STAD70 Statistics & Finance II
Assignment 3
Due date: March 25, 2022 (by 11:59pm). Late submissions will be penalized.
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Consider an n × n invertible square matrix A, and let u, v ∈ Rn be column vectors. Suppose that
1 + v⊤A−1v ̸= 0.
Show that the matrix A + uv⊤ is invertible and its inverse is
−1 A−1uv⊤A−1 A − 1+v⊤Au .
The point is that if the inverse of A is known (e.g. when A is an identity matrix), then the inverse of A + uv⊤ can be computed efficiently.
(b) As an extension, consider a k × k covariance matrix of the form Σ = ββ⊤ + D,
where β is k × m and D is a diagonal matrix (with positive entries). Show that Σ−1 = D−1 − D−1β(I + βD−1β)−1β⊤D−1.
Again the computation of the inverse is simplified.
Consider five stocks: 3M (MMM), (GS), AT&T (T), Boeing (BA) and Disney (DIS) . We use the S&500 (^GSPC) as a proxy of the market. Consider monthly returns of these assets from January 2000 to December 2022.
Given a training window, we estimate the covariance matrix in three ways:
(i) Σ1 is the sample covariance matrix.
(ii) Σ2 is the covariance matrix implied by the single index model.
(iii) Σ3 is the diagonal matrix whose (i,i)-entry is the sample variance of stock i.
We consider a rolling training window with 60 months. For each training window [t − 60 + 1,t] and each of the three methods, we estimate the global minimum variance portfolio and compute the return of the portfolio for period t. [For method (ii), derive and use an explicit formula of Σ2 using the result from Problem 1.] For each method, plot a density estimate of the realized returns and report the standard deviation of the return. Which method produces a portfolio with the smallest risk? Comment on your results.
Consider daily returns, over the year 2021, of the following (Dow Jones) stocks: 1
tickers_seq <- c("MMM", "AXP", "AMGN", "AAPL", "BA",
"CAT", "CVX", "CSCO", "KO", "DIS",
"DOW", "GS", "HD", "HON", "IBM",
"INTC", "JNJ", "JPM", "MCD", "MRK",
"MSFT", "NKE", "PG", "CRM", "TRV",
"UNH", "VZ", "V", "WBA", "WMT")
Consider the (daily) Fama-French factors that can be downloaded on the following website:
https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Research
(Be careful with the units.) Fit the Fama-French 3-factor model and comment carefully on the results, including its empirical performance over the simple single index model.
4. Consider the stocks in the S&P500 index:
https://en.wikipedia.org/wiki/List_of_S%26P_500_companies
By clicking the next to the GICS sector, you can sort the stocks according to the industries. Copy the table to e.g. Excel and prepare a table which contains 3 columns: ticker, name of the company, GICS sector; this will facilitate the computations in (a). We consider daily returns over the the sample period Jan 1, 2020 to Dec 31, 2021.
(a) Consider all stocks in the following industries: Utilities, Real Estate and Energy. Download the stock return data (explain what you do with missing data; one option is to drop the corresponding stocks). Order the stocks such that the vector of tickers correspond to
(utility stocks ... real estate stocks ... energy stocks)
Within each industry, group the stocks alphabetically according to the ticker. Plot a heat map for the empirical correlation matrix and comment on it.
(b) Perform PCA to the data (using the empirical covariance matrix). What is the minimum number of principal components so that over 70% of the total variance can be explained? Comment on the eigenvalues and whether the eigenvectors can be interpreted.
5. Consider the context of Theorem 9.13 (Characterization of the growth optimal portfolio) of the lecture notes. Let b∗ ∈ ∆n be such that
Argue that
E b∗⊤X ≤1, forallb∈∆n.
Xi =1,ifb∗i>0,
≤1, ifb∗i =0.
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