# Financial Risk Forecasting Chapter 8 Backtesting and stresstesting

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Financial Risk Forecasting Chapter 8 Backtesting and stresstesting Jon Danielsson London School of Economics 2015 To accompany Financial Risk Forecasting Published by Wiley 2011 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 1 of 80

2 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 2 of 80

3 Introduction Financial Risk Forecasting 2011,2015 Jon Danielsson, page 3 of 80

4 The focus of this chapter is on Backtesting Application of backtesting Significance of backtests Bernoulli coverage test Testing the independence of violations Joint test Loss-function-based backtests Expected shortfall backtesting Problems with backtesting Stress testing Financial Risk Forecasting 2011,2015 Jon Danielsson, page 4 of 80

5 Notation W T Testing window size T = W E +W T Number of observations in a sample η t = 0,1 Indicates whether a VaR violation occurs (i.e., η t = 1) υ i,i = 0,1 Number of violations (i = 1) and no violations (i = 0) observed in W T υ ij Number of instances where j follows i in W T Financial Risk Forecasting 2011,2015 Jon Danielsson, page 5 of 80

6 Backtesting Financial Risk Forecasting 2011,2015 Jon Danielsson, page 6 of 80

7 Forecasting VaR - Example Imagine you have 10 years of data, from 2004 to 2014 It is 2006 and you are trying to forecast risk for January 1st 2006 You will be using 2004 and 2005 to forecast daily VaR for the first day of 2006 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 7 of 80

8 The 500 trading days in 2004 and 2005 constitute the first estimation window W E is then moved up by one day to obtain the risk forecast for the second day of 2006, etc. Start End VaR forecast 1/1/ /12/2005 VaR(1/1/2006) 2/1/2004 1/1/2006 VaR(2/1/2006)... 31/12/ /12/2014 VaR(31/12/2014) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 8 of 80

9 What is backtesting? Backtesting is a procedure that can be used to compare various risk models It aims to take ex ante VaR forecasts from a particular model and compare them with ex post realized return (i.e., historical observations) Whenever losses exceed VaR, a VaR violation is said to have occurred Financial Risk Forecasting 2011,2015 Jon Danielsson, page 9 of 80

10 Usefulness of backtesting Useful in identifying the weaknesses of risk-forecasting models and providing ideas for improvement Not informative about the causes of weaknesses Models that perform poorly during backtesting should have their assumptions and parameter estimates questioned Backtesting can prevent underestimation of VaR and, hence, ensure that a bank carries sufficiently high capital However, it can also reduce the likelihood of overestimating VaR, which can lead to excessive conservatism Financial Risk Forecasting 2011,2015 Jon Danielsson, page 10 of 80

11 Assessment of VaR forecasts should ideally be done by tracking the performance of a model in the future using operational criteria However: as violations are only observed infrequently, a long period of time would be required Backtesting evaluates VaR forecasts by checking how a VaR forecast model performs over a period in the past Financial Risk Forecasting 2011,2015 Jon Danielsson, page 11 of 80

12 Estimation window (W E ): the number of observations used to forecast risk. If different procedures or assumptions are compared, the estimation window is set to whichever one needs the highest number of observations Testing window (W T ): the data sample over which risk is forecast (i.e., the days where we have made a VaR forecast) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 12 of 80

13 The entire sample size T is equal to the sum of W E and W T : t = 1 Entire data sample t = T Financial Risk Forecasting 2011,2015 Jon Danielsson, page 13 of 80

14 The entire sample size T is equal to the sum of W E and W T : t = 1 Entire data sample t = T t = 1 First estimation window t = W E VaR(W E +1) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 14 of 80

15 The entire sample size T is equal to the sum of W E and W T : t = 1 Entire data sample t = T t = 1 First estimation window t = W E VaR(W E +1) t = 2 Second estimation window t = W E +1 VaR(W E +2) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 15 of 80

16 The entire sample size T is equal to the sum of W E and W T : t = 1 Entire data sample t = T t = 1 First estimation window t = W E VaR(W E +1) t = 2 Second estimation window t = W E +1 VaR(W E +2) t = 3 Third estimation window t = W E +2 VaR(W E +3) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 16 of 80

17 The entire sample size T is equal to the sum of W E and W T : t = 1 Entire data sample t = T t = 1 First estimation window t = W E VaR(W E +1) t = 2 Second estimation window t = W E +1 VaR(W E +2) t = 3 Third estimation window t = W E +2 VaR(W E +3). Financial Risk Forecasting 2011,2015 Jon Danielsson, page 17 of 80

18 The entire sample size T is equal to the sum of W E and W T : t = 1 Entire data sample t = T t = 1 First estimation window t = W E VaR(W E +1) t = 2 Second estimation window t = W E +1 VaR(W E +2) t = 3 Third estimation window t = W E +2 VaR(W E +3). t = T W E Last estimation window t = T 1 VaR(T) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 18 of 80

19 VaR forecasts can be compared with the actual outcome: the daily 2001 to 2009 returns are already known Instead of referring to calendar dates (e.g., 1/1/2004), refer to days by indexing the returns, assuming 250 trading days per year: y 1 is the return on 1/1/1999 y 2,500 is the return on the last day, 31/12/2009 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 19 of 80

20 The estimation window W E is set at 500 days, and the testing window W T is therefore 2,000 days: t t +W E 1 VaR(t +W E ) VaR(501) VaR(502)... 1,999 2,499 VaR(2,500) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 20 of 80

21 VaR violation If the actual return on a particular day exceeds the VaR forecast, then the VaR limit is said to have been violated VaR violation: an event such that: η t = { 1, if y t VaR t 0, if y t > VaR t. υ 1 is the count of η t = 1 and υ 0 is the count of η t = 0, which are simply obtained by: υ 1 = Ση t υ 0 = W T υ 1. Financial Risk Forecasting 2011,2015 Jon Danielsson, page 21 of 80

