FRM PART I
FORMULA SHEET
BOOK CHAPTER FORMULA VARIABLES
𝜎𝑖 𝐶𝑜𝑣(𝑖, 𝑚) 𝜎𝑖𝑚
𝛽𝑖 = 𝜌(𝑖𝑚) = 2 = 2
𝜎𝑚 𝑖 𝜎𝑚 𝜎𝑚
𝑅𝑚 = expected market rate of return
CAPM formula
(expected return on 𝐸(𝑅𝑖 ) = 𝑅𝑓 + 𝛽𝑖 (𝑅𝑚 − 𝑅𝑓 ) 𝑅𝑓 = risk-free rate
asset i)
𝐸(𝑅𝑖 ) = expected return on asset i
𝐶𝑜𝑣(𝑖, 𝑚) 𝜎𝑖𝑚
𝜌𝑖𝑚 = =
𝜎𝑖 𝜎𝑚 𝜎𝑖 𝜎𝑚
𝑅𝑚 = expected market rate of return
Modern
Book 1 Portfolio 𝑅𝑓 = risk-free rate
Foundation of Theory (MPT) 𝜎𝑝
Capital market line 𝐸(𝑅𝑝 ) = 𝑅𝑓 + (𝑅 − 𝑅𝑓 ) 𝐸(𝑅𝑝 ) = portfolio expected return
Risk and the Capital 𝜎𝑚 𝑚
Management Asset Pricing
𝜎𝑝 = portfolio standard deviation
Model (CAPM)
𝜎𝑚 = market standard deviation
𝑅𝑖 = expected rate of return on asset i
𝑅𝑓 = risk-free rate
𝜎𝑝 𝐸(𝑅𝑝 ) = portfolio expected return
Capital allocation line 𝐸(𝑅𝑝 ) = 𝑅𝑓 + (𝑅 − 𝑅𝑓 )
𝜎𝑖 𝑖 𝜎𝑝 = portfolio standard deviation
𝜎𝑖 = standard deviation of asset
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FORMULA SHEET
𝜎(𝑅𝑃 ) = portfolio standard deviation
𝐸(𝑅𝑝 ) − 𝑅𝑓
Sharpe ratio Sharpe ratio = 𝐸(𝑅𝑝 ) = portfolio expected return
𝜎(𝑅𝑃 )
𝑅𝑓 = risk-free rate
𝛽𝑃 = portfolio beta
𝐸(𝑅𝑝 ) − 𝑅𝑓
Treynor ratio Treynor ratio = 𝐸(𝑅𝑝 ) = portfolio expected return
𝛽𝑃
𝑅𝑓 = risk-free rate
𝑅𝑃 = return on the portfolio
The tracking error (TE) 𝑇𝐸 = (𝑅𝑃 − 𝑅𝐵𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 )
𝑅𝐵𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 = return on the benchmark portfolio
𝑇 = target or required rate of return
𝑅𝑝 − 𝑇 1 𝑁 2
𝑆𝑅 = ∑ min(0, 𝑅𝑝𝑡 − 𝑇) = downside deviation,
Sortino ratio (SR) 1 𝑁 2 𝑁 𝑡=1
∑
𝑁 𝑡=1 min(0, 𝑅𝑝𝑡 − 𝑇) as measured by the standard deviation of negative
returns
𝐸(𝑅𝑃 − 𝑅𝐵 ) 𝑅𝑃 = return on the portfolio
Information ratio (IR) 𝐼𝑅 =
√𝑉𝑎𝑟(𝑅𝑃 − 𝑅𝐵 ) 𝑅𝐵𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 = return on the benchmark portfolio
𝑅𝑖 = rate of return on security 𝑖
𝐼1 − 𝐸(𝐼1 ) = difference between observed and
The Arbitrage expected values in factor k
Pricing Theory 𝑅𝑖 = 𝐸(𝑅𝑖 ) + 𝛽𝑖1 [𝐼1 − 𝐸(𝐼1 )] + ⋯ + 𝛽𝑖𝐾 [𝐼𝑘 − 𝐸(𝐼𝑘 )]
and Multifactor Return on a security + 𝑒𝑖 𝛽𝑖𝑘 = coefficient measuring the effect of changes
Models of Risk in a factor 𝐼𝑘
and Return
on the rate of return of security 𝑖
𝑒𝑖 =noise factor (i.e., the idiosyncratic factor).
