Historical simulation
Encyclopedia
Historical simulation in finance's value at risk
(VaR) analysis is a procedure for predicting value at risk by 'simulating' or constructing the cumulative distribution function
(CDF) of assets returns over time. Unlike most parametric
VaR models, Historical Simulation does not assume any distribution on the asset returns. Also, it is relatively easy to implement. However, there is a couple of shortcomings of historical simulation, and first of all is that it imposes a restriction on the estimation assuming asset returns are independent and identically-distributed (iid) which is not the case. From empirical evidence, it is known that asset returns are clearly not independent as they exhibit certain patterns such as volatility clustering
. Therefore it can be unrealistic to assume iid asset returns. Second restriction relates to time. Historical simulation, it applies equal weight to all returns of the whole period and this is inconsistent with the nature where there is diminishing predictability of data that are further away from the present. These two shortcomings leads economists and financial experts to further develop other non-parametric, semi-parametric and parametric models.
. The returns in this case are no longer assumed to be iid, rather there is an additional innovation term v is now assumed to be iid instead. This allows the means and variances to be 'filtered away', coupled with an empirically estimated CDF
, it becomes a more realistic model in predicting VaR.
Value at risk
In financial mathematics and financial risk management, Value at Risk is a widely used risk measure of the risk of loss on a specific portfolio of financial assets...
(VaR) analysis is a procedure for predicting value at risk by 'simulating' or constructing the cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
(CDF) of assets returns over time. Unlike most parametric
Parametric
Parametric may refer to:*Parametric equation*Parametric statistics*Parametric derivative*Parametric plot*Parametric model*Parametric oscillator *Parametric contract*Parametric insurance*Parametric feature based modeler...
VaR models, Historical Simulation does not assume any distribution on the asset returns. Also, it is relatively easy to implement. However, there is a couple of shortcomings of historical simulation, and first of all is that it imposes a restriction on the estimation assuming asset returns are independent and identically-distributed (iid) which is not the case. From empirical evidence, it is known that asset returns are clearly not independent as they exhibit certain patterns such as volatility clustering
Volatility clustering
In finance, volatility clustering refers to the observation, as noted by Mandelbrot , that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes." A quantitative manifestation of this fact is that, while returns themselves are...
. Therefore it can be unrealistic to assume iid asset returns. Second restriction relates to time. Historical simulation, it applies equal weight to all returns of the whole period and this is inconsistent with the nature where there is diminishing predictability of data that are further away from the present. These two shortcomings leads economists and financial experts to further develop other non-parametric, semi-parametric and parametric models.
Weighted historical simulation
Like the traditional historical simulation technique, weighted historical simulation applies decreasing weights to returns that are further away from the present, which overcomes the inconsistency of historical simulation with diminishing predictability of data that are further away from the present. However, weighted historical simulation still assumes iid asset returns.Filtered historical simulation
Filtered historical simulation is a semi-parametric technique in forecasting VaRValue at risk
In financial mathematics and financial risk management, Value at Risk is a widely used risk measure of the risk of loss on a specific portfolio of financial assets...
. The returns in this case are no longer assumed to be iid, rather there is an additional innovation term v is now assumed to be iid instead. This allows the means and variances to be 'filtered away', coupled with an empirically estimated CDF
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
, it becomes a more realistic model in predicting VaR.