22 Violation ratios The main tools used in backtesting are violation ratios, where the actual number of VaR violations are compared with the expected value. Violation ratio: VR = Observed number of violations Expected number of violations = υ 1 p W T If the violation ratio is greater than one the VaR model underforecasts risk If smaller than one the model overforecasts risk Financial Risk Forecasting 2011,2015 Jon Danielsson, page 22 of 80

23 One can make a judgment on the quality of the VaR forecasts by recording the relative number of violations or use more sophisticated statistical techniques We record the violations as η t, which takes the value 1 when a violation occurs and 0 otherwise The number of violations are collected in the variable υ, where υ 1 is the number of violations and υ 0 is the number of days without violations These two add up to make the testing period Financial Risk Forecasting 2011,2015 Jon Danielsson, page 23 of 80

24 Market risk regulations Financial institutions regulated under the Basel II Accords are required to set aside a certain amount of capital due to market risk, credit risk and operational risk It is based on multiplying the maximum of previous day 1%VaR and 60 days average VaR by a constant, Ξ, which is determined by the number of violations that happened previously: ( ) Market risk capital t = Ξ t max VaR 1% t,var 1% t +constant Financial Risk Forecasting 2011,2015 Jon Danielsson, page 24 of 80

25 VaR 1% t is average reported 1% VaR over the previous 60 trading days The multiplication factor Ξ t varies with the number of violations, υ 1, that occurred in the previous 250 trading days - the required testing window length for backtesting in the Basel Accords. This is based on three ranges for the number of violations, named after the three colors of traffic lights: 3, if υ 1 4 (Green) Ξ t = 3+0.2(υ 1 4), if 5 υ 1 9 (Yellow) 4, if 10 υ 1 (Red). Financial Risk Forecasting 2011,2015 Jon Danielsson, page 25 of 80

26 Estimation window length The W E length is determined by the choice of VaR model and probability level Different methods have different data requirements (EWMA requires c. 30 days, HS needs at least 300 for VaR 1%, and GARCH even more) When making comparisons, the estimation window should be sufficiently large to accommodate the most stringent data criteria Even within the same method, it may be helpful to compare different window lengths Financial Risk Forecasting 2011,2015 Jon Danielsson, page 26 of 80

27 Testing window length VaR violations are infrequent events: with a 1% VaR, a violation is expected once every 100 days, so that 2.5 violations are expected per year Therefore, the actual sample size of violations is quite small, causing difficulties for statistical inference We might need perhaps 10 violations for reliable statistical analysis, or 4 years of data Financial Risk Forecasting 2011,2015 Jon Danielsson, page 27 of 80

28 Violation ratios VR=1 is expected, but how can we ascertain whether any other value is statistically significant? A useful rule of thumb: If VR [0.8,1.2] it is a good forecast If VR < 0.5 or VR > 1.5 the model is imprecise Both bounds narrow with increasing testing window lengths As a first attempt, one could plot the actual returns and VaR together, thereby facilitating a quick visual inspection of the quality of the VaR forecast Financial Risk Forecasting 2011,2015 Jon Danielsson, page 28 of 80

29 Application of backtesting Financial Risk Forecasting 2011,2015 Jon Danielsson, page 29 of 80

30 Volatility and VaR: extreme example 140% EWMA \$110 Volatility 100% 60% 20% 20% 60% 100% \$90 \$70 \$50 \$30 \$10 VaR Financial Risk Forecasting 2011,2015 Jon Danielsson, page 30 of 80

31 Volatility Volatility and VaR: extreme example 140% 100% 60% 20% 20% 60% 100% EWMA MA \$110 \$90 \$70 \$50 \$30 \$10 VaR Financial Risk Forecasting 2011,2015 Jon Danielsson, page 31 of 80

32 Volatility and VaR: extreme example Volatility 140% 100% 60% 20% 20% 60% 100% EWMA MA HS \$110 \$90 \$70 \$50 \$30 \$10 VaR Financial Risk Forecasting 2011,2015 Jon Danielsson, page 32 of 80

33 Volatility and VaR: extreme example Volatility 140% 100% 60% 20% 20% 60% 100% EWMA MA HS GARCH \$110 \$90 \$70 \$50 \$30 \$10 VaR Financial Risk Forecasting 2011,2015 Jon Danielsson, page 33 of 80

34 10% Volatility and VaR for S&P 500 EWMA \$120 \$100 Volatility 5% \$80 0% \$60 VaR 5% \$40 \$20 10% Jan 98 Jan 02 Jan 06 Jan 10 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 34 of 80

35 10% Volatility and VaR for S&P 500 EWMA MA \$120 \$100 Volatility 5% \$80 0% \$60 VaR 5% \$40 \$20 10% Jan 98 Jan 02 Jan 06 Jan 10 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 35 of 80

36 Volatility 10% 5% 0% Volatility and VaR for S&P 500 EWMA MA HS \$120 \$100 \$80 \$60 VaR 5% \$40 \$20 10% Jan 98 Jan 02 Jan 06 Jan 10 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 36 of 80

37 Volatility and VaR for S&P 500 Volatility 10% 5% 0% EWMA MA HS GARCH \$120 \$100 \$80 \$60 VaR 5% \$40 \$20 10% Jan 98 Jan 02 Jan 06 Jan 10 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 37 of 80

38 Volatility and VaR for S&P 500, focus on % 2% EWMA \$40 \$35 Volatility 1% 0% 1% 2% 3% \$30 \$25 \$20 \$15 VaR Jan 03 Apr 03 Jul 03 Oct 03 Jan 04 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 38 of 80

39 Volatility and VaR for S&P 500, focus on Volatility 3% 2% 1% 0% 1% 2% EWMA MA \$40 \$35 \$30 \$25 \$20 VaR 3% \$15 Jan 03 Apr 03 Jul 03 Oct 03 Jan 04 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 39 of 80

40 Volatility and VaR for S&P 500, focus on % 2% EWMA MA HS \$40 \$35 Volatility 1% 0% 1% 2% 3% \$30 \$25 \$20 \$15 VaR Jan 03 Apr 03 Jul 03 Oct 03 Jan 04 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 40 of 80