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FORMULA SHEET
Variance/Covariance
𝑀2 − 𝑀
for a factor model with 𝑀+ 𝑀 = number of factors in the model
𝑀
M factors
Number of covariances 𝑛2 − 𝑛
𝑛 = number of variances
required 2
𝐸(𝑅𝑖 ) = expected return on stock 𝑖
𝑅𝑓 = risk-free interest rate
𝑆𝑀𝐵 = size factor
The Fama-French 𝐸(𝑅𝑖 ) = 𝑅𝑓 + 𝛽𝑖,𝑀𝐾𝑇 𝐸(𝑅𝑚 − 𝑅𝑓 ) + 𝛽𝑖,𝑆𝑀𝐵 𝐸(𝑆𝑀𝐵) 𝛽𝑖,𝑆𝑀𝐵 = factor-beta for the size factor
Model (FFM) + 𝛽𝑖,𝐻𝑀𝐿 𝐸(𝐻𝑀𝐿)
𝐻𝑀𝐿 = value factor
𝛽𝑖,𝐻𝑀𝐿 = factor-beta for the value factor
𝐸(𝑅𝑚 − 𝑅𝑓 ) = CAPM market factor
𝛽𝑖,𝑀𝐾𝑇 = factor-beta for the market-factor
Mutually exclusive 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0 𝑃(𝐴 ∩ 𝐵) = probability A intersection B
events 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) 𝑃(𝐴 ∪ 𝐵) = probability A union B
Book 2 Fundamentals 𝑃(𝐴 ∩ 𝐵) 𝑃(𝐴│𝐵) = probability of A given B
of Probability Conditional probability 𝑃(𝐴│𝐵) =
Quantitative 𝑃(𝐵) 𝑃(𝐴 ∩ 𝐵) = P(A|B)P(B)
Analysis
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
Independent events
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
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FORMULA SHEET
Conditional 𝑃(𝐴)𝑃(𝐵)
𝑃(𝐴|𝐵) = = 𝑃(𝐴)
Probability 𝑃(𝐵)
𝑃(𝐵|𝐴)𝑃(𝐴)
Bayes’ Theorem 𝑃(𝐴|𝐵) =
𝑃(𝐵)
𝑛!
𝑃(𝑋 = 𝑥) = 𝑝 𝑥 (1 − 𝑝)𝑛−𝑥 , 𝑥 = 0,1,2, … , 𝑛 𝑃(𝑋 = 𝑥) = probability mass function of X
𝑥! (𝑛 − 𝑥)!
