Algebra question about investment (related to economics and percentage)

Question

I have this school activity, I did do it, but my answers did not match! I would really really appreciate it if you could help me by explaining and giving me the answer. God bless you!

John, a beginner in investments was in doubt between 2 assets to invest.

1. The first would be a fund that offered security and reasonable volatility. This would expire after 6 years, paying 8% per year.
2. The second was an investment with greater risk but a greater expectation of return. It was a real estate fund that had been performing well in recent months.

John opted for the 2nd option and invested U$10,000.00 in this fund, acquiring 100 shares for U$100.00 each. After a year, the quotas devalued significantly and reached U$90 each.

He then decided to sell his shares and invest in a fixed-rate fund that expires in 5 years, giving up liquidity. Given this situation, what should be, approximately, the annual rate of this fund for the future value to be equaled to the first option?

a) 13%

b) 12,5%

c) 12%

d) 11,5%.

Answer 1

First we need to figure out how much John would have made had he invested in option number 1.

So we use the compound interest Formula A = P (1+r/n)^nt

A = final amount

P = Initial Balance

r = interest rate

n = number of times interest rate is applied per time period

t = number of time periods elapsed

So P = 10,000; r = 0.08 (8%); n = 1; t = 6 because the interest is annual and only occurs once a year

10,000(1+ 0.08/1)⁶ = $15,869

Ok so we know he would have had $15,869 had he invested in the first option.

We need to know how much he has left now.

He initially had 100 shares of $100 each but they lowered in value to $90 each after a year. He sold all 100 for $90 each which means he had $9000 left.

So now we need to use the formula again to figure out the annual rate that he would need to reach $15,869 in 5 years.

The easiest way is to work backwards. Enter each value r into the equation and solve for A.

Lets try 12%

A = 9000 (1+ 0.12/1)⁵

A = 15,861

Lets try 12.5%

A = 9000 (1 + 0.125/1)⁵

A = 16,218

We know 11% will be too low and 13% will be too high.

The question is a little finicky in asking "approximately...what annual rate...would be equal to the first option".

$15,861 is approximately right but is not quite equal to $15,869 but I believe that is the answer.

ANSWER 12%

If you have any questions please let me know!