Risk Analysis & Modelling

1. Risk Analysis & Modelling Lecture 2: The Mean-Variance Model

2. www.angelfire.com/linux/riskanalysisriskcoursehq@hotmail.com

3. The Mean-Variance Model • In this lecture we will introduce one of the most widespread models used to measure and analyse the nature of Market Risk • This model is the Nobel Prize winning Mean-Variance Model developed by Harry Markowitz in 1952 • Although over 60 years old this model is still in widespread use and it is an ideal way to introduce the power of quantitative risk modelling • It is the basis of the SCR calculation in Solvency II

4. Measuring Risk • Last week we made the statement that risk involves uncertainty over some future outcome • Even when we are not certain about the precise outcome that we will observe we often have an idea of the range of outcomes that might occur • We can place reasonable limits on our uncertainty

5. Limits on uncertainty We are uncertain about the precise outcome Current Position Although we are unsure precisely which outcome will occur we can describe a range of outcomes over which our uncertainty arises

6. Our thought experiment • Imagine you have £10,000 invested in a portfolio of 2 stocks • £6000 is invested in stock A and £4000 in stock B • You are uncertain about how much your portfolio will be worth in one month, however, as an expert investor you feel you can place limits on your uncertainty – you are not totally uncertain • You will do this by writing down a range of possible values for your portfolio and the probability or likelihood of these outcomes

7. Limits on Uncertainty Probability, Portfolio Value 10%, £11,000 (+10%) 45%, £10,500 (+5%) 20%, £10,000 (0%) £10,000 20%, £9,500 (-5%) 5%, £9,000 (-10%) 1 Month Time Horizon

8. Histogram of Outcomes

9. Your Friends Portfolio • Your friend has a portfolio of £5000 • After seeing your risk chart he decides to create one for his own portfolio • Using his knowledge of the assets in his portfolio he produces a histogram showing the possible changes in portfolio value and the probability of these changes • You note that because you are both using the proportional change in your portfolio values you can compare charts (even though your portfolios are of different sizes)

10. Friends “Riskier” Portfolio Greater spread of outcomes means greater risk

11. Which Portfolio Earns More? • You want to know which portfolio earns more on average or has the greater expected return On average we expect our portfolio to earn 1.75% over the 1 month, while we expect our friends portfolio to earn 3%

12. Which Portfolio Is Riskier? • While your friends portfolio has a greater expected return, we could argue that because the spread of outcomes on your friend’s portfolio is greater that the risk is greater • Although this can be ascertained visually it is useful to have a way of measuring this spread numerically • Statistics provides us with a simple means by which to do this: the variance (or the expected squared dispersion)

13. Definition of Variance • The expected return is defined as follows where E is the expectation operator (see appendix) and pr is the probability of that outcome The variance is defined as the expected squared difference The standard deviation is simply the square root of the variance

14. Calculating Variance • The variance on our portfolio is: The variance of the return on our portfolio is 0.002819 By similar calculation we can see that the variance on our friends portfolio is 0.00735 The variance of the return does not have any obvious intuitive meaning (unlike the expected return), but it can be used to compare risk or dispersion between portfolios

15. Calculating the Mean and Variance From Historical Observations • Normally we estimate the Mean and Variance of a phenomena from historical observations rather than hypothetical outcomes and probabilities • To estimate the average from a set of historical observations of size N we use the following formula (which is calculated using the AVERAGE formula in Excel): • To estimate the variance we can use the following formula** (which is calculated using the VARP formula in Excel):

16. Mean Variance Abstraction • The key insight of the Mean-Variance Model is that an asset or portfolio can be described using two numbers: the mean and variance of return • Although this does not capture the entire behaviour of a portfolio these 2 simple statistics directly relate to and quantify important aspects of a portfolio’s behaviour • They tell us how much profit we expect to make and how risky that profit is

17. Mean-Variance Abstraction Range or Distribution of Outcomes Simplification or Abstraction Mean (Measure of Expected Profit) Variance (Measure of Risk)