Binomial distribution |𝑥|
𝑛
𝐹𝑋 (𝑥) = ∑ ( ) 𝑝𝑖 (1 − 𝑝)𝑛−𝑖 𝐹𝑋 (𝑥) = cumulative distribution function of X
𝑖
𝑖=1
𝐸(𝑋) = 𝑛𝑝 𝐸(𝑋) = expectation of X
Univariate
Random 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝) 𝑉(𝑋) = variance of X
Variables
𝑃(𝑋 = 𝑥) = 𝑝 𝑥 (1 − 𝑝)1−𝑥 , 𝑥 = 0,1 ; 0 < 𝑝 < 1 𝑃(𝑋 = 𝑥) = probability mass function
0, 𝑦<0
Bernoulli distribution 𝐹𝑋 (𝑥) = {1 − 𝑝, 0 ≤ 𝑦 < 1 𝐹𝑋 (𝑥) = cumulative distribution function
1, 𝑦≥1
𝐸(𝑋) = 𝑝 𝐸(𝑋) = expectation of X
𝑉(𝑋) = 𝑝(1 − 𝑝) 𝑉(𝑋) = variance of X
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FORMULA SHEET
BOOK CHAPTER FORMULA VARIABLES
𝜎𝑖 𝐶𝑜𝑣(𝑖, 𝑚) 𝜎𝑖𝑚
𝛽𝑖 = 𝜌(𝑖𝑚) = 2 = 2
𝜎𝑚 𝑖 𝜎𝑚 𝜎𝑚
𝑅𝑚 = expected market rate of return
CAPM formula
(expected return on 𝐸(𝑅𝑖 ) = 𝑅𝑓 + 𝛽𝑖 (𝑅𝑚 − 𝑅𝑓 ) 𝑅𝑓 = risk-free rate
asset i)
𝐸(𝑅𝑖 ) = expected return on asset i
𝐶𝑜𝑣(𝑖, 𝑚) 𝜎𝑖𝑚
𝜌𝑖𝑚 = =
𝜎𝑖 𝜎𝑚 𝜎𝑖 𝜎𝑚
𝑅𝑚 = expected market rate of return
Modern
Book 1 Portfolio 𝑅𝑓 = risk-free rate
Foundation of Theory (MPT) 𝜎𝑝
Capital market line 𝐸(𝑅𝑝 ) = 𝑅𝑓 + (𝑅 − 𝑅𝑓 ) 𝐸(𝑅𝑝 ) = portfolio expected return
Risk and the Capital 𝜎𝑚 𝑚
Management Asset Pricing
𝜎𝑝 = portfolio standard deviation
Model (CAPM)
𝜎𝑚 = market standard deviation
𝑅𝑖 = expected rate of return on asset i
𝑅𝑓 = risk-free rate
𝜎𝑝 𝐸(𝑅𝑝 ) = portfolio expected return
Capital allocation line 𝐸(𝑅𝑝 ) = 𝑅𝑓 + (𝑅 − 𝑅𝑓 )
𝜎𝑖 𝑖 𝜎𝑝 = portfolio standard deviation
𝜎𝑖 = standard deviation of asset
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FORMULA SHEET
𝜎(𝑅𝑃 ) = portfolio standard deviation
𝐸(𝑅𝑝 ) − 𝑅𝑓
Sharpe ratio Sharpe ratio = 𝐸(𝑅𝑝 ) = portfolio expected return
𝜎(𝑅𝑃 )
𝑅𝑓 = risk-free rate
𝛽𝑃 = portfolio beta
𝐸(𝑅𝑝 ) − 𝑅𝑓
Treynor ratio Treynor ratio = 𝐸(𝑅𝑝 ) = portfolio expected return
𝛽𝑃
𝑅𝑓 = risk-free rate
𝑅𝑃 = return on the portfolio
The tracking error (TE) 𝑇𝐸 = (𝑅𝑃 − 𝑅𝐵𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 )
𝑅𝐵𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 = return on the benchmark portfolio
𝑇 = target or required rate of return
𝑅𝑝 − 𝑇 1 𝑁 2
𝑆𝑅 = ∑ min(0, 𝑅𝑝𝑡 − 𝑇) = downside deviation,
Sortino ratio (SR) 1 𝑁 2 𝑁 𝑡=1
∑
𝑁 𝑡=1 min(0, 𝑅𝑝𝑡 − 𝑇) as measured by the standard deviation of negative
returns
𝐸(𝑅𝑃 − 𝑅𝐵 ) 𝑅𝑃 = return on the portfolio
Information ratio (IR) 𝐼𝑅 =
√𝑉𝑎𝑟(𝑅𝑃 − 𝑅𝐵 ) 𝑅𝐵𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 = return on the benchmark portfolio
𝑅𝑖 = rate of return on security 𝑖
𝐼1 − 𝐸(𝐼1 ) = difference between observed and
The Arbitrage expected values in factor k
Pricing Theory 𝑅𝑖 = 𝐸(𝑅𝑖 ) + 𝛽𝑖1 [𝐼1 − 𝐸(𝐼1 )] + ⋯ + 𝛽𝑖𝐾 [𝐼𝑘 − 𝐸(𝐼𝑘 )]
and Multifactor Return on a security + 𝑒𝑖 𝛽𝑖𝑘 = coefficient measuring the effect of changes
Models of Risk in a factor 𝐼𝑘
and Return
on the rate of return of security 𝑖
𝑒𝑖 =noise factor (i.e., the idiosyncratic factor).
