3. Chapter Topics
3
Course 1
• Calculator & CFA & FRM
• Competition & Bloomberg
• The AM business & geometric vs arithmetic averages
• Trends in Asset allocation
• S&P 500 Prog. & Drawdown
…Syllabus
4. City Date Chapters Type
4
Rouen (123) 11.01
• Bank’s economic capital vs credit risk
• Factors used to calculate economic capital for credit risk:
probability of default, exposure & loss rate
• Expected loss (EL) & Unexpected loss (UL)
Présentiel synchrone
Rouen (123) 24.01
• Variance of default probability (binomial distribution)
• UL for a portfolio & UL contribution of each asset
• How economic capital is derived
• Model credit loss distribution
Présentiel synchrone
Rouen (123) 24.01
• Experts-based approaches, statistical-based models &
numerical approaches to predicting default
• Rating migration matrix & probability of default…
• Rating agencies’ assignment methodologies
Présentiel synchrone
Syllabus
Rouen (124) 09.02 • Overview of equities markets & explanation of Project Présentiel synchrone
Rouen (123) 13.03
• Rating agencies’ assignment methodologies
• Relationship between rating & probability of default
• Agencies’ ratings to internal experts-based rating systems
• Structural & reduced-form approaches to predicting
default
Présentiel synchrone
5. City Date Chapters Type
5
Rouen (123) 13.03
• Merton model & default probability & distance to default
• Z-score & linear discriminant analysis (LDA) to classify a
sample of firms by credit quality
• Logistic regression model to estimate default probability
Présentiel synchrone
Rouen (124) 15.03 • ...Bloomberg & equities markets Présentiel synchrone
Rouen (123) 26.03
• Exponential vs Poisson distributions
• Hazard rate & probability functions for default time &
conditional default probabilities…
• Unconditional & conditional default probability
Présentiel synchrone
Rouen (123) 02.04
• Risk-neutral default rates from spreads
• PROs of CDS market to estimate hazard rates
• CDS spread used to derive a hazard rate curve
• Spread risk & measurement using the mark-to-market and
spread volatility
Présentiel synchrone
…Syllabus
Rouen (123) 14.03
• Representing credit spreads
• Compute one credit spread given others when possible
• Spread ‘01
• Default risk & Bernoulli trial
Distanciel
6. City Date Chapters Type
6
Rouen (123) 09.04
• Extreme value theory (EVT) & risk management
• Peaks-over-threshold (POT) approach
Présentiel synchrone
Rouen (123) 10.04 • Generalized extreme value & POT
• Multivariate EVT for risk management
Distanciel
Rouen (124) 03.04 • ...Bloomberg & equities markets Présentiel synchrone
…Syllabus
Rouen (123) 05.04
• Bootstrap simulation & coherent risk measures
• Historical simulation using non-parametric density
estimation
Présentiel synchrone
Rouen (123) 09.04
• Age-weighted, volatility-weighted, correlation-weighted &
filtered historical simulation approaches
• PROs & CONs of non-parametric estimation methods
Présentiel synchrone
7. City Date Chapters Type
7
Rouen (123) 12.04
• Backtesting & backtesting VaR models
• Type I & Type II errors
Présentiel synchrone
Rouen (123) 15.04
• Conditional coverage in backtesting framework
• Basel rules for backtesting
Présentiel synchrone
Rouen (123) 16.04 • Case studies for Credit Risk & Market Risk Asynchrone
Rouen (124) 17.04 • Measurement of performance Présentiel synchrone
Rouen (124) 19.04 • Review exercises Présentiel synchrone
…Syllabus
8. Evaluation Weight Information
8
Quizzes 50% • Unannounced
Final exam (2 hours) 50% • Centre des examens (after Course 10)
…Syllabus
123
Project 40% • Groups of 3 students
Final exam (2 hours) 60% • Centre des examens (after Course 10)
124
9. 9
Temperature in Paris (in ℃)
Day June 2022 June 2023
1 17.0 21.0
2 19.0 16.5
3 19.0 19.0
4 23.0 20.5
5 18.5 20.5
6 18.0 20.0
7 20.0 21.5
8 18.0 22.5
9 17.5 24.5
10 19.0 25.0
11 21.0 23.0
12 19.5 23.0
13 18.5 24.0
14 21.0 23.5
15 24.0 23.5
16 24.0 24.0
17 27.0 25.5
18 29.0 24.0
Day June 2022 June 2023
19 21.0 21.0
20 19.0 23.0
21 21.0 23.5
22 21.5 21.5
23 24.0 22.5
24 21.5 23.5
25 16.5 27.5
26 18.0 20.5
27 19.5 21.0
28 21.0 21.5
29 21.0 22.0
30 16.0 20.0
Average 20.4 22.3
Standard deviation 3.0 2.2
Sources :
https://www.accuweather.com/fr/fr/paris/623/june-weather/623?year=2022
https://www.accuweather.com/fr/fr/paris/623/june-weather/623
10. 10
…Temperature in Paris (in ℃)
1
2
3
4
5
6
15 17 19 21 23 25 27 29
June 2022
𝐗 ̅=𝟐𝟎.𝟒
𝛔=𝟑.𝟎
!
