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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?

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2 Answers 2

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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.

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Answer

$\approx \$0$

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

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

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    $\begingroup$ 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 ? $\endgroup$
    – gregn3
    Commented May 21, 2020 at 17:11
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    $\begingroup$ @gregn3, thanks for pointing that out. Corrected now :) $\endgroup$ Commented May 21, 2020 at 17:30

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