Module Handbook

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Module MAT-61-12A-M-7

Specialization Actuarial and Financial Mathematics (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-61-12A-M-7 Specialization Actuarial and Financial Mathematics 9.0 CP (270 h)


CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. in WiSe
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Lecturers of the department Mathematics
Area of study [MAT-STO] Stochastics/Statistics/Financial Mathematics
Reference course of study [MAT-88.B84-SG] M.Sc. Actuarial and Financial Mathematics
Livecycle-State [NORM] Active


Each winter semester at least one of the following courses will be offered:[MAT-61-15-K-7], [MAT-61-20-K-7], [MAT-61-30-K-7] or [MAT-61-31-K-7].

The lecture offer for the specialization module planned for the following three semesters will be made available on the website of the master's programme "Actuarial and Financial Mathematics".

Module Part #A (Obligatory, 4.5 LP)

Type/SWS Course Number Title Choice in
Presence-Time /
SL SL is
required for exa.
PL CP Sem.
2V MAT-61-12-K-7
Interest Rate Theory
P 28 h 107 h - - PL1 4.5 WiSe
  • About [MAT-61-12-K-7]: Title: "Interest Rate Theory"; Presence-Time: 28 h; Self-Study: 107 h

Module Part #B (Obligation to choose, 4.5 LP)

One course of the list has to be chosen.
Type/SWS Course Number Title Choice in
Presence-Time /
SL SL is
required for exa.
PL CP Sem.
2V MAT-61-15-K-7
Continuous-time Portfolio Optimization
WP 28 h 107 h - - PL1 4.5 irreg.
2V MAT-61-20-K-7
Markov Switching Models and their Applications in Finance
WP 28 h 107 h - - PL1 4.5 irreg.
2V MAT-61-30-K-7
Risk Measures with Applications to Finance and Insurance
WP 28 h 107 h - - PL1 4.5 irreg.
2V MAT-61-31-K-7
Health, and Pension Insurance Mathematics
WP 28 h 107 h - - PL1 4.5 irreg.
  • About [MAT-61-15-K-7]: Title: "Continuous-time Portfolio Optimization"; Presence-Time: 28 h; Self-Study: 107 h
  • About [MAT-61-20-K-7]: Title: "Markov Switching Models and their Applications in Finance"; Presence-Time: 28 h; Self-Study: 107 h
  • About [MAT-61-30-K-7]: Title: "Risk Measures with Applications to Finance and Insurance"; Presence-Time: 28 h; Self-Study: 107 h
  • About [MAT-61-31-K-7]: Title: "Health, and Pension Insurance Mathematics"; Presence-Time: 28 h; Self-Study: 107 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 88150 ("Specialization Actuarial and Financial Mathematics")

Evaluation of grades

The grade of the module examination is also the module grade.


  • Basics of interest modelling (Bonds and linear products, swaps, caps and floors, bond options, rate of interest options, interest rate term structure curve, interest rates (short rates and forward rates)),
  • Heath–Jarrow–Morton framework (simple example: Ho-Lee model, general HJM drift condition, one- and multidimensional modelling),
  • Short rate models (general one factor-modelling, term structure equation, affine modelling of interest rate structure, Vasicek-, Cox-Ingersoll-Ross- and further models, option pricing model, model calibration),
  • Defaultable bonds (Merton model).
  • Introduction to portfolio optimization (problem statement),
  • Continuous-time portfolio problem: expected utility approach,
  • Martingale method for complete markets,
  • Stochastic control approach (HJB equation, verification theorems),
  • Portfolio-Optimization with restrictions (e.g. risk constraints, transaction costs),
  • Comparison with mean-variance analysis (Markowitz),
  • Portfolio optimization with financial derivatives,
  • Alternative methods.
  • Discrete-time and continuous-time Markov chains,
  • Hidden Markov models in discrete time,
  • Continuous time Markov switching models,
  • Parameter estimation and filtering,
  • Modelling financial asset prices,
  • Econometric properties of financial time series and model extensions,
  • Applications to portfolio optimization.
  • Preferences and expected utility,
  • Axiomatic introduction of risk measures,
  • Robust representation of convex and coherent risk measures,
  • Examples: Value at Risk, Average Value at Risk, Short case, worst case,
  • Extensions: Semi Dynamic, dynamic, distribution-free risk measures,
  • Estimation of risk measures,
  • Rating systems:
    • Score-based ratings,
    • Utility based ratings of financial products,
    • Risk-classes for insurance products,
  • Credit risk: Structural models and reduced form models,
  • Applications:
    • Risk-based insurance premiums,
    • Portfolio optimization under risk constraints,
    • Credit derivatives.
This lecture is based on the module [MAT-61-18-M-7]. It deals with dynamic models in life insurance mathematics and with life insurance products which allow for investment in the financial market. In addition, mathematical models and specific problems of pension plans and health insurance are addressed. The following topics are covered:

