# A risky investment

I invested $1000 into shares. On the first day the share price went up by 10%. On the second day it went down by 10%. This process continued - the share price would go up by 10% on odd days and down by 10% on even days. Without using a calculator (or a computer) can you predict how much money will I have after 10 years? ## 2 Answers Answer: You have not much money left. (About$0)

Reasoning:

Let your money on day $$n$$ be $$M_n$$. So $$M_0=1000$$
$$M_{2n+2}=\big(1-\frac1{10}\big)\big(1+\frac1{10}\big)M_{2n}=0.99M_{2n}$$
So the money you have went down. After $$10$$ years, you will not have much money left.

If we use a calculator:

Let a year be 365 days for convenience. Then we will have $$3650$$ days in $$10$$ years, so there will be $$1000\times(0.99)^{3650\div2}=0.0000108...$$. Of course, you will not want to invest this.

$$\approx \0$$

Reasoning

If value goes up by 10%, it means value becomes (1+10%) = 1.1 times of itself.... If value goes down by 10%, it means value becomes (1-10%) = 0.9 times of itself

Now we know that 10yrs = 365*10 = 3650 days(not taking leap years) So now the value you have is

$$1000*1.1*0.9*1.1*0.9... = 1000*\mathrm{1.1}^{3650/2}*\mathrm{0.9}^{3650/2}$$

which equals

$$1000*\mathrm{0.99}^{1825} \approx 0$$

• Are you sure your numbers are correct? (1-10%) = 0.99 , shouldn't that be (1-10%) = 0.90, and did you mean (1.1-10%) = 0.99 ? May 21 '20 at 17:11
• @gregn3, thanks for pointing that out. Corrected now :) May 21 '20 at 17:30