41 Volatility and VaR for S&P 500, focus on Volatility 3% 2% 1% 0% 1% EWMA MA HS GARCH \$40 \$35 \$30 \$25 VaR 2% 3% \$20 \$15 Jan 03 Apr 03 Jul 03 Oct 03 Jan 04 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 41 of 80

42 Volatility and VaR for S&P 500, focus on 2008 Volatility 10% 5% 0% EWMA \$100 \$80 \$60 VaR 5% \$40 10% Jul 08 Sep 08 Nov 08 Jan 09 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 42 of 80

43 Volatility and VaR for S&P 500, focus on 2008 Volatility 10% 5% 0% EWMA MA \$100 \$80 \$60 VaR 5% \$40 10% Jul 08 Sep 08 Nov 08 Jan 09 \$20 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 43 of 80

44 Volatility and VaR for S&P 500, focus on 2008 Volatility 10% 5% 0% EWMA MA HS \$100 \$80 \$60 VaR 5% \$40 10% \$20 Jul 08 Sep 08 Nov 08 Jan 09 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 44 of 80

45 Volatility and VaR for S&P 500, focus on 2008 Volatility 10% 5% 0% 5% EWMA MA HS GARCH \$120 \$100 \$80 \$60 \$40 VaR 10% \$20 Jul 08 Sep 08 Nov 08 Jan 09 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 45 of 80

46 Significance of Backtests Financial Risk Forecasting 2011,2015 Jon Danielsson, page 46 of 80

47 Focus on two issues: The number of violations (tested by the unconditional coverage) Clustering (tested by independence tests) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 47 of 80

48 Unconditional coverage property Ensures that the theoretical confidence level p matches the empirical probability of violation: For a 1% VaR backtest, we would expect to observe a VaR violation 1% of the time. If, instead, violations are observed more often, say 5% of the time, the VaR model is systematically underestimating risk at the 1% level In other words, the 1% VaR produced from the model is in reality the 5% VaR Financial Risk Forecasting 2011,2015 Jon Danielsson, page 48 of 80

49 Independence property Requires any two observations in the hit sequence to be independent of each other Intuitively, the fact that a violation has been observed today should not convey any information about the likelihood of observing a violation tomorrow If VaR violations cluster, we can predict a violation today if there was one yesterday, and therefore the probability of a loss exceeding 1% VaR today is higher than 1% In this case, a good VaR model would have increased the 1% VaR forecast following a violation Financial Risk Forecasting 2011,2015 Jon Danielsson, page 49 of 80

50 Pros and cons Unconditional coverage and independence are distinct and it is entirely possible that a VaR model satisfying one of them would not satisfy the other The main downside of these tests is that they rely on asymptotic distributions Given that violations are rare events, the effective sample size is relatively small and, therefore, tests may not be as robust as we would like One may be better off obtaining confidence bounds by means of simulations Financial Risk Forecasting 2011,2015 Jon Danielsson, page 50 of 80

51 Bernoulli coverage test η t is a sequence of Bernoulli-distributed random variables We then use the Bernoulli coverage test to ascertain the proportion of violations The null hypothesis for VaR violations is: H 0 : η B(p), where B stands for the Bernoulli distribution. Financial Risk Forecasting 2011,2015 Jon Danielsson, page 51 of 80

52 The Bernoulli density is given by: (1 p) 1 ηt (p) ηt,η t = 0,1. The probability p can be estimated by: ˆp = υ 1 W T The likelihood function is given by: L U (ˆp) = T t=w E +1 (1 ˆp) 1 ηt (ˆp) ηt = (1 ˆp) υ 0 (ˆp) υ 1 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 52 of 80

53 Denote this as the unrestricted likelihood function because it uses estimated probability ˆp in contrast to the likelihood function below where we restrict the probability value to p. Under the H 0, p = ˆp, so the restricted likelihood function is: L R (p) = T t=w E +1 = (1 p) υ 0 (p) υ 1 (1 p) 1 ηt (p) ηt Financial Risk Forecasting 2011,2015 Jon Danielsson, page 53 of 80

54 We can use an LR test to see whether L R = L U or, equivalently, whether p = ˆp: LR = 2(logL U (p) logl U (ˆp)) = 2log (1 ˆp)υu (ˆp) υ 1 (1 p) υ 0 (p) υ 1 asymptotic χ 2 (1) Choosing a 5% significance level for the test, the null hypothesis is rejected if LR > 3.84 The choice of significance level affects the power of the test: a low type I error implies a higher type II error and therefore a lower power for the test. Financial Risk Forecasting 2011,2015 Jon Danielsson, page 54 of 80

55 The Bernoulli coverage test is nonparametric in the sense that it does not assume a distribution for the returns and often provides good benchmarks for the assessment of accuracy of VaR models It does not have much power when sample sizes are small, like the one-year size specified in the Basel Accords Financial Risk Forecasting 2011,2015 Jon Danielsson, page 55 of 80

56 Testing the independence of violations It is also of interest to test whether violations cluster (i.e., whether all violations happen one after the other), indicating a sequence of losses since violations should theoretically spread out over time Calculate the probabilities of two consecutive violations (i.e., p 11 ) and the probability of a violation if there was no violation on the previous day (i.e., p 01 ) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 56 of 80

57 More generally, the probability that: p i,j = Pr(η t = i η t 1 = j) where i and j are either 0 or 1 The first-order transition probability matrix is defined as: ( ) 1 p01 p Π 1 = 01 1 p 11 p 11 The restricted likelihood function, where the transition matrix from the null hypothesis is used since the hit sequence is Bernoulli distributed, is: L R (Π 1 ) = (1 p 01 ) υ 00 p υ (1 p 11) υ 10 p υ where υ ij is the number of observations where j follows i. Financial Risk Forecasting 2011,2015 Jon Danielsson, page 57 of 80

58 Maximum likelihood (ML) estimates are obtained by maximizing L R (Π 1 ): υ 01 υ 00 ˆΠ 1 = υ 00 +υ υ υ 00 +υ υ υ 10 +υ 11 υ 10 +υ 11 Under the null hypothesis of no clustering, the probability of a violation tomorrow does not depend on today seeing a violation Financial Risk Forecasting 2011,2015 Jon Danielsson, page 58 of 80

