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FRM Level 1 Formulas – Quantitative Analysis

Estimated reading time: 3 minutes

 

Introduction

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Now…unto the formulas!

 

Exceptions

Exceptions = ( 1 –  Confidence Interval )  X  ( Amount of Days )

 

Portfolio’s Value-at-Risk

VaR  =   Value of Portfolio  X  [ E ( R ) – z*σ  ]

or

Portfolio Value-at-Risk

= ( 1-Day VaR )  X  ( n ½ ) , with n = the Number of Days

 

Semi-annual comparable yield

The semi-annual pay comparable yield for a given annual pay bond will be determined by the following relationship:

=  2 X  [  (  1 + YTM for Annual Pay Bond  )½   – 1  ]

 

Standard Deviation

The Standard Deviation of a given Sample Average is represented by:

= σ ( X ) / n½ 

Given that n is sufficiently large

 

GARCH

GARCH ( I , I ) Volatility Estimate = [  ω  +  α u2n−1   +   β σ2n−1   ] ½

 

Combinations

 nCr

=   n! / ( r! ) X ( n – r )!

(translation: The amount of ways we can select r out of n)

 

Standard Error

Standard Error of a Mean will be given:

= ( Standard deviation )  X  ( 1 / n½  )

To calculate the standard error of the sample mean:

We will divide the standard deviation of the sample by the square root of the sample size:

sx   = sd / ( n )0.5

 

Standard Deviation of Returns

Standard Deviation of Returns may be calculated through the formula:

Return Standard Deviation  = [  Σi ( xi – X )2  /  ( n – 1 ) ] 1/2

 

Basis

Basis  = ( Hedged Security’s Spot Price  –  Futures Contract Price used in Hedge )

 

Bayes’ Theorem

P (A / B) =  [ P ( B / A ) * P ( A )  ]  / P ( B )

or

P (AB) = P (B/A)  X  P (A)

 

Expectations

E (cX) = E (X) * c

E (X + Y) = E (X) + E (Y)

E (XY) = E (X) * E (Y)

(Assuming both X and Y are independent of each other)

 

Value of the z-statistic

The value of the z-statistic may be determined using the following formula:

( Sample Mean −  Hypothesized Mean ) /  ( Standard Deviation of Population  / ( Sample Size )½ 

or

z = (x – mean) / standard deviation

 

Sample Standard Deviation

Sample Standard Deviation  = ( Sample Variance ) ½ 

 

General Regression Equation

[  Yi  =  b0 +  b1 Xi  +  b2 X2 i  +  ei  ]

 

Coefficient of Determination

Coefficient of Determination =  ( Total Variation – Unexplained Variation )  /  Total Variation

or

Coefficient of Determination =  ( Explained Variation )  /  ( Total Variation )

 

Standard Error 

Standard Error  =  [ Sum of Squared Error  /  ( n – 2 ) ] ½  

 

Correlation Coefficient

Correlation Coefficient = Cov ( X , Y )  /  (  Standard Deviation X   *   Standard Deviation )

or

Correlation Coefficient  =  ( Coefficient of determination ) ½  

 

F-Statistic

F-Statistic  =  MSR / MSE

=   (  RSS / 1 )  /  [  SSE  /  (  n – k – 1  )  ]

 

Kurtosis

K =  SUM [ ( xi – μ )4  ]    /   σ4

 

Variance

Variance of X:  =  Sum of Squared Deviations in X / ( n – 1 )

 

Covariance

Covariance =  Sum of Product-of-Deviations / ( n – 1 )

 

Slope Coefficient

Slope Coefficient  =  Cov ( X , Y )  /  Var ( X )

 

Regression

The Regression’s Intercept Term, bo,

The Intercept Term ( bˆ0 )  =  Y  – ( X * bˆ) 1

 

Coefficient of Determination, R2

= ( Summation of Squares explained through Regression ) / ( Sum Total of Squares )

 

Standard Error of the Estimate

Standard Error of the Estimate ( SEE ) will be given:

=  [  RSS  /  ( n – k – 1 )  ] ½

Where:

n = The size of the Sample

k = The Number of Independent Variables

RSS = The Summation of e2 (or the Residual Sum of Squares)

 

Population Regression

Dependent Variable, Y =  (y Intercept) + (Slope Coefficient * Independent Variable) + Residual Term

 

Vasicek Model

The Vasicek model which defines a risk‐neutral process for r:

 dr= a(br )dt+ σdz,

Where:

a is a constant

b is a constant

σ is a constant

r represents the rate of interest

 

Chi Square Test

c2  =  ( n – 1 ) s2   /    σ2

 

F Test

F = S2a  /  S2b

Where:

S2a  is the Variance of Sample a

 S2is the Variance of Sample b

 

Bond Survival Rate

=  ( 1 –  Marginal Mortality Rate )

 

Total Variation

Total Sum of Squares =

( Sum of Squares Error/Residual Sum of Squares )

  +

( Sum of Squares Regression/Explained Sum of Squares )

 

That is,

TSS = RSS + ESS

 

Risk-Adjusted Return

The Risk-Adjusted Return that will be employed in the computation of RAROC will be given:

=  ( Expected Revenue)  – ( Expenses ) – ( Expected Loss )

 

Business Line RAROC

The Business Line RAROC will be:

=  Risk-Adjusted Return  /  Risk-Adjusted Capital

 

Variance

Var ( aX + bY ) = a2 Var (X) + b2 Var (Y) + 2ab * Cov (X, Y)

 

Summary

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