Financial Risk Management

Financial Risk Management Zvi Wiener Following P. Jorion,Financial Risk Manager Handbook FRM

Chapter 3Quantitative AnalysisFundamentals of Statistics Following P. Jorion 2001 Financial Risk Manager Handbook FRM

Statistics and Probability Estimation Tests of hypotheses Zvi Wiener

Returns Past spot rates S0, S1, S2,…, St. We need to estimate St+1. Random variable Alternatively we can do Zvi Wiener

Independent returns A very important question is whether a sequence of observations can be viewed as independent. If so, one could assume that it is drawn from a known distribution and then one can estimate parameters. In an efficient market returns on traded assets are independent. Zvi Wiener

Random Walk We could consider that the observations rt are independent draws from the same distribution N(, 2). They are called i.i.d. = independently and identically distributed. An extension of this model is a non-stationary environment. Often fat tails are observed. Zvi Wiener

Time Aggregation Zvi Wiener

FRM-99, Question 4 Random walk assumes that returns from one time period are statistically independent from another period. This implies: A. Returns on 2 time periods can not be equal. B. Returns on 2 time periods are uncorrelated. C. Knowledge of the returns from one period does not help in predicting returns from another period D. Both b and c. Zvi Wiener

FRM-99, Question 14 Suppose returns are uncorrelated over time. You are given that the volatility over 2 days is 1.2%. What is the volatility over 20 days? A. 0.38% B. 1.2% C. 3.79% D. 12.0% Zvi Wiener

FRM-99, Question 14 Zvi Wiener

FRM-98, Question 7 Assume an asset price variance increases linearly with time. Suppose the expected asset price volatility for the next 2 months is 15% (annualized), and for the 1 month that follows, the expected volatility is 35% (annualized). What is the average expected volatility over the next 3 months? A. 22% B. 24% C. 25% D. 35% Zvi Wiener

FRM-98, Question 7 Zvi Wiener

FRM-97, Question 15 The standard VaR calculation for extension to multiple periods assumes that returns are serially uncorrelated. If prices display trend, the true VaR will be: A. the same as standard VaR B. greater than the standard VaR C. less than the standard VaR D. unable to be determined Zvi Wiener

FRM-97, Question 15 Bad Question!!! “answer” is b. Positive trend assumes positive correlation between returns, thus increasing the longer period variance. Correct answer is that the trend will change mean, thus d. Zvi Wiener

Parameter Estimation Having T observations of an iid sample we can estimate the parameters. Sample mean. Equal weights. Sample variance Zvi Wiener

Parameter Estimation Note that sample mean is distributed When X is normal the sample variance is distributed Zvi Wiener

Parameter Estimation For large T the chi-square converges to normal Standard error Zvi Wiener

Hypothesis Testing Test for a trend. Null hypothesis is that =0. Since  is unknown this variable is distributed according to Student-t with T degrees of freedom. For large T it is almost normal. This means that 95% of cases z is in [-1.96, 1.96] (assuming normality). Zvi Wiener

Example: yen/dollar rate We want to characterize monthly yen/USD exchange rate based on 1990-1999 data. We have T=120, m=-0.28%, s=3.55% (per month). The standard error of the mean is approximately se(m)= s/T=0.32%. t-ratio is m/se(m) = -028/0.32=-0.87 since the ratio is less then 2 the null hypothesis can not be rejected at 95% level. Zvi Wiener

Example: yen/dollar rate Estimate precision of the sample standard deviation. se(s) = /(2T) = 0.229% For the null =0 this gives a z-ratio of z = s/se(s) = 3.55%/0.229% = 15.5 which is very high. Therefore there is much more precision in measurement of  rather than m. Zvi Wiener

Example: yen/dollar rate 95% confidence intervals around the estimates: [m-1.96 se(m), m+1.96 se(m)]=[-0.92%, 0.35%] [s-1.96 se(s), s+1.96 se(s)]=[3.1%, 4.0%] This means that the volatility is between 3% and 4%, but we cannot be sure that the mean is different from zero. Zvi Wiener

Regression Analysis Linear regression: dependent variable y is projected on a set of N independent variables x.  - intercept or constant  - slope  - residual Zvi Wiener

OLS Ordinary least squares assumptions are a. the errors are independent of x. b. the errors have a normal distribution with zero mean and constant variance, given x. c. the errors are independent across observations. Zvi Wiener

OLS Beta and alpha are estimated by Zvi Wiener

Since x and  are independent. Zvi Wiener

Residual and its estimated variance The quality of the fit is given by the regression R-square (which is the square of correlation (x,y)). Zvi Wiener

