Financial Risk Forecasting Chapter 5 Implementing Risk Forecasts

Transcription

1 Financial Risk Forecasting Chapter 5 Implementing Risk Forecasts Jon Danielsson 2017 London School of Economics To accompany Financial Risk Forecasting Published by Wiley 2011 Version 3.2, November 2017 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 1 of 74

2 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 2 of 74

3 The focus of this chapter is on Techniques for implementing risk forecasting, especially Historical simulation Risk measures and parametric methods Expected returns VaR with time-dependent volatility Financial Risk Forecasting 2011,2017 Jon Danielsson, page 3 of 74

4 Some concepts So far we have defined a sample to have a size T When it comes to what follows, it is useful to have a separate indication for number of observations used in the estimation, called estimation window, or W E The methods below fall into two main categories nonparametric No distribution of data assumed, no estimation, and hence no distributional parameters parametric Assume a distribution and estimate its parameters Financial Risk Forecasting 2011,2017 Jon Danielsson, page 4 of 74

5 Notation K Number of assets w K 1 vector of portfolio weights w k Portfolio weight on asset k X and Y Two different assets ϕ(.) Risk measure ϑ Portfolio value y k = {y t,k } T t=1 T 1 vector of returns on asset k y = T K T K matrix of historical returns Σ K K covariance matrix Financial Risk Forecasting 2011,2017 Jon Danielsson, page 5 of 74

6 Historical simulation Financial Risk Forecasting 2011,2017 Jon Danielsson, page 6 of 74

7 Historical simulation One asset Assumes that one of the observations in the estimation window will be the next days return, therefore Assume history repeats itself VaR is one of the observations in the estimation window, multiplied by the monetary value of the asset holdings, the portfolio value ϑ t ϑ t = number of stocks owned P t VaR t is the negative of the (T p) th smallest return, times ϑ t Financial Risk Forecasting 2011,2017 Jon Danielsson, page 7 of 74

8 500 days of the S&P % 1 % 0 % 1 % 2 % Financial Risk Forecasting 2011,2017 Jon Danielsson, page 8 of 74

9 500 days of the S&P % 1 % 0 % 1 % 2 % o VaR(99%) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 9 of 74

10 Sorted returns 2 % 1 % 0 % 1 % 2 % Financial Risk Forecasting 2011,2017 Jon Danielsson, page 10 of 74

11 Zoom in 1 % 1.5 % 2 % 2.5 % Financial Risk Forecasting 2011,2017 Jon Danielsson, page 11 of 74

12 Zoom in 1 % 1.5 % 2 % VaR(99%) 2.5 % Financial Risk Forecasting 2011,2017 Jon Danielsson, page 12 of 74

13 Zoom in 1 % VaR(95%) 1.5 % 2 % 2.5 % Financial Risk Forecasting 2011,2017 Jon Danielsson, page 13 of 74

14 Procedure Decide on a probability, p, e.g. 1% Have a sample of returns, y with length T, e.g Sort the y from the smallest to the largest, call that ys Take the (T p) th = ( ) th smallest value of ys, call that ys T p = ys 10 If you own one stock, and P t 1 = 1, then VaR is the 10 th smallest return, i.e. VaR t = ys 10 Otherwise have to multiply that by the number of stocks you own and their t 1 price VaR t = ys 10 P t 1 number of stocks Financial Risk Forecasting 2011,2017 Jon Danielsson, page 14 of 74

15 Multiple assets Take a vector of historical portfolio returns y is a T K matrix of returns w is a K 1 vector of portfolio weights We then get the time series vector of portfolio returns by y portfolio = yw And then you can simply treat the portfolio as if it were a single asset and apply HS Financial Risk Forecasting 2011,2017 Jon Danielsson, page 15 of 74

16 Expected shortfall estimation The expected losses conditional on VaR being violated Estimated by HS by taking the Mean of all observations less than or equal to -VaR Continuing with the single asset accent from above ES = i=1 ys i Financial Risk Forecasting 2011,2017 Jon Danielsson, page 16 of 74

