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18.642 | Fall 2024 | Undergraduate

# Topics in Mathematics with Applications in Finance

### Lecture 8

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Video description: This lecture provides a comprehensive overview of linear regression modeling, focusing on ordinary least squares (OLS) estimation, its mathematical formulation, and statistical properties. It discusses the derivation of the least squares estimator, the interpretation of the Hat matrix as a projection, the distributional assumptions under the normal linear model, inference using t- and F-tests, and model diagnostics including residual analysis and influence measures, concluding with extensions to generalized least squares for correlated errors.

Regression Analysis Slides (PDF)

Suggested Readings:

- Fama, Eugene F. and Kenneth R. French, “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33, no. 1 (February 1993): 3–56.
- Fama, Eugene F., and Kenneth R. French. “The Capital Asset Pricing Model: Theory and Evidence.” The Journal of Economic Perspectives 18, no. 3 (2004): 25–46. http://www.jstor.org/stable/3216805
- Lintner, John. “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets.” The Review of Economics and Statistics 47, no. 1 (1965): 13–37. https://doi.org/10.2307/1924119
- Lintner, John. “Security Prices, Risk, and Maximal Gains From Diversification.” The Journal of Finance 20, no. 4 (1965): 587–615. https://doi.org/10.2307/2977249
- Perold, André, F. “The Capital Asset Pricing Model.” Journal of Economic Perspectives 18, no.3 (2004): 3–24. https://doi.org/10.1257/0895330042162340
- Sharpe, William F. “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk.” The Journal of Finance 19, no. 3. (September 1964): 425–442. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x

### Lecture 9

Guest lecture with Stefan Andreev, Two Sigma Investments

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Video description: This lecture features a detailed guest lecture by Stefan Andreev on Principal Component Analysis (PCA) and its critical applications in quantitative finance, especially in modeling the U.S. bond market. The lecture explains PCA’s mathematical foundations, practical challenges, and how it helps identify key factors driving yield curve dynamics, enabling portfolio construction and risk management in highly correlated financial markets.

Principal Component Analysis Slides (PDF)

FedDataAnalysis Jupyter Notebook

FedBondYieldData_20230911.csv

## Course Info

##### Instructors

- Dr. Peter Kempthorne
- Dr. Vasily Strela
- Dr. Jake Xia

##### Departments

- Mathematics

##### As Taught In

Fall 2024

##### Level

Undergraduate

##### Topics

- Business
  - Finance
- Mathematics
  - Applied Mathematics
  - Probability and Statistics

##### Learning Resource Types

notes Lecture Notes
theaters Lecture Videos
assignment Problem Sets
assignment Activity Assignments
group_work Projects
Readings

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