Page History

• Faculty
• Field Description
• The Major in Mathematics
  • Program Learning Goals
  • Major Course Requirements
• The Major in Actuarial Science
  
  • Program Learning Goals
  • Major Course Requirements
• The Major in Financial Mathematics
  • Program Learning Goals
  • Major Course Requirements
• The Minor and Courses
• Courses in Mathematics (MTH)
• Department of Mathematics Web Site

Field Description

Mathematics has been described as the queen of all sciences. Understanding mathematics enables one to explain and analyze not only science and nature but almost all disciplines from archeology to zoology. Most recently, mathematics has become an indispensable tool in finance and other business related areas. To ensure that mathematics is available for students with varied backgrounds and different professional goals, the department offers courses at all levels. Advanced courses are designed to be taken by mathematics and actuarial science majors and those in related fields.

The Majors

• Mathematics
• Actuarial Science
• Financial Mathematics

Mathematics

The major in mathematics is designed to enable the student to enter the marketplace (industrial or educational) or to pursue further studies in mathematics or allied fields at the graduate level. Interested students should meet with an advisor in the Department of Mathematics as early as possible for assistance in formulating an appropriate course of study.

A student majoring in mathematics cannot declare a second major in statistics.

Program Learning Goals

Upon completion of the two-year sequence in calculus, students will be able to:

1. Differentiate and integrate a wide variety of algebraic and transcendental functions;
2. Apply such knowledge to a variety of verbal problems in economics, physics, and related rates;
3. Develop the Taylor series expansion for functions and compute the error terms occasioned by truncation of the series to a finite number of terms;
4. Use geometric vectors to prove theorems;
5. Deal with functions and surfaces (areas, volumes) in 3-dimensional space;
6. Use other (than Cartesian) coordinate systems, especially polar coordinates, in the study of graphs and, by change of variable, to facilitate certain integrations;
7. Follow subtle lines of reasoning, detect breaches of logic and validity, write sustained logical arguments;
8. List several approaches to the real number system, such as Dedekind cuts, the Bolzano–Weierstrass property, the nested-interval property, the existence of suprema and infima of bounded sets, the convergence of Cauchy sequences.

Upon completion of our courses in analysis beyond calculus, students will be able to:

1. Point out the analogies—the interplay and interconnections—between corresponding real-valued functions of a real variable and complex-valued functions of a complex variable;
2. Highlight some of the properties that follow from analyticity of functions on various domains;
3. Perform computations with complex numbers, evaluate contour integrals, evolve Laurent series of functions;
4. Show how metric spaces endowed with Euclidean and non-Euclidean metrics are particular examples of topological spaces;
5. Present properties of metrizable and nonmetrizable topological spaces as generalizations of properties that originate in the set of real numbers;
6. Explicate properties of connectedness and compactness in topological spaces.

Upon completion of our courses in algebra, students will be able to:

1. Trace the construction of the integral domain of rational integers and the fields of rational and complex numbers by successive refinements of, and additions to, the properties of a set;
2. Show how abstract initial conditions can be used to derive facts and features of a variety of algebraic structures;
3. Apply abstract algebra, which had its origins and motivation in number theory, back to number theory, to elucidate number-theoretic properties by placing them in a general (abstract) setting;
4. Prove theorems about groups, rings, fields, and other algebraic structures;
5. Account for the advantages of abstract formulations in mathematics;
6. Define the dimension of a vector space in terms of the (unique) number of vectors in a basis, accomplish basis-to-basis transformations, compute characteristic values and vectors, and enumerate some of the profound connections among the invertibility of matrices, systems of linear equations, determinants, linear independence, spanning sets and bases, rank, orthogonality.

Upon completion of our courses in geometry, students will be able to:

1. Discourse with authority on the impact and role of initial assumptions (postulates) on the structure of a geometrical system, mainly with reference to Lobachevskian and Riemannian geometry;
2. Cite facts (theorems) of Euclidean geometry that depend on the parallel postulate and hence are absent in neutral geometry;
3. Provide examples of finite and infinite incidence geometries and their isomorphisms;
4. Trace some of the history of geometry, especially as it concerns attempts to prove Euclid’s parallel axiom as a consequence of the other axioms;
5. Speak on difficulties encountered in endeavoring to establish the physical validity of a geometric theory – which the actual geometry of the universe is, given the homogeneity of space with respect to the parallel postulate; and of course
6. Compose mathematically correct proofs of geometric statements.

Upon completion of our other classes, students will be able to:

1. Solve differential equations using series expansions, Laplace transforms, and other standard techniques [differential equations];
2. Enunciate properties and applications of Eulerian, Hamiltonian, connected, cyclic, acyclic, planar, traversable, and other types of graphs [graph theory];
3. Approach combinatorics problems from two points of view which, when united, lead to solutions of problems in combinatorics using permutations, combinations, partitions, mathematical induction [combinatorics];
4. Trace the historical development of mathematics from antiquity to the present, including contributions to that cumulative subject from various cultures and countries [history of mathematics];
5. Stipulate properties and characteristics of whole numbers – divisibility, the division algorithm, Diophantine equations, unique factorization, the integers modulo n, Fermat’s theorem, Euler’s theorem, representation in different bases [theory of numbers];
6. Write computer programs in a high-level programming language to solve mathematical problems and verify their correctness, and invoke techniques of object-oriented programming to represent objects and their behaviors in code [algorithms, computers, and programming class].