18. Deriving the Portfolio’s Risk • Returning to our investment portfolio, its risk (or variance) and expected return is derived from the risk of the assets it contains • It is often useful to express the risk and return of the portfolio in terms of the assets it contains • It is often easier to quantify the risks of individual assets rather than the risks of the portfolio • We will derive some important results by expressing the statistical behaviour of the portfolio interms of the assets it contains

19. Relationship between the Assets and the Portfolio The behaviour of the portfolio is derived from the behaviour of the assets it contains – a portfolio containing assets with a high return will have a high return itself P Is it possible to derive the mean and variance of the returns on the portfolio in terms of the behaviour of the assets? B A

20. Mapping Assets to Portfolio • The known market value today of a portfolio of 2 assets is equal to the sum of their values: The unknown or random market value of the portfolio at some future time is equal to: The ~ above the variables simply denotes that they are random or uncertain variables which can take on a range of values

21. The relationship between the assets today and the future date can be expressed as: Where and denotes the random returns on asset A and asset B respectively If we want to express the portfolio’s return in terms of the return on assets we have:

22. We can rewrite this as: Where wa and wb are the proportion of the portfolios total value invested in asset A and asset B respectively These proportions are also called the investmentweights The usual restrictions on these investment weights is that they sum to 100% and cannot be negative For example, a £100 portfolio with £30 invested in A and £70 invested in B has wA is equal to 30% and wb is 70%

23. Example Calculation • The meaning of this mapping equation is fairly simple. • It simply states the obvious fact that the random behaviour of the return on the portfolio is directly related to the random behaviour of the assets in the portfolio • For example, if a portfolio contains 40% in stock A with a return of 5% and 60% in stock B with a return of 10% then the portfolio’s random return will be 8%:

24. Review Question • A portfolio contains two assets A and B • 30% of the portfolio is held in asset A and 70% in asset B • The return on Asset A is 10% and the return on Asset B is 20% • What is the return on the portfolio?

25. Implying the Portfolio’s Statistical Properties • We will now show it is possible to imply the mean and variance of return for the portfolio from the statistical properties of the assets contained in the portfolio • This will turn out to be extremely useful since it allows us to use simple mathematics to model and explore the nature of Risk in a Portfolio

26. Portfolio Expected Return • The expected return on the portfolio is equal to: So the expected return on the portfolio is equal to a weighted average of the returns on the assets

27. Review Question • A portfolio contains two assets A and B • 50% of the portfolio is held in asset A and 50% in asset B • Over the next year the expected or average return on asset A is 20% and the average return on asset B is expected to be 30% • What is the average or expected return on the portfolio over the next year?

28. Portfolio Variance • The portfolios variance can be defined as: Where cov is the covariance The intuition is that the variance of the portfolio depends not just on how the assets it contains move in isolation but also on how they move together or co-vary (measured by covariance). We note the relationship is quadratic with respect to the investment weights!

29. Covariance • Covariance measures the co-movement of 2 random variables and is defined as: A large positive covariance suggest that the random variables have a tendency to move together A large negative covariance suggests that the random variables have a tendency to move in the opposite direction to each other A covariance close to zero suggests that the random variables are independent in their movement about the mean

30. Covariance Diagram E(Y) E(X) Cov(X,Y) Covariance measures the simultaneous behaviour of two random variables about their centre. Do they have a tendency to move together, against each other or independently? Y X

31. Correlation • Correlation is a normalized measure of covariance • Correlation must be between –1 and +1 due to the Cauchy-Schwarz inequality (see appendix) • A strong positive correlation suggests that the two random variables move about their mean value in unison • A strong negative correlation suggests that the two random variable move in opposite directions about their mean value • Correlation is defined as:

32. Calculating Covariance from Historical Observations • To estimate the covariance from a set of historical observations for two random variables X and Y of size N we use the following formula**: • In Excel this can be calculated using the COVAR formula • The related correlation statistic can be calculated in Excel using the CORREL formula

33. Quadratic Relationship • Let us look at the relationship between the risk on a portfolio of 2 uncorrelated assets (zero covariance) both with a variance of 0.1 the risk of a portfolio containing those 2 assets • Our equation tells us that the variance of the return on the portfolio is: • If we graph this relationship out for different combinations of asset A and asset B we see something interesting…