Life Insurance Mathematics:

  • Dynamic models (Markov chain, continuous time),
  • Stochastic interest rates,
  • Products with investment in the financial market and guarantee funds,
  • Market consistent valuation.

Actuarial Mathematics for Pension Plans:

  • State diagrams and benefits,
  • Neuburger's model,
  • Estimation of decrement rates,
  • Premiums and actuarial reserves.

Health Insurance Mathematics:

  • Premium principles,
  • Reserves for increasing age and contract changes,
  • Profit participation and premium reductions ,
  • Risk assessment.

Competencies / intended learning achievements

Upon successful completion of the module the students understand the fundamentals of the theory of interest rate products and modelling of interest rate markets. They are able to understand the deep relations in the theory of interest rate modelling and they know to critically apply analytical valuation techniques for interest rate products.

In addition, they have acquired in-depth knowledge of specific concepts and methods in other areas of financial and actuarial mathematics, such as methods for solving stochastic control problems (stochastic control, duality approach), Markov switching models, the theory of risk measurements or advanced topics of life insurance mathematics. They have learned to apply these methods and are able to critically assess the implementation and application of the theoretical results.

They have gained a precise and independent handling of terms, propositions and methods of the lecture. They understand proofs presented in the lecture and are able to reproduce and explain them. They can in particular outline the conditions and assumptions that are necessary for the validity of the statements and how these are to be interpreted in the context of actuarial and financial mathematics.


  • T. Björk: Arbitrage Theory in Continuous Time,
  • D. Brigo, F. Mercurio: Interest Rate Models – Theory and Practice,
  • N. Branger, C. Schlag: Zinsderivate – Modelle und Bewertung.
  • I. Karatzas, S.E. Shreve: Methods of Mathematical Finance,
  • R. Korn: Optimal Portfolios,
  • R. Korn, E. Korn: Option Pricing and Portfolio Optimization - Modern Methods of Financial Mathematics,
  • H. Pham: Continuous-time Stochastic Control and Optimization with Financial Applications.
A. Bain, D. Crisan: Fundamentals of Stochastic Filtering,

O. Cappé, E. Moulines, T. Rydén: Inferences in Hidden Mrkov Models,

R.J. Elliott, L. Aggoun, J.B. Moore: Hidden Markov Models – Estimation and Control,

S. Frühwirth-Schnatter: Finite Mixture and Markov Switching Models,

J.R. Norris: Markov Chains,

R.S. Tsay: Analysis of Financial Time Series.

H. Föllmer, A. Schied: Stochastic Finance: An Introduction in Discrete Time,

L. Rüschendorf: Mathematical Risk Analysis.

  • M. Koller: Stochastic Models in Life Insurance,
  • H. Milbrodt, M. Helbig: Mathematische Methoden der Personenversicherung.

Requirements for attendance (informal)

Further requirements depending on the choice of the further specialized course, see the respective course description.


Requirements for attendance (formal)


References to Module / Module Number [MAT-61-12A-M-7]

Course of Study Section Choice/Obligation
[MAT-88.B84-SG] M.Sc. Actuarial and Financial Mathematics Specialisation Actuarial and Financial Mathematics [P] Compulsory