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Chapter 2: Problem 17

Compound Interest. Old Time Savings Bank pays 5 percent interest on its savings accounts. If you deposit \(\$ 1,000\) in the bank and leave it there, how much interest will you carn in the first year? The second year? The tenth year?

Short Answer

Expert verified
The interest earned in the first year will be \$50, in the second year will be \$52.50, and in the tenth year will be \$49.94.

Step by step solution

01

Understanding the Problem

The problem states that a deposit of $1,000 is made at an interest rate of 5 percent per annum. Therefore, the principal \(P = \$1,000\), the annual interest rate \(r = 5\% = 0.05\), and it is compounded annually \(n = 1\). The formula for compound interest is \(A = P(1 + \frac{r}{n})^{nt}\), where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount, \(r\) is the annual interest rate (in decimal form), \(n\) is the number of times that interest is compounded per unit \(t\), and \(t\) is the time the money is invested for (in years). In this problem, interest is compounded annually, so \(n = 1\).
02

First Year Interest

We substitute the values into the equation to find out how much money is in the account after the first year \(t = 1\): \(A = \$1,000(1 + (0.05/1))^{1*1} = \$1,050\). The interest earned in the first year is therefore \$1,050 - \$1,000 = \$50.
03

Second Year Interest

We do the same for the second year \(t = 2\): \(A = \$1,000(1 + (0.05/1))^{1*2} = \$1,102.50\). The interest earned in the second year is therefore \$1,102.50 - \$1,050 = \$52.50. Notice how the interest earned in the second year is greater than the first due to the added 'interest on interest' effect.
04

Tenth Year Interest

For the tenth year \(t = 10\), the calculations follow similarly: \(A = \$1,000(1 + (0.05/1))^{1*10} = \$1,628.89\). The interest earned in the tenth year is therefore \$1,628.89 - \$1,578.95 (the amount from the ninth year) = \$49.94. Over time, thanks to compounding interest, the initial principal has increased by over 60%.

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Key Concepts

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

Principle Amount
The principle amount is the initial sum of money put into an investment before any interest accumulates. It serves as the base on which interest is calculated. For instance, in our exercise, the principle amount is \( \$1,000 \). It's crucial to understand that the final amount of money you'll have after a certain period depends heavily on the principle amount; the larger it is, the more interest you'll earn, provided the interest rate and other factors remain constant. A useful tip is to always double check the principle amount before proceeding with compound interest calculations as it affects the total return on investment.
Annual Interest Rate
The annual interest rate is the percentage increase in the principle amount over one year. In the textbook exercise, this rate is given as 5%, or \(0.05\) when expressed as a decimal, which is necessary for calculations. This rate is critical as it can significantly influence the growth of the investment. A higher interest rate typically means more earnings on your principle, while a lower rate indicates slower growth. It's pivotal to not confuse the annual interest rate with the rate applied over different time periods, which can appear if the compounding frequency is more than once per year.
Time Value of Money
The time value of money is a financial concept recognizing that a sum of money is worth more now than the same sum in the future due to its potential earning capacity. This core concept ensures that every dollar you have today is more valuable than one promised in the future because of the ability to earn interest. In the context of our exercise, the money deposited grows over time, and thanks to compound interest, the effect of time on the value of money is accentuated. This is demonstrated as the amount of money earned in interest increases each year with the same initial interest rate.
Compounding Frequency
Compounding frequency refers to how often the interest is calculated and added to the principle amount. Common compounding frequencies include annually, semi-annually, quarterly, monthly, or even daily. In our example, the compounding frequency is annual, meaning the interest is calculated and added to the original balance each year. If the frequency were semi-annual, the interest would be calculated and added twice a year, which could result in more interest earned over time, due to what's called 'compounding interest' or 'interest on interest'. The compounding effect is the key reason why starting to invest early in life can have such a pronounced impact on wealth accumulation.

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Most popular questions from this chapter

Annuity Value. Your landscaping company can lease a truck for \(\$ 8,000\) a year (paid at yearend) for 6 years. It can instead buy the truck for \(\$ 40,000\). The truck will be valueless after 6 years. If the interest rate your company can earn on its funds is 7 percent, is it cheaper to buy or lease?

EAR versus APR. You invest \(\$ 1,000\) at a 6 percent annual interest rate, stated as an APR. Interest is compounded monthly. How much will you have in 1 year? In 1.5 years?

Retirement Savings. You believe you will spend \(\$ 40,000\) a year for 20 years once you retire in 40 years. If the interest rate is 5 percent per year, how much must you save each year until retirement to meet your retirement goal?

Real versus Nominal Rates. You will receive \(\$ 100\) from a savings bond in 3 years. The nominal interest rate is 8 percent. a. What is the present value of the proceeds from the bond? b. If the inflation rate over the next few years is expected to be 3 percent, what will the real value of the \(\$ 100\) payoff be in terms of today's dollars? c. What is the real interest rate? d. Show that the real payoff from the bond (from part b) discounted at the real interest rate (from part \(c\) ) gives the same present value for the bond as you found in part a.

Calculating Interest Rate. You borrow \(\$ 1,000\) from the bank and agree to repay the loan over the next year in 12 equal monthly payments of \(\$ 90 .\) However, the bank also charges you a loan-initiation fee of \(\$ 20,\) which is taken out of the initial proceeds of the loan. What is the effective annual interest rate on the loan taking account of the impact of the initiation fee?
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