Introduction
Value of a risk (VaR) is widely defined as probabilistic measure of market risk. Financial analysts use VaR to predict the future value of the market portfolio. Previous scholars who invented this idea of VaR did not use the name value of a risk. However, names like dollar at risk (DaR), income at risk (IaR), capital at risk (CaR), and earning at risk were used in place of VaR. The name ‘value at risk’ originated within JP Morgan before 1985.
Historically, value of a risk was developed though two parallel line. One of them was called Portfolio theory and the other capital adequacy computations. Leavens, 1945 was the first person to write on a portfolio of bonds over some horizon. He explained that, at the end of the horizon, each bond of the portfolio would mature or default and be worthless. In his explanations, he assumed that events of default were independent, and he found out that the value of portfolio at the end of the given horizon had a binomial distribution. In his work he never mentioned the term VaR; however, he talked about the spread of probability between gain and losses. In his calculations, he considered standard deviation of portfolio as an influential factor in determining bond value at the end of the horizon.
Later in 1952 Markowitz and Roy published separate but similar work on VaR measures. In their work, they developed means of choosing portfolio that optimize reward after a given period of time and at a given level of risk. They proposed VaR measures that made use of covariance between market risk factors in order to reflect diversification and edging effects. In their prediction, they used different VaR metrics. Markowitz used variance of simple return which incorporated the covariance matrix for risk factors, but Roy used a metric of shortfall risk, which in cooperated, the mean vector and covariance matrix for risk factors. The unavailability of processing power limited the use of VaR to the theoretical description, which was only, used as the basis of new emerging theories. Later technological advancement and market changes have provided the means which effectively use VaR and also expanded the application of VaR on more assets. The technological advancements supported the development of sophisticated modeling techniques in VaR, for example, Kenneth Garbade developed a sophisticated modeling technique in 1980’s which was applicable across US debt market. In the model, the price of each bond was predicted on the basis of price sensitivity and the portfolio market values were taken to be normally distributed. In 1990’s JP Morgan publicized ‘value at risk’ to professionals in financial sectors where he clearly gave the risk metrics. This led to recognition of VaR measures by Basle committee. The committee used the acquired knowledge on securities to advise and authorize banks to perform regular capital calculations using VaR.
The VaR Debate
The release of risk metrics and adoption of VaR measures brought a lot of criticism among security analyst in early 1990’s. This criticism was based on: the inconsistent results produced by different methods of implementing VaR, perception that VaR is conceptually flawed and the fact that wide spread use of value at risk involves systematic risks.
Some critics of VaR on the basis of inconsistent results, forexample, Bider in 1995 used Monte Carlo and historical VaR measures to calculate sixteenth value of given portfolios. He found that, the sixteenth measure of each portfolio produced inconsistent results. Other critics on different basis used logical methods to justify why they considered VaR flawed. Despite the criticism of VaR and justification of the criticism, today VaR is used in predicting the value of bonds and securities more accurately than before due to technological advancement and availability of more information on securities and bonds.
Present information on value at risk (VaR)
The use of VaR is more profound in economics and finance. Scholars in the two fields generally define VaR as the maximum loss. This loss cannot be exceeded with a given probability, over a given period of time and in a given confidence interval. Therefore, security houses and investment banks use VaR to measure the market risk of assets portfolios.
Mathematically, Var is defined by the formulae given as VaRα = inf{ lϵⱤ: P (L> l)≤ 1- α} = inf{ lϵⱤ : FL (l)≥α . This formulae is explained in words as follows given a confidence interval between 0 and 1 the VaR of the portfolio at given confidence interval α is the one given by the smallest number l such that the probability that L is more than l (small L) is less or equal to (1- α). In the computation of VaR, properties of monotonicity, homogeneity, and sub additivity and translation invariance are assumed throughout the calculation. In monotonicity, it is assumed that if a portfolio Z1 has better value than Z2 in all scenarios then the risk of Z1 should always be less than that of Z2. In sub-additivity, it is assumed that the risk of two portfolios together cannot get worse be worse than adding the two risks separately. In Positive homogeneity, it is assumed that doubling portfolio leads to automatic doubling of the risk. Then in translation invariance, if aϵⱤ and Ⱬ€L, then p (Ⱬ+a) = p (Ⱬ)-a where a is taken as insurance to portfolio Ⱬ. Therefore, the risk of Ⱬ+a is less than that of Ⱬ and this difference is represented by the insurance money paid (a).
- Cov(X,Y) = Cov(Y,X).
- Cov(X,X) = Var(X) (ie. The covariance of X with itself is the variance).
- Cov(αX,Y) = αCov(X,Y) , where α,.
Conclusion
The concept of VaR has evolved over time since 1945. This is attributable to technological and market changes. The basic line of the concept is use of probability in determining the value of securities over a given period of time. Despite several criticism of the method, financial analysis and economists across the globe have adopted the method in their day to day activities.
End notes
1.Garbade, Kenneth D. Assessing risk and capital adequacy for Treasury securities,
Topics in Money and Securities Markets, 22, New York: Bankers Trust (1986).
2.Leavens, Dickson H. Diversification of investments, Trusts and Estates, 80,
(1945) 469-473.
3.Markowitz, Harry, M. Portfolio Selection, Journal of Finance, 7 (1) (1952), 77-91.
Corrigan, Gerald .Remarks before the 64th annual mid-Winter meeting of the New
4.New York State Bankers Association, January 30, Waldorf-Astoria, New York City: Federal Reserve Bank of New York (1992).