Major Course Requirements

Actuarial Science

The field of actuarial science applies mathematical principles and techniques to problems in the insurance industry. Progress in the field is generally based upon completion of examination given by the Society of Actuaries. The Baruch College major is designed to prepare students to pass the P, FM, IFM (formerly MFE), LTAM (formerly MLC), and STAM (formerly C) exams offered by the Society of Actuaries. Classes are offered which fulfill current VEE (Validation by Educational Experience) requirements in economics, finance, and statistics. Students interested in this highly structured program are urged to meet with an advisor in the Department of Mathematics as early as possible for assistance in formulating an appropriate course of study.

A student majoring in actuarial science cannot minor in mathematics or declare a second major in statistics.

Program Learning Goals

Upon completion of the required core courses in actuarial mathematics, students will be able to:

1. Examine and solve problems dealing with discrete and continuous probability distributions.
2. Recognize when a specific probability distribution is applicable.
3. Determine an appropriate distribution to model a specific scenario in a risk-management context.
4. Compute equivalent interest and discount rates (both nominal and effective).
5. Write an equation of value for a set of cash flows. Estimate effective compound yield rates for the set of cash flows using a simple interest approximation.
6. Calculate present and future values for various types of annuities and perpetuities such as annuities-due, perpetuities-due, annuities-immediate, perpetuities-immediate, arithmetic or geometric annuities, and non-level annuities.
7. Determine the payment amount for a loan with a specific repayment structure.
8. Find the outstanding balance immediately after a payment on a loan.
9. Calculate the amount of principal and amount of interest in a payment for an amortized loan.
10. Perform an amortization on a coupon bond.
11. Compute yield rates for a callable bond at each of the call dates.
12. Calculate values, duration, and convexity for both zero-coupon bonds and coupon bonds.
13. Use first-order approximation methods based on duration to estimate the change in present value of a portfolio based on changes in interest rates.
14. Construct an investment portfolio to immunize a set of liability cash flows.
15. Calculate minimal variance portfolios with and without constraints.
16. Perform pricing and hedging of European and American type derivative securities in the context of one- and multi-period binomial models.
17. Construct arguments based on the no-arbitrage principle, and devise arbitrage strategies when this principle is violated.
18. Price European derivative securities in the context of the Black-Scholes model.
19. Derive a put-call parity relation, and use it for pricing and hedging.

Upon completion of elective courses in actuarial mathematics, students will be able to:

1. Find closed-form solutions to ordinary and partial differential equations derived from financial models.
2. Derive the celebrated Black-Scholes formula by solving the Black-Scholes PDE.
3. Compute values of European, American, and exotic options using finite difference numerical methods.
4. Download options market data and use it as input for codes generating implied volatility surfaces.
5. Describe and classify different kinds of short-term insurance coverage.
6. Explain the role of rating factors and exposure in pricing short-term insurance.
7. Create new families of distributions by applying the technique of multiplication by a constant, raising to a power, exponentiation, or mixing.
8. Calculate various measures of tail weight and interpret the results to compare tail weights.
9. Calculate risk measures, including Value at Risk and Tail Value at Risk, and explain their properties, uses, and limitations.
10. Calculate premiums using the pure premium and loss ratio methods.
11. Use Maximum Likelihood Estimation and Bayesian Estimation to estimate parameters for severity, frequency, and aggregate distributions for individual, grouped, truncated, or censored data.
12. Use hypothesis tests (e.g., Chi-square goodness-of-fit, Kolmogorov-Smirnov, and likelihood ratio tests) and score-based approaches (e.g., the Schwarz-Bayesian Criterion, the Bayesian Information Criterion, and the Akaike Information Criterion) to perform model selection on a collection of data.
13. Apply credibility models such as the Buhlmann and Buhlmann-Straub models, and explicate the relationship between these models and Bayesian models.
14. Explain the concepts of random sampling, statistical inference and sampling distribution.
15. State and use basic sampling distributions.
16. Describe and apply the main methods of estimation including matching moments, percentile matching, and maximum likelihood.
17. Describe and apply the main properties of estimators including bias, variance, mean squared error, consistency, efficiency, and UMVUE.
18. Construct confidence intervals for unknown parameters, including the mean, differences of two means, variances, and proportions.
19. Analyze data using basic statistical inference tools like confidence intervals and hypothesis testing for the population mean.
20. Apply tools such as analysis of variance, tests of significance, residual analysis, model selection, and predication in both the simple and multiple regression models.
21. Demonstrate proficiency in some basic programming skills in SAS and the time-series Forecasting interactive system. Perform time-series analysis using these tools.
22. Identify patterns in data such as trend or seasonality. Incorporate these patterns into the time-series analysis of the data, and perform error analysis of the data.
23. Explain K-means and hierarchical clustering, and interpret the results of a cluster analysis.