34. Risk for Various Portfolios 100% in A and 0% in B 0% in A and 100% in B Minimum Risk Containing 50% of A and 50% of B

35. Interpreting the Curve • What this curve is telling us is that the risk (measured in terms of variance) of a combination of asset A and asset B is lower than the individual risks of the two assets • In this case it is possible for a portfolio of 50% of A and 50% of B to have half the risk of the individual assets • This result is not entirely intuitive, and at first sight might seem nonsensical • However this is an extremely important result • The interpretation of this curvature in the risk of the portfolio is that it represents diversification

36. What if we have perfect correlation? • We can show that the quadratic relation reduces to a linear relationship when the correlation between assets become perfect (ie no diversification effect) • From the definition of correlation we see:

37. Using our equation for 2 assets: We can see in the absence of diversification (perfect correlation between assets) the relation between the risk of the portfolio and the investment weights becomes linear This result explains the effect we saw when we artificially set the correlation between assets to 1 - making them perfectly correlated

38. Diversification • Diversification is extremely important to financial institutions • The insurance industry is based on the concept of diversification or pooling of risk • Risk is diversified between insurance policies – the risk of a portfolio of policies is less than the sum of the risks on the individual policies • One of the powerful abilities of the mean-variance framework is that it brings diversification into a simple quantitative framework • Diversification arises when the assets (or liabilities) making up the portfolio move independently • By moving independently their movements can offset each other

39. Diversification and Solvency II • The regulatory authorities place solvency capital requirements on insurance companies • The level of capital that an insurance company has to hold depends on the potential losses that might occur on its underwriting and investment portfolios • One of the weaknesses of the traditional Actuarial Methods used in the Insurance Industry is they measure risks in isolation rather than across the business as a whole • Solvency II allows insurance companies to take into account the effect of diversification across different lines of business and across different asset classes

40. Larger Portfolios • The equations we developed can be used to derive the statistical properties of a portfolio of just 2 assets • For a modest portfolio of 30 assets the equation describing the relationship the portfolio variance and the assets contained would contain 465 variance and covariance terms! • Financial institutions hold portfolios containing thousands of assets • In order to perform these calculations on portfolios of this scale we need to use matrices

41. 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)

42. Matrix Multiplication Review Question • Multiply these 3 matrices together: * * = (1,2) (2,2) (2,1) • What is the size/dimension of the resulting matrix? • What are its element(s)

43. Hint! * = (1,2) (1,2) (2,2) = * * * (1,2) (1,2) (2,2) (2,2) (2,1)

44. Expected Return Vector • The relationship between the expected return on a portfolio and the statistical properties of the assets it contains can be expressed as follows: E(rP)= WT.E Where W is the weight vector (matrix with one column) and E is the expected return vector E = W =

45. We can see that this is equivalent to the earlier equation by multiplying out the matrices = *

46. Covariance Matrix • The relationship between the variance of a portfolio and the statistical properties of the assets it contains can be expressed as Var(rP)= WT.C.W Where W is the weight vector and C is the covariance matrix C = W =

47. Again, we can see that this is equivalent to the earlier equation by multiplying out the matrices * * = * =

48. The Covariance Matrix • We will be using the covariance matrix extensively – it is the main tool we will use to describe the nature of uncertainty or risk • You can think of it as a specially ordered table containing the covariances and variances of random variables • It contains information not only on how individual assets behave but also how they behave together (via their covariance) • It is a semi-positive definite matrix (ie WT.C.W >=0 for every non-zero column vector W - no negative variances! This is equivalent to saying the Eigen Values of the covariance matrix should be greater than or equal to zero)

49. B A A Identical Terms B Covariance of a random variable with itself is its variance Covariance Matrix for 2 Assets A B C A B C Covariance Matrix for 3 Assets

50. Correlation Matrix • A matrix closely related to the covariance matrix is the correlation matrix • As you can probably deduce from the name it contains correlations rather than covariances! • Correlations are normalised covariances whose value ranges between -1 and 1 • The definition of correlation is: Where px,y is the correlation between x and y and sx and sy are the standard deviations of x and y.