Quantitative Techniques
- The plot
In the late 2008 during the financial crisis, the exchange rate of the Australian dollar against the Japanese Yen was very low and was very high against the UK pound.
The US Treasury bill of rate during this was on downward trend and it was approaching zero.
- The returns for each asset have been calculated in the excel using the formulae
where Pt is the rate at time t
- The histogram for each of the returns is shown below
- The histograms on the returns from the exchange rate for the Australian dollar against the Japanese Yen
The mean=0.00237694
Median=0
Variance=1.09710006
Standard deviation=1.04742544
Skewness=0.37218026
Kurtosis=7.21318797
Looking at the coefficient of kurtosis for these data, the curve is more peaked than the normal curve, what is called leptokurtic.
- The histograms on the returns from the exchange rate for the Australian dollar against the UKP
The mean= -0.0107221
Median=0
Variance= 0.53947729
Standard deviation= 0.73449118
Skewness= 0.05875106
Kurtosis= 4.49591754
Looking the coefficient of kurtosis for these data the curve is more peaked than the normal curve, what is called leptokurtic.
- the returns from the exchange rate for the Australian dollar against the USD
| meanR3= | -0.0069154 |
| MEDIANR3= | 0 |
| MODE= | 0 |
| VARIANCE= | 0.57776257 |
| SD= | 0.76010695 |
| SKEWNESS= | 0.28171271 |
| KURTOSIS= | 8.27673924 |
Looking the coefficient of kurtosis for these data the curve is more peaked than the normal curve, what is called leptokurtic.
- The hypothesis is stated us
The test statistics is given by
reject the null hypothesis when
For the means of the returns for the Australian dollar against other currencies, since the t-value is less than the critical value, we reject the null hypothesis at 5% level of significance and conclude that the means are statistically significance from zero.
- The hypothesis is stated as :
The test statistics is given by
reject the null hypothesis when
Since the t- values are less than the critical values, we reject the null hypothesis at 5% level of significance and conclude that the means for the returns are significantly different from each other.
- The correlation matrix for the returns is given by
- The assumption for independence is not realistic because the returns are highly correlated.
The appropriate test statistic is a chi square test, which is used to test for independence.
- If were to choose a maximum of two portfolio, I will choose the ones with the highest expected returns. That is Australian dollar against the USD, and Australian dollar against the Japanese Yen.
The covariance matrix is given by
The inverse of covariance matrix is
Now we multiply the c and the excess expected return
*
The weights are
W1=-0.0593
W2=1.063
The optimum expected return will be
=0.002014
