The University of Southampton
Courses

# MATH2040 Financial Mathematics

## Module Overview

The compound interest model is developed in detail, and is applied to mortgage and commercial loans, to consumer credit transactions, to the valuation of securities, to the appraisal of investment projects, and to the measurement of investment performance. The investment and risk characteristics of the standard asset classes available for investment purposes are also considered, as is the issue of matching of assets to liabilities. The term structure of interest rates and its representation through yield curves is also covered. In addition, the module provides an introduction to simple stochastic interest rate models and to the no-arbitrage pricing of forward contracts

### Aims and Objectives

#### Module Aims

To provide students with a solid grounding in compound interest theory and experience of its application to the analysis of financial transactions.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Use a generalised cash-flow model to describe financial transactions
• Apply discounted cash flow techniques to the valuation of securities, including the effects of taxation.
• Demonstrate an understanding of the term structure of interest rates
• Describe the main investment and risk characteristics of the standard asset classes available for investment purposes
• Calculate the discounted mean term or volatility of an asset or liability and analyse whether an asset-liability position is matched or immunized
• Demonstrate an understanding of the nature and use of simple stochastic interest rate models
• Calculate the forward price and value of a forward contract using no-arbitrage pricing
• Take into account the time value of money by using the concepts of compound interest and discounting
• Demonstrate how interest rates and discount rates change when the underlying time period is altered
• Calculate the present value and accumulated value of a cash flow of equal or unequal payments, at a specified rate of interest, and at a real rate of interest, assuming a given rate of inflation.
• Define and use standard compound interest functions
• Analyse straightforward compound interest problems, and solve resulting equations of value, including for the implied rate of return
• Describe how a loan may be repaid by regular instalments of interest and capital.
• Apply discounted cash flow techniques to investment project appraisal.
• Evaluate investment performance using a variety of measures.

### Syllabus

• Simple and compound interest. Time value of money. Rate of interest, rate of discount, and force of interest. Accumulated values and discounted values. Accumulation and discounting of a (possibly infinite) cash flow to a given time, where both the rate of cash flow and the force of interest may be time-varying. • Relationships between rates of interest and discount over different time periods. Nominal rates, effective rates, rates payable multiple times per annum. • Definition of the standard compound interest functions and relationships between them. • Generalised cash flow modelling. Equation of value for a cash flow problem, and methods of solution. • Loans. Equation of value corresponding to periodic repayment of a loan. Interest and capital content of annuity payments where the annuity is used to repay a loan. Consumer credit transactions. Annual Percentage Rate of Charge (APR). • Net present value (NPV), accumulated profit, and internal rate of return (IRR) for investment projects. • Investment project appraisal using NPV and IRR. Real rate of return in presence of inflation. • Measurement of investment performance. • Ordinary shares. Constant dividend growth model of share valuation. Fixed-interest securities. Present value and redemption yield for a fixed-interest security, including effects of taxation. • Yield curves and the term structure of interest rates. • Investment and risk characteristics of standard asset classes (Government fixed-interest securities, other fixed-interest securities, equities, etc.) available for investment purposes. • Discounted mean term, volatility, convexity. Matching of assets and liabilities, immunization. • Simple stochastic interest rate models. Mean, variance, and distribution function for the accumulated amount of an initial investment, and applications. • Forward interest rates. Forward contracts. The concept of no-arbitrage pricing and its use in determining the fair value of a forward contract.

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem classes, workshops, assigned problems and solutions, class tests and solutions, office hours, and private study.

TypeHours
Teaching54
Independent Study96
Total study time150

Hull covers the part of the syllabus relating to derivative securities, though this is also covered by Garrett..

BUTCHER, M.V. and NESBITT, C.J. (1971). Mathematics of Compound Interest.

BROVERMAN, S.A. (2010). Mathematics of Investment and Credit.

Kellison, Butcher and Nesbitt, and Broverman all cover similar ground..

KELLISON, S.G (2008). Theory of Interest.

HULL, J.C., (2014). Options, Futures, and Other Derivatives.

GARRETT, S.J. (2013). An Introduction to the Mathematics of Finance: A Deterministic Approach.

McCUTCHEON, J.J. and SCOTT, W.F., (1986). An Introduction to the Mathematics of Finance.

Garrett is an essential text and covers all of the syllabus. Students should obtain a copy. Problems will be assigned from this text..

McCutcheon and Scott covers most of the syllabus, is a good second choice, and, like Garrett, has a large number of good problems..

### Assessment

#### Summative

MethodPercentage contribution
Class Test 10%
Class Test 10%
Exam  (3 hours) 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

(MATH1024 Introduction to Probability and Statistics AND MATH1056 Calculus) OR ECON1011 Quantitative Modelling in Economics

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

A limited number of course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.