59 Then, p 01 = p 11 = p and the estimated transition matrix is simply: ( ) 1 ˆp ˆp ˆΠ 0 = 1 ˆp ˆp where υ 01 +υ 11 ˆp = υ 00 +υ 10 +υ 01 +υ 11 The unrestricted likelihood function according to null hypothesis uses the estimated transition matrix and is: L U (ˆΠ 0 ) = (1 ˆp) υ 00+υ 10ˆp υ 01+υ 11 Financial Risk Forecasting 2011,2015 Jon Danielsson, page 59 of 80

60 Likelihood ratio test The likelihood ratio test: ( LR = 2 logl U (ˆΠ 0 ) logl R (ˆΠ 1 ) ) asymptotic χ 2 (1) This test does not depend on true p and only tests for independence Financial Risk Forecasting 2011,2015 Jon Danielsson, page 60 of 80

61 The main problem with tests of this sort is that they must specify the particular way in which independence is breached However, there are many possible ways in which the independence property is not fulfilled: The test will have no power to detect departures from independence if the likelihood of VaR being violated today depends on whether VaR was violated 2 days ago not on yesterdays VaR being violated. Financial Risk Forecasting 2011,2015 Jon Danielsson, page 61 of 80

62 Testing S&P 500: comparison of the four VaR models January 30, 1998 to December 31, 2009 (W T = 3000) with crisis Model Coverage test Independence test Test statistic p-value Test statistic p-value EWMA MA HS GARCH Financial Risk Forecasting 2011,2015 Jon Danielsson, page 62 of 80

63 Testing S&P 500: comparison of the four VaR models January 30, 1998 to November 1, 2006 (W T = 2000) - without crisis Model Coverage test Independence test Test statistic p-value Test statistic p-value EWMA MA HS GARCH Financial Risk Forecasting 2011,2015 Jon Danielsson, page 63 of 80

64 Joint test We can jointly test whether violations are significantly different from those expected and whether there is violation clustering by constructing the following test statistic: LR(joint) = LR(coverage)+LR(independence) χ 2 (2) The joint test has less power to reject a VaR model which only satisfies one of the two properties Individual tests should be used if the user has some prior knowledge of weaknesses in the VaR model. Financial Risk Forecasting 2011,2015 Jon Danielsson, page 64 of 80

65 Loss-function-based backtests An alternative backtest could be constructed based on a general loss function: l(var t (p),y t ) It would take into consideration the magnitude of VaR violation. An example of such a loss function is: l(y,x) = 1+(y x) 2,y x Otherwise, it takes the value zero If y indicates returns and x negative VaR, then we would typically calculate sample average loss as: ˆL = 1 W T T t=w E +1 l(y t,var t (p)) Financial Risk Forecasting 2011,2015 Jon Danielsson, page 65 of 80

66 Pros and cons Advantage of loss-function-based backtests The functional form of the loss function is flexible and can be tailored to address specific concerns Disadvantages of loss-function-based backtests In order to determine whether ˆL is too large relative to what is expectedan explicit assumption about P/L distribution needs to be made. A null hypothesis for ˆL should be mandatory Financial Risk Forecasting 2011,2015 Jon Danielsson, page 66 of 80

67 Given the uncertainty over the true P/L distribution, a finding that ˆL is too large relative to what is expected could be due to: Inaccuracy in the VaR model An inaccurately assumed distribution In other words, the test is a joint test of the VaR model and the P/L distribution. This is problematic Loss-function-based backtests may consequently be better suited for discriminating among competing VaR models than judging the absolute accuracy of a single model Financial Risk Forecasting 2011,2015 Jon Danielsson, page 67 of 80

68 Expected Shortfall Backtesting Financial Risk Forecasting 2011,2015 Jon Danielsson, page 68 of 80

69 It is harder to backtest expected shortfall ES than VaR because we are testing an expectation rather than a single quantile Fortunately, there exists a simple methodology for backtesting ES that is analogous to the use of violation ratios for VaR For days when VaR is violated, normalized shortfall NS is calculated as: NS t = y t ES t where ES t is the observed ES on day t Financial Risk Forecasting 2011,2015 Jon Danielsson, page 69 of 80

70 From the definition of ES, the expected Y t, VaR is violated, is: E[Y t Y t < VaR t ] ES t = 1 Therefore, average NS, denoted by NS, should be one and this forms our null hypothesis: H 0 : NS = 1 In what follows, we opted to implement just the EWMA and HS versions, since the other two (MA and GARCH) are quite similar Financial Risk Forecasting 2011,2015 Jon Danielsson, page 70 of 80

71 The reliability of any ES backtest procedure is likely to be much lower than that of VaR backtest procedures: With ES, we are testing whether the mean of returns on days when VaR is violated is the same as average ES on these days. Much harder to create formal tests to ascertain whether normalized ES equals one or not than the coverage tests developed above for VaR violations Hence, backtesting ES requires many more observations than backtesting VaR In instances where ES is obtained directly from VaR, and gives the same signal as VaR (i.e., when VaR is subadditive), it is better to simply use VaR Financial Risk Forecasting 2011,2015 Jon Danielsson, page 71 of 80

72 Problems with Backtesting Financial Risk Forecasting 2011,2015 Jon Danielsson, page 72 of 80

73 Backtesting assumes that there have been no structural breaks in the data throughout the sample period: But financial markets are continually evolving, and new technologies, assets, markets and institutions affect the statistical properties of market prices Unlikely that the statistical properties of market data in the 1990s are the same as today, implying that a risk model that worked well then might not work well today Financial Risk Forecasting 2011,2015 Jon Danielsson, page 73 of 80

74 Data mining and intellectual integrity: Backtesting is only statistically valid if we have no ex ante knowledge of the data in the testing window If we iterate the process, continually refining the risk model with the same test data and thus learning about the events in the testing window, the model will be fitted to those particular outcomes, violating underlying statistical assumptions Financial Risk Forecasting 2011,2015 Jon Danielsson, page 74 of 80

