Risk Analysis & Modelling

1. Risk Analysis & Modelling Lecture 1: Introduction

2. www.angelfire.com/linux/riskanalysisRiskCourseHQ@Hotmail.com

3. What is Risk? • The definition of risk is the “Possibility of Loss” • When we observe risk we observe uncertainty about the possible outcome • Even if we are uncertain about the exact outcome, we can often think of a range of possibilities that might occur • Risk arises from some of the uncertain outcomes being less favourable than others • On this course we will be focusing on the quantitative evaluation of the risks faced by insurance companies

4. Risk And Future Outcomes Best Outcome ? Current Position Worst Outcome Future Position Time

5. Types of Financial Risk • Financial institutions face many risks • We will focus on the types of risk of interest to insurance companies that can be analysed in a quantitative fashion • For the purpose of study it is useful to categorise types the risks faced by these institutions • Some of the types of risk we will study are: Market Risk, Underwriting Risk, Reserving Risk and Credit Risk. • The meaning of these categories will become apparent when we study the techniques used to model them • As with most broad concepts their definition can a bit fuzzy!

6. Market Risk • Market Risk arises from owning an asset whose market value or price changes over time • Market Risk can be directly observed through movements in the market price of the asset • An example of market risk would be the uncertainty of your wealth decreasing if you had £100,000 invested in the FTSE-100 over the next year

7. Market Risk Diagram Market Asset Price ? Market Asset Price Initial Wealth Market Asset Price Future Wealth

8. Underwriting Risk • When an Insurance Company sells a policy it Underwrites or Insures the policy holder against some specified loss • Whether this insured loss will occur and its cost are often unknown when the policy is sold • These unknown quantities lead to Underwriting Risk which describes the uncertainty surrounding the levels of claims the insurance company will incur

9. Underwriting Risk on an Insurance Policy Large Claim Insurance Policy ………………… ………………… ………………… ………………… ………………… Average Claim ? Small Claim No Claim

10. Reserve Risk • Insurance Companies are not just uncertain about the amounts they will have to pay but there is also uncertainty about the timing of these payments • For certain classes of insurance (such as liability insurance) there can be a delay of a number of years between an accident or loss occurring and a claim payment being made • The insurance company is uncertain about the total value of these future claims, but tries to set aside capital to pay these claims in a Loss Reserve • The Reserve Risk describes the possibility that this capital set aside will be insufficient to meet the actual claims (ultimate claims) eventually paid by the insurer

11. Reserve Risk Ultimate Claim Loss Reserve Less Than Loss Reserve Loss Reserve Ultimate Claim ? Equal To Loss Reserve Loss Reserve Ultimate Claim Greater

12. Credit Risk • Credit Risk arises from a counter-party in an agreement being unable or unwilling to meet their financial obligations • For example, the credit risk on a bank loan is the possibility that the borrower can no longer pay the amount due. When the borrower cannot pay a default is said to have occurred. • Insurance companies face Credit Risk when the reinsurance company it has ceded business to cannot meet its obligations to pay claims. • Credit risk is harder to measure and quantify than market risk.

13. Credit Risk Diagram Full Repayment Creditor ? Partial Repayment Debtor No Repayment

14. Other Types of Risk • In the literature you will find definitions for 100s of types of risk • Enterprise Risk, Operational Risk, Political Risk, Strategic Risk, Legal Risk and so on • We could spend the rest of the lecture going through all these types, however most are not of interest to us since we cannot model them quantitative in a meaningful way • And this course is about Modelling Risk.

15. What is Modelling • The definition of Modelling is: “To produce a representation or simulation of “ • Risk Modelling is to simulate Risk • The components or building blocks of our models will be numerical and statistical techniques • A crucial part of the modelling process is to build a simplified, abstraction of reality • Good models are simple models which capture important elements of the real world we wish to examine • Bad models are complex models which contain a lot of irrelevant detail – complexity leads to error

16. Modelling Involves Simplification Modelling Process: What is Important, What Can We Simplify? Interpreting Model: What Does It Mean? Simplified Model of the Real World Complicated Real World Phenomena

