QA-3 FRM-GARP Sep-2001 Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/ ~mswiener/zvi.html Quantitative Analysis 3
Dec 21, 2015
QA-3 FRM-GARP Sep-2001
Zvi Wiener
02-588-3049http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
Quantitative Analysis 3
QA-3 FRM-GARP Sep-2001
Fundamentals of Statistics
Following Jorion 2001
Financial Risk Manager Handbook
Zvi Wiener - QA3 slide 3http://www.tfii.org
Statistics and Probability
Estimation
Tests of hypotheses
Zvi Wiener - QA3 slide 4http://www.tfii.org
Returns
Past spot rates S0, S1, S2,…, St.
We need to estimate St+1.
Random variable 1
1
t
ttt S
SSr
Alternatively we can do
1
lnt
tt S
SR
ttt
tt
t
tt rr
S
SS
S
SR
1ln1lnln1
1
1
Zvi Wiener - QA3 slide 5http://www.tfii.org
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 - QA3 slide 6http://www.tfii.org
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 - QA3 slide 7http://www.tfii.org
Time Aggregation
12010
1
1
2
0
1
1
2
0
202 lnlnlnln RR
S
S
S
S
S
S
S
S
S
SR
)()()( 120102 RERERE
),(2)()()( 1201122
012
022 RRCovRRR
)(2)(
)(2)(
012
022
0102
RR
RERE
Zvi Wiener - QA3 slide 8http://www.tfii.org
Time Aggregation
TRR
TRERE
T
T
)()(
)()(
122
1
TRRT )()( 1
Zvi Wiener - QA3 slide 9http://www.tfii.org
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 - QA3 slide 10http://www.tfii.org
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 - QA3 slide 12http://www.tfii.org
FRM-98, Question 7Assume 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 - QA3 slide 13http://www.tfii.org
FRM-98, Question 7
22223
22
2113 35.015.015.0
%24236.0313
av
Zvi Wiener - QA3 slide 14http://www.tfii.org
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 - QA3 slide 15http://www.tfii.org
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 - QA3 slide 16http://www.tfii.org
Parameter Estimation
Having T observations of an iid sample we can estimate the parameters.
Sample mean.
T
iix
T 1
1
Equal weights.
Sample variance
T
iix
T 1
22 ˆ1
1ˆ
Zvi Wiener - QA3 slide 17http://www.tfii.org
Parameter Estimation
)1(~ˆ)1( 2
2
2
TT
Note that sample mean is distributed
T
N2
,~ˆ
When X is normal the sample variance is distributed
Zvi Wiener - QA3 slide 18http://www.tfii.org
Parameter Estimation
1
2,~ˆ 422
TN
For large T the chi-square converges to normal
Standard errorT
se2
1~)ˆ(
Zvi Wiener - QA3 slide 19http://www.tfii.org
Hypothesis Testing
Tz
/ˆ
0ˆ
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 - QA3 slide 20http://www.tfii.org
Example: yen/dollar rateWe 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 - QA3 slide 21http://www.tfii.org
Example: yen/dollar rateEstimate 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 - QA3 slide 22http://www.tfii.org
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 - QA3 slide 23http://www.tfii.org
Regression Analysis
Linear regression: dependent variable y is
projected on a set of N independent variables x.
Ttxy ttt ,,1,
- intercept or constant
- slope
- residual
Zvi Wiener - QA3 slide 24http://www.tfii.org
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 - QA3 slide 25http://www.tfii.org
OLS
Beta and alpha are estimated by
xy ˆˆ
T
tt
T
ttt
xxT
yyxxT
1
2
1
)(1
1
))((1
1
Zvi Wiener - QA3 slide 26http://www.tfii.org
),(),( xxCovxyCov
)(),( 2 xxxCov
Since x and are independent.
)(
),(2 x
xyCov
Zvi Wiener - QA3 slide 27http://www.tfii.org
T
tt
ttttt
T
xyyy
1
22 ˆ2
1)ˆ(
ˆ
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)).
T
tt
T
tt
yyR
1
2
1
2
2
)(
ˆ
1
Zvi Wiener - QA3 slide 28http://www.tfii.org
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
)(
)(
)(
)(1
)()()(
2
2
2
22
2222
yy
x
xy
R2
Zvi Wiener - QA3 slide 29http://www.tfii.org
Linear Regression
To estimate the uncertainty in the slope coefficient
we use
2
22
)(
)ˆ()ˆ(
xxt
It is useful to test whether the slope coefficient is
significantly different from zero.
Zvi Wiener - QA3 slide 30http://www.tfii.org
Matrix Notation
TNTNT
N
T xx
xx
y
y
11
1
1111
Xy
yXXX TT 1)( 122 ))(()( XX T
Zvi Wiener - QA3 slide 31http://www.tfii.org
ExampleConsider ten years of data on INTC and S&P 500, using total rates of returns over month.
S&P500
INTC
Zvi Wiener - QA3 slide 32http://www.tfii.org
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%
xy
probability
Zvi Wiener - QA3 slide 33http://www.tfii.org
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:
53.1229.0
1349.1
)ˆ(
1ˆ
s
z
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 - QA3 slide 34http://www.tfii.org
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 - QA3 slide 35http://www.tfii.org
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 - QA3 slide 36http://www.tfii.org
Autoregression
Here k is the k-th order autoregression
coefficient.
tktkt xy
Zvi Wiener - QA3 slide 37http://www.tfii.org
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 - QA3 slide 38http://www.tfii.org
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 - QA3 slide 39http://www.tfii.org
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 - QA3 slide 40http://www.tfii.org
FRM-99, Question 2027,14 ba
a-14 b-27 (a-14)(b-27)3 -5 -150 -1 0-2 4 -8-1 2 -2
-25
Cov(a,b) = -25/4 = -6.25Why not -25/3??
Zvi Wiener - QA3 slide 41http://www.tfii.org
FRM-99, Question 6Daily 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 equalC. The two sets of returns tend to be almost equal in magnitude but opposite in sign.D. None of the above.
Zvi Wiener - QA3 slide 42http://www.tfii.org
FRM-99, Question 10You 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 - QA3 slide 43http://www.tfii.org
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 - QA3 slide 44http://www.tfii.org
FRM-00, Question 125If 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 - QA3 slide 45http://www.tfii.org
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 - QA3 slide 46http://www.tfii.org
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 - QA3 slide 47http://www.tfii.org
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.