75 Stresstesting Financial Risk Forecasting 2011,2015 Jon Danielsson, page 75 of 80

76 The purpose of stresstesting is to create artificial market outcomes in order to see how risk management systems and risk models cope with the artificial event, and to assess the ability of a bank to survive a large shock The gap that stress testing aims to fill is model failure to encounter rare situations that could cause a severe loss, since backtesting relies on recent historical data Financial Risk Forecasting 2011,2015 Jon Danielsson, page 76 of 80

77 Scenario analysis The main aim is to come up with scenarios that are not well represented in historical data but are nonetheless possible and detrimental to portfolio performance We then revalue the portfolio in this hypothetical environment and obtain an estimate of maximum loss This procedure then allows the bank to set aside enough capital for such an eventuality Financial Risk Forecasting 2011,2015 Jon Danielsson, page 77 of 80

78 Examples of historical scenarios Scenario Period Stock market crash October 1987 ERM crisis September 1992 Bond market crash April 1994 Asian currency crisis Summer 1997 LTCM and Russia crisis August 1998 Global crisis Eurozone crisis Since 2010 Scenarios can be classified into two broad types: Simulating shocks that have never occurred or are more likely to occur than historical data suggest Simulating shocks that reflect permanent or temporary structural breakswhere historical relationships do not hold Financial Risk Forecasting 2011,2015 Jon Danielsson, page 78 of 80

79 Issues in scenario analysis The results indicate that banks should set aside a large amount of capital to absorb the worst case loss, which is usually not practical But there are other actions that a bank can take such as buying insurance for the events in question and changing the composition of the portfolio to reduce exposure to a particular market variable The lack of a mechanism to reliably judge the probability of a stress test scenario since they are both highly subjective and difficult to pinpoint The potential for feedback effects, or endogenous risk since the focus is on institution-level risk, where feedback effects are disregarded Financial Risk Forecasting 2011,2015 Jon Danielsson, page 79 of 80

80 Scenario analysis and risk models It is straightforward to integrate scenario analysis with risk models in use. Assume a risk manager has assigned a probability ξ to a particular scenario and the potential loss arising from it. The probability can be purely a subjective judgment from the managers experience or could be derived from a model Financial Risk Forecasting 2011,2015 Jon Danielsson, page 80 of 80

### Effective Techniques for Stress Testing and Scenario Analysis

Effective Techniques for Stress Testing and Scenario Analysis Om P. Arya Federal Reserve Bank of New York November 4 th, 2008 Mumbai, India The views expressed here are not necessarily of the Federal Reserve

### Contents. List of Figures. List of Tables. List of Examples. Preface to Volume IV

Contents List of Figures List of Tables List of Examples Foreword Preface to Volume IV xiii xvi xxi xxv xxix IV.1 Value at Risk and Other Risk Metrics 1 IV.1.1 Introduction 1 IV.1.2 An Overview of Market

### Reducing Active Return Variance by Increasing Betting Frequency

Reducing Active Return Variance by Increasing Betting Frequency Newfound Research LLC February 2014 For more information about Newfound Research call us at +1-617-531-9773, visit us at www.thinknewfound.com

### Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

### An Aggregate Reserve Methodology for Health Claims

An Aggregate Reserve Methodology for Health Claims R = P M M Unpaid Paid F F Upaid Paid Where: R = The estimated claim reserve P = Observed paid claims M Paid, M Unpaid = Portion of exposure basis that

### Financial Risk Forecasting Chapter 3 Multivariate volatility models

Financial Risk Forecasting Chapter 3 Multivariate volatility models Jon Danielsson 2016 London School of Economics To accompany Financial Risk Forecasting www.financialriskforecasting.com Published by

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### Testing on proportions

Testing on proportions Textbook Section 5.4 April 7, 2011 Example 1. X 1,, X n Bernolli(p). Wish to test H 0 : p p 0 H 1 : p > p 0 (1) Consider a related problem The likelihood ratio test is where c is

### OPTIMAL PORTFOLIO SELECTION WITH A SHORTFALL PROBABILITY CONSTRAINT: EVIDENCE FROM ALTERNATIVE DISTRIBUTION FUNCTIONS. Abstract

The Journal of Financial Research Vol. XXXIII, No. 1 Pages 77 10 Spring 010 OPTIMAL PORTFOLIO SELECTION WITH A SHORTFALL PROBABILITY CONSTRAINT: EVIDENCE FROM ALTERNATIVE DISTRIBUTION FUNCTIONS Yalcin

### The Best of Both Worlds:

The Best of Both Worlds: A Hybrid Approach to Calculating Value at Risk Jacob Boudoukh 1, Matthew Richardson and Robert F. Whitelaw Stern School of Business, NYU The hybrid approach combines the two most

### Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk

Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day

### Likelihood Approaches for Trial Designs in Early Phase Oncology

Likelihood Approaches for Trial Designs in Early Phase Oncology Clinical Trials Elizabeth Garrett-Mayer, PhD Cody Chiuzan, PhD Hollings Cancer Center Department of Public Health Sciences Medical University

### Models for Count Data With Overdispersion

Models for Count Data With Overdispersion Germán Rodríguez November 6, 2013 Abstract This addendum to the WWS 509 notes covers extra-poisson variation and the negative binomial model, with brief appearances

### Volatility Models in Risk Management

Volatility Models in Risk Management The volatility models described in this book are useful for estimating Value-at-Risk (VaR), a measure introduced by the Basel Committee in 1996. In many countries,

### Least Squares Estimation

Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

### Dr Christine Brown University of Melbourne

Enhancing Risk Management and Governance in the Region s Banking System to Implement Basel II and to Meet Contemporary Risks and Challenges Arising from the Global Banking System Training Program ~ 8 12

### An Alternative Route to Performance Hypothesis Testing

EDHEC-Risk Institute 393-400 promenade des Anglais 06202 Nice Cedex 3 Tel.: +33 (0)4 93 18 32 53 E-mail: research@edhec-risk.com Web: www.edhec-risk.com An Alternative Route to Performance Hypothesis Testing

### Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics

Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics Jan Krhovják Basic Idea Behind the Statistical Tests Generated random sequences properties as sample drawn from uniform/rectangular

### Statistics in Retail Finance. Chapter 6: Behavioural models

Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics:- Behavioural

### Robust Conditional Variance Estimation and Value-at-Risk. Cherif Guermat and Richard D.F. Harris

Robust Conditional Variance Estimation and Value-at-Risk Cherif Guermat and Richard D.F. Harris School of Business and Economics University of Exeter December 000 Abstract A common approach to estimating

### Financial Assets Behaving Badly The Case of High Yield Bonds. Chris Kantos Newport Seminar June 2013

Financial Assets Behaving Badly The Case of High Yield Bonds Chris Kantos Newport Seminar June 2013 Main Concepts for Today The most common metric of financial asset risk is the volatility or standard

### Validating Market Risk Models: A Practical Approach

Validating Market Risk Models: A Practical Approach Doug Gardner Wells Fargo November 2010 The views expressed in this presentation are those of the author and do not necessarily reflect the position of

### Tail-Dependence an Essential Factor for Correctly Measuring the Benefits of Diversification

Tail-Dependence an Essential Factor for Correctly Measuring the Benefits of Diversification Presented by Work done with Roland Bürgi and Roger Iles New Views on Extreme Events: Coupled Networks, Dragon

### Validation of Internal Rating and Scoring Models

Validation of Internal Rating and Scoring Models Dr. Leif Boegelein Global Financial Services Risk Management Leif.Boegelein@ch.ey.com 07.09.2005 2005 EYGM Limited. All Rights Reserved. Agenda 1. Motivation

### Optimal Risk Management Before, During and After the 2008-09 Financial Crisis

Optimal Risk Management Before, During and After the 2008-09 Financial Crisis Michael McAleer Econometric Institute Erasmus University Rotterdam and Department of Applied Economics National Chung Hsing

### Understanding Margins. Frequently asked questions on margins as applicable for transactions on Cash and Derivatives segments of NSE and BSE

Understanding Margins Frequently asked questions on margins as applicable for transactions on Cash and Derivatives segments of NSE and BSE Jointly published by National Stock Exchange of India Limited

### Multinomial and Ordinal Logistic Regression

Multinomial and Ordinal Logistic Regression ME104: Linear Regression Analysis Kenneth Benoit August 22, 2012 Regression with categorical dependent variables When the dependent variable is categorical,

### OBJECTIVE ASSESSMENT OF FORECASTING ASSIGNMENTS USING SOME FUNCTION OF PREDICTION ERRORS

OBJECTIVE ASSESSMENT OF FORECASTING ASSIGNMENTS USING SOME FUNCTION OF PREDICTION ERRORS CLARKE, Stephen R. Swinburne University of Technology Australia One way of examining forecasting methods via assignments

### EVALUATING THE PERFORMANCE CHARACTERISTICS OF THE CBOE S&P 500 PUTWRITE INDEX

DECEMBER 2008 Independent advice for the institutional investor EVALUATING THE PERFORMANCE CHARACTERISTICS OF THE CBOE S&P 500 PUTWRITE INDEX EXECUTIVE SUMMARY The CBOE S&P 500 PutWrite Index (ticker symbol

### Overview. October 2013. Investment Portfolios & Products. Approved for public distribution. Investment Advisory Services

Equity Risk Management Strategy Overview Approved for public distribution October 2013 Services Portfolios & Products Equity Risk Management Strategy* Tactical allocation strategy that seeks to adjust

### STATISTICAL SIGNIFICANCE AND THE STANDARD OF PROOF IN ANTITRUST DAMAGE QUANTIFICATION

Lear Competition Note STATISTICAL SIGNIFICANCE AND THE STANDARD OF PROOF IN ANTITRUST DAMAGE QUANTIFICATION Nov 2013 Econometric techniques can play an important role in damage quantification cases for

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### MAKE VOLATILITY YOUR ASSET. LJM PRESERVATION & GROWTH FUND

MAKE VOLATILITY YOUR ASSET. LJM PRESERVATION & GROWTH FUND MARKETS CAN BE ROCKY. HARNESS VOLATILITY WITH LIQUID ALTERNATIVES. Liquid alternatives are mutual fund investments that derive value from sources

### Is the Basis of the Stock Index Futures Markets Nonlinear?

University of Wollongong Research Online Applied Statistics Education and Research Collaboration (ASEARC) - Conference Papers Faculty of Engineering and Information Sciences 2011 Is the Basis of the Stock

### Understanding Margins

Understanding Margins Frequently asked questions on margins as applicable for transactions on Cash and Derivatives segments of NSE and BSE Jointly published by National Stock Exchange of India Limited

Market Risk Control Andrew Threadgold, Group Head of Market Risk June 4, 2007 Key messages Strong risk management and control culture Comprehensive framework of market risk controls Highly qualified risk

### PITFALLS IN TIME SERIES ANALYSIS. Cliff Hurvich Stern School, NYU

PITFALLS IN TIME SERIES ANALYSIS Cliff Hurvich Stern School, NYU The t -Test If x 1,..., x n are independent and identically distributed with mean 0, and n is not too small, then t = x 0 s n has a standard

### How to assess the risk of a large portfolio? How to estimate a large covariance matrix?

Chapter 3 Sparse Portfolio Allocation This chapter touches some practical aspects of portfolio allocation and risk assessment from a large pool of financial assets (e.g. stocks) How to assess the risk

### ALM Stress Testing: Gaining Insight on Modeled Outcomes

ALM Stress Testing: Gaining Insight on Modeled Outcomes 2013 Financial Conference Ballantyne Resort Charlotte, NC September 18, 2013 Thomas E. Bowers, CFA Vice President Client Services ZM Financial Systems

### FINANCIAL ECONOMICS OPTION PRICING

OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.

