Formulas Risk Management

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Chapter 7: How traders manage their risks

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ΔP: Change PF value

ΔS: Change variable value

Δσ: Change in volatility 

Δr: Change interest rate

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Delta-neutral Portfolio

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Chapter 8: Interest rate risk

Parallel yield curve shifts

One interest-dependent instrument

Duration

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ci: cash flow at ti

y: yield

Dollar duration

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Δy: Change yield

Key relationship

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Modified duration

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m: compounding frequency

Convexity

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Dollar duration

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Portfolio interest-dependent instruments

Small parallel shift

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Duration

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Xi: PV ith asset

P: PV all assets

Dollar duration

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Nonparallel yield curve shifts

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Chapter 9: VaR

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Impact of autocorrelation

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VaR conversion

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Incremental VaR: VaR = ∑iVaR

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xi = invested ith subportfolio

Aggregating VaRs

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Backtesting

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Kupiec’s Two-Tailed Test

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n: Trials

m: # of exceptions

p: prob of exception

Chapter 10: Volatility

Simplified variance

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Power Law

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K,a: constants

ARCH model

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a: weight observation i

Volatility weighting schemes

EMWA

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GARCH (1,1)

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VL: Long-run variance rate

: Weight VL[pic 44]

Ljung-Box statistic

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m: # of observations

ck: autocorrelation for a lag of k

K: # of lags

Volatility forecast GARCH (1,1)

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Volatility term structures

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V(0): σ2n

Impact of volatility changes

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Chapter 11: Correlations and Copulas

Correlation two variables

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V1: Variable 1

V2: Variable 2

Covariance

Cov(V1,V2) = E(V1V2)-E(V1)E(V2)

Monitoring correlations between 2 variables:

Xi=(Xi-Xi-1)/Xi-1        Yi=(Yi-Yi-1)/Yi-1

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EMWA

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Garch(1,1)

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Variance-Covariance Matrix

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Bivariate Normal Distribution

V1, V2 normal with mean:

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Multivariate Normal Distribution

Generating random sample:

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Factor Models

One factor: [pic 63]

Multifactor:

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Factor Copula Model

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Vasicek’s Model

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Q: Probability Default Time T

X: Confidence Level (0.999)

Possible loss:

Investment * WCDR*(1-recovery rate)

Chapter 12: Basel I, II and Solvency II

Cook ratio

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Derivatives

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Netting

Without netting

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With netting

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Net replacement ratio (NRR)

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Credit equivalent amount

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1996 Amendment

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Total capital = (credit risk RWA + Market risk RWA) *0.08

Basel II

Total capital required = 0.08*(credit risk RWA + market risk RWA + operational risk RWA)