Archive for June, 2008

Stop It!: Stop-Loss Results for Bullish Scenarios

Thursday, June 12th, 2008

In my blog on May 27th we looked at different types of common stop/loss strategies and discussed when and how to use them. I also mentioned that I was going to test out several of the more common strategies under various market conditions. After a software snafu on the part of the software vendor (if I’m to be a beta tester, I’d like to get something for my efforts!–grrr), I finally was able today to finish my long simulation.

The Set-Up
What I did was to look at three different portfolios of 10 stocks each–one composed of stocks priced less than $10, one composed of stocks priced between $10 and $30, and the other of stocks priced between $30 and $100. My simulation consisted of finding stocks breaking to new 250 day highs, which is a good strategy in a bull market. I ran my simulations at two different times when the market was on a bull run, from 9/26/06 to 2/20/07, and from 3/19/07 to 6/4/07. (Simulation and management account information are described at the end of the article.*)

The following stops were used: Trailing stops between 5%-50%, Gain/Loss stops with Gains set at 1000% (literally no stop on the gain side) and losses varying from 5%-25%, a ratchet stop, and finally, no stop at all. I would have loved to test out the parabolic SAR, but my program couldn’t do that and it would have meant weeks of manual input. (As it was, this took me long enough.)

The following table shows the results. The first row of each simulation shows the profit or loss over the time period. The second row is the number of trades which each trade being equal to a buy and a sell for a total of two commissions (what is known as a round-turn in futures parlance). The win/loss percentage is in the third row followed by the maximum portfolio draw-down in the fourth row.

The Results
The first thing to glean from this table is that it could use more data. While we can’t pinpoint the exact best stop/loss to use in each case, we can note certain tendencies.
Low-priced Stocks (<$10): In the case of small stocks, we see that trailing stops didn’t do nearly as well as the G/L stop. An absolute loss of 5%-10% seems to work best while also keeping the number of trades to a minimum. Note that the use of trailing stop losses involves significantly more trades, and although commission costs are included in the profit/loss figures, executing this number of trades would take a lot of time on the part of the trader. Trailing stops also resulted in larger portfolio draw-downs. The ratchet stop did well in one scenario and poorly in the other. The reason for the poor performance was just bad luck–selling one stock at one particular time had the effect of the program selecting poor performers down the line (sort of a snow-ball effect). (This shows you why you need more than two simulations to get reliable data.)
Mid-Priced Stocks ($10 – $30): Contrary to the small stocks. the trailing stops fared better here especially in limiting draw-downs in the second market scenario.
High-Priced Stocks ($30 – $100): Here, absolute losses between 5%-15% out-performed the trailing losses, with the 10% G/L edging out the others. **

Summary
It’s clear that we need more data points to identify which stops to use. Setting stop/losses is as much of a science as it is an art, and setting the most appropriate ones depends on many variables, such as market type, stock price, stock selection criteria, etc. The above table makes it clear that setting the right stop/loss results in better returns, but it’s also interesting to note that not setting a stop/loss isn’t that bad, but one could expect that since we’re buying strong stocks in a strong market which is probably is not the case for other market scenarios. One thing that surprised me was how well the ratchet stop did (except in one instance). You can also see how poorly the 5% trailing loss did in general and the inordinate amount of trades it requires.

The bottom line here is that if you have to pick one type of stop to use, I’d go with a strict percentage loss between 5%-15% followed by a ratchet stop. Or, you can fly by the seat of your pants–use no stop/loss criteria and exit when your stock breaks some type of resistance or starts to sell off. (See “Other Stop Loss Methods” at the end of my May 27th blog.)

When I have another chunk of time, I’ll look at setting stops under bear market conditions.

