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## 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 ) / n^{½}

Given that n is sufficiently large

## GARCH

GARCH ( I , I ) Volatility Estimate = [ ω + α u^{2}_{n−1} + β σ^{2}_{n−1} ] ^{½}

## Combinations

^{n}C_{r}

= 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 / n^{½} )

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:

s_{x} = sd / ( n )^{0.5}

## Standard Deviation of Returns

Standard Deviation of Returns may be calculated through the formula:

Return Standard Deviation = [ Σ_{i} ( x_{i} – 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

[ Y_{i} = b_{0} + b_{1 }X_{i} + b_{2 }X^{2}_{ i} + e_{i } ]

## 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 _{Y })

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 } ] / σ^{4}

## 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, b_{o},

The Intercept Term ( bˆ_{0 }) = Y – ( X * bˆ) ^{1}

## Coefficient of Determination, R^{2}

= ( 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 e^{2} (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*(*b*− *r *)*dt*+ *σdz*,

Where:

*a is a constant *

*b is a constant *

*σ is a *constant

*r *represents the rate of interest

## Chi Square Test

c^{2} = ( n – 1 ) s^{2} / σ^{2}

## F Test

F = S^{2}_{a} / S^{2}_{b}

Where:

S^{2}_{a} is the Variance of Sample a

S^{2}_{b }is 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 ) = a^{2} Var (X) + b^{2} Var (Y) + 2ab * Cov (X, Y)

## Summary

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