Financial Risk Forecasting Chapter 1 Financial markets, prices and risk

Transcription

1 Financial Risk Forecasting Chapter 1 Financial markets, prices and risk Jon Danielsson 2017 London School of Economics To accompany Financial Risk Forecasting Published by Wiley 2011 Version 3.1, October 2017 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 1 of 103

2 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 2 of 103

3 The focus of this chapter Statistical techniques for analyzing prices and returns in financial markets Stock market indices, e.g. the S&P 500 Prices, returns and volatilities Three stylized facts of financial returns: 1. Volatility clusters 2. Fat tails 3. Nonlinear dependence See Appendix A for more detailed discussion on the statistical methods See Appendices B and C for introduction to R and Matlab Financial Risk Forecasting 2011,2017 Jon Danielsson, page 3 of 103

4 Notation T Sample size t = 1,...,T A particular observation period (e.g. a day) P t Price at time t R t = Pt P t 1 P t 1 Simple return Y t = log Pt P t 1 Continuously compounded return y t A sample realization of Y t σ Unconditional volatility σ t Conditional volatility K Number of assets w K 1 vector of portfolio weights ν Degrees of freedom of the Student-t ι Tail index d dividends Financial Risk Forecasting 2011,2017 Jon Danielsson, page 4 of 103

5 Prices, returns and indices Financial Risk Forecasting 2011,2017 Jon Danielsson, page 5 of 103

6 Total returns USA Equities Bonds profit loss Financial Risk Forecasting 2011,2017 Jon Danielsson, page 6 of 103

7 Total returns Austria Equities Bonds profit loss Financial Risk Forecasting 2011,2017 Jon Danielsson, page 7 of 103

8 Total returns South Africa Australia USA New Zealand Sweden Canada Denmark Finland UK Netherlands Switzerland Ireland Spain Portugal France Belgium Austria Equities Bonds Financial Risk Forecasting 2011,2017 Jon Danielsson, page 8 of 103

9 Stock indices A stock market index shows how a representative portfolio of stock prices changes over time A price-weighted index weighs stocks based on their prices A stock trading at $100 makes up 10 times more of total than a stock trading at $10 A value-weighted index weighs stocks according to the total market value of their outstanding shares Impact of change in stock price proportional to overall market value Financial Risk Forecasting 2011,2017 Jon Danielsson, page 9 of 103

10 Stock indices The most widely used index is the Standard & Poor s 500 (S&P 500) largest 500 traded companies in the US Examples of value-weighted indices: S&P 500, FTSE 100 (UK), TOPIX (Japan) Examples of price-weighted indices: Dow Jones Industrial Average (US), Nikkei 225 (Japan) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 10 of 103

11 Prices and returns Denote prices by P t. Usually we are more interested in the return we make on an investment Definition Return The relative change in the price of a financial asset over a given time interval, often expressed as a percentage There are two types of returns: 1. Simple (R) 2. Compound (Y) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 11 of 103

12 Simple returns Definition A simple return is the percentage change in prices R t = P t P t 1 P t 1 Including dividends R t = P t P t 1 +d P t 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 12 of 103

13 Continuously compounded returns Definition The logarithm of gross return ( ) Pt Y t = log(1+r t ) = log = log(p t ) log(p t 1 ) P t 1 P t+1 = P t e R Financial Risk Forecasting 2011,2017 Jon Danielsson, page 13 of 103

14 Simple and continuous The difference between R t and Y t is not large for daily returns As the time between observations goes to zero, so does the difference between the two measures: lim t 0 Y t = R t log(1000) log(990) = = log(1000) log(800) = = Financial Risk Forecasting 2011,2017 Jon Danielsson, page 14 of 103

15 Symmetry Continuous returns are symmetric ( ) ( ) log =log Simple are not Financial Risk Forecasting 2011,2017 Jon Danielsson, page 15 of 103

16 Simple returns are Used for accounting purposes Investors are usually concerned with simple returns Continuously compounded returns have some advantages Mathematics is easier (e.g. how returns aggregate over many periods, used in Chapter 4) Used in derivatives pricing, e.g. the Black Scholes model Financial Risk Forecasting 2011,2017 Jon Danielsson, page 16 of 103

17 Issues for portfolios R t,portfolio return on a portfolio Weighted sum of returns of individual assets: R t,portfolio = K k=1 w k R t,k = w R t While ( ) Pt,portfolio Y t,portfolio = log P t 1,portfolio K k=1 ( ) Pt,k w k log P t 1,k Because the log of a sum does not equal the sum of logs Financial Risk Forecasting 2011,2017 Jon Danielsson, page 17 of 103

