Cliff has some money he wants to invest for his retirement. He is offered two options:
1. 10%/a simple interest
2. 5%/a compounded annually
Under what conditions should he choose the first option? Explain.
To compare the two investment options, we need to consider the time horizon, the amount of money Cliff wants to invest, and his expected return.
1. Simple Interest (10% per annum):
With simple interest, the interest is calculated only on the initial investment amount, without compounding over time. This means that the interest earned each year remains the same.
Let's say Cliff wants to invest $10,000 for a period of 5 years.
Year 1: $10,000 + (10% of $10,000) = $10,000 + $1,000 = $11,000
Year 2: $11,000 + (10% of $10,000) = $11,000 + $1,000 = $12,000
Year 3: $12,000 + (10% of $10,000) = $12,000 + $1,000 = $13,000
Year 4: $13,000 + (10% of $10,000) = $13,000 + $1,000 = $14,000
Year 5: $14,000 + (10% of $10,000) = $14,000 + $1,000 = $15,000
After 5 years, Cliff's investment will grow to $15,000.
2. Compounded Annually (5% per annum):
With compound interest, the interest is calculated on both the initial investment and any previously earned interest, compounding over time. The interest earned each year will increase as the investment grows.
Using the same $10,000 investment for 5 years at a compounded interest rate of 5% annually:
Year 1: $10,000 + (5% of $10,000) = $10,000 + $500 = $10,500
Year 2: $10,500 + (5% of $10,500) = $10,500 + $525 = $11,025
Year 3: $11,025 + (5% of $11,025) = $11,025 + $551.25 = $11,576.25
Year 4: $11,576.25 + (5% of $11,576.25) = $11,576.25 + $578.81 = $12,155.06
Year 5: $12,155.06 + (5% of $12,155.06) = $12,155.06 + $607.75 = $12,762.81
After 5 years, Cliff's investment will grow to approximately $12,762.81.
Comparing the results, it is clear that the compounded interest option with a lower interest rate of 5% outperformed the simple interest option with a higher rate of 10% over the 5-year period.
Therefore, choosing the compounded interest option might be a better choice for Cliff's retirement investment.
To compare the two investment options, we need to understand how simple interest and compound interest work. Let's go step by step:
1. Simple interest: Simple interest is calculated based on the initial principal amount only. The formula to calculate simple interest is:
Simple Interest = Principal × Rate × Time
2. Compound interest: Compound interest is calculated not only on the initial principal amount but also on the accumulated interest over time. The formula to calculate compound interest is:
Compound Interest = Principal × (1 + Rate)^Time - Principal
Now, let's analyze each option:
1. 10% simple interest:
If Cliff chooses this option, he will earn a fixed 10% interest on the initial principal amount every year. The interest earned will remain the same every year, regardless of how much the principal grows or changes.
2. 5% compound interest:
If Cliff chooses this option, his interest will compound annually at a rate of 5%. This means that each year, the interest earned will be added to the principal, and subsequent interest will be calculated on the new higher principal amount.
To compare the two options, we could consider different factors, such as the growth of the principal over time, the total interest earned, and the final amount at retirement. Additionally, we would need information about how much money Cliff is investing and the time period until his retirement.
Once we have those details, we can use the formulas mentioned above to calculate the outcomes for both options. This will allow us to compare and make an informed decision about which option might be more beneficial for Cliff's retirement investment.