17. The need for computer based models • Even though the models we will be building will be simplified for the purpose of teaching they will involve millions of calculations • Far too many calculations to make with a calculator, pen and paper! • Risk modelling is a practical science and this course shows you how to build working risk models not just talk about them! • Along with every concept we cover on the course we will also learn the numerical and computing techniques necessary to apply that concept in practice

18. Computers and Modelling • When you buy a low end 2.8 Ghz computer with two Cores you are purchasing a machine that can make 5.6 billion calculations per second! • Gigahertz (Ghz) stands for billions of cycles per second, and one cycle is roughly one calculation • So each Core can make approximately 2.8 billion calculations per second • If you were to perform 5.6 billion calculations on a calculator at 1 calculation every 3 seconds without rest it would take you over 500 years! • With this almost “unlimited” calculating power the problem is frequently not the complexity of the model but whether or not an individual has the technical skills to describe the model to the computer

19. Microsoft Excel • We will use Microsoft Excel to build and explore the various risk models we will study • Excel is a spreadsheet software package and is probably the primary tool used in the financial industry to make calculations • We will be using some of the advanced features of Excel combined with VBA (Visual Basic for Applications) programming techniques* • Excel is one of the simplest and widely used tools with which to develop computer based risk models – although it has some limitations • VBA provides an ideal introduction to the world of programming - which is extremely useful and a lot easier than many people realise!

20. Spreadsheets and Matrices

21. What is a spreadsheet? • A spreadsheet can be thought of as a giant table which can contain numbers and formula. • The spreadsheet is made up of cells which are identified by their column (represented by a sequence of letters A,B,C,D…. ,AA,AB..) and row (represented by a number 1,2,3,4…) • The best way to learn about spreadsheets is to play about with them….

22. Elements of a Spreadsheet Numeric values Column Identifiers Formula; add A1 and A2 Row Identifiers

23. Copying and Pasting Numbers and Formula • To avoid having to retype numbers and formula we can copy and paste values • One important point to note is that when copying and pasting formula in Excel the row and column references for the input cells are shifted • This will turn out to be a very useful, time saving feature in many of the models we will build • We can instruct Excel not to adjust the formula by placing a ‘$’ sign infront of the elements of the formula

24. Copy & Pasting Values Formula and values copied and pasted 2 rows to the right and 2 rows down, notice how the formula adjusts

25. Matrices • Matrices are ordered blocks or tables of numbers • The numbers are organised into rows and columns much like a spread sheet • Many of the operations that can be performed on numbers can be performed on matrices such as addition, subtraction and multiplication • There are some “special” matrix operations such as inversion and transposition • Matrices are a very important practical tool for performing large sets of calculations • Matrices are an important conceptual tool which allows us to generalise calculations for problems of different size

26. A 2 by 2 Matrix Columns 1 2 1 Rows 2 (2,2) matrix • This is a 2 by 2 matrix because it has 2 rows and 2 columns. • The matrix contains a total of 4 elements. • It is a Square matrix because it has the same number of rows and columns • The element at row 2 and column 1 is 9

27. Matrix Notation • When we wish to denote a matrix we will use a bold capital letter: A • A is a matrix • We denote the size or dimensions of the matrix by giving the rows and columns of the matrix in brackets: A is (Rows, Columns) • When we wish to denote the element of a matrix we will use a capital letter with two subscripts: Ar,c • Where r is the row of the element and c is the column of the element of matrix A

28. An Example of the Notation C1 C2 C3 R1 A = R2 R3 A is (3,3) A is a square matrix

29. Matrix Operations We have seen that a matrix is an ordered block of numbers Like numbers the meaning of a matrix is defined by the values they represent and the order of those values Matrices like numbers become really useful when we start performing operations on them such as multiplication, addition and subtraction. Matrices and their operations are particularly useful because they allow us to describe large blocks of calculations simultaneously The meaning of the operation depends on the matrices they are applied to

30. Matrix Addition and Subtraction • Two Matrices A and B of dimensions (rA,cA) and (rB,cB) can be added or subtracted if and only if rA =rB and cA =cB (columns of A equal columns B and rows A equal rows B) • C = A + B or C = A - B then C is the same dimension as A and B: (rA ,cA) and (rB ,cB) • Ci,j= Ai,j + Bi,j the element at row i column j for C is equal to the sum of the elements at row i and column j in A and B. • The meaning of the addition of 2 matrices depends on the data that they store and the order of that data.

