Financial Risk Management
Quantitative Analysis Fundamentals of Statistics Following P. Jorion, Financial Risk Management Chapter 3
Daniel HERLEMONT
Statistics and Probability Estimation Tests of hypotheses
Daniel HERLEMONT
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Returns Past spot rates S0, S1, S2,…, St. We need to estimate St+1. Random variable
rt =
S t − S t −1 S t −1
S Alternatively we can do Rt = ln t S t −1 S S − S t −1 = ln (1 + rt ) ≈ rt Rt = ln t = ln1 + t S S t −1 t −1 Daniel HERLEMONT
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.
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Random Walk
We could consider that the observations rt are independent draws from the same distribution N(µ, σ2). 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.
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Time Aggregation
S R02 = ln 2 S0
S S S S = ln 2 1 = ln 2 + ln 1 S1 S1 S 0 S0
= R01 + R12
E ( R02 ) = E ( R01 ) + E ( R12 )
σ 2 ( R02 ) = σ 2 ( R01 ) + σ 2 ( R12 ) + 2Cov ( R01 , R12 ) E ( R02 ) = 2 E ( R01 )
σ 2 ( R02 ) = 2σ 2 ( R01 ) Daniel HERLEMONT
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Time Aggregation
E ( RT ) = E ( R1 )T
σ 2 ( RT ) = σ 2 ( R1 )T σ ( RT ) = σ ( R1 ) T
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Time Aggregation
µT = µ annualT σ T = σ annual T Assuming normality
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Time Aggregation
Assume that yearly parameters of CPI are: mean = 5%, standard deviation (SD) = 2%. Then daily mean and SD of CPI changes are:
1 µd = µ y = 0.02% 250 1 σd =σ y = 0.1265% 250 Daniel HERLEMONT
Time Aggregation with Correlation Consider a process with autocorrelation The variance of 2 days return
higher than in the IID case if ρ > 0 : trending market lower than in the IID case if ρ > 0 : mean reverting market
VaR book p. 136 Daniel HERLEMONT
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The Effect of the mean at various horizons
ref; VaR, p. 138
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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.
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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%
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FRM-99, Question 14
σ ( R20 ) = 10σ ( R10 )
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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%
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FRM-98, Question 7
σ 13 = σ 12 + σ 22 + σ 32 = 0.15 2 + 0.15 2 + 0.35 2
σ av =
σ 13 3
= 0.236 ≈ 24%
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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
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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.
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Parameter Estimation Having T observations of an iid sample we can estimate the parameters. Sample mean.
1 T µˆ = ∑ xi T i =1 Equal weights.
1 T ( xi − µˆ )2 σˆ = ∑ T − 1 i =1 2
Sample variance
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Parameter Estimation
Note that sample mean is distributed
σ2 µˆ ~ N µ , T
When X is normal the sample variance is distributed
(T − 1)σˆ 2
σ2
~ χ 2 (T − 1)
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Parameter Estimation
For large T the chi-square converges to normal
σˆ 2 ~ N σ 2 , σ 4 Standard error
2 T −1
se(σˆ ) ~ σ
1 2T
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Hypothesis Testing
Test for a trend. Null hypothesis is that µ=0.
z=
µˆ − 0 σˆ / T
Since σ 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). Daniel HERLEMONT
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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.
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Example: yen/dollar rate Estimate precision of the sample standard deviation. se(s) = σ/√(2T) = 0.229% For the null σ=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 σ rather than m.
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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. Daniel HERLEMONT
Regression Analysis
Linear regression: dependent variable y is projected on a set of N independent variables x.
y t = α + β xt + ε t ,
t = 1, K, T
α - intercept or constant β - slope ε - residual
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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.
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OLS
Beta and alpha are estimated by
1 T ( xt − x )( yt − y ) ∑ T − 1 t =1 βˆ = 1 T ( xt − x ) 2 ∑ T − 1 t =1
αˆ = y − βˆ x
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Cov ( y, x) = Cov (α + β x + ε , x)
= βCov( x, x) = βσ 2 ( x) Since x and ε are independent.
β=
Cov ( y, x) σ 2 ( x)
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Residual and its estimated variance
ε t = y t − yˆ t = y t − α − β xt σ (εˆ ) = 2
1 T 2 εˆt ∑ T − 2 t =1
The quality of the fit is given by the regression R-square (which is the square of correlation ρ(x,y)). T
∑ εˆ R2 = 1−
2 t
t =1 T
∑(y
t
− y) 2
t =1 Daniel HERLEMONT
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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
σ 2 ( y ) = β 2σ 2 ( x) + σ 2 (ε ) R2
β 2σ 2 ( x) σ 2 (ε ) 1= + 2 σ 2 ( y) σ ( y)
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Linear Regression To estimate the uncertainty in the slope coefficient we use
σ ( βˆ ) = 2
σ 2 (εˆ)
∑ (x
t
− x)2
It is useful to test whether the slope coefficient is significantly different from zero.
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Matrix Notation
y1 x11 L x1N β1 ε1 M = M M + M M yT xT 1 K xTN β N ε T
y = Xβ + ε
β = ( X T X ) −1 X T y σ 2 ( β ) = σ 2 (ε )( X T X ) −1 Daniel HERLEMONT
Example Consider ten years of data on INTC and S&P 500, using total rates of returns over month. INTC
S&P500
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y = α + βx + ε Coeff. Estimate SE
T-stat
P-value
α
0.0168
0.0094
1.78
0.77
β
1.349
0.229
5.9
0.00
R-square
0.228
SE(y)
10.94%
SE(ε)
9.62%
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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: βˆ − 1 1.349 − 1 z= = = 1.53 0.229 s ( βˆ ) 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.
Daniel HERLEMONT
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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 β and σ) the results are still valid. Problems arise when X include lagged dependent variables - this can cause bias.
Daniel HERLEMONT
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.
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Autoregression
yt = α + β k xt −k + ε t Here βk is the k-th order autoregression coefficient.
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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.
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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.
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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
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FRM-99, Question 20
a = 14, b = 27 a-14 3 0 -2 -1
b-27 -5 -1 4 2
(a-14)(b-27) -15 0 -8 -2 -25
Cov(a,b) = -25/4 = -6.25 Why not -25/3?? Daniel HERLEMONT
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.
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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.
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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.
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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.
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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.
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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.
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