# Very unusual problem - any help highly appreciated

• Nov 16th 2009, 01:13 AM
gillchandler1
Very unusual problem - any help highly appreciated
Suppose there is a stock (Stock A) and an index (Index A).

Stock A's price=$150 Index A's level=2450 The current ratio of Stock A/Index A Level = 150/2450 = 0.06122 Now supposing you look on the historic ratio chart and think that the ratio will increase from the current 0.06122 to say 0.07500 (ie you think stock A will outperform the index). Just from knowing the expected ratio (0.07500) and the current ratio level (0.06122), would it be possible to calculate by how much % stock A outperformed the index by ? Thanks a lot for all your help. • Nov 17th 2009, 05:48 AM HallsofIvy Quote: Originally Posted by gillchandler1 Suppose there is a stock (Stock A) and an index (Index A). Stock A's price=$150
Index A's level=2450

The current ratio of Stock A/Index A Level = 150/2450 = 0.06122

Now supposing you look on the historic ratio chart and think that the ratio will increase from the current 0.06122 to say 0.07500 (ie you think stock A will outperform the index).

Just from knowing the expected ratio (0.07500) and the current ratio level (0.06122), would it be possible to calculate by how much % stock A outperformed the index by ?

Thanks a lot for all your help.

Right now you have A(0)/I(0)= .06122 and you believe that at some time in the future, A(t)/I(t) will be 0.07500. That tells you that (A(t)/I(t)) divided by A(0)/I(0) will be [tex]\frac{A(t)/I(t)}{A(0)/I(0)}= \frac{A(t)}{A(0)}{I(0)/I(t)}= .07500/.06122= 1.225. If we think of A= A(t)/A(0) as measuring how well A has "performed" and I= I(t)/I(0) as measuring how well the index has "performed", then we have A/I= 1.225 so we could say that, in some sense, A has "outperformed" the index by 22.5%. I'm not sure that is a useful "sense".
• Nov 17th 2009, 06:13 AM
Hello gillchandler1

Welcome to Math Help Forum!
Quote:

Originally Posted by gillchandler1
Suppose there is a stock (Stock A) and an index (Index A).

Stock A's price=$150 Index A's level=2450 The current ratio of Stock A/Index A Level = 150/2450 = 0.06122 Now supposing you look on the historic ratio chart and think that the ratio will increase from the current 0.06122 to say 0.07500 (ie you think stock A will outperform the index). Just from knowing the expected ratio (0.07500) and the current ratio level (0.06122), would it be possible to calculate by how much % stock A outperformed the index by ? Thanks a lot for all your help. The answer is that there isn't a fixed percentage difference, but you can calculate one if you know the other. Suppose that the stock is originally valued at $$\displaystyle S and increases by \displaystyle s%; and the index, valued at$$\displaystyle I$, increases by $\displaystyle i$%. Then the values will increase by factors of $\displaystyle \left(1+\frac{s}{100}\right)$ and $\displaystyle \left(1+\frac{i}{100}\right)$ respectively. The ratio of the new values will then be
$\displaystyle \frac{\left(1+\dfrac{s}{100}\right)S}{\left(1+\dfr ac{i}{100}\right)I}=\left(\frac{100+s}{100+i}\righ t)\left(\frac{S}{I}\right)$
So if the original ratio, $\displaystyle \frac{S}{I}$, was $\displaystyle 0.06122$, and the new one is $\displaystyle 0.075$, we have
$\displaystyle \left(\frac{100+s}{100+i}\right)\left(0.06122\righ t)=0.075$

$\displaystyle \Rightarrow \left(\frac{100+s}{100+i}\right)=\frac{0.075}{0.06 122}=1.225$
Re-arranging this equation gives:
$\displaystyle s=1.225i+22.5$
which will give the value of $\displaystyle s$ for a given value of $\displaystyle i$, enabling a comparison to be made. For example:
If $\displaystyle i = 0$ (i.e. the index does not rise, $\displaystyle s = 22.5$; i.e the stock rises by $\displaystyle 22.5$%.

If $\displaystyle i = 5$ (i.e. the index rises by $\displaystyle 5$%), $\displaystyle s = 28.625$; the stock rises by $\displaystyle 28.625$%, outperforming the index by $\displaystyle 23.625$%.

... and so on.