R2 R square If the fit is excellent and the errors are zero, R2=1. If the fit is poor, the sum of squared errors will beg as large as the sum of deviations of y around its mean, and R2=0. Alternatively Zvi Wiener

Linear Regression To estimate the uncertainty in the slope coefficient we use It is useful to test whether the slope coefficient is significantly different from zero. Zvi Wiener

Matrix Notation Zvi Wiener

Example Consider ten years of data on INTC and S&P 500, using total rates of returns over month. INTC S&P500 Zvi Wiener

probability Coeff. Estimate SE T-stat P-value  0.0168 0.0094 1.78 0.77  1.349 0.229 5.9 0.00 R-square 0.228 SE(y) 10.94% SE() 9.62% Zvi Wiener

The beta coefficient is 1.35 and is significantly positive. It is called systematic risk it seems that it is greater than one. Construct z-score: It is less than 2, thus we can not say that Intel’s systematic risk is bigger than one. R2=23%, thus 23% of Intel’s returns can be attributed to the market. Zvi Wiener

Pitfalls with Regressions OLS assumes that the X variables are predetermined (exogenous, fixed). In many cases even if X is stochastic (but distributed independently of errors and do not involve  and ) the results are still valid. Problems arise when X include lagged dependent variables - this can cause bias. Zvi Wiener

Pitfalls with Regressions Specification errors - not all independent (X) variables were identified. Multicollinearity - X variables are highly correlated, eg DM and gilden. X will be non invertible, small determinant. Linear assumption can be problematic as well as stationarity. Zvi Wiener

Autoregression Here k is the k-th order autoregression coefficient. Zvi Wiener

FRM-99, Question 2 Under what circumstances could the explanatory power of regression analysis be overstated? A. The explanatory variables are not correlated with one another. B. The variance of the error term decreases as the value of the dependent variable increases. C. The error term is normally distributed. D. An important explanatory variable is excluded. Zvi Wiener

FRM-99, Question 2 D. If the true regression includes a third variable z that influences both x and y, the error term will not be conditionally independent of x, which violates one of the assumptions of the OLS model. This will artificially increase the explanatory power of the regression. Zvi Wiener

FRM-99, Question 20 What is the covariance between populations a and b: a 17 14 12 13 b 22 26 31 29 A. -6.25 B. 6.50 C. -3.61 D. 3.61 Zvi Wiener

FRM-99, Question 20 a-14 b-27 (a-14)(b-27) 3 -5 -15 0 -1 0 -2 4 -8 -1 2 -2 -25 Cov(a,b) = -25/4 = -6.25 Why not -25/3?? Zvi Wiener

FRM-99, Question 6 Daily returns on spot positions of the Euro against USD are highly correlated with returns on spot holdings of Yen against USD. This implies that: A. When Euro strengthens against USD, the yen also tends to strengthens, but returns are not necessarily equal. B. The two sets of returns tend to be almost equal C. The two sets of returns tend to be almost equal in magnitude but opposite in sign. D. None of the above. Zvi Wiener

FRM-99, Question 10 You want to estimate correlation between stocks in Frankfurt and Tokyo. You have prices of selected securities. How will time discrepancy bias the computed volatilities for individual stocks and correlations between these two markets? A. Increased volatility with correlation unchanged. B. Lower volatility with lower correlation. C. Volatility unchanged with lower correlation. D. Volatility unchanged with correlation unchanged. Zvi Wiener

FRM-99, Question 10 The non-synchronicity of prices does not affect the volatility, but will induce some error in the correlation coefficient across series. Intuitively, this is similar to the effect of errors in the variables, which biased downward the slope coefficient and the correlation. Zvi Wiener

FRM-00, Question 125 If the F-test shows that the set of X variables explains a significant amount of variation in the Y variable, then: A. Another linear regression model should be tried. B. A t-test should be used to test which of the individual X variables can be discarded. C. A transformation of Y should be made. D. Another test could be done using an indicator variable to test significance of the model. Zvi Wiener

FRM-00, Question 125 The F-test applies to the group of variables but does not say which one is most significant. To identify which particular variable is significant or not, we use a t-test and discard the variables that do not display individual significance. Zvi Wiener

FRM-00, Question 112 Positive autocorrelation of prices can be defined as: A. An upward movement in price is more likely to be followed by another upward movement in price. B. A downward movement in price is more likely to be followed by another downward movement. C. Both A and B. D. Historic prices have no correlation with future prices. Zvi Wiener

Financial Risk Management