*All accounts began with $100,000 initial investment in a 50% margin account. Trade commissions were $9.95/trade. No dividends were collected and account interest was included. Each portfolio maintained 10 stocks that were automatically selected from a list generated according to the parameters of my break-out strategy. The strategy parameters were as follows: 1. Only stocks that traded on the major indices (NYSE, NASDAQ, and AMEX) were considered; 2. Average daily volume must be greater than 200,000 shares; 3. The volume on the break-out day had to be greater than the average daily volume; and 4. A stock break-out was defined as one breaking to a new 250 day high. Closing day prices were used in all calculations and no duplicate holdings were allowed.

**Blog Update (6/13/08): I changed the wording of the stock categories from Small Stocks to Low-Priced Stocks, Medium Stocks to Mid-Priced Stocks, and Large Stocks to High-Priced Stocks. Prof. Pat pointed out to me that my original wording could be construed as company size which is not always the case, so I changed the wording to more accurately reflect reality.

The VWAP–Volume Weighted Average Price

Wednesday, June 11th, 2008

I’m in the middle of my stop/loss research that I began a couple of weeks ago, before my software program crumped out and a trip back to my homeland forced me to table it. But instead of neglecting today’s blog, I thought I’d introduce the concept of the VWAP. No, this isn’t a type of gourmet sandwich or new German automobile. It’s an acronym that stands for volume-weighted average price. I know it sounds like a mouthful, but the VWAP is a useful tool that traders and investors can add to their trading arsenal.

What is the VWAP?
The VWAP for a particular stock is a simple mathematical formula computed by adding up the dollars traded for every transaction (price times the number of shares traded) and then dividing by the total number of shares traded in that time period (usually in a day). VWAPs are mostly used by institutions and pension funds which is why the typical retail investor (i.e., you) probably isn’t familiar with it. You can find it in some of the more powerful charting programs. I use Qcharts, a subscription-based charting program, and one of its features includes a hot list of 100 stocks that are currently trading at prices above their VWAPS and another list trading below them.

What does the VWAP tell us?
Although the VWAP sounds a bit esoteric, it has some very practical uses. Some traders use it as a pricing indicator. If a stock is trading below its VWAP, then it’s considered a bargain and that’s when a bullish investor would buy it. (The reverse situation would apply to a short trade.) Others use it as an overall indicator of a stock’s direction. For example, stocks trading above their VWAPs are those that people–institutions especially–are buying. This is considered a bullish sign. On the other hand, a stock trading below its VWAP is considered to be bearish, especially if it’s been trading below it for some time. These make excellent shorting candidates. Looking at my hotlists right now, I’m seeing Ensco Int’l. (ESV) and James River Coal (JRCC) near the top of the Above VWAP list. The charts of both stocks are making new highs. On the short side, Baidu (BIDU), Intuitive Surgical (ISRG), and Deckers Outdoor (DECK) are heading up the Below VWAP list. Although none of these stocks are even close to hitting new lows, their charts are all in a bearish trend.

The other way I like to use my VWAP lists is to look at the values on each of the list to determine the overall tone of the market. Stocks trading above their VWAPs have VWAP values above zero. A value above +150 or so I consider to be very bullish. Values above +300 indicate extreme over-bought conditions and almost always signal that a market turnaround is imminent. Conversely, stocks trading below their VWAPs have VWAP values less than zero. Through many years of daily observation, I’ve found that the magnitude of the negative VWAPs are typically not as great as the positive ones, so that values around -100 I consider to be quite bearish and those below -200 to show an extremely oversold market.

To properly deduce market tone, I look at VWAP values in the middle of my hot lists which is the fiftieth stock from the top. This is more representative of the total market since the values at the top of the list are generally quite skewed. Here’s a few guidelines that correlate with market tone:
1. If the fiftieth value on the Above VWAP list is much greater in magnitude than the corresponding one on the Below VWAP list, the market is in an uptrend since there is more buying activity. The greater the difference between the two values, the more bullish the market. In highly bullish markets it’s not unusual to see a positive VWAP of +150 to +200 and negative VWAPs in the single digits.
2. If the positive VWAP value (the one at the fiftieth position) falls and the corresponding negative VWAP begins to increase (in absolute magnitude terms), the market is changing direction and heading downwards. (Vice versa for the other direction.) This is good to note because this type of shift is sometimes an early indication of changing market winds and can be a useful tool for day traders.

