An asset A has volatility estimated as σA = 0.2 An asset B has volatility estimated as σB = 0.4 The assets have covariance = σAB = -0.33 State the hypothesis of correlation and test this at a 10% significance level
So firstly I worked out the correlation coefficient which I believe to be -4.125 I did this by
p=$-0.33/(0.2*0.4)$
adding to this I am told by the teacher I need to revere engineer the formula for d using a value taken from the Normal table. However I’m even unsure what my normal table value is
We can test the correlation using the fact that when the correlation is null then the statistic
$$ t = r \sqrt{\frac{n-2}{1-r^2}} $$
(where $r=\frac{\sigma_{AB}}{\sigma_A \sigma_B}$ is the sample correlation coefficient and $n$ is the sample size)
is approximately distributed as student's T with $n-2$ degrees of freedom if the samples are large enough.
You would need to plug in your value of $n$, and find the critical $t$ value that would give you the desired significance.
Then you could invert the formula as $r = - \frac{t}{\sqrt{n-2+t^2}}$, and if your estimated $r$ is below that threshold then the null hypothesis of no correlation would be falsified.
For example, if $n=42$ then the critical $t$ value for $n-2 = 40$ degrees of freedom and $10\%$ significance level would be $1.684$ according to this table.
Thus the critical $r$ value would be $r = - \frac{t}{\sqrt{n-2+t^2}} = - \frac{1.684}{\sqrt{40+1.684^2}} = - 0.03931$. Since the estimated $r$-value is $r = -4.125 < - 0.03931$ we would reject the null hypothesis and confirm that the assets are negatively correlated.