Problem 95 - Precalculus with Limits

Question

The effective yield of an investment plan is the percent increase in the balance after 1 year. Find the effective yield for each investment plan. Which investment plan has the greatest effective yield? Which investment plan will have the highest balance after 5 years? (a) 7\(\%\) annual interest rate, compounded annually (b) 7\(\%\) annual interest rate, compounded continuously (c) 7\(\%\) annual interest rate, compounded quarterly (d) 7.25\(\%\) annual interest rate, compounded quarterly

Step-by-Step Solution

Answer

Plan (d) with a 7.43\% annual interest rate compounded quarterly has the greatest effective yield and will have the highest balance after 5 years.

Step 1: Calculate annual yield for Plan a

Calculate the annual yield for the first investment plan using the formula: \(A = P(1 + r/n)^{nt}\), where P is the principal amount (initial investment), r is the annual interest rate (in decimal), n is the number of times that interest is compounded per unit t, t is the time the money is invested for. Here, \( P=1\), \(r=0.07\), \(n=1\), and \(t=1\). So, \(A=1(1+0.07/1)^{1*1}=1.07\). The effective yield is \(A-1=0.07\) or \(7\%\).

Step 2: Calculate annual yield for Plan b

The formula for continuous compounding is \(A = Pe^{rt}\), where e is approximately 2.71828 (the base of the natural logarithm). Here, \(P=1\), \(r=0.07\), and \(t=1\). So, \(A=1*e^{0.07*1}=1.0725\). The effective yield is \(A-1=0.0725\) or \(7.25\%\).

Step 3: Calculate annual yield for Plan c

Use the first formula again, but with \(n=4\) for quarterly compounding. So, \(A=1(1+0.07/4)^{4*1}=1.0718\). The effective yield is \(A-1=0.0718\) or \(7.18\%\).

Step 4: Calculate annual yield for Plan d

Here, \(r=0.0725\), and \(n=4\). So, \(A=1(1+0.0725/4)^{4*1}=1.0743\). The effective yield is \(A-1=0.0743\) or \(7.43\%\).

Step 5: Compare yields and calculate future balances

By comparison, Plan (d) has the greatest yield at 7.43\%. To find the balance after 5 years, use the respective formulas with \(t=5\). After calculation, it will be found that Plan (d) will also have the highest balance.

Key Concepts

Compound Interest

When it comes to growing your investments, compound interest is a term you'll frequently encounter. It's the process where the interest earned on a sum of money is reinvested so that in subsequent periods, interest is then earned on the initial principal and the accumulated interest. This creates a snowball effect where your money grows at an accelerating rate over time, making it a powerful tool for increasing wealth.

In the context of our investment plans, compound interest plays a pivotal role, particularly in how it's calculated during different compounding periods. For example, when the interest is compounded annually as in Plan (a), the formula used is \(A = P(1 + r/n)^{nt}\), and after one year, we would calculate the effective yield by subtracting the initial amount from the total amount after interest has been applied. The result is a percentage that shows the growth of the investment due to compound interest.

Continuous Compounding

Continuous compounding is an extreme case of compound interest where we imagine that the interest is calculated and added an infinite number of times in any given period. The formula that mathematically represents continuous compounding is \(A = Pe^{rt}\), with \(e\) representing the mathematical constant approximately equal to 2.71828. This formula is crucial for calculating the effective yield for plans like Plan (b) within our exercise.

Under continuous compounding, money can grow faster than with standard compound interest because the reinvestment occurs incessantly, giving the principal virtually no downtime before it starts earning additional interest. Consequently, Plan (b) has a slightly higher effective yield compared to when it's compounded just annually or quarterly.

Interest Rate Compounding Periods

The frequency of compounding—how often interest is added to the principal—can significantly impact the growth of an investment. Common compounding periods include annual, semi-annual, quarterly, monthly, daily, or continuously. The general rule is that the more frequently interest is compounded, the greater the amount of interest will be earned on an original investment over a set period of time.

In our textbook problem, we explore the different compounding periods specifically with Plans (a), (c), and (d). To calculate the effective yield for quarterly compounding as seen in Plans (c) and (d), we adjust the formula to reflect four compounding periods per year \(n=4\), resulting in the investment growing faster than it would if compounded annually. This difference in compounding frequency helps to illustrate why Plan (d) with a 7.25% interest rate compounded quarterly provides the greatest effective yield and why it will accumulate the highest balance after 5 years.