P&L exceeding VaR limits is not an uncommon phenomenon. In fact, (accurate) VaR models should be designed such that there is some expectation of P&L exceeding the VaR (the number of breaches being proportionate to the level of confidence). However, in practice, whenever P&L exceeds VaR it is prudent to investigate why the breach has occurred. For OSFI-regulated financial institutions,…

# Market Risk

# Upper Bound for RER under VaR

In continuation of some previous posts on residual estimation risk (RER), we establish an upper bound for RER when the risk measure is VaR for any arbitrary error distribution , where the error distribution is defined as the difference between an actual loss distribution and a loss estimator (see [1] for more details). Asymmetric Error Distribution For an arbitrary…

# Sensitivities for Make-whole Callable Bonds

A callable bond is a bond that provides the issuer with the right to exchange the bond for its call value in cash. There are a few ways one can value a callable bond by extending the expression of a vanilla bond: Yield-to-x: Calculate the yield-to-maturity, yield(s)-to-call, and/or yield-to-worst and take the lowest of this set. Revalue the bond with…

# Residual Estimation Risk

My 3-month hiatus has ended thanks to the Canadian long weekend. In these past few months I haven't had much time time to commit to posting new content, primarily because I started a new job. This has put my research interests on hold for the time being. So while I will likely not post as frequently as I have in…

# Risk and the Retail Investor

Risk can be elicited in many ways, shapes, and forms. Unfortunately, the typical retail investor is incognisant of how risk affects them financially. It is not surprising why this is the case. In an industry where there are buyers and sellers of financial products, it is more advantageous for the seller to focus on the rewards, rather than the risks…

# VARs in SAS: VAR, TVAR, and RVAR

In a previous post, various risk measures were introduced and their functions were provided in R. Since the code to calculate VAR, TVAR, and RVAR is quite simple for the historical simulation approach, I've decided to provide these same functions in SAS. And to call on these functions, given an arbitrary dataset "portfolio" with column "balance" and confidence level of 95% (or…

# VARs in R: VAR, TVAR, and RVAR

Risk measures come in many varieties. Some are mean-based, some are median-based. Some are coherent, some are not. However, most risk measures of interest satisfy law-invariance, translation invariance, positive homogeneity, and monotonicity. Three examples of such are Value-at-Risk (VAR), Tail Value-at-Risk (TVAR) (or Expected Shortfall), and Restricted Value-at-Risk (RVAR). VAR is defined as It represents the maximum value a distribution can attain within confidence level .…