## Financial Mathematics for Actuarial Science 2

 Unit code: MATH20951 Credit Rating: 10 Unit level: Level 2 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

Prerequisite

#### Additional Requirements

MATH20951 pre-requisites

For students on the Actuarial Science and Mathematics programme only.

#### Aims

The unit aims to provide further instruction in simple financial transactions as used in actuarial science and the mathematics involved.

#### Overview

The unit covers methods of describing and assessing simple investments under a range of assumptions.

#### Learning outcomes

On successful completion of this module students will be able to:

• Determine the cashflow from a description of a project and calculate discounted values, net present values, accumulated profits and yields from the cashflow. Use these indicators to compare projects.
• Calculate discounted payback periods for cashflows (discrete and/or continuous) and use this to determine the accumulated profit of projects when different interest rates are charged. Calculate inflation indices and find the real yield of cashflows with inflation.
• Compute and compare indicators of investment performance (money weighted rate of return, time weighted rate of return and linked internal rate of return).
• Prove the existence of yields for simple cashflows using the intermediate value theorem.
• Price a fixed interest security or determine its yield, including the effects of income tax, capital gains tax and variants on the payment schedules for these. Be able to derive and use Makeham’s formula.
• Derive monotonicity properties of yields and price with respect to redemption dates and interest rate. Use this information to advise on investments with optional redemption dates.
• Define arbitrage and use the Law of One Price to determine the price of a forward contract by the construction of portfolios.
• Define the n-year spot rate and forward rates. Determine the relationships between these so as to find one set of yields or spot rates given others. Define and calculate par yields and price bonds with time structure.
• Use Taylor’s Theorem to define and calculate the effective duration, discounted mean term and convexity of assets and liabilities. Assess the effect of small changes in the interest rate on pension funds and determine whether a fund can be immunised.
• Determine the mean and variance of accumulating under random interest rates. Be able to calculate the mean and variance of annuities under random interest rates.
• Use the Law of Large Numbers to justify looking at lognormal distributions. Use lognormal distributions to assess the likely performance of investment strategies by giving the probability of meeting a given success criterion.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework; one in-class test, weighting within unit 20% each.
• Examination at end of semester 1, 2 hours weighting within unit 80%.

#### Syllabus

This unit explores some further simple financial topics from a mathematical point of view.

• The role of finance within actuarial science,
• Appraisal and Comparison of Projects,
• Description of Investments,
• Compound Interest Problems : Fixed and uncertain income, Real Rates of interest, index linked bonds, Capital Gains Tax,
• Arbitrage. The No-Arbitrage Assumption. Forward Contracts,
• Term Structure of Interest Rates. Discrete and continuous time rates. Duratio, Convexity and Immunisation.
• Stochastic Interest Rate Models.

#### Recommended reading

• Core Reading : Subject CT1, Financial Mathematics. Produced by the Actuarial Profession
• JJ McCutcheon and WF Scott, An Introduction to the Mathematics of Finance. Heinemann, 1986

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  The in-class test also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Paul Glendinning - Unit coordinator