Common Objectives – Actuarial and Financial Mathematics

Upon completion of the required finance courses for the actuarial science and financial mathematics majors, students will be able to:

1. Expound on the governance of corporations.
2. Outline the operation of financial markets and institutions.
3. Measure corporate performance.
4. Analyze risk and return. Determine the opportunity cost of capital.
5. Perform capital budgeting using various techniques.
6. Compute the present and future values of investments with multiple cash flows.
7. Describe the mechanisms that cause fluctuation of bond yields.
8. Calculate internal rate of return.
9. Perform and interpret scenario analysis for a proposed investment.
10. Calculate financial break-even points.
11. Determine relevant cash flows for a proposed project.
12. Determine a firm’s overall cost of capital.

Major Course Requirements

Financial Mathematics

This major is first and foremost a course of study in mathematics, with a focus on the computational tools and techniques needed to thrive in the financial engineering industry. In today’s specialized world, a sophisticated level of mathematical understanding is an essential competitive edge. As this program includes courses in Economics and Finance, students who would usually not consider a traditional mathematics major will find this program especially attractive. Interested students are urged to contact the Department of Mathematics as early as possible.  The student will be assigned an advisor who will aid in formulating an appropriate course of study.

A student majoring in financial mathematics cannot minor in mathematics or declare a second major in statistics.

Program Learning Goals

Upon completion of the major in Financial Mathematics, students will be able to:

1. Perform linear algebraic calculations such as matrix multiplication and inversion, solving systems of linear equations, Gram-Schmidt orthogonalization, Cholesky decomposition, computation of eigenvalues and eigenvectors.
2. Obtain exact and numerical solutions to differential equations arising in finance such as the Black-Scholes model and its corresponding partial differential equation.
3. Compute implied asset price volatilities for European and American options from options market data.
4. Compute empirical volatilities from asset price time series using GARCH-type models.
5. Apply the fundamental notions of probability theory – including continuous and discrete random variables, expected value and variance, conditional expectation, multivariate distributions, the law of large numbers, the central limit theorem, and moment-generating functions – to settings in finance where randomness arises, such as in the modelling of asset prices and interest rates.
6. Apply the basic properties of martingales.
7. Calculate minimum variance portfolios in a Markowitz and CAPM setting.
8. Calculate call and put stock option values using a binomial model.
9. Calculate call and put option values using the Black-Scholes model.
10. Compute expectation for random variables and probabilities of events pertaining to Brownian motion.
11. Compute expectations of functions of Ito processes using the Ito formula.
12. Apply stochastic calculus to financial situations.
13. Apply the theory of Markov chains to appropriate settings. Examples include: the computation of invariant distributions, the implementation of the Hastings-Metropolis algorithm, and Gibbs sampling.
14. Apply the theory of arrival processes to settings such as corporate default models.
15. Apply the theory of Brownian motion and related continuous-time stochastic processes such as the Ornstein-Uhlenbeck process to model the evolution of correlated asset values over time as well as the evolution of the Treasury yield curve over time.
16. Use tools of statistical inference in the context of financial data. These tools include Bayesian estimation, maximum likelihood estimation, multiple regression analysis, confidence intervals, the t- and F-distributions for determining statistical significance, and analysis of variance.
17. Implement Black-Karasinski and Hull-White and related lattice-based interest rate models to value fixed-income derivative securities like options on bonds, interest rate swaps, caps, floors, and swaptions.
18. Build simulative interest-rate models based on continuous-time stochastic processes to value fixed-income derivative securities.
19. Build elementary computer programs in Python and C++ to simulate stochastic processes.
20. Use these models to calculate a fixed-income security’s duration, convexity, and key-rate duration for hedging purposes.

Common Objectives – Actuarial and Financial Mathematics

Upon completion of the required finance courses for the actuarial science and financial mathematics majors, students will be able to:

1. Expound on the governance of corporations.
2. Outline the operation of financial markets and institutions.
3. Measure corporate performance.
4. Analyze risk and return. Determine the opportunity cost of capital.
5. Perform capital budgeting using various techniques.
6. Compute the present and future values of investments with multiple cash flows.
7. Describe the mechanisms that cause fluctuation of bond yields.
8. Calculate internal rate of return.
9. Perform and interpret scenario analysis for a proposed investment.
10. Calculate financial break-even points.
11. Determine relevant cash flows for a proposed project.
12. Determine a firm’s overall cost of capital.

Major Course Requirements

The Minor

The minor in mathematics provides students with a background in the various theories and uses of mathematics. The minor requires the completion of MTH 3006, MTH 3010, MTH 3020, MTH 3030, or MTH 3050, and any other 3- or 4- or 5-credit mathematics course numbered 3000 or higher with the exception of

This minor is not open to students who are pursuing a major in statistics.

Courses in Mathematics (MTH)