4. a. Someone in the 36 percent tax bracket can earn 9 percent annually on her investments
in a tax-exempt IRA account. What will be the value of a one-time $10,000 investment
in 5 years? 10 years? 20 years?
b. Suppose the preceding 9 percent return is taxable rather than tax-deferred and the taxes
are paid annually. What will be the after-tax value of her $10,000 investment after 5, 10,
and 20 years?
You have a fairly large portfolio of U.S. stocks and bonds. You meet a financial planner at a
social gathering who suggests that you diversify your portfolio by investing in emerging market
stocks. Discuss whether the correlation results in Exhibit 3.10 support this suggestion
a. To calculate the value of the one-time $10,000 investment in a tax-exempt IRA account after a certain number of years, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount after investing for a certain number of years
P = the initial investment amount ($10,000)
r = the annual interest rate (9%)
n = the number of times interest is compounded per year (assume annually)
t = the number of years
For 5 years:
A = 10,000(1 + 0.09/1)^(1*5)
A = 10,000(1.09)^5
A ≈ 15,245.04
For 10 years:
A = 10,000(1 + 0.09/1)^(1*10)
A = 10,000(1.09)^10
A ≈ 23,674.57
For 20 years:
A = 10,000(1 + 0.09/1)^(1*20)
A = 10,000(1.09)^20
A ≈ 59,837.97
b. If the 9% return is taxable and the taxes are paid annually, we need to consider the after-tax value of the investment. Let's assume that the tax rate is 36%.
To calculate the after-tax value, we need to subtract the taxes paid on the investment returns. The formula can be modified as:
A = [P(1 + r/n)^(nt)] - (Tax Rate * Investment Returns)
For 5 years:
A = [10,000(1 + 0.09/1)^(1*5)] - (0.36 * (10,000(1 + 0.09/1)^(1*5) - 10,000))
A ≈ 12,457.62
For 10 years:
A = [10,000(1 + 0.09/1)^(1*10)] - (0.36 * (10,000(1 + 0.09/1)^(1*10) - 10,000))
A ≈ 19,243.82
For 20 years:
A = [10,000(1 + 0.09/1)^(1*20)] - (0.36 * (10,000(1 + 0.09/1)^(1*20) - 10,000))
A ≈ 45,977.76
Therefore, the after-tax value of the $10,000 investment after 5, 10, and 20 years would be approximately $12,457.62, $19,243.82, and $45,977.76 respectively.
To calculate the value of a one-time $10,000 investment over different time periods and with different tax scenarios, we need to consider the annual interest rate and the tax bracket.
a. Tax-exempt IRA account:
Given that the person is in the 36 percent tax bracket and can earn 9 percent annually on investments in a tax-exempt IRA account, we can calculate the future value of the investment after 5, 10, and 20 years.
To calculate the future value of the investment, we can use the compound interest formula, which is:
Future Value (FV) = Present Value (PV) * (1 + interest rate)^n
Where:
- PV is the present value or initial investment ($10,000 in this case).
- Interest rate is the annual interest rate divided by 100 (9 percent in this case).
- n is the number of years.
For 5 years:
FV = $10,000 * (1 + 0.09)^5
For 10 years:
FV = $10,000 * (1 + 0.09)^10
For 20 years:
FV = $10,000 * (1 + 0.09)^20
b. Taxable investment with annual tax payments:
If the preceding 9 percent return is taxable and taxes are paid annually, the after-tax value of the investment will be different.
To calculate the after-tax value, we need to calculate the tax paid each year and subtract it from the investment growth. The tax paid each year can be calculated by multiplying the annual return by the tax bracket.
For 5 years:
FV = $10,000 * (1 + 0.09 - 0.09 * 0.36)^5
For 10 years:
FV = $10,000 * (1 + 0.09 - 0.09 * 0.36)^10
For 20 years:
FV = $10,000 * (1 + 0.09 - 0.09 * 0.36)^20
By plugging the numbers into the respective formulas, you can calculate the values for each scenario and time period.
P = Po + Po*r*t.
P = 10000 + 10000*0.09*5 = $14,500. =
Value after 5 years.