18 S&P 500 index Great Depression Internet Bubble US Civil War 2007 crisis 10 5 Nifty fifty Financial Risk Forecasting 2011,2017 Jon Danielsson, page 18 of 103

19 40 % 30 % 20 % 10 % 0 % 10 % S&P 500 returns US Civil War Great Depression Asian crisis 20 % 30 % 1987 crash 2008 crisis Financial Risk Forecasting 2011,2017 Jon Danielsson, page 19 of 103

20 S&P 500 statistics 1929 to 2016, daily returns Mean % Standard error 1.143% Min 20.47% Max 16.60% Skewness Kurtosis Note how small mean is compared to the s.e. (volatility) But mean grows at rate T and the volatility at T Financial Risk Forecasting 2011,2017 Jon Danielsson, page 20 of 103

21 Three stylized facts Present in most financial returns Volatility clusters Fat tails Nonlinear dependence Financial Risk Forecasting 2011,2017 Jon Danielsson, page 21 of 103

22 Software Excel is useless for we are trying to do here Four main software choices 1. Julia 2. Python (Numpy) 3. R 4. Matlab Go to Financial Risk Forecasting 2011,2017 Jon Danielsson, page 22 of 103

23 Data Financial data can be obtained from many sources, e.g. 1. wrds.wharton.upenn.edu (CRSP) 2. Bloomberg 3. finance.google.com 4. finance.yahoo.com (often best but does not always work) Best to save data as a CSV file and import that into R/Matlab Get SP-500 CSV file from Financial Risk Forecasting 2011,2017 Jon Danielsson, page 23 of 103

24 Matlab sp500 = csvread( sp 500.csv,1); plot (sp500 (:,2)) y = diff ( log(sp500 (:,2))); plot (y) mean(y) std (y) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 24 of 103

25 R sp500 = read. csv( sp 500.csv ) plot (sp500 [,2], type= l ) y = diff ( log(sp500 [,2])) plot (y, type= l ) mean(y) sd(y) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 25 of 103

26 Volatility Financial Risk Forecasting 2011,2017 Jon Danielsson, page 26 of 103

27 Volatility The standard deviation/error of returns Two concepts of volatility: Unconditional volatility is volatility over an entire time period (σ) Conditional volatility is volatility in a given time period, conditional on what happened before (σ t ) Clear evidence of cyclical patterns in volatility over time, both in the short run and the long run Financial Risk Forecasting 2011,2017 Jon Danielsson, page 27 of 103

28 Daily volatility Calculations σ = 1 N 1 N (y i µ) 2 i=1 Annualised N 1 N (y i µ) 2 i=1 Matlab std (y) sqrt (250) std(y) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 28 of 103

29 40 % Cycles in volatility SP % 20 % 10 % 0 % 1920 s 1940 s 1960 s 1980 s 2000 s Financial Risk Forecasting 2011,2017 Jon Danielsson, page 29 of 103

30 40 % Cycles in volatility SP % 20 % mean 10 % 0 % Financial Risk Forecasting 2011,2017 Jon Danielsson, page 30 of 103

31 80 % Cycles in volatility SP % 40 % 20 % mean Financial Risk Forecasting 2011,2017 Jon Danielsson, page 31 of 103

32 Volatility Returns Simulated volatility clusters 8 high low Financial Risk Forecasting 2011,2017 Jon Danielsson, page 32 of 103

33 Volatility clusters Volatility changes over time in a way that is partially predictable Volatility clusters Engle (1982) suggested a way to model this phenomenon His autoregressive conditional heteroskedasticity (ARCH) model is discussed in Chapter 2 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 33 of 103

34 Autocorrelations Correlations measure how 2 variables (x,y) move together Corr(x,y) = 1 N 2 N (x µ x )(y µ y ) i=1 Autocorrelations measure how a single variable is correlated with itself 1 lag ˆβ 1 = Corr(x 1,...,N 1,x 2,...,N ) N lags ˆβ i = Corr(x 1,...,N i,x i+1,...,n ) Matlab autocorr (y,20) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 34 of 103

