Whenever I feel that the stock market is about to go into a bear market, I would run a stress test on my portfolio. The method I use is Value-at-Risk (VaR), which measures the amount of loss at a certain confidence level or probability of occurrence. For example, a VaR at 99% confidence level (which is equivalent to 1% probability of occurrence) of $1,000 would mean that the loss would not exceed $1,000 99% of the time. VaR has a holding period tagged to it, such as daily, monthly or yearly. A daily VaR at 99% confidence level (henceforth written as "VaR@99%") of $1,000 would mean that the loss for any one day would not exceed $1,000 99% of the time. As losses could accumulate over time, the longer the holding period, the larger is the VaR for the same confidence level. Assuming that share price changes follow a normal distribution, the monthly VaR would be the daily VaR multiplied by the square root of the no. of trading days in the month. If a month has 22 trading days, the monthly VaR would be SQRT(22) or 4.69 times of the daily VaR. Similarly, the yearly VaR would be SQRT(12) or 3.46 times of the monthly VaR.

To compute the daily VaR@99% for a particular stock, you'll need to compute the % change in stock price for each day for that stock for a sufficiently long period, say, 10 years. Based on the daily % changes, find out the % change that is not exceeded 99% of the time. Multiply this by the value of the stock that you hold and you'll get the daily VaR for that stock. The monthly stock prices can also be used for this computation, for which you'll get the monthly VaR.

You can do the above computation for every stock in your portfolio. However, the VaR for the portfolio is not equal to the sum of the VaR of all stocks in the portfolio. The VaR for the portfolio is less that the sum of the VaR of the stocks. This is due to the effects of diversification, as not all stocks will go down by the same extent at the same time. To compute the VaR for a portfolio, you'll need to sum up the value of the portfolio for each day, then compute the % change in value of the portfolio for each day and find out the % change that is not exceeded 99% of the time. The figure below shows how the VaR for individual stocks and portfolios are computed.

VaR Computation for Individual Stocks & Portfolios |

An alternative approach for computing the VaR for a particular stock would be to find out the beta of the stock relative to the Straits Times Index (STI). By definition, a beta of say, 1.2, would mean that the stock price changes is 1.2 times of the index changes. The same approach could be used for a portfolio.

**Practical Experience**

So, is VaR a good stress test? I ran a stress test on my portfolio (excluding cash & Singapore Government Securities bonds) in Jan 08, a few months before the Global Financial Crisis (GFC) happened. The yearly VaR@99.5% was computed to be a 24% drop in portfolio value. I also back-tested the portfolio based on the decline experienced during the dot-com bust, from the peak in Jan 00 till the trough in Sep 01. The estimated decline was 26%, which was very similar to the yearly VaR@99.5% (note: no theoretical relationship exists between the 2 sets of computation).

So, was the actual decline in portfolio value close to that predicted by the VaR and back-testing computations? At the depth of the GFC in Feb 09, the actual decline was a staggering 51%, or 2.1 times that expected from the yearly VaR@99.5%, which by definition, is only exceeded once in every 200 years!

So, what went wrong? Firstly, the GFC was a very rare black swan event whose frequency of occurrence was smaller than the 0.5% probability used in the VaR computation (after all, how often do the big investment banks such as Lehman Brothers collapse?). Having said that, studies have shown that black swan events do happen with greater frequency than that expected by normal distribution.

Secondly, there were a number of REITs and Business Trusts (BTs) in my portfolio. These were non-existent prior to 2003 and therefore had never experienced a major bear market decline. Although they were assumed to track the STI through their betas in the VaR computation, the period in which their betas were computed did not include a bear market decline. The decline of the REITs and BTs far exceeded that estimated by the STI index changes.

Thirdly, there were also some bank preference shares in the portfolio. These usually had very little volatility except on the days they went ex-dividend. However, because the GFC was financial in nature, even the preference shares of our local banks, which are among the safest in the world, fell more than expected.

As shown above, despite conscious attempts at diversification into preference shares, REITs and BTs in addition to shares, in times of crisis, most types of investment will go down together. Studies have shown that the correlation between different classes of investments tends towards one during crises.

**Conclusion**

Considering that VaR analysis and back-testing failed dramatically during the GFC, would I still carry out such analysis in future? Despite the various shortcomings discussed above, VaR analysis and back-testing have the following benefits:

- They provide a sense of the loss expected at the depth of a bear market, especially for a buy-and-hold investor. As rational as we all try to be, emotions play a significant role in investors' psychology and the eventual returns achieved. By estimating the amount of loss in advance of the decline, an investor is better able to prepare himself psychologically for the decline and make rational decisions at the depth of the bear market.
- They allow adjustments in the portfolio to be made. If the estimated loss is too much to bear, an investor can adjust his portfolio to something more palatable.
- They identify the stocks that are likely to decline the most. Based on this information, an investor can determine if he should still keep those stocks in his portfolio.

So, yes, I will still carry out VaR and back-testing analysis during times of market pressure, but will not take the figures computed at face value. Recognise VaR for what it is, which is the expected loss for an event with a certain probability of occurrence. The actual decline may or may not have the same probability of occurrence and hence the actual loss will differ from the VaR figures computed. It's just like investing; despite taking losses from time to time, we're still investing for the long run :)

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