Conclusion
I’ve been watching the positive and negative VWAPs for over the past hour, just before the market closes and I did see the VWAPs shift from a positive value of +110 and a negative value of -88 to a positive value of +76 and a negative value of -117. This shift corresponded to a six point decline in the S&P.

Many of you may not have access to this indicator but if you do, you now know a few ways to take advantage of it. VWAPs are especially good indicators of market tone that day traders of S&P futures might find to be particularly beneficial. Good luck and good VWAPping!

Using Puts to Pay at the Pump: More Oil Stocks

Tuesday, June 10th, 2008

Yesterday we looked at a way we could use petroleum giant Exxon-Mobil to help pay for our skyrocketing fuel bill. We did this by selling puts to generate income according to two methods:
1. Writing front-month options with strikes closer to the current price.
2. Writing puts with further out strike prices, say 6 months to 2 years.

Whichever method one selects depends on the options pricing and the expertise and commitment of the investor.

Today I’d like to expand yesterday’s discussion and look at other candidates from the oil sector as well as touch on other income generating methods. I looked at the charts of the top companies across all the industries in the oil sector–equipment makers, drillers, exploration companies, field services, plus several others–and selected the ones that I felt had the most compelling charts. I’m telling you this so you’ll know that the stocks that I’m about to recommend are selected from all industries and don’t all fall into one category. I did this for diversification purposes just in case you want to write puts on several stocks. I also tried to select companies with lower stock prices, and in the few instances where the stock is more expensive, I suggest writing a bull-put credit spread instead.

Okay, enough banter. Here’s my list of stocks and how to play them (in no particular order).

Note: If there’s a large difference between the bid and the ask price, my suggestion is to place a limit order someplace between the two. You’ll have a better chance of getting filled if your price is biased towards the bid side.

The High-Priced Spreads
Aegean Marine Petrol (ANW): This company provides fueling services to ships in port and at sea. Its profits along with its stock chart has been steadily increasing. The company’s recent purchase of four new tankers indicates that it expects its business to keep on expanding, at least for the next couple of years. The stock trades in an upward trending, narrow range, and for the past month it’s been bouncing off its 30 day moving average (dma). It’s currently trading right at its 30dma around $40/share, so if you like this stock, now would be a could time to sell a put position. It has major support levels at $27.50 and $35. Here are the puts I like right now:
Very Risky: Jun40: $1.45 bid x $1.60 ask, Open Interest (OI) = 555. Options expiration is only 11 days away, and since this option is at-the-money (ATM), I would suggest it only to the most hale and hearty–or those who would love to buy the stock at the strike price.
Moderately Risky: Sep35: $2.10 bid x $2.25 ask, OI = 349. Less risky is the Sep30: $0.75 bid x $0.85 ask, OI = 464. You could also take the more conservative approach and sell a bull-put credit spread with a total credit of $1.35 per contract.
Least Risky: Dec30: $1.60 bid x $1.75 ask, OI = 634. Dec 25: $0.60 bid x $0.70 ask, OI = 80. You can also combine the two into a bull-put credit spread for a price of $1.00 – $1.05.

Helmerich Payne (HP): This company engages in drilling oil and gas wells for others (it doesn’t own any of its own). It also sports one of the very best charts in the sector with its price doubling in just the last seven months to around $65. It’s risen so fast and furiously that I think it might be due for a much-needed breather, especially since a Goldman Sachs analyst recently upgraded it last week. (Uh, why did it take him so long?) The company is contracted to continue its drilling efforts for the next several years so it should be keeping itself quite busy. The chart shows minor support levels at $60 and $50 with major support in the $43-$46 range. The stock has a tendency to bounce off of its 30 dma and 50 dma, and I’d wait until it hits one of those before selling puts.
Moderately Risky: Sep55: $2.15 x $2.30, OI = 389. Less risky is Sep50: $1.10 x $1.25, OI = 252. Bull-put credit spread using these options for a credit of $1.05 – $1.10.
Conservative: Jan45: $1.60 x $1.70, OI = 2058.

