STAD70 Statistics & Finance II
March 14, 2022. Time allowed: 110 minutes.
Academic integrity statement. Academic integrity is a fundamental value of learning and schol- arship at the UofT. Participating honestly, respectfully, responsibly, and fairly in this academic community ensures that your UofT degree is valued and respected as a true signifies of your individual academic achievement. By submitting this midterm you agree with the following: You will not commit academic misconduct, and am aware of the penalties that may be imposed if you commit an academic offence.
There are 2 parts in the midterm. Total points: 100
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Part I. Theoretical.
Write your answers on a piece of paper, then take pictures/scan to submit. You may also use a tablet and upload directly.
1. (10 points) Consider two independent random variables X and Y . Assume E[X] = E[Y ] = 0, Var(X) = σX2 and Var(Y ) = σY2 are strictly positive, and the skewness of both X and Y is zero. Let their kurtosis be respectively KX and KY . Derive an expression for the kurtosis of X + Y .
2. (20 points) Consider the single period mean-variance framework, where the covariance matrix of the asset return is Σ. Suppose we want to minimize the variance of the portfolio, but we also want that the optimized portfolio is not too far away from the equal-weighted portfolio n1 1, where n is the number of assets and 1 = (1, . . . , 1)⊤ ∈ Rn. One formulation of this problem is
1⊤ λ12 min w Σw+ w− 1 ,
w∈Rn:w⊤1=1 2 2 n
where λ > 0 is a fixed tuning parameter and ∥x∥2 = x⊤x. Solve this problem using Langrange multiplier, and argue that when λ → ∞ the solution tends to the equal-weighted portfolio n1 1.
3. (15 points) Consider an n × n covariance matrix of the form Σ= na11⊤ +σ2I,
where a, σ2 > 0.
(a) (5 points) Verify that Σ is strictly positive definite.
(b) (5 points) Show that 1 = (1,…,1)⊤ is an eigenvector, and find the corresponding eigenvalue.
(c) (5 points) You are given that the direction 1 corresponds to the first principal com- ponent (for a random vector with covariance matrix Σ). Find the proportion of total variance explained by the first principal component.
Part II. Data analysis.
Submit a single R file (containing all codes) and a separate text file (e.g. MS word, markdown, or plain text file) which contains your answers. You don’t have to export the graphs as images. Comment your codes. The R file must be self-contained. In particular, it must contain commands to load the necessary packages.
To begin, load the data in the file midterm data.RData by using the code load(“midterm data.RData”).
4. (15 points) The object R q4 contains a sequence of hypothetical asset returns (with length 500).
(a) (6 points) Report the following summary statistics: sample mean, standard deviation, sknewness and kurthosis. Plot a density estimate and compare it with the density of a normal distribution whose mean and standard deviation match those of the data.
(c) (9 points) Can the series be adequately modelled by a Gaussian white noise process? Plot the sample acf, perform at least two relevant statistical tests and report their p-values. Comment on the results.
5. (20 points) The object R q5 contains the weekly returns of S&P500 (first column) and n = 28 US stocks from January 2005 to December 2021.
(a) (10 points) Divide the sample period equally into two parts. For each half, fit a single index model (where the market portfolio is the S&P500, and without the risk-free return). Produce the following plots:
– Plot of betas (y-axis) versus the tickers (x-axis), for both periods. Use red for the first period and blue for the second period.
– Scatterplot of the betas in the second period against the betas in the first period. Report the correlation between the two variables.
(b) (10 points) For each period, compute the estimated covariance matrix based on the single index model. Compare the eigenvalues of the two matrices and comment on the stationarity of the covariance matrix over time.
6. (20 points) Consider the data set R q5 in Problem 5, but without using the market returns, i.e., use
R_q6 <- R_q5[, 2:29]
(a) (10 points) Perform PCA to the data and report the smallest number of principal components such that the proportion of explained total variance is higher than 80%.
(b) (10 points) Using the first principal component, consider the 1-dimensional projec- tion of the data. Compare it with the market return R q5[, 1] and report your observations.
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