### An introduction to Value-at-Risk Learning Curve September 2003

An introduction to Value-at-Risk Learning Curve September 2003 Value-at-Risk The introduction of Value-at-Risk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk

### A Sound Approach to Stock Investment for Individual Investors Unmasking Common Fallacies

A Sound Approach to Stock Investment for Individual Investors Unmasking Common Fallacies By Masahiro Shidachi Financial Research Group 1. Introduction As interest rates in Japan continually mark new historical

### Building and Interpreting Custom Investment Benchmarks

Building and Interpreting Custom Investment Benchmarks A White Paper by Manning & Napier www.manning-napier.com Unless otherwise noted, all fi gures are based in USD. 1 Introduction From simple beginnings,

### A Review of Cross Sectional Regression for Financial Data You should already know this material from previous study

A Review of Cross Sectional Regression for Financial Data You should already know this material from previous study But I will offer a review, with a focus on issues which arise in finance 1 TYPES OF FINANCIAL

### Industry Environment and Concepts for Forecasting 1

Table of Contents Industry Environment and Concepts for Forecasting 1 Forecasting Methods Overview...2 Multilevel Forecasting...3 Demand Forecasting...4 Integrating Information...5 Simplifying the Forecast...6

### VaR-x: Fat tails in nancial risk management

VaR-x: Fat tails in nancial risk management Ronald Huisman, Kees G. Koedijk, and Rachel A. J. Pownall To ensure a competent regulatory framework with respect to value-at-risk (VaR) for establishing a bank's

### Diversifying with Negatively Correlated Investments. Monterosso Investment Management Company, LLC Q1 2011

Diversifying with Negatively Correlated Investments Monterosso Investment Management Company, LLC Q1 2011 Presentation Outline I. Five Things You Should Know About Managed Futures II. Diversification and

### Review of Likelihood Theory

Appendix A Review of Likelihood Theory This is a brief summary of some of the key results we need from likelihood theory. A.1 Maximum Likelihood Estimation Let Y 1,..., Y n be n independent random variables

### Technical Report. Value at Risk (VaR)

Technical Report On Value at Risk (VaR) Submitted To: Chairman of Department of Statistics University of Karachi MUNEER AFZAL EP-042922 BS. Actuarial Sciences & Risk Management Final Year (Last Semester)

### Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

### Lecture 12/13 Bond Pricing and the Term Structure of Interest Rates

1 Lecture 1/13 Bond Pricing and the Term Structure of Interest Rates Alexander K. Koch Department of Economics, Royal Holloway, University of London January 14 and 1, 008 In addition to learning the material

### Risk Management for Fixed Income Portfolios

Risk Management for Fixed Income Portfolios Strategic Risk Management for Credit Suisse Private Banking & Wealth Management Products (SRM PB & WM) August 2014 1 SRM PB & WM Products Risk Management CRO

### Market Risk for Single Trading Positions

Chapter 6 Market Risk for Single Trading Positions Market risk is the risk that the market value of trading positions will be adversely influenced by changes in prices and/or interest rates. For banks,

### Volatility modeling in financial markets

Volatility modeling in financial markets Master Thesis Sergiy Ladokhin Supervisors: Dr. Sandjai Bhulai, VU University Amsterdam Brian Doelkahar, Fortis Bank Nederland VU University Amsterdam Faculty of

### Chapter 4: Vector Autoregressive Models

Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Reject Inference in Credit Scoring. Jie-Men Mok

Reject Inference in Credit Scoring Jie-Men Mok BMI paper January 2009 ii Preface In the Master programme of Business Mathematics and Informatics (BMI), it is required to perform research on a business

### Regime Switching Models: An Example for a Stock Market Index

Regime Switching Models: An Example for a Stock Market Index Erik Kole Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam April 2010 In this document, I discuss in detail

### Bayesian logistic betting strategy against probability forecasting. Akimichi Takemura, Univ. Tokyo. November 12, 2012

Bayesian logistic betting strategy against probability forecasting Akimichi Takemura, Univ. Tokyo (joint with Masayuki Kumon, Jing Li and Kei Takeuchi) November 12, 2012 arxiv:1204.3496. To appear in Stochastic

### Sales forecasting # 2

Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univ-rennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting

### Statistical Machine Learning

Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

### The Trip Scheduling Problem

The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems

### Relationship among crude oil prices, share prices and exchange rates

Relationship among crude oil prices, share prices and exchange rates Do higher share prices and weaker dollar lead to higher crude oil prices? Akira YANAGISAWA Leader Energy Demand, Supply and Forecast

### I.e., the return per dollar from investing in the shares from time 0 to time 1,

XVII. SECURITY PRICING AND SECURITY ANALYSIS IN AN EFFICIENT MARKET Consider the following somewhat simplified description of a typical analyst-investor's actions in making an investment decision. First,

### FIN 683 Financial Institutions Management Market Risk

FIN 683 Financial Institutions Management Market Risk Professor Robert B.H. Hauswald Kogod School of Business, AU Overview Market risk: essentially a price risk a FI might need to liquidate a position

### CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

### Testing Simple Markov Structures for Credit Rating Transitions

Testing Simple Markov Structures for Credit Rating Transitions Nicholas M. Kiefer and C. Erik Larson OCC Economics Working Paper 2004-3 Abstract Models abound that analyze changes in credit quality. These

### Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

### Yao Zheng University of New Orleans. Eric Osmer University of New Orleans

ABSTRACT The pricing of China Region ETFs - an empirical analysis Yao Zheng University of New Orleans Eric Osmer University of New Orleans Using a sample of exchange-traded funds (ETFs) that focus on investing

### LDA at Work: Deutsche Bank s Approach to Quantifying Operational Risk

LDA at Work: Deutsche Bank s Approach to Quantifying Operational Risk Workshop on Financial Risk and Banking Regulation Office of the Comptroller of the Currency, Washington DC, 5 Feb 2009 Michael Kalkbrener

### Implementing Reveleus: A Case Study

Implementing Reveleus: A Case Study Implementing Reveleus: A Case Study Reveleus is a state of the art risk management tool that assists financial institutions in measuring and managing risks. It encompasses