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, FRM PART I
FORMULA SHEET
Variance/Covariance
𝑀2 − 𝑀
for a factor model with 𝑀+ 𝑀 = number of factors in the model
𝑀
M factors
Number of covariances 𝑛2 − 𝑛
𝑛 = number of variances
required 2
𝐸(𝑅𝑖 ) = expected return on stock 𝑖
𝑅𝑓 = risk-free interest rate
𝑆𝑀𝐵 = size factor
The Fama-French 𝐸(𝑅𝑖 ) = 𝑅𝑓 + 𝛽𝑖,𝑀𝐾𝑇 𝐸(𝑅𝑚 − 𝑅𝑓 ) + 𝛽𝑖,𝑆𝑀𝐵 𝐸(𝑆𝑀𝐵) 𝛽𝑖,𝑆𝑀𝐵 = factor-beta for the size factor
Model (FFM) + 𝛽𝑖,𝐻𝑀𝐿 𝐸(𝐻𝑀𝐿)
𝐻𝑀𝐿 = value factor
𝛽𝑖,𝐻𝑀𝐿 = factor-beta for the value factor
𝐸(𝑅𝑚 − 𝑅𝑓 ) = CAPM market factor
𝛽𝑖,𝑀𝐾𝑇 = factor-beta for the market-factor
Mutually exclusive 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0 𝑃(𝐴 ∩ 𝐵) = probability A intersection B
events 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) 𝑃(𝐴 ∪ 𝐵) = probability A union B
Book 2 Fundamentals 𝑃(𝐴 ∩ 𝐵) 𝑃(𝐴│𝐵) = probability of A given B
of Probability Conditional probability 𝑃(𝐴│𝐵) =
Quantitative 𝑃(𝐵) 𝑃(𝐴 ∩ 𝐵) = P(A|B)P(B)
Analysis
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
Independent events
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
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FORMULA SHEET
Conditional 𝑃(𝐴)𝑃(𝐵)
𝑃(𝐴|𝐵) = = 𝑃(𝐴)
Probability 𝑃(𝐵)
𝑃(𝐵|𝐴)𝑃(𝐴)
Bayes’ Theorem 𝑃(𝐴|𝐵) =
𝑃(𝐵)
𝑛!
𝑃(𝑋 = 𝑥) = 𝑝 𝑥 (1 − 𝑝)𝑛−𝑥 , 𝑥 = 0,1,2, … , 𝑛 𝑃(𝑋 = 𝑥) = probability mass function of X
𝑥! (𝑛 − 𝑥)!
Binomial distribution |𝑥|
𝑛
𝐹𝑋 (𝑥) = ∑ ( ) 𝑝𝑖 (1 − 𝑝)𝑛−𝑖 𝐹𝑋 (𝑥) = cumulative distribution function of X
𝑖
𝑖=1
𝐸(𝑋) = 𝑛𝑝 𝐸(𝑋) = expectation of X
Univariate
Random 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝) 𝑉(𝑋) = variance of X
Variables
𝑃(𝑋 = 𝑥) = 𝑝 𝑥 (1 − 𝑝)1−𝑥 , 𝑥 = 0,1 ; 0 < 𝑝 < 1 𝑃(𝑋 = 𝑥) = probability mass function
0, 𝑦<0
Bernoulli distribution 𝐹𝑋 (𝑥) = {1 − 𝑝, 0 ≤ 𝑦 < 1 𝐹𝑋 (𝑥) = cumulative distribution function
1, 𝑦≥1
𝐸(𝑋) = 𝑝 𝐸(𝑋) = expectation of X
𝑉(𝑋) = 𝑝(1 − 𝑝) 𝑉(𝑋) = variance of X
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