Linear VaR measures employ linear transformations. Many names have been used to describe linear VaR measures, such as parametric, variance-covariance, closed-form, or delta-normal VaR measures. There are shortcomings with most of these names. While linear VaR measures are parametric, so are most VaR measures. While linear VaR measures use variances and covariances, so do all VaR measures, with the exception of historical VaR measures. While some linear VaR measures employ a delta remapping, most do not.

Also, while a normal assumption is common with linear VaR measures, it is by no means universal. Linearity describes the one characteristic that is common to all linear transformations: they are applicable to portfolios whose portfolio mapping function is a linear polynomial. Such portfolios include portfolios of equities, physical commodities, or futures. The market value of such portfolios depends linearly upon applicable key factors. Other portfolios are so nearly linear that they can reasonably be approximated with a linear polynomial.

These include portfolios of forwards (including foreign exchange forwards) and most non-callable debt. Linear VaR measures are generally not applicable to portfolios that hold options or instruments with embedded options. These include callable bonds, mortgage-backed securities and many structured notes. The Analytic Variance-Covariance Method can be simpler to estimate since we do not need the entire distribution of factor values. It specifies distributions and payoff profiles (e. g. , normal and linear) and decomposes securities into simpler transactions/buckets.

It then derives the Variances/Covariances of standard transactions. VaR is then calculated based on the standard definition of variance. Analytic Method is intuitively simpler and does not require any pricing models but it is not conducive to sensitivity analysis and cannot handle non-linear payoff profiles such as options. Historical Simulation involves identifying factors that affect market values of securities in the portfolio. It simulates future values of the factors using historical data. Then the simulated factors are used to estimate the value of the portfolio several times.

Finally, a histogram is created of the portfolio’s expected change in value and the relevant probability levels for the VaR calculation are identified (e. g. , the change in portfolio that occurs at the lowest 1% of the distribution). Historical Simulation does not assume specific distributions for the securities and uses real-world data but it requires pricing models for all instruments and allows limited sensitivity analysis. Monte Carlo Simulation involves the same steps as in the Historical Simulation method except that we use Monte Carlo techniques to obtain the simulated Factor values.

It can value complex derivatives properly by generating a large number of trials to determine the distribution of possible outcomes. It is often used to deal with extreme cases and predict future results. These characteristics make it the most appropriate method of dealing with non-linear returns and measuring risk in bond portfolios that use embedded options. 8 Monte Carlo Simulation makes it easier to do sensitivity analysis but requires the analyst to specify asset distributions as well as pricing models.

Empirical Tests have proved that to date, tests of the three methods suggest that the approaches can yield similar results when portfolio payoffs are linear, 95% confidence level is used and there are not many large outliers in the historical data set. The biggest differences can occur between the 2 simulation approaches and the analytic method when non-linear payoffs are a significant share of the portfolio and they do not cancel out (e. g. , long a large number of put options), large number of outliers in the historical data set, 99% or higher confidence level is used.

If the portfolio has linear (or weakly non-linear) payoffs, then the Analytic method might be best. If the portfolio has non-linear payoffs, then the two Simulation methods are better. If stress-testing and sensitivity analysis are needed, then Monte Carlo Simulation is the preferred method. However, it can be very complex to remove all possible arbitrage opportunities from the simulation. 9 In some respects, VaR is a natural progression from earlier portfolio theory including, however, many different aspects and improvements.

Although the variance-covariance analysis has the same theoretical basis as Portfolio Theory, the other two approaches of VaR, historical simulation and Monte Carlo simulation do not. One of its advantages is that it interprets risk in bond portfolios in terms of the maximum likely loss. Another advantage of the VaR approach is that it can be applied to a much broader range of risk problems by taking into account the correlation between them and present greater flexibility in terms of the approaches applied to different circumstances.

Perhaps the greatest advantage is that VaR approaches are better at accommodating statistical problems such as non-normal returns. By doing so, it provides management of firms with more accurate details on risks leading to more informed and better risk management. It also provides a consistent, integrated treatment of risks across the institution, leading to greater risk transparency and a more consistent treatment of risks across the firm. Its approaches enhance decision-making by supplementing new operational decision rules to guide investment, hedging and trading decisions.

Moreover, systems based on VaR methodologies can be used to measure other risks such as credit, liquidity and cashflow risks, as well as the market risks measured by VaR systems proper. This leads to a more integrated approach o the management of different kinds of risks, and to improved budget planning and better strategic management. It gives insight to firms on how to comply with capital adequacy regulations whilst retaining their portfolios to minimise the burden that such regulations impose on them.

As with most statistical approaches for dealing with risk in bond portfolios, VaR methods also have weaknesses and limitations. One of them is that all VaR systems are backward-looking. As they are based on, the assumption that what has taken place in the past will occur in the future, they attempt to forecast likely future losses using past data. Although its assumptions provide it with flexibility, on the other hand, they inherit the risk that these assumptions might not stand in any given circumstances, hence give irrelevant results.

If the VaR approach is used to evaluate risk in bond portfolios, including embedded options, it would not be wise to use a model that assumes normal returns and should be replaced by one that allows for non-normality. Finally, another limitation is that the method is not foolproof. It is only to be used by experts who know how to use the tools that go with it.

1 Dr Terry Watsham, The Role and Structure of Interest Rates in the Economy, Lecture Notes on Bond Pricing, p41, University of Brighton,2004. 2 Fabozzi, Bond Markets: Analysis and Strategies, 4th ed, Prentice Hall, Chapter 3,p46.3 Dr Terry Watsham, Risk in Bonds, p9, University of Brighton,2004. 4 Dr Terry Watsham, Risk in Bonds, Properties of Duration, p29, University of Brighton,2004.

5 Fabozzi, Bond Markets: Analysis and Strategies, 4th ed, Prentice Hall, Chapters 4,14,17. 6 Watsham, T. J, Options and Futures in International Portfolio Management, p290, London : Chapman ; Hall, 1992. 7 Watsham, T. J, International Portfolio Management, Chapter 6, pp 126-130, London : Longman, c1993 1993. 8 Beyond Value at Risk, Kevin Dowd, Ch. 1,3,4,5 John Willey ; Sons,1998. 9 Value at Risk, Philippe Jorion, Ch. 7,9, McGraw Hill Int, 2nd edition,2001.