### 3. Nonparametric methods

3. Nonparametric methods If the probability distributions of the statistical variables are unknown or are not as required (e.g. normality assumption violated), then we may still apply nonparametric tests

### Multiple Testing. Joseph P. Romano, Azeem M. Shaikh, and Michael Wolf. Abstract

Multiple Testing Joseph P. Romano, Azeem M. Shaikh, and Michael Wolf Abstract Multiple testing refers to any instance that involves the simultaneous testing of more than one hypothesis. If decisions about

### A Guide to the Insider Buying Investment Strategy

Mar-03 Aug-03 Jan-04 Jun-04 Nov-04 Apr-05 Sep-05 Feb-06 Jul-06 Dec-06 May-07 Oct-07 Mar-08 Aug-08 Jan-09 Jun-09 Nov-09 Apr-10 Sep-10 Mar-03 Jul-03 Nov-03 Mar-04 Jul-04 Nov-04 Mar-05 Jul-05 Nov-05 Mar-06

### CAB TRAVEL TIME PREDICTI - BASED ON HISTORICAL TRIP OBSERVATION

CAB TRAVEL TIME PREDICTI - BASED ON HISTORICAL TRIP OBSERVATION N PROBLEM DEFINITION Opportunity New Booking - Time of Arrival Shortest Route (Distance/Time) Taxi-Passenger Demand Distribution Value Accurate

### The Empirical Approach to Interest Rate and Credit Risk in a Fixed Income Portfolio

www.empirical.net Seattle Portland Eugene Tacoma Anchorage March 27, 2013 The Empirical Approach to Interest Rate and Credit Risk in a Fixed Income Portfolio By Erik Lehr In recent weeks, market news about

### The Emperor has no Clothes: Limits to Risk Modelling

The Emperor has no Clothes: Limits to Risk Modelling Jón Daníelsson Financial Markets Group London School of Economics www.riskresearch.org September 2001 Forthcoming: Journal of Money and Finance Abstract

### ESTIMATION PORTFOLIO VAR WITH THREE DIFFERENT METHODS: FINANCIAL INSTITUTION RISK MANAGEMENT APPROACH

«ΣΠΟΥΔΑΙ», Τόμος 54, Τεύχος 2ο, (2004) / «SPOUDAI», Vol. 54, No 2, (2004), University of Piraeus, pp. 59-83 ESTIMATION PORTFOLIO VAR WITH THREE DIFFERENT METHODS: FINANCIAL INSTITUTION RISK MANAGEMENT

Stress Testing Trading Desks Michael Sullivan Office of the Comptroller of the Currency Paper presented at the Expert Forum on Advanced Techniques on Stress Testing: Applications for Supervisors Hosted

### Algorithmic Trading Session 1 Introduction. Oliver Steinki, CFA, FRM

Algorithmic Trading Session 1 Introduction Oliver Steinki, CFA, FRM Outline An Introduction to Algorithmic Trading Definition, Research Areas, Relevance and Applications General Trading Overview Goals

### Setting the scene. by Stephen McCabe, Commonwealth Bank of Australia

Establishing risk and reward within FX hedging strategies by Stephen McCabe, Commonwealth Bank of Australia Almost all Australian corporate entities have exposure to Foreign Exchange (FX) markets. Typically

### Poisson Models for Count Data

Chapter 4 Poisson Models for Count Data In this chapter we study log-linear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the

### Volatility in the Overnight Money-Market Rate

5 Volatility in the Overnight Money-Market Rate Allan Bødskov Andersen, Economics INTRODUCTION AND SUMMARY This article analyses the day-to-day fluctuations in the Danish overnight money-market rate during

### Testing Market Efficiency in a Fixed Odds Betting Market

WORKING PAPER SERIES WORKING PAPER NO 2, 2007 ESI Testing Market Efficiency in a Fixed Odds Betting Market Robin Jakobsson Department of Statistics Örebro University robin.akobsson@esi.oru.se By Niklas

### Basics of Statistical Machine Learning

CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar

### Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

### Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

### Risk Visualization: Presenting Data to Facilitate Better Risk Management

Track 4: New Dimensions in Financial Risk Management Risk Visualization: Presenting Data to Facilitate Better Risk Management 1:30pm 2:20pm Presenter: Jeffrey Bohn Senior Managing Director, Head of Portfolio

### ARE STOCK PRICES PREDICTABLE? by Peter Tryfos York University

ARE STOCK PRICES PREDICTABLE? by Peter Tryfos York University For some years now, the question of whether the history of a stock's price is relevant, useful or pro table in forecasting the future price

### Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans

Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Challenges for defined contribution plans While Eastern Europe is a prominent example of the importance of defined

### Pricing and Strategy for Muni BMA Swaps

J.P. Morgan Management Municipal Strategy Note BMA Basis Swaps: Can be used to trade the relative value of Libor against short maturity tax exempt bonds. Imply future tax rates and can be used to take

### Test of Hypotheses. Since the Neyman-Pearson approach involves two statistical hypotheses, one has to decide which one

Test of Hypotheses Hypothesis, Test Statistic, and Rejection Region Imagine that you play a repeated Bernoulli game: you win \$1 if head and lose \$1 if tail. After 10 plays, you lost \$2 in net (4 heads

### AT&T Global Network Client for Windows Product Support Matrix January 29, 2015

AT&T Global Network Client for Windows Product Support Matrix January 29, 2015 Product Support Matrix Following is the Product Support Matrix for the AT&T Global Network Client. See the AT&T Global Network

### Multivariate Analysis of Ecological Data

Multivariate Analysis of Ecological Data MICHAEL GREENACRE Professor of Statistics at the Pompeu Fabra University in Barcelona, Spain RAUL PRIMICERIO Associate Professor of Ecology, Evolutionary Biology

### Java Modules for Time Series Analysis

Java Modules for Time Series Analysis Agenda Clustering Non-normal distributions Multifactor modeling Implied ratings Time series prediction 1. Clustering + Cluster 1 Synthetic Clustering + Time series

### Sample Size and Power in Clinical Trials

Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance

### The Delta Method and Applications

Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and