Quantitative Analysis

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1 EduPristie FRM I \ Quatitative Aalysis EduPristie

2 Momets distributio Samplig Testig Correlatio & Regressio Estimatio Simulatio Modellig EduPristie FRM I \ Quatitative Aalysis 2

Momets distributio Samplig Testig Correlatio & Regressio Estimatio Simulatio Modelig Mea Variace Skewess Kurtosis Mea: X i 1 i Mode: Value that occurs most

of squared deviatios from mea Sample variace 2 i 1 (X - X ( -1) mea Populatio variace N 2 i 1 Stadard deviatio = Variace Var(ax+by)=a 2 Var(x)+ b 2

0 σ 2 of retur of stock Q=225.0 Cov (P,Q) =53.2 Curret Holdig $1 m i P. New Holdig: shiftig $ 1 millio i Q ad keepig USD 3 millio i stock P.

5 old σ = 100 = 10 Reductio = 5% Leptokurtic: More peaked tha ormal (fat tails); kurtosis>3 Platykurtic: Flatter tha a ormal; kurtosis<3 Kurtosis of Normal = 3

3 Momets distributio Samplig Testig Correlatio & Regressio Estimatio Simulatio Modelig Mea Variace Skewess Kurtosis Mea: X i 1 i Mode: Value that occurs most frequetly Media: Midpoit of data arraged i ascedig/ descedig order s Avg. of squared deviatios from mea Sample variace 2 i 1 (X - X ( -1) mea Populatio variace N 2 i 1 Stadard deviatio = Variace Var(ax+by)=a 2 Var(x)+ b 2 Var(y)+2abCov(x,y) i N ) 2 (X - ) i 2 Positively: mea> media> mode Negatively: mea< media< mode Skewess of Normal = 0 σ 2 of retur of stock P= σ 2 of retur of stock Q=225.0 Cov (P,Q) =53.2 Curret Holdig $1 m i P. New Holdig: shiftig $ 1 millio i Q ad keepig USD 3 millio i stock P. What %age of risk (σ), is reduced? σ P = *w 2 σ A2 + (1-w) 2 σ B2 +2w(1-w)Cov(A,B)] w= 0.75 c 2 = 100*(0.75) *(0.25) 2 +2*0.25*0.75*53.2 σ P = 9.5 old σ = 100 = 10 Reductio = 5% Leptokurtic: More peaked tha ormal (fat tails); kurtosis>3 Platykurtic: Flatter tha a ormal; kurtosis<3 Kurtosis of Normal = 3 Excess Kurtosis = Kurtosis - 3 If distributios of returs from fiacial istrumets are leptokurtotic. How does it compare with a ormal distributio of the same mea ad variace? Leptokurtic refers to a distributio with fatter tails tha the ormal, which implies greater kurtosis EduPristie FRM I \ Quatitative Aalysis 3

Momets distributio Samplig Testig Correlatio & Regressio Estimatio Simulatio Modelig Properties P(A) = # of fav.

B)P(B) P(A B)=P(A)*P(B)If A,B idepedet Coutig priciples No. of ways to select r out of objects: C r =!/[r!* (-r)!] No.

Assume a costat quarterly default rate. What is the probability that ABC will ot have defaulted by April 1, 2004?

subsidiary will default if the paret defaults, but the paret will ot ecessarily default if the subsidiary defaults.

5% Biomial Oly 2 possible outcomes: failure or success.

The 1-year margial default prob. of each bod is 5.93%. If spread of default prob. is eve over the year, Calculate prob.

