# Need Help with this %'s problem

• nando94

#### nando94

Bob invests his life savings in the stock market. If he loses 20% the first year, what annual return, to the nearest percent, must he earn over the next two years so that his annual return for the whole three year period is 10%?

a. 6%
b. 15%
c. 29%
d. 36%
e. 45%

here is the answer. I am getting lost after knowing that he has 80% after the first year.

29%

If you put the initial capital as C, then at the end of the first year he has 0.8C=C-0.2C (as you said)

Let's say that k is the annual return for the next two years.

Can you write, in terms of k and C, the interest at the end of the second and third year?

That is, after the first year, the interest is -0.2C, where -0.2 is the tax of interest that year.

You will get a quadratic equation in k, and solving you get the answer.

If "x" is the initial investment then after the first years he's got 0.8x

At the end of 3 years however he requires 1.1^3 x to achieve his target. So he has to go from 0.8x to 1.1^3 x in two years.

If the multiplying factor per year is "a" (btw, a = 1+r), then he requires 0.8 a^2 x = 1.1^3 x

Solve the above for "a".

Bob invests his life savings in the stock market. If he loses 20% the first year, what annual return, to the nearest percent, must he earn over the next two years so that his annual return for the whole three year period is 10%?

a. 6%
b. 15%
c. 29%
d. 36%
e. 45%

here is the answer. I am getting lost after knowing that he has 80% after the first year.

29%

This is a multiple choice question, so as such you don't have to find the answer, just indentify which one of the options is the answer.

We could compare the market to a simple investment.
With compound interest we gain a little more than with simple interest. The shorter the compounding period, the [slightly] better the gain. [Interest paid monthly does better than interest paid annually]. but really large differences take quite a long time period to show up

If your reduced funds were invested, what Simple interest rate would achieve your required return in 2 more years. As a compounding interest, you could expect the required return to be slightly less than that.

Don't forget you want to average a +10% return for each year of the whole investment.

Note: Had the Options offered all been within 1% of each other, you would clearly be required to calculate a more accurate answer - but it still may be quicker to calculate the final value with each of the offered rates to see which one works, rather than reverse solve an exponential equation.