31. Matrix Addition an Example + = = • We can add the 2 matrices because they are of the same dimension (3,3) • The resulting matrix is of dimension (3,3)

32. Matrix Subtraction an Example - = = • We can subtract the 2 matrices because they are of the same dimension (3,2) • The resulting matrix is of dimension (3,2)

33. Matrix Subtraction Review Question • Subtract Matrix A from Matrix B ie (B-A) A = B = Note that both A and B have a single column and are therefore a special type of matrix called a vector

34. Matrix Subtraction Review Question • Subtract Matrix A from Matrix B = = - Note that both A and B have a single column and are therefore a special type of matrix called a vector

35. An Invalid Matrix Addition + = = • We cannot add the 2 matrices because they are of different dimensions (3,3) and (2,2) • The resulting matrix is invalid and has no meaning

36. Array Formula In Excel • All of the matrix operations we will be performing in Excel will be a special class of formula called “Array Formula” • Array Formula are different from normal Excel formula in that they work on ranges or arrays rather than individual cells • When entering an Array Formula an output range is selected before the formula is typed. The formula is entered by pressing Ctrl-Shift-Enter • Curly Brackets appear about the formula once Ctrl-Shift-Enter is pressed • The Array Formula represents a “block” made up of more than one cell.

37. Matrix Addition In Excel Formula adds the 2 matrices “A3:C5” and “E3:G5” :“A3:C5+E3:G5” notice the curly brackets that appear after the formula is entered with Ctrl-Shift-Enter Input Ranges Output Range Selected

38. Matrix Subtraction In Excel Formula subtracts the 2 matrices “A3:C5” and “E3:G5” : “A3:C5-E3:G5” notice the curly brackets indicating an array formula Input Ranges/Matrices Output Range Selected

39. Matrix Multiplication • Matrix Multiplication is a more complex operation and can involve many calculations. • Matrix Multiplication is one of the key calculations we will use in our risk models • Two Matrices A and B of dimensions (rA,cA) and (rB,cB) can multiplied if and only if cA =rB • C = A * B then C is of dimension (rA ,cB) • Ci,j= S Ai,k * Bk,j for k = 1 to N where N =cA =rB • Matrix multiplication is not commutativeA*B doesn’t equal B*A

40. Matrix Multiplication an Example 1 A B = * (1,2) (2,1) C C = (1,1) (1,1) • We can multiply these 2 matrices because the columns of A (2) is equal to the rows of B (2) • The resulting matrix C has a number of rows equal to 2 (since A has 2 rows) and columns equal to 2 (since B has 2 columns)

41. Matrix Multiplication an Example 1a * = (1,2) (3,1) C C = (1,1) (1,1)

42. Matrix Multiplication an Example 2 A B = * (1,2) (2,2) C = (1,2)

43. Matrix Multiplication an Example 2 A B = * (2,2) (2,2) C C = (2,2) (2,2)

44. Matrix Multiplication an Example 3 B A = * (1,2) (2,2) C C = (1,2) (1,2) • We can multiply these 2 matrices because the columns of A (2) is equal to the rows of B (2) • The resulting matrix C has a number of rows equal to 1 (since A has 1 rows) and columns equal to 2 (since B has 2 rows)

45. Matrix Multiplication Review Question • Multiply these 2 matrices together: * = (1,2) (2,2) • What is the size/dimension of the resulting matrix? • What are its element(s)

46. Matrix Multiplication Review Question = * (1,2) (2,2) =

47. Matrix Multiplication In Excel Formula multiplies matrices “A3:C5” by “E3:G5”: “MMULT(A3:C5, E3:G5)” and outputs to the result to the selected range I3:K5. MMULT is a special built in Excel Function. Output Range Selected Input Ranges/Matrices

48. The Transpose of a matrix • When a matrix is transposed its rows and columns are interchanged • If A is of dimension (rb ,cb) then AT is of dimension (cb ,rb) where ATj,i= Ai,j • Sometimes matrices need to be transposed before they are multiplied or added

49. Matrix Transposition an Example A = AT = • The columns of matrix A become the rows of AT • The matrix is rotated when it is transposed

50. Matrix Transpose Review Question • Transpose the following matrix: • What is the size/dimenion of the resulting matrix? • What are its element(s)