17 Importance of sample size The most extreme observations fluctuate a lot more than observations that are less extreme Therefore, the bigger the sample the more precise the estimation of HS should be The downside is that old data may not be all that representative And if there is a structural break in the data (like in 2007) the VaR forecasts take longer to adjust to structural changes in risk As a general rule Minimum recommended sample size: 3 p Financial Risk Forecasting 2011,2017 Jon Danielsson, page 17 of 74

18 Issues No model assumptions needed In the absence of structural breaks HS tends to perform well It captures nonlinear dependence directly But performs badly when data has structural breaks This can be seen in the discussion about backtesting in Chapter 8 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 18 of 74

19 Estimation Use the stock data from chapter 3 Use the last 1000 days for the estimation (W E = 1000) p = 0.01 w = c(0.4,0.6) Portfolio value is $1000 Use Amazon for one stock calculation Financial Risk Forecasting 2011,2017 Jon Danielsson, page 19 of 74

20 Matlab estimation prices = csvread ( amzn goog. csv,1); y1 = diff ( log ( prices (:,2))); y2 = diff ( log ( prices (:,3))); portfolio = 1000 w = [ ] p = 0.01 WE = 1000 y1 = y1(( length (y1) WE+1):end ); y2 = y2(( length (y2) WE+1):end ); y = [ y1 y2] w ; ys = sort (y1 ); VaRAMZN = ys(we p) portfolio ys = sort (y ); VaRPortfolio = ys(we p) portfolio Financial Risk Forecasting 2011,2017 Jon Danielsson, page 20 of 74

21 R estimation price = read. csv( amzn goog. csv ) y1 = diff ( log ( price [,2])) y2 = diff ( log ( price [,3])) portfolio = 1000 w = c (0.4,0.6) p = 0.01; WE = 1000 y1 = tail (y1,we) y2 = tail (y2,we) VaRAMZN = sort (y1 )[p WE] portfolio y = cbind (y1,y2) % % w VaRPortfolio = sort (y )[p WE] portfolio Financial Risk Forecasting 2011,2017 Jon Danielsson, page 21 of 74

22 Parametric methods Financial Risk Forecasting 2011,2017 Jon Danielsson, page 22 of 74

23 Parametric In this section we abstract from time, and assume that we are calculating VaR and ES on day t conditional on A return density on day t (perhaps the normal or Student-t) f( ) With distribution And parameters θ Normal θ = (σ,µ) Student-t θ = (σ,µ,ν) F(x) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 23 of 74

24 Recall The definition of VaR is p =Pr[Q VaR(p)] = VaR(p) f q (x)dx Profit and loss when we own one stock Q t = P t P t 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 24 of 74

25 VaR for simple returns and holding one stock The definition of simple returns is R t = P t P t 1 P t 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 25 of 74

26 VaR for simple returns and holding one stock The definition of simple returns is R t = P t P t 1 P t 1 The definition of VaR (we own one stock) p =Pr[Q t VaR(p)] Financial Risk Forecasting 2011,2017 Jon Danielsson, page 26 of 74

27 VaR for simple returns and holding one stock The definition of simple returns is R t = P t P t 1 P t 1 The definition of VaR (we own one stock) p =Pr[Q t VaR(p)] =Pr(P t P t 1 VaR(p)) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 27 of 74

28 VaR for simple returns and holding one stock The definition of simple returns is R t = P t P t 1 P t 1 The definition of VaR (we own one stock) p =Pr[Q t VaR(p)] =Pr(P t P t 1 VaR(p)) =Pr(P t 1 R t VaR(p)) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 28 of 74

29 VaR for simple returns and holding one stock The definition of simple returns is R t = P t P t 1 P t 1 The definition of VaR (we own one stock) p =Pr[Q t VaR(p)] =Pr(P t P t 1 VaR(p)) =Pr(P t 1 R t VaR(p)) ( ) Rt =Pr σ VaR(p) P t 1 σ Financial Risk Forecasting 2011,2017 Jon Danielsson, page 29 of 74

