Table of Contents
Markowitz Portfolio
Modern portfolio theory assumes that investors can construct optimal portfolios offering the maximum possible expected return for a given level of risk. It is assumed that historical mean performance is the best estimator for future (expected) performance. The efficient portfolio is the portfolio that has
- highest reward for a given level of risk OR
- lowest level of risk for a given return.
For this project (03-markowitz_optimized.ipynb), we pick
5 stocks (AAPL, WMT, TSLA, GE, AMZN, DB) and download their historical
stock data (2012-2023) using yfinance. We collect the
closing prices into a dataframe and calculate the log daily returns for
each stock. A given array of weights (summing to 1) determines a portfolio. We
denote by \(w_i\) the weight of the
i-th stock, \(r_i\) the return of the
i-th stock and \(\mu_i\) the expected
return (mean). Then the expected return of the
portfolio is given by
\[ \begin{align*} \mu_{PF} = \mathbb E\left[\sum_i w_i r_i\right]& = \sum_i w_i \mathbb E(r_i)\\ &=\sum_i w_i\mu_i \\ &=\underline{w}^T\underline{\mu} \end{align*} \]
The expected portfolio variance (risk) is given by
\[ \begin{align*}\sigma^2_{PF} = \mathbb{E}[(r_i-\mu_i)^2] &= \sum_i\sum_j w_i w_j \sigma_{ij}\\ &=\underline{w}^T\cdot M\cdot \underline{w},\end{align*} \] where \(M\) is the covariance matrix (which contains the relationship between all the stocks in the portfolio) and \(w_i\) are the weights as above.
To find the best portfolio, we follow a Monte-Carlo simulation
approach. First, we generate several (10,000) random weights to get
random portfolios. So, we define a generate_portfolios
function that retruns an array of returns and variances for each
portfolio. Using this, we generate an Efficient Frontier plot
color-encoded on the Sharpe ratio (which is a measure of excess return
(risk premium) per unit of standard deviation in an asset).
Finally, to get the optimal portfolio with the highest Sharpe ratio
(practically minimizing risk while maximizing expected returns), we
first define a function called statistics which returns an
array of portfolio return, volatility and the ratio return/volatility
for a given set of weights and log daily returns. Then, we use
scipy.optimize on the negative of the Sharpe ratio output
of statistics (as the optimization package only finds
minimum and we want a max). The function optimize_portfolio
finally returns the optimum weights. The star below denotes the optimal
portfolio. 
The output is [0.332 0.322 0.178 0. 0.169 0. ]. So,
based on the collected historical data, to achieve the best Sharpe ratio
of 0.914, we should have 36% of stocks in AAPL, 32.2% in WMT, 17.8% in
TSLA, 16.9% in AMZN, and none in GE and DB. But since the best Sharpe
ratio is less 1, it is best to look for a different set of stocks.
Natural gas storage futures
This project is about commodity storage contracts. It was part of the JPMorgan Chase & Co. Quantitative Research virtual “internship”. The ultimate goal was to prepare a prototype storage contract for natural gas. To price the contract, we are given historical (monthly) data (from JPM&C Forage). We use it to estimate the future gas price at any date.
Before we make a forecasting model, time series analysis on the historical data is done in the draft notebook. From the time series plots, the data shows clear trend and seaonality, but we nonetheless check for stationarity of the data by performing an Augmented Dickey-Fuller test. We also check for autocorrelation and seasonality.
In this notebook,
we do seasonal_decompose from the statsmodel
package to conclude that our time series is
multiplicative with trend and seasonality. So, we use
Holt-Winters Exponential Smoothing for forecasting
prices. We get an NRMSE of 0.0189. The notebook ends with a price
forecast function. It uses interpolated price forecast data to return
the estimated price at any given date (in the past and up to 1 year in
the future).
Lastly, we devise a prototype contract pricing model in this notebook. The input parameters involved were injection dates, withdrawal dates, prices on those dates, rate at which gas can be injected/withdrawn, maximum volume that can be stored, and storage costs. With certain assumptions (eg. no transport delay, and interest rates are zero) and the predicted prices from above, we write a function that takes these inputs and gives back the value of the contract.
Probability of Default and FICO scores
The second part of the Forage project was to estimate the probability
of default for a borrower given a loan dataset
(02-loan_data.csv).
Looking at the histograms of the features, it appears that they follow a Gaussian (normal) distribution and we check if each feature, given a specific class of the target variable (default=0,1), follows a Gaussian distribution. As this turns out to be the case, we train a Gaussian Naive Bayes and we achieve a 97.8% accuracy on a test set.
Next, in the FICO brackets notebook, we give a detailed Python code to strategically bucket customers with various FICO scores (i.e., apply quantization). Context: we are told that the hypothetical architecture we are working in requires categorical data. So, we need to create a rating map that maps the FICO score of the borrowers to a rating where a lower rating signifies a better credit score. We end up deploying a log-likelihood optimization.