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FRM Level 1 Formulas – Foundations of Risk Management

Estimated reading time: 3 minutes

 

Preface

Below are FRM formulas for the level 1 Foundations of Risk Management segment.

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Correlation coefficient between two Securities:

Correlation coefficient =

(Covariance between Security A and Security B) /

(Standard Deviation of Security A * Standard Deviation of Security B)

 

Variance of Two Securities

Variance =

(WeightA)2 * (Standard Deviation of A)2    +

(WeightB)2 * (Standard Deviation of B)2    +

(2 * Correlation CoefficientAB * Standard Deviation of A * Standard Deviation of B * WeightA * WeightB )

 

Perfect Positive Correlation

ρ = 1

 

Perfect Negative Correlation

ρ = -1

 

Standard Deviation

= (Variance)1/2

 

Risk Adjusted Ranking

Based on the Morningstar Rating System, an investment fund’s risk adjusted ranking may be calculated by:

(Fund Return / Average Peer Return ) – (Fund Risk / Average Peer Risk)

 

Sharpe Ratio

= (Rp – Rf) / σp

Where:

Rp   is the Portfolio Return

Rf    is the Treasury-Bill Returns (or the Risk Free Rate)

σp    is the Portfolio Standard Deviation of Return

 

Sortino Ratio

= (Return on Portfolio – Minimum Accepted Return)Standard Deviation of Returns Below Minimum Accepted Return

 

Treynor Measure

= (Return on Portfolio – Risk free rate) / Portfolio Beta

 

Jenson’s Alpha

= Return on Portfolio – CAPM predicted Return

 

Information Ratio

= (Return on Portfolio – Benchmark Return) / Tracking Error

 

A portfolio’s beta using the respective weightings:

Portfolio beta = w1R1 + w2 R2 + w3 R3

 

Portfolio Expected Return    

= Rf + R(Rm – Rf)

 

Expected Return

Capital Asset Pricing Model, CAPM

We recall the CAPM formula:

Expected Return, ER = Rf + β*(Rm – Rf)

Now the part of the formula “(Rm – Rf)” is actually termed the “risk premium”

Thus, expected return on the security will be:

= Risk Free Rate + [ Security’s Beta *( Equity Risk Premium) ]

 

Adjusted Exposure

= Outstandings + (Unused Portion of Commitments) * Usage Given Default

 

Expected Loss

= AE * EDF * LGD

Where:

AE is the Adjusted Exposure

EDF is the Probability of Default

LGD is the Loss Given Default = ( 1 – Recover Rate %)

 

Expected Loss is defined as:

Exposure * (1 – Recovery Rate) * (Probability of Default)

 

Basis

Basis = Spot price of hedged asset – Futures price of contract.

 

Market Coefficient of Variation

Market Coefficient of Variation = Standard Deviation of Market / Market Expected Return

 

Standard Deviation of Market

Standard Deviation of Market = Market Coefficient of Variation * Market Expected Return

 

Standard deviation of a two-stock portfolio

Standard deviation of a two-stock portfolio, we may use the formula:

s = [WA2σA 2 + WB 2σB 2 + 2WAWBσAσBrA,B]1/2

 

Correlation Coefficient

For two assets ‘A’ and ‘B’

Correlation coefficient = CovarianceAB / [(Standard deviationA * Standard deviationB)]

Rearranged, we have:

CovarianceAB = (Correlation coefficient) * [(Standard DeviationA * Standard DeviationB)]

 

Covariance

CovarianceAB = (Correlation coefficient) * [(Standard DeviationA * Standard DeviationB)]

 

Total Risk

Total Risk = Market Risk + Firm Specific Risk

or

Total Risk = Systematic Risk + Unsystematic Risk

Note:

Market risk can also be called Systematic risk or Undiversifiable risk

Firm Specific Risk can also be called Unsystematic risk

 

Expected Return

Expected Return = Stock’s Alpha + (Stock’s Beta * % Market Movement)

 

Excess Return

Excess Return = Expected Return – Risk Free Rate

 

Correlation Coefficient

Correlation coefficient and covariance of two assets ‘A’ and ‘B’

Correlation coefficient = CovarianceAB / (Standard deviationA * Standard deviationB)

 

Standard Deviation of Market

Standard Deviation of Market = Market Coefficient of Variation * Market Expected Return

 

Beta

The calculation of beta on a given Security A and the market

Beta = Covariance between Security A & the Market / Variance of Market Return

 

Value at Risk

The Value at Risk (risk adjusted) performance measure can be stated as follows:

( Portfolio Return – RFR ) /  (Portfolio Value at Risk / Initial Portfolio Value )

Where RFR = Risk Free Rate

 

Capital Markets Line

The Capital Markets Line is from the given equation:

RF + [E(RM) − RF] / σM

Where:

Intercept = RF             

Slope = [E(RM) − RF] / σM

 

Expected Residual Return

Expected Residual Return = Information Ratio * Residual Risk

 

Covariance for Markets A and B

Covariance formula for Markets A and B:

Cov (A, B) = βA,1 βB,1 σ2F1 + βA,2 βB,2 σ2F2 + (βA,1 βB,2 + βA,2 βB,1) * Cov (F1, F2)

 

Equation of a Straight Line

The equation of a straight line: y = m(x) + c, also expressed as:

Y = Slope (x)  +  Intercept

 

Summary

We hope that this post was useful! It’s the first in a series we have lined up.

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