QUESTION
Question 1 (1 + 1 + 1 + 1 = 5 marks)
For the variable Value find the a) mean,
b) median,
c) range,
d) standard deviation and
e) variance.
Question 2 (4 marks)
Estimate with 95% confidence the average age of a house.
Question 3 (5 marks)
Using an appropriate hypothesis test, at a 5% level of significance, can you conclude that there is any difference between the average value between houses that have a fireplace (Fireplace = 1) and houses that don’t have a fireplace (Fireplace = 0)?
Question 4 (1 + 1 + 4 = 6 marks)
(a) Estimate the regression equation to predict the Value (y) from the number of Rooms (x).
(b) Determine the coefficient of determination and interpret its value.
Does the data provide evidence that the slope of the equation from part a) is positive? Perform an appropriate hypothesis test using a 5% level of significance
SOLUTION
Question 1: (i). For the given data, the Mean of the variable “Value” is 191.2
(ii). For the given data set the Median of the variable “Value” is 188.3
(iii). For the given data set, the Standard Deviation of the variable “Value” is 33.44
(iv). For the given data set, the Variance of the variable “Value” is 1118.29
(v). For the given data set, the Range of the variable “Value” is 205.
Question 2: The Mean-Variance analysis is suitable when returns are distributed normally or the preferences of the investors are quadratic. In the applications that are real, basically returns are not normally allocated and utilities are non-quadratic. The practice of using mean-variance analysis (Levy and Markowitz, 1979) is justified by reflecting that this analysis can be observed as a second order approximation of Taylor-series of standard utility functions, including the exponential utility and the power utility. Hence, the dependability of this analysis is jointly based on the degree of non-normality of the returns data and the nature of the function of non-quadratic utility. The estimates of second order are extremely correlated to exact values of power and exponential utility functions over a broad range of limited values for mutual funds. This result was extended by Hlawitschka (1994) to explain that the ranking of funds of mean-variance is highly correlated to the ranking on the basis of the true utility function, and that approximations of third or even higher order do not essentially improve the rank correlation. There for the estimate of the given data comes out to 181.7
Question 3: This type of analysis simplifies if it is expected that there is one asset that has zero risk, which means it would be a risk free asset, the usage of this theorem would be beneficial. Risk-free asset mean, that there is certain amount of return is fixed on this kind of asset, meaning expected return can never be zero or negative in case of choosing a risk-free asset.
Question 4 : (a) The Estimate the regression equation to predict the Value (y) from the number of Rooms (x) is 205
(b) The coefficient of determination is 265
(c) Under this division, the single period result is extended to a multi-period horizon. This segment is secondary and not important as one-period section. It is not compulsory to first compute the one-period mean-variance analysis to calculate multi-period analysis. One can directly switch to this segment. But, if the requirement is to extend the one-period mean-variance analysis result, both the sections are needed to be related. This is the basic section of analysis. In this segment, the standard mean-variance is developed for log returns for single period log. In this section, the result cannot be extended to a multi-period environment, even if it is required. To do this, one has to switch over to the next segments of mean-variance analysis.
REFERENCES
Markowitz, Harry M., 1952, Portfolio selection, Journal of Finance 7, 77-91
Levy, H., Markowitz, H.M., 1979, Approximating expected utility by a function of mean and variance, American Economic Review 69, 308–317.
Hlawitschka, W., 1994, The empirical nature of Taylor-series approximations to expected utility, American Economic Review 84, 713–719
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