30 So p =Pr[Q t VaR(p)] ( ) Rt =Pr σ VaR(p) P t 1 σ Denote the distribution of standardized returns, Rt /σ, by F R ( ) The inverse distribution is F 1 R (p) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 30 of 74

31 So p =Pr[Q t VaR(p)] ( ) Rt =Pr σ VaR(p) P t 1 σ Denote the distribution of standardized returns, Rt /σ, by F R ( ) The inverse distribution is F 1 R (p) So VaR t (p) = σf 1 R (p)p t 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 31 of 74

32 VaR for continuously compounded returns Definition Y t = logp t logp t 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 32 of 74

33 VaR for continuously compounded returns Definition Y t = logp t logp t 1 p = Pr(P t P t 1 VaR(p)) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 33 of 74

34 VaR for continuously compounded returns Definition Y t = logp t logp t 1 p = Pr(P t P t 1 VaR(p)) = Pr ( P t 1 ( e Y t 1 ) VaR(p) ) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 34 of 74

35 VaR for continuously compounded returns Definition Y t = logp t logp t 1 p = Pr(P t P t 1 VaR(p)) = Pr ( ( P t 1 e Y t 1 ) VaR(p) ) ( ( Yt = Pr σ log VaR(p) ) ) 1 +1 P t 1 σ when VaR(p) P t 1 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 35 of 74

36 Denote the distribution of standardized returns (Y t /σ) by F y ( ) The inverse distribution is F 1 y (p) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 36 of 74

37 Denote the distribution of standardized returns (Y t /σ) by F y ( ) The inverse distribution is F 1 y (p) So VaR(p) = (exp(f 1 y (p)σ) 1)P t 1 Not very convenient but Financial Risk Forecasting 2011,2017 Jon Danielsson, page 37 of 74

38 How different is from By approximation e F 1 y (p)σ 1 σf 1 R (p) It depends on the sampling frequency and volatility Recall the discussion in the first section of Chapter 1 Where simple returns are approximately same as compound returns at the daily frequency, therefore F R F y Suppose the distribution is the normal and p = 0.01 F 1 (p) = 2.3 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 38 of 74

39 So For daily returns the volatility is also low (perhaps 0.01) so Therefore exp( ) 1 = For small F 1 y (p)σ the VaR for holding one unit of asset is: VaR(p) σf 1 y (p)p t 1 meaning The VaR for continuously compounded returns is approximately the same as the VaR using simple returns Financial Risk Forecasting 2011,2017 Jon Danielsson, page 39 of 74

40 When there is more than one asset In the two assets case σ 2 portfolio = ( w 1 w 2 ) ( σ 11 σ 12 σ 12 σ 22 And generally in the K asset case Then as before σ 2 portfolio = w Σw )( w1 w 2 ) VaR(p) = σ portfolio F 1 (p)p t 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 40 of 74

41 VaR when returns are normally distributed ϑ = 1, σ = 1, p = 0.05 VaR = Φ 1 (0.05) = 1.64 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 41 of 74

42 VaR when returns are normally distributed ϑ = 1, σ = 1, p = 0.05 VaR = Φ 1 (0.05) = 1.64 If σ 1, then the VaR is: VaR = σ1.64 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 42 of 74

43 VaR when returns are normally distributed ϑ = 1, σ = 1, p = 0.05 VaR = Φ 1 (0.05) = 1.64 If σ 1, then the VaR is: VaR = σ1.64 If portfolio value is not equal to 1: VaR = σ1.64ϑ Financial Risk Forecasting 2011,2017 Jon Danielsson, page 43 of 74

44 VaR under the Student-t distribution Advantage of Student-t VaR over normal is fat tails ν indicates how fat tails are When ν = the Student-t becomes normal Financial Risk Forecasting 2011,2017 Jon Danielsson, page 44 of 74

45 VaR when returns are Student-t ϑ = 1000,σ = 0.01,p = 0.05 ν F 1 VaR $ $ $22.4 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 45 of 74

46 Expected shortfall under normality Given VaR ES = VaR(p) = ϑσ φ(f 1 (p)) p xf VaR (x)dx Financial Risk Forecasting 2011,2017 Jon Danielsson, page 46 of 74

