Jul 31

Value at Risk (VaR): 2007 FRM. Part 1

by David Harper, CFA, FRM, CIPM

FRM | Risk |

var_foundation1

2007 FRM Learning Outcomes (related)

  • LO 7.1 Discuss reasons for the widespread adoption of VAR as a measure of risk.
  • LO 7.2 Define value at risk (VaR)...
  • LO 7.4 Discuss assumptions underlying VaR calculations…

Why did VaR Became Popular: Innovation, commercial and regulatory

Value at Risk (VaR) gained traction due to need (inferior traditional risk metrics), a regulatory impetus and commercial acceptance. Reasons for the widespread adoption of VaR include:

  • INNOVATION: Before VaR, the common measure of risk was beta. Beta is the single risk factor in the capital asset pricing model (CAPM; you can think of the CAPM as a special case of the multi-factor model: but it has only one factor, and that factor is beta). Beta has suffered many challenges, critics, and limitations; people were ready for a better risk metric.
  • COMMERCIAL: J.P. Morgan is widely credited with advancing VaR by deploying its "open architecture metric" called RiskMetrics
  • REGULATORY: in 1998, the Bank for International Settlements (BIS)  started to allow banks to use internal models such as VaR in order to calculate their capital requirements. Why did they do this? In large part because the original Basel Accord was too blunt; its four risk buckets were too "homogenous." The encouraged regulatory arbitrage.

 JP Morgan later said about the introduction of RiskMetrics in 1994, "we took the bold step of revealing the internal risk management methodology…and a free data set…At the time, there was little standardization in the marketplace".

 

What is VAR? A summary, statistical measure of probable loss over time (T) with confidence level (alpha)

VaR is the statistical answers to the question, "how much could we lose, in a given time frame with some degree of confidence?" VaR gives the worst expected loss, over a time horizon, given some confidence level. It does not give the worst-case scenario.

Note: the user must choose both a time horizon and a confidence level; typical confidence levels are 95% and 99%.

19VaR has shortcomings (in addition to some common limitations like model risk, implementation risk):

  • As above, VaR gives no information about loss in excess of VaR. This is the domain of extreme value theory (EVT).
  • VaR says nothing about the distribution of losses below the VaR (itself)
  • VaR is subject to sampling variation
  • VaR is not a coherent risk measure (it is not sub-additive)

 

The assumptions behind VaR

VaR makes these assumptions:

  • Stationarity: the (shape of the) probability distribution is constant over time
  • Random walk: tomorrow's outcome is independent of today's outcome
  • Non-negative: requirement: assets cannot have negative value
  • Time consistent: what is true for a single period is true for multiple periods; e.g., assumptions about a single week can be extended to a year
  • Normal: expected returns follow a normal distribution

A couple of these crop up often in the FRM. In particular, stationarity and normality. Empirically, nobody seems to believe in stationarity. If we are taking about a normal distribution, stationarity says the mean and volatility and constant. If not, they are time-varying. (Jorion thinks both tend to be time varying, Allen implicates volatility as time-varying). Normality is an especially dubious assumption; many research studies have proven that asset returns are not normally distributed.

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