Suppose you invest $100,000 in a mutual fund for 10 years. The fund earns 6% pretax per year, makes no annual distributions (and thus there is no income to be taxed each year) and you sell the fund at the end of the 10 years. You pay a 20% tax on capital gains and a 40% tax on ordinary income. What is the pre-tax total dollar accumulation at the end of 10 years? What is the after-tax total dollar accumulation at the end of 10 years? Suppose instead you invest the $100,000 in preferred stock paying 6% per annum with the dividend taxed at 20% per year (as dividends receive their own tax preferred rate). Assume the after-tax dividends are reinvested in the preferred stock. What is the after-tax total dollar accumulation at the end of 10 years? What do you conclude from comparing the above alternatives?

Let's calculate the pre-tax and after-tax total dollar accumulation for the mutual fund and the preferred stock investment after 10 years.

Mutual Fund Investment: Pre-tax Accumulation: For the mutual fund the investment grows at a compounded rate of 6% per year. 

The future value (FV) of the investment can be calculated using the compound interest formula: [ FV = P(1 + r)^n ] where P = principal amount ($100,000), r = annual interest rate (6%, or 0.06) and n = number of years (10).

[ FV = 100,000(1 + 0.06)^{10} ] 

[ FV = approx 100,000(1.79085) ] 

[ FV = approx 179,085 ]

After-tax Accumulation: For the after-tax accumulation, you'll have to consider the capital gains tax of 20%. Since the investment makes no annual distributions, the tax will only be applied at the end of 10 years when the mutual fund is sold.

Capital Gain = $179,085 - $100,000

 = $79,085 

Capital Gains Tax = 20% of $79,085 

= $15,817

The after-tax total dollar accumulation is: 

After-tax Total = Pre-tax Total - Capital Gains Tax 

After-tax Total = $179,085 - $15,817 

After-tax Total approx $163,268.

Preferred Stock Investment: For this scenario, dividends are taxed annually. The after-tax return is calculated by subtracting the dividend tax from the yearly dividend, and then reinvesting the after-tax dividend each year.

 Annual Dividend = P \times r  

 Annual Dividend = 100,000 \times 0.06 

 Annual Dividend = 6,000 

The after-tax dividend each year is: 

After-tax Dividend = Annual Dividend \times (1 - Tax Rate) 

After-tax Dividend = 6,000 \times (1 - 0.20) 

After-tax Dividend = 6,000 \times 0.80 

 After-tax Dividend = 4,800 

Now, the after-tax dividend is reinvested each year at the same 6% rate of return. To find the future value of these reinvestments over 10 years, you can use the future value of an annuity formula. The formula for the future value of an annuity due (since we assume reinvestment at the start of the year) is:

 FV annuity due = PMT \times 

frac {(1 + r)^n - 1}{r}

right \times (1 + r) 

where PMT is the after-tax dividend being reinvested each year ($4,800).

Applying this formula:

FV{annuity due = 4,800 \times 

frac{(1 + 0.06)^{10} - 1}{0.06} 

times 1.06 

 FV = approx 62,712 

So the after-tax total dollar accumulation for the preferred stock investment is approximately $62,712.

Comparing the above alternatives, the mutual fund investment yields a higher after-tax total dollar accumulation than the preferred stock investment.

The difference in accumulation is mainly due to the tax treatment. The mutual fund investment benefits from tax deferral since the returns are not taxed until the investment is liquidated after 10 years, whereas the preferred stock dividends are taxed each year, reducing the amount available to be reinvested.