47 R probabilities qt(0.05,1000) qt(0.05,5) qt(0.05,3) for ( df in c(1000,5,3)) cat(round( qt(0.05, df) ,1), \n ) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 47 of 74

48 Matlab probabilities tinv (0.05,1000) tinv (0.05,5) tinv (0.05,3) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 48 of 74

49 VaR and ES in R sigma=0.01 portfolio=1000 p=0.01 sigma portfolio qnorm(p) sigma portfolio dnorm(qnorm(p))/p Financial Risk Forecasting 2011,2017 Jon Danielsson, page 49 of 74

50 VaR and ES in Matlab sigma=0.01 portfolio=1000 sigma portfolio norminv(p) sigma portfolio normpdf(norminv(p))/p ans = Financial Risk Forecasting 2011,2017 Jon Danielsson, page 50 of 74

51 Expected returns Financial Risk Forecasting 2011,2017 Jon Danielsson, page 51 of 74

52 Expected returns Is it reasonable to assume µ = 0? Given that statistical uncertainty is more than 10% in most VaR calculations, VaR calculation is only significant to one digit Mean is smaller than that For S&P 500: µ = 0.019%,σ = 1.15% Financial Risk Forecasting 2011,2017 Jon Danielsson, page 52 of 74

53 VaR with mean The definition of VaR is p =Pr[Q VaR(p)] = VaR(p) f q (x)dx Profit and loss when we own one stock, and mean is zero, E(Q) = 0 Q t = P t P t 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 53 of 74

54 If mean is not zero, E(Q) 0, then the definition of VaR is Pr[Q +E(Q) VaR(p)] = p so VaR(p) = σf 1 (p) µ Note how the mean has a minus in front of it A positive mean pulls the distribution to the right Making VaR smaller Financial Risk Forecasting 2011,2017 Jon Danielsson, page 54 of 74

55 Time aggregation of VaR with mean If the returns are IID, then both mean and variance aggregate at the same rate Mean and variance over T days is equal to T times mean and variance over one day Which implies that the volatility aggregates at the square root of time The T-period VaR is therefore: VaR(Tday) = σ(tday)f 1 (p) µ(tday) And T < T = Tσ(1day)F 1 (p) Tµ(1day) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 55 of 74

56 so... The assumption µ = 0 is relatively harmless as the error is small at the daily level. Daily S&P 500: µ = 0.019%,σ = 1.15% Annual: µ = 4.87,σ = 18.2 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 56 of 74

57 VaR when returns do not have zero mean $ 400 $ 300 portfolio value=$1000 VaR $ 200 $ 100 $ Day Financial Risk Forecasting 2011,2017 Jon Danielsson, page 57 of 74

58 VaR when returns do not have zero mean $ 400 mean=0 mean= $ 300 portfolio value=$1000 VaR $ 200 $ 100 $ Day Financial Risk Forecasting 2011,2017 Jon Danielsson, page 58 of 74

59 Issues in including the mean It is much more difficult to estimate the mean than the variance And unless necessary, should be avoided And for VaR over one day or 10 days is not necessary in most cases Financial Risk Forecasting 2011,2017 Jon Danielsson, page 59 of 74

60 Estimation issues Financial Risk Forecasting 2011,2017 Jon Danielsson, page 60 of 74

61 VaR with time dependent volatility When using conditional volatility models, like EWMA and GARCH First update volatility And then calculate one day ahead VaR Financial Risk Forecasting 2011,2017 Jon Danielsson, page 61 of 74

62 GARCH(1,1) is: Normal GARCH ˆσ 2 t+1 = ω +αy 2 t +βˆσ 2 t Software packages (R and Matlab) usually return the last volatility of the sample (ˆσ t ), and the parameters One then needs to update the volatility by using ˆσ t and the model parameters So the VaR at t +1 is ˆσ 2 t+1 = ˆω + ˆαy 2 t + ˆβˆσ 2 t VaR t+1 = ˆσ t+1 Φ 1 (p)ϑt Financial Risk Forecasting 2011,2017 Jon Danielsson, page 62 of 74