35 If autocorrelations are statistically significant there is evidence for predictability The coefficients of an autocorrelation function (ACF) give the correlation between observations and lags We will test both returns (y), predictability in mean (price forecasting or alpha) And returns squared (y 2 ), predictability in volatility Financial Risk Forecasting 2011,2017 Jon Danielsson, page 35 of 103

36 The LB test for autocorrelations Joint significance of autocorrelation coefficients (ˆβ 1, ˆβ 2,..., ˆβ N ) can be tested by using the Ljung-Box (LB) test J N = T(T +2) N i=1 ˆβ i 2 T N χ2 (N) Matlab [h, pvalue, stat]=lbqtest (y,20) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 36 of 103

37 S&P to 2015 ACF of daily returns ACF Lags Financial Risk Forecasting 2011,2017 Jon Danielsson, page 37 of 103

38 S&P to 2015 ACF of squared daily returns ACF Lags Financial Risk Forecasting 2011,2017 Jon Danielsson, page 38 of 103

39 LB tests for S&P 500 Daily returns N LB statistic, 21 lags p-value 22, , < Daily returns squared T LB statistic, 21 lags p-value 22,752 12,633.0 < ,500 4,702.1 < Financial Risk Forecasting 2011,2017 Jon Danielsson, page 39 of 103

40 Fat tails Financial Risk Forecasting 2011,2017 Jon Danielsson, page 40 of 103

41 Definition Fat tails A random variable is said to have fat tails if it exhibits more extreme outcomes than a normally distributed random variable with the same mean and variance The mean variance model assumes normality Financial Risk Forecasting 2011,2017 Jon Danielsson, page 41 of 103

42 Fat tails The tails are the extreme left and right parts of a distribution If the tails are fat, there is a higher probability of extreme outcomes than one would get from the normal distribution with the same mean and variance Also implies that there is a lower probability of non-extreme outcomes Probabilities are between zero and one so the area under the distribution is one Financial Risk Forecasting 2011,2017 Jon Danielsson, page 42 of 103

43 The Student t distribution The degrees of freedom, (ν), of the Student t distribution indicate how fat the tails are ν = implies the normal ν < 2 superfat tails For a typical stock 3 < ν < 5 The Student t is convenient when we need a fat tailed distribution Financial Risk Forecasting 2011,2017 Jon Danielsson, page 43 of 103

44 0.4 Tails 0.3 probability outcome Financial Risk Forecasting 2011,2017 Jon Danielsson, page 44 of 103

45 0.4 Tails 0.3 probability outcome Financial Risk Forecasting 2011,2017 Jon Danielsson, page 45 of 103

46 normal Student t(2) Tails probability outcome Financial Risk Forecasting 2011,2017 Jon Danielsson, page 46 of 103

47 normal Student t(2) chi 2 Tails probability outcome Financial Risk Forecasting 2011,2017 Jon Danielsson, page 47 of 103

48 normal Student t(2) chi 2 Tails Zoom probability outcome Financial Risk Forecasting 2011,2017 Jon Danielsson, page 48 of 103

49 Probability of extreme outcomes If S&P 500 returns were normally distributed, the probability of a one-day drop of 23% would be ! The table below gives probabilities of different returns assuming normality Returns above or below Probability 1% % % % % % Financial Risk Forecasting 2011,2017 Jon Danielsson, page 49 of 103

50 Max and min of S&P 500 returns Per decade, daily returns 10 % 0 % 10 % 20 s 30 s 40 s 50 s 60 s 70 s 90 s 00 s 20 % 80s 10 s Financial Risk Forecasting 2011,2017 Jon Danielsson, page 50 of 103

51 10 % Max and min of S&P 500 returns Per year, daily returns 0 % 10 % % Financial Risk Forecasting 2011,2017 Jon Danielsson, page 51 of 103

52 0.5 Empirical density vs. normal S&P 500 daily returns, 2000 to Density Outcomes Financial Risk Forecasting 2011,2017 Jon Danielsson, page 52 of 103

53 Empirical density vs. normal S&P 500 daily returns, 2000 to 2015 Returns Normal Cumulative probability data higher than normal data lower data higher data lower than normal Outcomes Financial Risk Forecasting 2011,2017 Jon Danielsson, page 53 of 103

54 0.15 Empirical density vs. normal S&P 500 daily returns, 2000 to 2015 Returns Normal Cumulative probability Outcomes Financial Risk Forecasting 2011,2017 Jon Danielsson, page 54 of 103

