VaR and Me

In the early nineties, I was a graduate student at Princeton University. My mentor John was not eager to direct me on my research yet, so I had to read all kinds of literature by myself, looking for an interesting topic for my dissertation. Of course, I still have to give myself some restrictions, that is, to choose within the scope of his research interests, including decision theory, optimization, asset allocation, parallel algorithms and so on.

Reading these documents, a doubt began to come to my mind, that is, why should financial risks be measured by volatility? Of course volatile stocks have high future uncertainty in their prices and are likely to fall sharply and cause losses. Therefore, it seems understandable to use volatility to describe risks. My question is that volatility considers both the upward and downward changes in stock prices. That is to say, the uncertainty of how a stock rises is also included in the volatility, but for stock holders, what are the risks if a stock goes up? I think this definition cannot accurately describe financial risks.

What are investors most concerned about? It’s a loss! If I had ten thousand dollars in stocks, my biggest loss would be ten thousand dollars, and 100% certain the loss would not exceed this. But what about the probability of losing 500K and 800K? These are the risk issues that I care about.

So I think, if I redefine the risk as the possibility of a certain loss that I can bear, and then choose an investment plan that can minimize this possibility, will I get a better portfolio than what the existing investment decision theory gives me?

What was the investment decision theory at that time? It’s a long story. Here I just want to talk about Markwitz’s modern portfolio theory, because this is the foundation. This theory assumes that investors are risk averse investors. If two assets have the same expected return, investors will choose the less risky one. Only on the premise of obtaining higher expected returns, investors will take greater risks. In other words, if an investor wants to obtain greater returns, he must accept greater risks. A rational investor will choose the portfolio with the least risk among several portfolios with the same expected return. Another situation is that if several investment portfolios have the same investment risk, investors will choose the one with the highest expected return. Such a portfolio is called the best portfolio (Efficient Portfolio).

There is no problem with this theory. The problem is how to define risk. Markwitz defines risk as the statistical variance of the portfolio. As for me, I want to define this risk as the probability of loss at a certain level.

I started to do some research, and soon discovered that these two definitions are equivalent, when the financial assets in the portfolio obey an ideal probability distribution, that is, the normal distribution in statistics. This is not surprising. The normal distribution has a very “perfect” mathematical assumption. It only needs two parameters to determine the entire distribution. The two parameters Markwitz uses are average and variance, which are the return and volatility of the portfolio. The normal distribution determined by these two parameters also specifies the probability of a certain loss that I am concerned about. Conversely, if I choose the two parameters as the average rate of return and the probability of a certain loss, I can also determine a normal distribution, including its volatility.

This equivalence did not make me unhappy, because it means that I can still get all the results of current theories by changing a parameter. But I feel my parameter is a better risk measure. More importantly, if the financial assets in the portfolio are not normally distributed, the two methods will have different results. So I tried other different distributions, and the results were indeed different. The most interesting thing is that if the probability distribution of these assets cannot be written with formulas at all, then the results have to be simulated by Monte Carlo methods.

I talked to Tami who is a senior student of John, and she said it was good idea, so I went to talk to John. But John was not impressed. He said it was too simple, and did not involve much mathematics, not enough to write a doctoral dissertation. Then he gave me the topic he had thought of, which was stochastic optimization in parallel algorithms. So I started to work on this more complicated topic for a few years, then graduated and went to work for a company on Wall Street. At that time, my work and interest also shifted to stochastic differentiation and option pricing.

After another two years, I changed my job to another company and started to be a risk manager. One day we were asked to adopt and calculate “VaR: Value at Risk”. The moment I saw it, I know it is what I wanted to study a few years before.

“Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses. For a given portfolio, time horizon, and probability p, the p VaR can be defined informally as the maximum possible loss during that time after excluding all worse outcomes whose combined probability is at most p. This assumes mark-to-market pricing, and no trading in the portfolio.”

VaR methodologies were developed by some people on Wall Street, roughly about the time when I thought about it at Princeton University.

Development was most extensive at J. P. Morgan, which published the methodology and gave free access to estimates of the necessary underlying parameters in 1994. This was the first time VaR had been exposed beyond a relatively small group of quants. Two years later, the methodology was spun off into an independent for-profit business now part of RiskMetrics Group (now part of MSCI).

In 1997, the U.S. Securities and Exchange Commission ruled that public corporations must disclose quantitative information about their derivatives activity. Major banks and dealers chose to implement the rule by including VaR information in the notes to their financial statements.

Worldwide adoption of the Basel II Accord, beginning in 1999 and nearing completion today, gave further impetus to the use of VaR. VaR is the preferred measure of market risk, and concepts similar to VaR are used in other parts of the accord.