in finance, representing the amount of money an investment will grow to over a period of time when compounded at a certain interest rate. It answers the question, "How much will my money be worth in the future?" In our exercise, we start with a predetermined future value of $500.

Calculating the present value is essentially the reverse process of determining future value. It asks, "How much money should I invest today to reach this future amount, given specific compounding conditions and a set nominal interest rate?" Therefore, knowing your financial goals and the future values you would like to reach is crucial when planning your investments.
Effective Interest Rate
The effective interest rate offers a clearer picture than the nominal rate. It reflects the actual amount of interest earned or paid after accounting for compounding. The more frequently the interest is compounded, the higher the effective interest rate will be. This reveals how compounding periods directly affect investment growth or loan costs.

The exercise demonstrates that even with the same nominal interest rate, the effective interest rate becomes larger with more frequent compounding periods like quarterly or monthly, compared to semiannual compounding. Therefore, understanding the effective interest rate is essential for making informed financial decisions, whether it's for saving, investing, or taking loans.

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Most popular questions from this chapter

Present and future values for different interest rates Find the following values. Compounding/discounting occurs annually. a. An initial \(\$ 500\) compounded for 10 years at 6 percent. b. \(\quad\) An initial \(\$ 500\) compounded for 10 years at 12 percent. c. The present value of \(\$ 500\) due in 10 years at 6 percent. d. The present value of \(\$ 1,552.90\) due in 10 years at 12 percent and also at 6 percent. e. Define present value, and illustrate it using a time line with data from part d. How are present values affected by interest rates?

Paying off credit cards Simon recently received a credit card with an 18 percent nominal interest rate. With the card, he purchased a new stereo for \(\$ 350.00\). The minimum payment on the card is only \(\$ 10\) per month. a. If he makes the minimum monthly payment and makes no other charges, how long will it be before he pays off the card? Round to the nearest month. b. If he makes monthly payments of \(\$ 30,\) how long will it take him to pay off the debt? Round to the nearest month. c. How much more in total payments will he make under the \(\$ 10\) -a-month plan than under the \(\$ 30\) -a-month plan?

Loan amortization and EAR You want to buy a car, and a local bank will lend you \(\$ 20,000 .\) The loan would be fully amortized over 5 years \((60\) months), and the nominal interest rate would be 12 percent, with interest paid monthly. What would be the monthly loan payment? What would be the loan's EAR?

Present value of an annuity Find the present ralues of these ordinary anmuities. Discounting occurs once a year. a. \(\quad \$ 400\) per year for 10 years at 10 percent. b. \(\$ 200\) per year for 5 years at 5 percent. c. \(\$ 400\) per year for 5 years at 0 percent. d. Rework parts a, b, and c assuming that they are annuities due.

\(\mathrm{PV}\) and a lawsuit settlement It is now December \(31,2005,\) and a jury just found in favor of a woman who sued the city for injuries sustained in a January 2004 accident. She requested recovery of lost wages, plus \(\$ 100,000\) for pain and suffering, plus \(\$ 20,000\) for her legal expenses. Her doctor testified that she has been unable to work since the accident and that she will not be able to work in the future. She is now \(62,\) and the jury decided that she would have worked for another 3 years. She was scheduled to have earned \(\$ 34,000\) in \(2004,\) and her employer testified that she would probably have received raises of 3 percent per year. The actual payment will be made on December 31 , \(2006 .\) The judge stipulated that all dollar amounts are to be adjusted to a present value basis on December \(31,2006,\) using a 7 percent annual interest rate, using compound, not simple, interest. Furthermore, he stipulated that the pain and suffering and legal expenses should be based on a December, \(31,2005,\) date. How large a check must the city write on December \(31,2006 ?\)
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