55 Non normality and fat tails Three observations: 1. Peak is higher than normal 2. Sides are lower than normal 3. Tails are much thicker (fatter) than normal Financial Risk Forecasting 2011,2017 Jon Danielsson, page 55 of 103

56 Identification of fat tails Two main approaches for identifying and analyzing tails of financial returns: statistical tests and graphical methods The Jarque-Bera (JB) and the Kolmogorov-Smirnov (KS) tests can be used to test for fat tails QQ plots allow us to analyze tails graphically by comparing quantiles of sample data with quantiles of reference distribution An alternative graphical method for detecting fat tails is plotting sequential moments Financial Risk Forecasting 2011,2017 Jon Danielsson, page 56 of 103

57 Jarque Bera test The Jarque Bera (JB) test is a test for normality and may point to fat tails if rejected The JB test statistic is Matlab [h,p]=jbtest (y) T 6 Skewness2 + T 24 (Kurtosis 3)2 χ 2 (2) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 57 of 103

58 Kolmogorov-Smirnov test Based on minimum distance estimation comparing sample with a reference distribution, like the normal Matlab [h,p] = kstest (y) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 58 of 103

59 QQ plots A QQ plot (quantile-quantile plot) compares the quantiles of sample data against quantiles of reference distribution Used to assess whether a set of observations has a particular distribution Can also be used to determine whether two datasets have the same distribution Matlab qqplot (y) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 59 of 103

60 Sample quantiles Daily S&P 500 returns vs. Normal QQ plot, 1989 to 2015 more extremes than normal Theoretical quantiles more extremes than normal Financial Risk Forecasting 2011,2017 Jon Danielsson, page 60 of 103

61 Daily S&P 500 returns vs. Normal Many observations seem to deviate from normality and the QQ-plot has clear S shape Indicates that returns have fatter tails than normal, but how much fatter? We can use Student-t with different degrees of freedom as reference distribution (fewer degrees of freedom give fatter tails) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 61 of 103

62 Daily S&P 500 returns vs. Student t(5) 0.10 Sample quantiles Theoretical quantiles Financial Risk Forecasting 2011,2017 Jon Danielsson, page 62 of 103

63 Daily S&P 500 returns vs. Student t(4) 0.10 Sample quantiles Theoretical quantiles Financial Risk Forecasting 2011,2017 Jon Danielsson, page 63 of 103

64 Daily S&P 500 returns vs. Student t(3) 0.10 Sample quantiles Theoretical quantiles Financial Risk Forecasting 2011,2017 Jon Danielsson, page 64 of 103

65 Sequential moments Thickness of tail measured by tail index, ι, where lower ι indicates thicker tails (formal definition in Chapter 9) For Student-t, the tail index corresponds to degrees of freedom Sequential moments makes use of sample moments of data, where the m-th centered moment is E[(X µ) m ] = which can only be calculated if m < ι (x µ) m f(x)dx Financial Risk Forecasting 2011,2017 Jon Danielsson, page 65 of 103

66 Sequential moments Tail thickness can be analyzed graphically by plotting moments of the data as more observations are added: 1 t t i=1 x t m For example, if 3rd moment converges but 5th moment does not, the tail index should be between 3 and 5 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 66 of 103

67 Sequential moments from Student-t(4) observations Moments th 3rd Number of observations Financial Risk Forecasting 2011,2017 Jon Danielsson, page 67 of 103

68 Sequential moments from Student-t(4) observations 5th 3rd Moments Number of observations Financial Risk Forecasting 2011,2017 Jon Danielsson, page 68 of 103

69 Sequential moments for oil price returns 400 January 1990 to August th 3rd Moments Number of observations Financial Risk Forecasting 2011,2017 Jon Danielsson, page 69 of 103

70 Nonlinear dependence Financial Risk Forecasting 2011,2017 Jon Danielsson, page 70 of 103

71 Correlations Correlations are a linear concept y = αx +ǫ Then α is proportional to the correlation between x and y A different way to say that is linear dependence The relationship between the two variables is always the same regardless of the magnitude of the variables Under the normal distribution, dependence is linear Key assumption for the mean variance model Financial Risk Forecasting 2011,2017 Jon Danielsson, page 71 of 103

72 Nonlinear dependence Nonlinear dependence (NLD) implies that dependence between variables changes depending on some factor. In finance, perhaps according to market conditions Example: Different returns are relatively independent during normal times, but highly dependent during crises If returns were jointly normal, correlations would decrease for extreme events, but empirical evidence shows exactly the opposite Assumption of linear dependence does not hold in general Financial Risk Forecasting 2011,2017 Jon Danielsson, page 72 of 103