𝐗 − 𝛔 = 𝟏𝟕. 𝟒 !
𝐗 + 𝛔 = 𝟐𝟑. 𝟒
68.2% of all observations
should lie within !
𝐗 ± 𝟏𝛔
95.4% of all observations
should lie within !
𝐗 ± 𝟐𝛔
99.7% of all observations
should lie within !
𝐗 ± 𝟑𝛔
11. 11
…Temperature in Paris (in ℃)
!
𝐗 − 𝛔 = 𝟐𝟎. 𝟏 !
𝐗 + 𝛔 = 𝟐𝟒. 𝟓
1
2
3
4
5
6
15 17 19 21 23 25 27 29
June 2023
𝐗 ̅=𝟐𝟐.𝟑
𝛔=𝟐.𝟐 68.2% of all observations
should lie within !
𝐗 ± 𝟏𝛔
95.4% of all observations
should lie within !
𝐗 ± 𝟐𝛔
99.7% of all observations
should lie within !
𝐗 ± 𝟑𝛔
12. • You flip a coin. What is the probability of having a tail ?
• ~ Bernoulli random variable
• Pr(X = x) = p = 50%
• You flip a coin 6 Xmes. What is the probability of having exactly 4 tails ?
• T, T, T, T, T, T
• T, T, T, T, T, H
• T, T, T, T, H, T
• …
• H, H, H, H, H, H
12
Pr (X = 4) = 23.4%
Careful here…
same thing !
f(x) vs F(x)
13. • …You flip a coin 6 times. What is the probability of having exactly 4 tails ?
• T, T, T, T, T, T
• T, T, T, T, T, H
• T, T, T, T, H, T
• …
• H, H, H, H, H, H
• ~ Binomial random variable à Pr(X = x)
• =
𝐧
𝐱
𝐩𝐱
𝟏 − 𝐩 (𝐧2𝐱)
• =
𝐧!
𝐱! 𝐧2𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧2𝐱)
• =
𝟔!
𝟒! 𝟔2𝟒 !
𝟎. 𝟓𝟒 𝟏 − 𝟎. 𝟓 (𝟔2𝟒) = 𝟐𝟑. 𝟒%
13
• Binomial random variable : number of
successes in n Bernoulli trials
• Probability (p) of success : constant
for all trials
• Trials : independent
…f(x) vs F(x)
15. • …You flip a coin 6 times. What is the probability function?
15
0%
5%
10%
15%
20%
25%
30%
35%
0 1 2 3 4 5 6
Pr(X = x)
…f(x) vs F(x)
f(x)
16. • …You flip a coin 6 times. What is the CUMULATIVE function?
16
0%
20%
40%
60%
80%
100%
0 1 2 3 4 5 6
Pr(X ≤ x)
…f(x) vs F(x)
F(x)
17. • Discrete vs continuous random variables
17
Bernoulli Binomial
• Probability
function
• Pr(X = x)
p
𝐧!
