Alan inherited $100,000 with the stipulation that he "invest it to financially benefit his family." Alan and his wife Alice decided they would invest the inheritance to help them accomplish two financial goals: purchasing a Park City vacation home and saving for their son Cooper’s education. Vacation Home Cooper’s Education
Initial investment $50,000 $50,000
Investment horizon 5 years 18 years

Alan and Alice have a marginal income tax rate of 32 percent (capital gains rate of 15 percent) and have decided to investigate the following investment opportunities.

Required:
Determine the two annual after-tax rate of return.

the question is missing the information about potential investments, so I looked for a similar one:

                                                                    5 Years  18 Years

Corporate bonds                                        5.75%  4.75%  

(ordinary interest taxed annually)

Dividend-paying stock                                 3.50%   3.50%  

(no appreciation and dividends are taxed at 15%)  

Growth stock                                              FV $65,000FV $140,000  

Municipal bond (tax-exempt)                3.20%  3.10%  

Alan and Alice should invest in growth stocks since they yield the highest after tax return:

5 years:

FV of growth stocks = $65,000

taxable gain = $65,000 -$50,000 = $15,000 x 15% = $2,250

net gain = $15,000 - $2,250 = $12,750

to determine the yield rate we can use the future value formula:

62,750 = 50,000 x (1 + r)⁵

(1 + r)⁵ = 62,750 / 50,000 = 1.255

⁵√(1 + r)⁵ = ⁵√1.255

1 + r = 1.046

r = 4.6% after tax yield per year

18 years:

FV of growth stocks = $140,000

taxable gain = $140,000 -$50,000 = $90,000 x 15% = $13,500

net gain = $90,000 - $13,500 = $76,500

to determine the yield rate we can use the future value formula:

126,500 = 50,000 x (1 + r)¹⁸

(1 + r)¹⁸ = 126,500 / 50,000 = 2.53

¹⁸√(1 + r)¹⁸ = ¹⁸√2.53

1 + r = 1.053

r = 5.3% after tax yield per year