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
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