Extensions of VaR
Conditional Value at Risk (CVaR) Rockafellar and Uryasev introduced conditional value-at-risk (CVaR) in 2000. CVaR is a tail risk metric that quantifies the amount of the expected losses beyond the VaR cutoff point at a specific confidence level. It is…
Advantages and Limitations of VaR
The following are some of the advantages and limitations of the VaR. Advantages Easy to understand: VaR is a single number that approximates the amount of risk in a portfolio. VaR is presented either as a percentage of the value…
Estimating VaR
Parametric Method The parametric method obtains a VaR estimate by using the formula below: $$ VaR_p=\mu-\alpha_p\sigma $$ Where: \(VaR_p\) is the estimated VaR of portfolio \(p\). \(\mu\) is the mean of the portfolio or the expected return of the portfolio….
Methods of Estimating VaR
Parametric (Variance-Covariance) Method The parametric method is also called the variance-covariance method. This method looks at the price changes of an investment over a lookback period and computes a portfolio’s maximum loss using probability theory. It uses the standard deviation…
Value at Risk (VaR)
Value at Risk (VaR) measures the probability of underperformance by providing a statistical measure of downside risk. In the case of a continuous random variable, VaR can be computed as follows: $$ VaR\left(X\right)=-t \text{ where } P\left(X < t\right)=p $$…
Potential Benefits of Multiple Risk Dimensions When Modeling Asset Returns
Benefits of Multifactor Models to Investors Multifactor models help investors understand the comparative risk exposures of equity, fixed income, among other asset returns. This is done by performing a granular risk and return attribution on the actively managed portfolios. Multifactor…
Uses of Multifactor Models and Interpreting the Output of Analyses Based on Multifactor Models
Uses of Multifactor Models Multifactor models are mainly used for return attribution, risk attribution, and portfolio construction. Return Attribution Return attribution is a set of techniques used in the identification of the excess return of a portfolio, relative to its…
Active Risk, Tracking Risk and Information Ratio
Active Risk Active return refers to the return on the portfolio above the return on the benchmark. That is, $$ \text{Active return} = R_P-R_b $$ Active risk, also known traditionally as tracking error or tracking risk, is a risk that…
Macroeconomic Factor Models, Fundamental Factor Models, and Statistical Factor Models
Multifactor Model A multifactor model attempts to explain the observed historical return of security returns in the form of the equation below: $$ R_i=a_i+b_{i,1}l_1+b_{i,2}l_2+\ldots+b_{i,L}l_L+C_i $$ Where: \(R_i\) is the return on security \(i\). \(a_i\) and \(C_i\) are the constant and…
Calculating Expected Returns from the Arbitrage Pricing Model
Single-factor Model The single-factor model assumes that there is just one macroeconomic factor, for example, the return on the market. Therefore: $$ R_i=E(R_i)+\beta_iF+\varepsilon_i $$ Where: \(E(R_i)\) is the expected return on stock \(i\). \(R_i\) is the return for stock \(i\)….
