Problem 53 - College Algebra

Question

Use the compound interest formulas \(A-P\left(1+\frac{r}{n}\right)^{n t}\) and \(A-P e^{n}\) to solve Exercises \(53-56 .\) Round answers to the nearest cent. Find the accumulated value of an investment of \(\$ 10,000\) for 5 years at an interest rate of \(5.5 \%\) if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Step-by-Step Solution

Answer

The future value of an investment of $10,000 compounded: (a) semiannually is approx. $13,749.11, (b) quarterly is approx. $13,804.21, (c) monthly is approx. $13,861.03, and (d) continuously is approx. $13,879.47 (rounded to the nearest cent).

Step 1: Calculate for semiannual compounding

Putting the given values P=$10,000, r=0.055, n=2 (semiannually), and t=5 in the compound interest formula \(A = P \left(1+ \frac{r}{n} \right)^{nt}\), we can solve for A which is our accumulated value.

Step 2: Calculate for quarterly compounding

Putting the given values P=$10,000, r=0.055, n=4 (quarterly), and t=5 in the compound interest formula, we solve for A.

Step 3: Calculate for monthly compounding

Putting the given values P=$10,000, r=0.055, n=12 (monthly), and t=5 in the compound interest formula, we solve for A.

Step 4: Calculate for continuous compounding

Putting the given values P=$10,000, r=0.055, and t=5 in the formula for continuous compounding \(A = Pe^{rt}\), we solve for A.

Key Concepts

Compound Interest Formula

Understanding the compound interest formula is essential when delving into the world of investments and savings. This formula is the key to calculate how much money you'll accumulate over a period of time when the interest is applied on both the initial principal and the interest already earned. The general compound interest formula is \[A = P\left(1+ \frac{r}{n}\right)^{nt}\],where

• \(A\) represents the future value of the investment/loan, including interest
• \(P\) is the principal amount (the initial amount of money)
• \(r\) is the annual interest rate (in decimal form)
• \(n\) is the number of times that interest is compounded per year
• \(t\) is the time the money is invested or borrowed for, in years

Imagine you're planting a tree. The principal amount (\(P\)) is the seed you plant, the rate (\(r\)) is the fertility of the soil, and the number of times interest compounds (\(n\)) is like how frequently you water the tree. Time (\(t\)) is how long you let the tree grow. The fruit of this tree is the compound interest, which can be much more bountiful compared to simple interest where your tree is only watered once at the end of every year.

Accumulated Value Calculation

The accumulated value of an investment is simply the amount of money you end up with after a certain period, considering the compound interest. This is different from the starting principal as it includes the interest that’s been added to the original amount over time. To calculate the accumulated value when dealing with different compounding periods, you just plug the appropriate values into the compound interest formula we discussed above. For example, when money is compounded semiannually, \(n = 2\), and for quarterly, \(n = 4\). The compounded monthly scenario uses \(n = 12\), and for continuous compounding, we use the formula \(A = Pe^{rt}\), as interest is added an infinite number of times. Each of these different compounding scenarios will yield a different accumulated value, which means the frequency of compounding can have a significant effect on how much you earn from your investment over time. So it’s vital to understand the specifics of your investment terms regarding how often the interest is calculated and added to your principal.

Time Value of Money

The time value of money is a financial concept that describes the idea that money available now is worth more than the same amount in the future due to its potential earning capacity. This is why when you invest, you can earn interest, and as time goes by, your money can grow as it earns more interest. The compound interest formula takes the time value of money into account by incorporating not just how much and at what rate your money will grow, but also how often the interest will be applied. In the exercise we discussed, a $10,000 investment grows differently over 5 years depending on how the interest is compounded—whether it's semiannually, quarterly, monthly, or continuously. The more frequently interest is compounded, the more opportunity your money has to grow, illustrating the profound effect of time on the value of money. This is an important principle to understand for anyone looking to save or invest because it can significantly impact investment decisions, savings plans, and the overall management of personal finances.