qa-3 frm-garp sep-2001 zvi wiener 02-588-3049 mswiener/zvi.html quantitative analysis 3

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


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


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.


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.


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


Time Aggregation

TRR

TRERE

T

T

)()(

)()(

122

1

TRRT )()( 1


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.


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%


FRM-99, Question 14

)(10)( 1020 RR


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%


FRM-98, Question 7

22223

22

2113 35.015.015.0

%24236.0313

av


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


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.


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



)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



1

2,~ˆ 422

TN

For large T the chi-square converges to normal

Standard errorT

se2

1~)ˆ(


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).


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.


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.


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.


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


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.


OLS

Beta and alpha are estimated by

xy ˆˆ

T

tt

T

ttt

xxT

yyxxT

1

2

1

)(1

1

))((1

1


),(),( xxCovxyCov

)(),( 2 xxxCov

Since x and are independent.

)(

),(2 x

xyCov


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


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


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.


Matrix Notation

TNTNT

N

T xx

xx

y

y

11

1

1111

Xy

yXXX TT 1)( 122 ))(()( XX T


ExampleConsider ten years of data on INTC and S&P 500, using total rates of returns over month.

S&P500

INTC


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


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.


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.


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.


Autoregression

Here k is the k-th order autoregression

coefficient.

tktkt xy


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.


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.


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


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??


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.


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.


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.


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.


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.



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.

qa-3 frm-garp sep-2001 zvi wiener 02-588-3049 mswiener/zvi.html quantitative analysis 3

Documents

wiener qa3 slide

time aggregation slide

zvi wiener

independent returns

predicting returns

important question

bad question

time periods