73 Evidence of nonlinear dependence Daily returns for Microsoft, Morgan Stanley, Goldman Sachs and Citigroup May 5, June 12, 2015 MSFT MS GS MS 46% GS 46% 81% C 37% 65% 63% August 1, August 15, 2007 MSFT MS GS MS 93% GS 82% 94% C 87% 93% 92% Financial Risk Forecasting 2011,2017 Jon Danielsson, page 73 of 103

74 Implications of fat tails Non normality and fat tails have important consequences in finance Assumption of normality may lead to a gross underestimation of risk However, the use of non-normal techniques is highly complicated, and unless correctly used, may lead to incorrect outcomes Financial Risk Forecasting 2011,2017 Jon Danielsson, page 74 of 103

75 Volatility and fat tails Volatility is the a correct measure of risk if and only if the returns are normal If they follow the Student-t or any of the fats, then volatility will only be partially correct as a risk measure We discuss this in more detail in Chapter 4 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 75 of 103

76 The quant crisis of 2007 Many hedge funds using quantitative trading strategies ran into serious difficulties in June 2007 The correlations in their assets increased very sharply So they were unable to get rid of risk Financial Risk Forecasting 2011,2017 Jon Danielsson, page 76 of 103

77 Goldman Sachs s flagship Global Alpha fund (summer of 2007) We were seeing things that were 25 standard deviation moves, several days in a row, said David Viniar, Goldmans chief financial officer. There have been issues in some of the other quantitative spaces. But nothing like what we saw last week. Financial Risk Forecasting 2011,2017 Jon Danielsson, page 77 of 103

78 Lehmans (summer of 2007) Wednesday is the type of day people will remember in quantland for a very long time, said Mr. Rothman, a University of Chicago Ph.D. who ran a quantitative fund before joining Lehman Brothers. Events that models only predicted would happen once in 10,000 years happened every day for three days. Financial Risk Forecasting 2011,2017 Jon Danielsson, page 78 of 103

79 Volatility and fat tails Goldman s 25 sigma event under the normal has a probability of Age of the universe is estimated to be days while the earth is days old Goldman expected to suffer a one day loss of this magnitude less than one every universes Or perhaps the distributions were really not Gaussian Financial Risk Forecasting 2011,2017 Jon Danielsson, page 79 of 103

80 Copulas Financial Risk Forecasting 2011,2017 Jon Danielsson, page 80 of 103

81 Exceedance correlations Exceedance correlations show the correlations of (standardized) stock returns X and Y as being conditional on exceeding some threshold, i.e. { Corr[X,Y X Q X (p) and Y Q Y (p)], for p 0.5 ρ(p) = Corr[X,Y X > Q X (p) and Y > Q Y (p)], for p > 0.5 where Q X (p) and Q Y (p) are the p-th quantiles of X and Y given a distributional assumption Can be used to detect NLD Financial Risk Forecasting 2011,2017 Jon Danielsson, page 81 of 103

82 Quantile correlation Exceedance plot Bivariate normal and Student-t 0.6 Normal ρ= Normal ρ=0.7 Student t(3) ρ= Probability Financial Risk Forecasting 2011,2017 Jon Danielsson, page 82 of 103

83 Quantile correlation Empirical exceedance plot Disney and IBM daily returns, January 1986 to June Data Normal Student t(3) Probability Financial Risk Forecasting 2011,2017 Jon Danielsson, page 83 of 103

84 Copulas and nonlinear dependence How do we model nonlinear dependence more formally? One approach is multivariate volatility models (see Chapter 3) Alternatively we can use copulas, which allow us to create multivariate distributions with a range of types of dependence Financial Risk Forecasting 2011,2017 Jon Danielsson, page 84 of 103

85 Intuition behind copulas A copula is a convenient way to obtain the dependence structure between two or more random variables, taking NLD into account We start with the marginal distributions of each random variable and end up with a copula function The copula function joins the random variables into a single multivariate distribution by using their correlations Financial Risk Forecasting 2011,2017 Jon Danielsson, page 85 of 103

86 Intuition behind copulas The random variables are transformed to uniform distributions using the probability integral transformation The copula models the dependence structure between these uniforms Since the probability integral transform is invertible, the copula also describes the dependence between the original random variables Financial Risk Forecasting 2011,2017 Jon Danielsson, page 86 of 103

