## Saturday 27 July 2013

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:
1. 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.
2. 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.
3. 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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## Sunday 21 July 2013

### The Great SGX Sale on REITs

As usual, whenever there is a Great Singapore Sales, everybody rushes into it. But when SGX holds a sales, everybody runs away from it.

The latest SGX sales is on REITs/ Business Trusts (BTs), which have fallen by about 14% from end Apr 13 till end Jun 13 on average. The decline on some individual REITs/ BTs is even more, with Suntec falling by nearly 20%. The trigger for this decline is an impending end to the 3rd round of Quantitative Easing and rise in interest rates in US. It should also be highlighted that prior to this decline, REITs had risen by an average of 36% from end May 12 till end Apr 13. So, is this decline rational and overdone?

Let's us look at the historical yields of REITs and BTs for some answers. The chart below plots the median yields of REITs and BTs, as well as that of the short-term (3-month), medium-term (10-year) and long-term (30-year) Singapore Government Securities (SGS) benchmark bonds. The chart also plots the movement of share prices as represented by the Straits Times Index. (See REIT Yield Statistics on how the yields were derived and important points to note for this statistics).

 Historical Median REIT/ BT Yield (Jun 13)

Prior to the Global Financial Crisis (GFC), the median yield of REITs hovered between 4% and 6%. This spiked up during the GFC between May 08 and Apr 10. If we discount the spike in yields during the GFC, the normal range of yields should be between 4% and 7.5%. Just before the latest decline, the median yield was at 5.14% in end Apr 13, which is within the normal range.

The chart below further plots the cumulative distribution of the historical median yields since Sep 03. It also includes a breakdown for the various sectors. The squares represent the median yields in end Apr 13 before the decline while the triangles represent that in end Jun 13.

 Cumulative REIT Yield Distribution (Jun 13)

The median yield for all REITs before the decline in end Apr is around the 37th percentile, meaning the historical median yields exceed the prevailing median yield 63% of the time. This means that REITs were overvalued in end Apr, but not excessively so. Nevertheless, the median yields for individual REIT sectors indicate that they could be overvalued. For example, the median yield for the Healthcare sector was reaching new highs while that of the Retail sector was exceeded 80% of the time historically. So, some amount of decline was not unexpected.

After the decline in end Jun, the median yield for all REITs is around the 62nd percentile, indicating some undervaluation. The valuation for the various individual sectors have also become more attractive, hovering between the 47th to 66th percentiles for most sectors. The only sector that still indicate a high valuation is the Healthcare sector, whose median yield is around the 6th percentile level. So, for most sectors, the decline has led to more attractive valuations, but the valuation might not be sufficiently attractive for every investor.

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## Sunday 7 July 2013

### The Irrelevance of Time Diversification

Time diversification is the holding of an investment over a long period of time and thereby achieving a diversification of annualised return over time. It is often said that time diversification helps to improve your investment returns. At one point in time, I was an ardent fan of time diversification. However, after thinking further about it, I realised that time diversification, while not totally irrelevant, is quite irrelevant. Let's take a closer look on the case of time diversification.

Using the historical returns of the Straits Times Index from end-1984 till end-2012 as the base data, we can construct the annualised return over different holding periods.

 Annualised Returns Over Different Holding Periods

As can be seen in the figure above, the annualised return over a 1-year holding period can vary greatly from +78% to -49%. Anybody holding shares during the worst 1-year period would have suffered a great loss. Over a 2-year period, the worst annualised return improves to -23%. This worst annualised return continues to improve as the holding period increases, eventually reaching a positive figure when the holding period reaches 20 years. This evidence is often used to prove that over long periods of time, share investments can yield positive returns, irrespective of when you start your first investment. The first part of the statement "over long periods of time, share investments can yield positive returns" is true and is where time diversification is relevant. But the second part of the statement "irrespective of when you start your first investment", is one of the reasons why time diversification is irrelevant. I can think of 3 reasons why time diversification is not relevant.

Firstly, consider the worst annualised return of the 1-year and 2-year holding periods. On the surface, the worst annualised return of -23% over a 2-year holding period appears much better than the worst annualised return of -49% over a 1-year holding period. But if you consider the eventual returns for an initial capital of \$100,000 at the end of the respective holding periods, it will look as shown in the table below:

 Holding Period 1-Year 2-Year Worst Annualised Return -49.4% -23.2% At End of Year 0 \$100,000 \$100,000 1 \$50,586 \$76,810 2 \$58,997

At the end of the 1-year holding period, the eventual return is -49%. But at the end of the 2-year holding period, the eventual return is -41%. The annualised return of -23% over a 2-year holding period translates to an eventual return of -41% at the end of the 2-year period. This is not much different from the eventual return of -49% for a 1-year holding period. It is cold comfort to the investor by telling him that the worst annualised return is halved when his holding period is doubled (in the example above). The only comfort he has, if any, is the speed at which the money is lost, over a 2-year period instead of a 1-year period. In essence, time diversification is averaging the eventual return.

Secondly, consider the best case where the worst annualised return is a positive 3.7% in a 20-year holding period. This is just 4.7% off the best annualised return in the same holding period. At the end of the 20-year period, an initial capital of \$100,000 would translate to \$206,354. This is a very respectable return, considering that this is based on a worst annualised return for the holding period. However, what is the eventual return based on the best annualised return of 8.4%, which is just 4.7% better? The eventual return is \$506,421, which is 2.5 times better than the worst eventual return of \$206,354. Again, this is cold comfort to the investor that he cannot do worse than \$206,354, when he could possibly get 2.5 times more had he made his investment at the best entry time. (The median eventual return is \$329,129, which is 59% better than the worst eventual return).

The third and last reason why time diversification is irrelevant is: we only have 1 life; there is ONLY 1 holding period applicable to everyone of us regardless of whether or when you invest.

So, to conclude, time diversification is relevant in the sense that over long periods of time, share investments can yield positive returns. It is irrelevant in the sense that it is the eventual return, not the average annualised return, that counts.

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