Ultra Petroleum (UPL) & PetroHawk Energy (HK): Both companies are involved in domestic oil and gas exploration, drilling, and production. Although they sport similar charts, I prefer UPL because its rise has been steadier and less dramatic than HKs, despite Jim Cramer’s fawning all over the stock (HK) yesterday. UPL is trading down from its all-time high set a few days ago, and judging from the topping tail that it put in on that day, it looks like it’ll be heading south for a while. I’d wait until it bounces off of its 40dma that its been using as minor support before jumping in. It’s currently trading around $95, and shows major support levels at $80, $75, $70, $63-$66, and $60. It has a very solid support level at $75, and I’d either sell puts at or below that level or else sell a 75/70 credit spread for the longer term.
Conservative (UPL): Jan75: $4.00 x $4.30, OI = 4811. Jan70: $2.95 x $3.20, OI = 5825. The credit spread using these options would be from $1.05 – $1.15.

Weatherford International (WFT): The company provides equipment and services used for drilling and production of oil and gas wells. Yes, this company is similar to HP but I like its chart and I like the drillers since they have solid contracts for the next couple of years. WFT is also cheaper than HP ($45 versus $65). WFT has minor support at $40, and major support at $35-$36 and $30. The stock put in a topping tail along with most of the other oil stocks several days ago. It’s still trending down and I’d wait until the price stabilizes before I’d write any puts. Note that the options on this stock are much more liquid than the others meaning that you’ll be able to get better fills.
Risky: Jul42.50: $1.55 x $1.65, OI = 2013.
Conservative: 10Jan35 (2010 Leaps): $4.20 x $4.90, OI = 496, or even more conservative is 10Jan30: $2.50 x $3.10, OI = 232. Again, the most conservative approach (if you’re not interested in buying the stock) is the credit spread of the above at $1.75 – $1.95.

The El Cheapo Stocks
These are two good stocks that are cheap enough to own.

Gran Tierra Energy (GTE): I’ve been following this stock since it was trading around $1.50 which was just last October. Since then, it changed its stock symbol (from GTRE) and climbed up over 500% to $6.50. Did I buy any at $1.50? Um, no. But I just might sell some puts to see if I can get it at a good price. The company is engaged in oil and gas exploration and production and has properties in South America. The chart is indicating that it’s taking a break, so I’d wait before I do anything and jump in if it gets back down to the $6 neighborhood. One good strategy would be to sell puts at the $5 strike in the hopes of buying the stock. If and when the stock is put to you, you can then use it to generate more income by selling covered calls. Nifty, huh?
Moderately Risky: Nov5: $0.65 x $1.05, OI = 186. This option is rather thinly traded but you’ll probably get filled if you keep placing a limit order between the bid and the ask.

Harvest National Resources (HNR): This is yet another exploration company engaged in developing oil and gas properties in South America, Africa, Indonesia, and China. The climbing price of oil bodes well for all oil exploration companies, including this one. The stock suffered a serious decline losing over 30% of its value in the last two weeks in April. But since then, it’s made a remarkable comeback and is trading at $11, just two bucks shy of its pre-fall level. Company insiders have been buying the stock which is something they wouldn’t be doing if they didn’t feel it was a good investment. The stock has fairly strong support levels at $8, $9, and $10. The stock is still rising but it needs to make it past $12 resistance. If it can’t do that, then I’d wait until it pulls back, perhaps to $10.50 or even $10.
Only decent play: Dec10: $0.70 x $0.85, OI = 271.

