When I was considering what to do with my $10,000 Lottery winnings, my broker suggested I invest half of it in gold, the value of which was growing by 7% per year, and the other half in certificates of deposit (CDs), which were yielding 3% per year, compounded every 6 months. Assuming that these rates are sustained, how much will my investment be worth in 13 years? (Round your answer to the nearest cent.)
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To calculate the future value of your investments, we need to calculate the future value of gold and the future value of CDs separately and then add them together.
1) Future Value of Gold:
The value of gold is growing by 7% per year. We can use the formula:
Future Value = Present Value * (1 + Annual Interest Rate)^Number of Years
Here, the Present Value is $10,000 and the Annual Interest Rate is 7% or 0.07. The Number of Years is 13.
Future Value of Gold = $10,000 * (1 + 0.07)^13 = $21,137.63 (rounded to the nearest cent).
2) Future Value of CDs:
The CDs are yielding 3% per year, compounded semi-annually. We can use the formula:
Future Value = Present Value * (1 + (Annual Interest Rate / Number of Compounding Periods))^Number of Compounding Periods * Number of Years
Here, the Present Value is also $10,000. The Annual Interest Rate is 3% or 0.03. The Number of Compounding Periods per year is 2 (since it is compounded semi-annually). The Number of Years is 13.
Future Value of CDs = $10,000 * (1 + (0.03 / 2))^2*13 = $13,968.48 (rounded to the nearest cent).
3) Total Future Value:
To find the total future value, we add the future value of gold and the future value of CDs.
Total Future Value = $21,137.63 + $13,968.48 = $35,106.11 (rounded to the nearest cent).
Therefore, your investment will be worth approximately $35,106.11 in 13 years.
To calculate the future value of your investment, you can use the formula for compound interest:
Future Value = Principal Amount × (1 + Interest Rate / Number of Compounding Periods)^(Number of Compounding Periods × Number of Years)
Let's break down the calculation for each investment option:
1. Gold:
You invested half of your $10,000 winnings, which is $5,000. The interest rate is 7% per year. Since it is not mentioned how frequently the interest is compounded, we assume it is compounded annually.
Future Value of Gold Investment = $5,000 × (1 + 7% / 1)^(1 × 13)
= $5,000 × (1 + 0.07)^13
= $5,000 × (1.07)^13
≈ $11,234.49
2. Certificates of Deposit (CDs):
You also invested half of your $10,000 winnings, which is $5,000. The interest rate is 3% per year, compounded semi-annually (every 6 months).
To convert the annual interest rate to a semi-annual rate, we need to divide it by 2. So the interest rate for each 6-month period is 3% / 2 = 1.5%.
The number of compounding periods is calculated as the number of years multiplied by the frequency of compounding. In this case, it is 13 years × 2 compounding periods per year = 26 compounding periods.
Future Value of CD Investment = $5,000 × (1 + 1.5% / 2)^(2 × 13)
= $5,000 × (1 + 0.015)^26
≈ $6,918.15
Now, to find the total value of your investments in 13 years, you sum the future values of the two investments:
Total Future Value = Future Value of Gold Investment + Future Value of CD Investment
≈ $11,234.49 + $6,918.15
≈ $18,152.64
So, if the rates are sustained, your investment will be worth approximately $18,152.64 in 13 years.
5000*(1+.07*13) + 5000(1+.03/2)^(2*13)
= $16,913.55