𝐱! 𝐧 − 𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧$𝐱)
• Density
function
• Pr(X = x)
Discrete
f(x) =
𝟏
𝛔 𝟐𝛑
𝐞2𝟏/𝟐
𝐱"𝛍
𝛔
𝟐
Normal Student Lognormal Chi-squared
ConCnuous - cannot count !
• Also called Z-distribution
• N(𝛍, 𝛔𝟐
)
…f(x) vs F(x)
18. • …Discrete vs continuous random variables
18
f(x) rather complicated
• Also called t-distribution
• Defined by a single parameter - degree of freedom (df)
• Fatter tails than standard normal distribution
…f(x) vs F(x)
Bernoulli Binomial
• Probability
function
• Pr(X = x)
p
𝐧!
𝐱! 𝐧 − 𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧$𝐱)
• Density
function
• Pr(X = x)
Discrete
Normal Student Lognormal Chi-squared
ConCnuous - cannot count !
19. • …Discrete vs continuous random variables
19
f(x) =
𝟏
𝐱𝛔 𝟐𝛑
𝐞
2𝟏/𝟐
𝐥𝐧(𝐱)"𝛍
𝟐𝛔𝟐
𝟐
…f(x) vs F(x)
Bernoulli Binomial
• Probability
function
• Pr(X = x)
p
𝐧!
𝐱! 𝐧 − 𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧$𝐱)
• Density
function
• Pr(X = x)
Discrete
Normal Student Lognormal Chi-squared
ConCnuous - cannot count !
20. • …Discrete vs continuous random variables
20
f(x) rather complicated
…f(x) vs F(x)
Bernoulli Binomial
• Probability
function
• Pr(X = x)
p
𝐧!
𝐱! 𝐧 − 𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧$𝐱)
• Density
function
• Pr(X = x)
Discrete
Normal Student Lognormal Chi-squared
ConCnuous - cannot count !
21. • …Discrete vs continuous random variables
21
• Distribution of asset returns Distribution of asset prices
…f(x) vs F(x)
Bernoulli Binomial
• Probability
function
• Pr(X = x)
p
𝐧!
𝐱! 𝐧 − 𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧$𝐱)
• Density
function
• Pr(X = x)
Discrete
Normal Student Lognormal Chi-squared
ConCnuous - cannot count !
22. • You roll a 7-faces die, 100 times
• What is the probability of face 4 showing 34 times?
• Pr(X=4, 34 times) =
𝟏𝟎𝟎!
𝟑𝟒! 𝟏𝟎𝟎2𝟑𝟒 !
(
𝟏
𝟕
)𝟑𝟒 𝟏 −
𝟏
𝟕
(𝟏𝟎𝟎2𝟑𝟒)
= 𝟎. 𝟎𝟎𝟎𝟎𝟒%
22
f(x) = Pr(X = x) F(x) = Pr(X≤ 𝐱) Mean Variance
Binomiale Discrete
𝐧!
𝐱! 𝐧 − 𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧2𝐱) Pr(X = x) +
Pr(X = x - 1) + …
np np(1-p)
Binomial, Poisson & N-distributions
23. • …You roll a 7-faces die, 100 times
• What is the probability of face 4 showing 34 times?
23
𝐏 𝐗 = 𝐱 =
𝛌𝐱𝐞2𝛌
𝐱!
=
𝟏𝟒. 𝟐𝟗𝟑𝟒
𝐞2𝟏𝟒.𝟐𝟗
𝟑𝟒!
= 𝟎. 𝟎𝟎𝟎𝟒%
Poisson Discrete
𝛌𝐱𝐞2𝛌
𝐱!
Pr(X = x) +
Pr(X = x - 1) + …
𝛌 𝛌
= np
= 100*1/7 = 14.29
= 𝛌
Use if time/space
…Binomial, Poisson & N-distributions
f(x) = Pr(X = x) F(x) = Pr(X≤ 𝐱) Mean Variance
Binomiale Discrete
𝐧!
𝐱! 𝐧 − 𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧2𝐱) Pr(X = x) +
Pr(X = x - 1) + …
np np(1-p)
24. • …You roll a 7-faces die, 100 times
• What is the probability of face 4 showing 34 times?
