Bayes’ Formula
P A|B
P B|A PB
PA
Basic Statistics
Expected Value E X P x x P x x ... P x x
Variance σ EX μ
Covariance Cov X, Y E X E X Y E Y
Correlation
ρ
Cov X,Y σσ
Skewness& kurtosis
Skewness
EX μ σ
Positive skewness: Mode < Median < Mean
Negative skewness: Mode > Median > Mean
Kurtosis
EX μ EX μ
Excess kurtosis = sample kurtosis – 3
Sums of Random Variables If X and Y are any random variables: EX Y EX EY If X and Y are independent: Var X Y Var X Var Y If X and Y are not independent: Var X Y Var X Var Y 2Cov X, Y
α ER
R β ER R
Arbitrage Pricing Theory
ER R β R β R
⋯ βR
R β E R Rf ⋯ β E R Rf
专业来自 101%的投入
1
FRM Part I Easy Sheet
2
专业来自 101%的投入
FRM Part I Easy Sheet
QUANTITATIVE METHODS
专业来自 101%的投入
3
the sampling distribution of the sample mean
approaches a normal probability distribution with
mean μ and a variance σ2/n equal to as the sample
size becomes large (n≥30).
X~N
μ,
σ n
Measure of Central tendency
Population mean:
μ
∑X N
Sample mean:
X
∑X n
Measurement of dispersion
Population variance :
Xμ
Sortino Ratio
ER
1 T
∑
R
其中 R MAR
MAR MAR
Information Ratio IR
ER ER σR R
Tracking Error R R
∑
(Method 1) (Method 2)
(N is the number of return periods measured)
Jensen’s Alpha:
Student’s t‐distribution: Student’s t‐distribution: is symmetric; fatter tails; Degrees of freedom=(n‐1).
Central Limit Theorem:
When selecting simple random samples of size n from a population with a mean μ and a finite variance σ2,
are normally distributed. Standardized Normal Distribution Z X σ μ ∼ N 0,1 Confidence Interval 68% of observations fall within 1σ 90% of observations fall within 1.65σ 95% of observations fall within 1.96σ 99% of observations fall within 2.58σ
pk
PX k
λ k!
e
EX λ
DX λ
λ np
Normal Distribution: Completely described by mean and variance (μ, σ2) It is symmetric with skewness measure of 0, i.e.,
mean = mode = median Kurtosis = 3 Linear combinations of normal random variables
Capital Market Line (CML):
ER
R
ER R σ
σ
CAPM(SML)
E RP Rf E RM Rf β，其中
,
β
FRM Part I Easy Sheet
Measures of Performannor Ratio TR
ER R β
FUNDATIONS OF RISK MANAGEMENT
Portfolio Management Theory
Portfolio Return: ER ωER ωER
Portfolio Variance: σP2 ωA2 σA2 ωB2 σB2 2ρωAωBσAσB
σ ω σ ω σ 2ω ω cov A, B
Common Probability Distributions
Continuous Uniform Distribution:
0 for x a
Fx
x b
a a
for a
x
b
1 for x b
Binomial distribution px Cp 1 p E X np D X np 1 p
Poisson Distribution
σ
N
Population standard deviation :
XX
σ
N
Sample variance:
s2
∑ni 1 Xi X 2
n1
Sample standard deviation:
XX
s
n1
Sampling & Estimation
Sampling Distribution: Probability distribution of all possible sample statistics computed from a set of equal‐size samples randomly drawn from the same population. The sampling distribution of the mean is the distribution of estimates the mean.