87 Theory of copulas Suppose X and Y are two random variables representing returns of two different stocks, with densities f and g: X f and Y g Together, the joint distribution and marginal distributions are represented by the joint density h: (X,Y) h We focus separately on the marginal distributions (F,G) and the copula function C, which combines them into the joint distribution H Financial Risk Forecasting 2011,2017 Jon Danielsson, page 87 of 103

88 Theory of copulas We want to transform X and Y into random variables that are distributed uniformly between 0 and 1, removing individual information from the bivariate density h Theorem 1.1 Let a random variable X have a continuous distribution F, and define a new random variable U as: U = F(X) Then, regardless of the original distribution F: U Uniform(0,1) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 88 of 103

89 Theory of copulas Applying this transformation to X and Y we obtain: U = F(X) and V = G(Y) Using this we arrive at the following theorem Theorem 1.2 Let F be the distribution of X, G the distribution of Y and H the joint distribution of (X,Y). Assume that F and G are continuous. Then there exists a unique copula C such that: H(X,Y) = C(F(X),G(Y)) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 89 of 103

90 Theory of copulas In applications we are more likely to use densities: h(x,y) = f(x) g(y) C(F(X),G(Y)) The copula contains all dependence information in the original density h, but none of the individual information Note that we can construct a joint distribution from any two marginal distributions and any copula, and we can also extract the implied copula and marginal distributions from any joint distribution Financial Risk Forecasting 2011,2017 Jon Danielsson, page 90 of 103

91 The Gaussian copula One example of a copula is the Gaussian copula Let Φ( ) denote the normal (Gaussian) distribution and Φ 1 ( ) its inverse Let U,V [0,1] be uniform random variables and Φ ρ ( ) the bivariate normal with correlation coefficient ρ Then the Gaussian copula function can be written as: C(U,V) = Φ ρ (Φ 1 (U),Φ 1 (V)) This function allows us to join the two marginal distributions into a single bivariate distribution Financial Risk Forecasting 2011,2017 Jon Danielsson, page 91 of 103

92 Application of copulas To illustrate we use the same data on Disney and IBM as used before By comparing a scatterplot for simulated bivariate normal data with one for the empirical data, we see that the two do not have the same joint extremes Financial Risk Forecasting 2011,2017 Jon Danielsson, page 92 of 103

93 Prices & returns Data/Code Volatility Fat tails NLD Issues Copulas Gaussian scatterplot 10 Asset Asset 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 93 of 103

94 Empirical scatterplot Daily Disney and IBM returns, January 1986 to June IBM both crash together Disney Financial Risk Forecasting 2011,2017 Jon Danielsson, page 94 of 103

95 Application of copulas We estimate two copulas for the data, a Gaussian copula and a Student-t copula The copulas can be drawn in three dimensions Financial Risk Forecasting 2011,2017 Jon Danielsson, page 95 of 103

96 Fitted Gaussian copula Daily Disney and IBM returns, January 1986 to June Financial Risk Forecasting 2011,2017 Jon Danielsson, page 96 of 103

97 Fitted Student-t copula Daily Disney and IBM returns, January 1986 to June Financial Risk Forecasting 2011,2017 Jon Danielsson, page 97 of 103

98 Application of copulas It can be difficult to compare distributions by looking at three-dimensional graphs Contour plots may give a better comparison Financial Risk Forecasting 2011,2017 Jon Danielsson, page 98 of 103

99 Contours of Gaussian copula Daily Disney and IBM returns, January 1986 to June IBM Disney Financial Risk Forecasting 2011,2017 Jon Danielsson, page 99 of 103

100 Contours of Student-t copula Daily Disney and IBM returns, January 1986 to June IBM Disney Financial Risk Forecasting 2011,2017 Jon Danielsson, page 100 of 103

101 Clayton s copula As noted earlier, there are a number of copulas available One widely used is the Clayton copula, which allows for asymmetric dependence Parameter θ measures the strength of dependence We estimate a Clayton copula for the same data as before Financial Risk Forecasting 2011,2017 Jon Danielsson, page 101 of 103

102 Contours of Clayton s copula, θ = Asset Asset 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 102 of 103

103 Contours of Clayton s copula, θ = Daily Disney and IBM returns, January 1986 to June IBM Disney Financial Risk Forecasting 2011,2017 Jon Danielsson, page 103 of 103