24
𝛍 = 𝛌
𝛔𝟐 = 𝛌
Normale Contin.
𝟏
𝛔 𝟐𝛑
𝐞
2𝟎.𝟓
𝐱2𝛍
𝛔
𝟐
Integral or
z-table
𝛍 𝛔𝟐
…Binomial, Poisson & N-distributions
Poisson Discrete
𝛌𝐱𝐞2𝛌
𝐱!
Pr(X = x) +
Pr(X = x - 1) + …
𝛌 𝛌
f(x) = Pr(X = x) F(x) = Pr(X≤ 𝐱) Mean Variance
Binomiale Discrete
𝐧!
𝐱! 𝐧 − 𝐱 !
𝐩𝐱
𝟏 − 𝐩 (𝐧2𝐱) Pr(X = x) +
Pr(X = x - 1) + …
np np(1-p)
26. 26
• Poisson distribution
• f x =
<*="+
>!
• µ = λ
• σ? = λ
• Exponential distribution
• Actions occur independently at a constant rate per unit of time or length
• > 0 & skew > 0, declines steadily to the right & asymptotic
• f x = λ ∗ e2<>
• F x = 1 − e2<>
• µ =
@
<
• σ? =
@
<,
Poisson vs exponential distributions
27. • …File S&P 500.xls
• 4 moments of distributions
27
• Skewness • Whether distribution is symmetrically shaped or lopsided
• Dispersion • How far returns are dispersed from their center
• Central tendency • Where returns are centered
• Kurtosis • Whether extreme outcomes are likely
…Population, samples & moments
28. • …File S&P 500.xls
• …4 moments of distributions
28
• Range, mean absolute deviation,
variance & standard deviation
• Skew > 0 : frequent small losses & few
extreme gains
• Proportion of probability in the tails
• Leptokurtic, mesokurtic or platykurtic
• Central tendency • Where returns are centered
• Dispersion • How far returns are dispersed from their center
• Skewness • Whether distribution is symmetrically shaped or lopsided
• Kurtosis • Whether extreme outcomes are likely
• Arithmetic & geometric & weighted
means, median, mode, quartiles,
quintiles, deciles & percentiles
…Population, samples & moments
29. • …File S&P 500.xls
• …4 moments of distributions
29
• Kurtosis
• Skewness
• Dispersion • Variance
• Most widely used measure
of dispersion
• 𝛔𝟐
=
∑𝐢.𝟏
𝐍 𝐗𝐢2𝛍 𝟐
𝐍
• 𝐬𝟐
=
∑𝐢.𝟏
𝐍 𝐗𝐢2!
𝐗 𝟐
𝐧2𝟏
Population Sample
• Central tendency • Arithmetic mean
• 𝛍 =
∑𝐢.𝟏
𝐍 𝐗𝐢
𝐍
• !
𝐗 =
∑𝐢.𝟏
𝐍 𝐗𝐢
𝐍
…Population, samples & moments
31. • …File S&P 500.xls
• Descriptive statistics
31
What can you say about the
distribution of returns?
…Population, samples & moments
32. • …File S&P 500.xls
• …Descriptive statistics
32
Moyenne 0.63%
Erreur-type 0.16%
Médiane 0.91%
Mode #N/A
Écart-type 5.38%
Variance de l'échantillon 0.29%
Kurstosis (Coefficient d'aplatissement) 7.77
Coefficient d'asymétrie 0.1166
Plage 69%
Minimum -30%
Maximum 39%
Somme 7.28
Nombre d'échantillons 1 152
Arithmetic
= Kurtosis()
= Skewness()
Kurtosis, in Excel, is
“Excess kurtosis (vs 3)” !
What can you say about the
distribution of returns?
…Population, samples & moments
33. • …File S&P 500.xls
• https://www.statskingdom.com/shapiro-wilk-test-calculator.html
33
What can you say about the
distribution of returns?
…Population, samples & moments
34. Note : some of the study material comes
either from CFA Institute, Schweser or
GARP
