The questions below should be carried out in Python or R. You’ll need to use the Excel attached for question III. Submissions are evaluated based on correctness, analytical thinking, documentation, structure and clarity of the code/model, as well as the style. of the output and data visualization. Please state all your assumptions that you’d like us to consider.
I. Stock Return Regression
Load info.txt and performance.csv into Python or R. The file performance.csv has columns ID, Date, and Performance, which are monthly returns for each instrument identified by ID and each Date since January 1, 1990. The file info.txt has static information about each instrument identified by ID, including Name, Currency, and (Geographic) Focus.
a). Join input files performance.csv and info.txt, and produce a data frame with Date, Index1, Index2, Index3 and Stock A. The values are monthly returns.
b). Split the dataset into training and testing per 80/20 split. Use the training dataset, regress the Stock A returns against all 3 indices together, and interpret the results. Please set seed to make reproducible outputs.
c). What might be the potential problems with this regression, and how to identify the problems?
d). Use Lasso model to re-run the regression. Interpret the new coefficients, and explain why it’s different from the results from linear regression.
e). Use the fitted models from b) and d), predict Stock A returns in testing dataset, and plot the fitted values against the actual values. Which model provides a better fit, and what metric do you use to measure the fitness?
II. Optimal Investing
a). Download stock price data for 10+ equities from free online sources.
b). Use these stocks to compute a mean-variance efficient frontier. While mean-variance optimization is built into a lot of software packages, using a generic optimization package with the correct objective function and constraints is preferred here. How did you compute the expected return for each stock? The covariance matrix? What start/end date did you use for the return series? What frequency are the returns? Visualize the covariance matrix. Report all the expected returns and covariances as annualized quantities.
c). Explain what the efficient frontier means. Which portfolio would you invest in and why?
d). Add short-sale constraints and plot the constrained efficient frontier together with the one from
b). Explain any differences you see.
e). If you wanted to invest in 200 equities, explain any difficulties in computing the covariance matrix and how these can be overcome.
III. Case Study - See attached Excel
This is an open-ended case study. We ask that you make assumptions where you see fit and specify all assumptions made in the sheet. Don't stress it - we know you may not be an expert in private markets, so it's ok if it isn't perfect. We kindly ask that you just do what you can to the best of your ability.
IV. Correlated Contribution Times
Say you invest in a PE fund that will invest your commitment equally in 10 companies. Ignore management fees. Let τ1, . . ., τ10 denote the time when each company is purchased.
a). Assume τi ~iid exp(λ) for λ = 1/2.5. Simulate the contributions for one fund. Plot the cumulative contributions. What is the interpretation of λ?
b). Now simulate 100 funds and compute the average cumulative contribution curve. Plot it. What is this curve?
c). In practice, contribution times may be dependent. If I find one excellent company to buy, it’s more likely that I will soon find another excellent company to purchase. Find a model that will allow you to simulate dependent contribution times. Describe the model and the intuition behind the dependence. How did you set the parameters? Simulate one fund and plot it.
d). Simulate 100 funds and compute the average cumulative contribution curve. How does it differ from b)?
e). For the models in a) and c), compute the maximum contribution in any one calendar year across many simulations. Plot the distribution of the maximum. LPs need to hold cash in order to meet these contributions. Under which model does the client need to hold less cash?