4 Momets distributio Samplig Testig Correlatio & Regressio Estimatio Simulatio Modelig Properties P(A) = # of fav. Evets/ # of Total Evets 0 < P(A) <1, P(A c )=1-P(A) P(AUB)=P(A)+P(B)-P(A B) =P(A)+P(B) If A,B mutually exclusive P(A B)= P(A B)/P(B) P(A B)=P(A B)P(B) P(A B)=P(A)*P(B)If A,B idepedet Coutig priciples No. of ways to select r out of objects: C r =!/[r!* (-r)!] No. of ways to arrage r objects i places: P r =!/(-r)! ABC was ic. o Ja 1, Its expected aual default rate of 10%. Assume a costat quarterly default rate. What is the probability that ABC will ot have defaulted by April 1, 2004? P(No Default Year) = P(No default i all Quarters) = (1-PDQ1)*(1-PDQ2)*(1-PDQ3)*(1-PDQ4) PDQ1=PDQ2=PDQ3=PDQ4=PDQ P(No Def Year) = (1-PDQ) 4 P(No Def Quarter) = (0.9) 4 = 97.4% Sum rule ad Bayes' Theorem P( B) A B) A c B) c c P( B) B / A)* P( A) B / A )* P( A ) P(A/B)*P(B) P(B/A) c c P(A/B)*P(B) A/B )*P(B ) The subsidiary will default if the paret defaults, but the paret will ot ecessarily default if the subsidiary defaults. Calculate of a subsidiary & paret both defaultig. Paret has a PD =.5% subsidiary has PD of.9% P(P S) = P(S/P)*P(P) = 1*0.5 = 0.5% Biomial Oly 2 possible outcomes: failure or success. P(x)= C x *p x *(1-p) -x Biomial Radom Variable E(X)=*p Var(X)=*p*(1-p)=*p*q Discrete A portfolio cosists of 17 ucorrelated bods. The 1-year margial default prob. of each bod is 5.93%. If spread of default prob. is eve over the year, Calculate prob. of exactly 2 bods defaultig i first moth? 1-moth default rate =1- ( )1/12 = Ways to select 2 bods out of 17 = 17 C 2 = 17*16/2 P(Exactly 2 defaults) = (17*16/2)*( )2*( )15 = 0.325% Poisso Fix the expectatio λ=p. P(x)=λ x e -λ /x! if x>=0 P(x)=0 otherwise Cotiuous AB The umber of false fire alarms i a suburb of Housto averages 2.1 per day. What is the (apprximate) probability that there would be 4 false alarms o 1 day? P(X=x) = (λ x e -x )/x! X= 2.1, x = 4 P(2.1) = 0.1 EduPristie FRM I \ Quatitative Aalysis 4

Momets distributio Samplig Testig Correlatio & Regressio Estimatio Simulatio Modelig Discrete Cotiuous AB Cotiuous uiform distributio

For x <=a F(x)=(x-a)/(b-a) For a<x<b F(x)=1 For x >=b The R.V.

Data 99.7% of Data -4-3 -2-1 0 1 2 3 4 If Z is a stadard ormal R.V. A evet X is defied to happe if either -1< Z < 1 or Z > 1.5.

-1 +1 1.5 The sum of areas show i two figures Area 1 = 1-2*(1- N(1)) = 1-2*(0.1587) Area 2 = 0.0668, Total Area = 0.

5 Momets distributio Samplig Testig Correlatio & Regressio Estimatio Simulatio Modelig Discrete Cotiuous AB Cotiuous uiform distributio Normal Distributio (ND) Outcome oly betwee [a, b] P(outside a & b) = 0 Cumulative desity fuctio (cdf) for Uiform distributio: F(x)=0 For x <=a F(x)=(x-a)/(b-a) For a<x<b F(x)=1 For x >=b The R.V. X with desity fuctio f(x) = 1 / (b - a) for a < x < b, ad 0 otherwise, is said to have a uiform distributio over (a, b). Calculate its mea. a Sice the distributio is uiform, the mea is the ceter of the distributio, which is the average of a ad b = (a+b)/2 68% of Data 95% of Data 99.7% of Data If Z is a stadard ormal R.V. A evet X is defied to happe if either -1< Z < 1 or Z > 1.5. What is the prob. of evet X happeig if N (1) =0.8413, N (0.5) = ad N (-1.5) = , where N is the CDF of a stadard ormal variable The sum of areas show i two figures Area 1 = 1-2*(1- N(1)) = 1-2*(0.1587) Area 2 = , Total Area = Stadardized RV is ormalized mea = 0, σ = 1 Z-score: # of σ a give observatio is from populatio mea. Z=(x-µ)/σ At a particular time, the market value of assets of the firm is $100 M ad the market value of debt is $80 M. The stadard deviatio of assets is $ 10 M. What is the distace to default? z = (A-K)/σ A = (100-80)/10 = 2 EduPristie FRM I \ Quatitative Aalysis 5

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Quantitative Analysis

Quantitative Analysis EduPristie www.edupristie.com Modellig Mea Variace Skewess Kurtosis Mea: X i = i Mode: Value that occurs most frequetly Media: Midpoit of data arraged i ascedig/ descedig order s Avg. of squared deviatios