Hello gillchandler1

Welcome to Math Help Forum! 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.

Grandad