Market Risk Measurement and Management Flashcards
The relationship between arithmetic rt and geometric Rt returns using a Taylor’s series expansion for the natural log
Empirical and estimated VaR
Lognormal VaR
Delta-normal VaR
- Delta-normal VaR assumes that:
- returns are normally distributed (or multivariate normal)
- [delta] the portfolio/position exposures, to the risk factor(s), are expressed linearly
Sprectral risk measure
A spectral risk measure is a risk measure given as a weighted average of outcomes where bad outcomes are included with larger weights
Age-weighted historical simulation (Boudoukh, Richardson and Whitelaw approach (BRW))
- The data is weighted using a weighting function
- w(1) = w(1)
- w(2) = λw(1)
- w(3) = λ2w(1)
- …
Volatility-weighted historical simulation (Hull and White (HW))
- rt,i = the return of asset i on day t
- σt,i = the estimated volatility on day t
- σT,i = the most recent estimation of volatility
Filtered historical simulation advantages
- It enables us to combine the non-parametric attractions of HS with a sophisticated (e.g., GARCH) treatment of volatility, and so take account of changing market volatility conditions.
- It is fast, even for large portfolios.
- As with the earlier HW approach, FHS allows us to get VaR and ES estimates that can exceed the maximum historical loss in our data set.
- It maintains the correlation structure in our return data without relying on knowledge of the variance-covariance matrix or the conditional distribution of asset returns.
- It can be modified to take account of autocorrelation or past cross-correlations in asset returns.
- It can be modified to produce estimates of VaR or ES confidence intervals by combining it with an OS or bootstrap approach to confidence interval estimation.
- There is evidence that FHS works well.
Risk axiomes
Average correlation between n assets
Market risk consists of four types of risk
- Equity risk
- Interest-rate risk
- Currency risk
- Commodity risk
Credit migration
The risk that the credit quality of a debtor decreases
Joint probability of default for two binomial events
Standard deviation of a binomially distributed variable X
CVaR credit value-at-risk
Measures the maximum loss of a portfolio of correlated debt with a certain probability for a certain timeframe
DV01

Arbitrage-free models
Models that take the initial term structure as given
Equilibrium models
Models that start with assumptions about the interest rate process and about the risk premium demanded by the market for bearing interest rate risk and then derive the risk-neutral process
Risk-neutral process with drift
- λ = drift term
- dr denotes the change in the rate over a small time interval dt, measured in years
- σ denotes the annual basis-point volatility of rate changes
- dw denotes a normally distributed random variable with a mean of zero and a standard deviation of dt1/2
Risk-neutral process with time-dependant drift (Ho-Lee model)
- λ = drift term
- The drift of the interest rate process is presumed to be time-varying.
- No long-run equilibrium value is defined in the Ho-Lee model.
Risk-neutral process with time-dependant volatility
Standard deviation of the change in rate
Tree approximating a risk-neutral process with drift
Tree approximating a risk-neutral process with time-dependant drift