Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose payments were made at the end of each quarter into an ordinary annuity earning interest at the rate of \(10 \% /\) year compounded quarterly. If the future value of the annuity after \(5 \mathrm{yr}\) is \(\$ 50,000\), what was the size of each payment?

Step-by-Step Solution

Answer

The size of each payment made at the end of each quarter into the ordinary annuity is approximately $1,966.90.

Step 1: Identify the variables involved

Here, we have: Future value of the annuity (FV) = $50,000 Annual interest rate (R) = 10% = 0.1 (in decimal form) Number of years (n) = 5 Number of compounding periods per year (k) = 4 (quarterly) We need to calculate the size of each payment, which we will denote by (PMT).

Step 2: Adjust the interest rate and number of periods

First things first, we need to convert the annual interest rate to the equivalent quarterly interest rate, and the number of years into the number of total quarters. Quarterly interest rate (r) = R / k Number of quarters (t) = n * k Plug in the values and find r and t: r = 0.1 / 4 = 0.025 t = 5 * 4 = 20

Step 3: Use the future value of an ordinary annuity formula

The future value (FV) of an ordinary annuity is found using the following formula: \(FV = PMT \times \frac{(1+r)^t - 1}{r}\) We need to isolate PMT to solve for it: \(PMT = \frac{FV \times r}{(1+r)^t - 1}\) Now, plug in the values we calculated in previous steps and solve for PMT: \(PMT = \frac{50000 \times 0.025}{(1+0.025)^{20} - 1} \approx 1966.90\)

Step 4: Conclusion

The size of each payment made at the end of each quarter into the ordinary annuity is approximately $1,966.90.

Key Concepts

Compounded Interest

Compounded interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. It's a fundamental concept in finance that shows how money can grow over time when it's reinvested.
For example, suppose you have a savings account with a quarterly compounded interest rate of 10% per year. If you start with \(1,000 and the interest is compounded quarterly, this means that every quarter, the bank will apply a 2.5% interest rate (as 10% divided by 4 quarters) not only to your original \)1,000 but also to any interest that your money has earned during the previous quarters.
In the context of our exercise, the compounded interest is what makes the future value of an annuity grow over time. Since our annuity compounds quarterly, each of the annuity payments earns interest, which, in turn, earns more interest in subsequent quarters. This compounded effect leads to substantially more savings as compared to simple interest, where interest is earned only on the principal amount.

Annuity Payments

Annuity payments refer to regular fixed payments made or received over a certain period. An ordinary annuity involves payments made at the end of each period, such as monthly rents or quarterly insurance payments.
When you're calculating the future value of an ordinary annuity, as in the given textbook exercise, you're determining how much all of these payments will be worth after accounting for the compounding interest over the entire period of the annuity.

Understanding the Payments

Annuity payments are determined by several factors, including the interest rate, the frequency of compounding, the number of payments, and the total time span of the annuity. For our exercise, by calculating these payments, we can plan financial goals like retirement savings, where you know how much you will have by contributing a certain amount quarterly, considering the compounding effect over time.

Time Value of Money

The time value of money is a principle that suggests money available now is worth more than the same amount in the future because of its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received.
This concept underpins the calculations we use to determine the future value of annuity payments. The payments that are made early in the annuity's term have more time to accumulate interest than payments made later on. Thus, each payment can be viewed as an individual investment that grows over time.

Application in Annuities

In our exercise, understanding the time value of money helps us comprehend why the future value of an annuity is more than the sum of the payments made. It's because those payments are not just stored but are actively earning interest, and each quarter's interest earnings will increase the overall future value of the annuity.