63 R portfolio=1000 p=0.01 g = garchfit( garch (1,1),y, include.mean=f, trace=f) omega = g@fit $matcoef [1,1] alpha = g@fit $matcoef [2,1] beta = g@fit $matcoef [3,1] sigma2 = omega + alpha tail (y,1)ˆ2 + beta tail (g@h. t,1) VaR = sqrt (sigma2) qnorm(p) portfolio omega alpha1 beta e e e 01 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 63 of 74

64 Matlab portfolio=1000 p=0.01 [ parameters, lik,ht]=tarch (y,1,0,1); omega = parameters (1) alpha = parameters (2) beta = parameters (3) sigma2 = omega + alpha y(end)ˆ2 + beta Ht(end) VaR = sqrt (sigma2) norminv (p) portfolio parameters = Financial Risk Forecasting 2011,2017 Jon Danielsson, page 64 of 74

65 Estimating variances and the Student-t Normal θ = (σ,µ) Student-t θ = (σ,µ,ν) These parameters are defined by the distribution In general, the sample variance ˆσ 2 = 1 T 1 T (x i ˆµ) 2 Is not the same as the population variance, except for the normal The population parameters need to be estimated from fitting the distribution i=1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 65 of 74

66 Student-t variance For a standard Student-t (area under density is one) normal, σ = 1 While the sample variance is: ν ν 2 If ν 2 variance of Student-t is not defined Financial Risk Forecasting 2011,2017 Jon Danielsson, page 66 of 74

67 r=rt (1000,3) plot (r) sd(r) library (QRM) # Sigma is variance fit.norm(r) mu Sigma R simulaton test fit. st (r) nu mu sigma Financial Risk Forecasting 2011,2017 Jon Danielsson, page 67 of 74

68 r = trnd (3,1000,1); plot (r) std (r) Matlab simulaton test fitdist (r, Normal ) mu = [ , ] sigma = [ , ] fitdist (r, tlocationscale ) mu = [ , ] sigma = [ , ] nu = [ , ] Financial Risk Forecasting 2011,2017 Jon Danielsson, page 68 of 74

69 R Student-t GARCH g=garchfit( garch (1,1),y, cond= std, include.mean=f, trace=f) Log Likelihood : omega alpha1 beta1 shape e e e e+00 omega = g@fit $matcoef [1,1] alpha = g@fit $matcoef [2,1] beta = g@fit $matcoef [3,1] nu = g@fit $matcoef [4,1] sigma2 = omega + alpha tail (y,1)ˆ2 + beta tail (gt@h.t,1) e 05 VaR = sqrt (sigma2) qt(p,nu) portfolio Financial Risk Forecasting 2011,2017 Jon Danielsson, page 69 of 74

70 Matlab Student-t GARCH [ parameters, lik,ht]=tarch (y,1,0,1, STUDENTST ); lik e+04 parameters = omega = parameters (1) alpha = parameters (2) beta = parameters (3) nu= parameters (4) sigma2 = omega + alpha y(end)ˆ2 + beta Ht(end) e 05 VaR = sqrt (sigma2) tinv (p,nu) portfolio Financial Risk Forecasting 2011,2017 Jon Danielsson, page 70 of 74

71 Inconsistency The VaR from R is and for Matlab Which is correct? It is almost impossible to identify There is a small difference in the parameter estimates, especially ν is a little higher for Matlab (so its inverse will be lower) And σ T is a a little higher for R And that this enough to account for the differences Financial Risk Forecasting 2011,2017 Jon Danielsson, page 71 of 74

72 Stressed VaR (and ES) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 72 of 74

73 Typical VaR (and ES) The risk calculations above are based on using the most recent history If today is t then days t W E,...,t 1 To calculate risk (VaR or ES) today Financial Risk Forecasting 2011,2017 Jon Danielsson, page 73 of 74

74 Stressed VaR (and ES) The idea behind stressed VaR and ES Often indicated by S-VaR and S-ES Is to use the most traumatic ( most volatile) historical estimation window instead of the most recent For example July 2008 to June 2009 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 74 of 74