Conclusion
These are my picks but if you have a particular oil company that you love to hate, then go with that. You’ll get much more satisfaction at the pump. Remember, too, about royalty energy trusts which I mentioned as a way of generating income in my February 19th blog. You can apply this method of selling puts on any of the optionable trusts that you care to buy. The optionable trusts are the following: HGT, BTE, BPT, SJT, PBT, and PWE. They all sport dividend yields of 8-12%.

Using Puts to Pay at the Pump

Monday, June 9th, 2008

I just returned from a week spent in America’s Dairyland, and woke up today wondering what I was going to write about. Being out of the loop for a week has rusted my little grey cells, so I switched on CNBC in hopes of finding some inspiration. Lo and behold I happened to twig onto a short piece on options plays. One of the guests offered an interesting way that the consumer could subsidize his gas purchases by selling out-of-the-money (OTM) puts on Exxon-Mobil (XOM). That struck me as a droll idea and a fun way to get revenge on those “evil” oil companies. (I don’t share the view that oil companies are all evil; the current high price of oil is more a result of speculation on the part of those “evil” pension funds and others.) So, how does this concept of put selling work and are there other oil sector stocks that we can use for our nefarious purposes?

Selling Puts
Let’s recap our knowledge of puts. Buying a put option gives the buyer the right, but not the obligation to sell a stock (or a future) at a specified price called the strike price on or before the expiration date which is usually the third Friday of the expiration month.* In contrast, the seller of a put option assumes the obligation of buying the stock from the put seller at the strike price. Why is this an attractive investment? Because the put seller immediately gets the premium obtained from the sale of the put placed into her brokerage account which can be used to subsidize gas purchases. In exchange for the cash, the investor is taking on the risk that the stock might drop at or below the strike price at which time she will be forced to purchase the stock. In order to do this, you must be cleared for this level of options activity (ask your broker) and have the necessary funds available should you be forced to buy the stock. You should also be happy to own the stock at the strike price. (For more information on the mechanics of this strategy, see Recipe #6: Put Pot Pie.)

The Exxon exxample
Would I pick Exxon to pay for my gas? Let’s consider the chart which shows that it’s been channeling between $80 and $95 for the past year. Currently, it’s trading right in the middle of its channel around $88, so if I were playing it, I’d wait until it traded below $82 before selling puts. Remember, the lower the price of the stock, the higher the premium you’ll collect.


Now which strikes to sell? The weekly chart of XOM shows support levels at $5 increments from $80 down to its major support level at $55. (This means that the support levels are roughly near $80, $75, $70, $65, $60, and $55.) The strike price you select should be based on your tolerance for risk and where you perceive the price of oil will be in the future. Keep in mind, too, that the further away the expiration date, the riskier the play, so if you want to write an option for a year out or more, go for strikes that are further out-of-the-money.

Two ways to play it
You can sell puts on a monthly basis at strike prices that are closer to the current price, or sell longer term options that are further out of the money. Which method you choose is based on the amount of time you have to devote to selecting and watching monthly options as well as on the options premiums themselves. In the case of Exxon, the July 70 Put is selling for $0.10, the Oct 70 Put for $0.80, the 09Jan 70 Put for $1.80, and the 10Jan 70 Put for $5.00.

If I were to play Exxon, I’d probably choose the latter, long-term method. I’d wait until the stock dropped below $82 and sell any of the January 2010 leaps in the $55-$70 strike range. Then, since Exxon has a tendency to channel, I’d buy back the option when the price hits the $95 level and shows signs of rolling over. This pattern can be done every few months while collecting at least a few bucks per contract. For example, if you had sold the 10Jan 70 Put on January 22, you would have collected about $10/contract thus adding $1000/contract into your account. On April 24, the stock almost touched $95, and if you had sold your option then when it was trading for $4, you could have pocketed $600/contract. For all of you traveling salespeople in gas-guzzling vehicles, you’d probably need to sell more than one contract to cover your fuel bill.

I’m not sure that I’d be in Exxon for the very long term since its chart, along with many others in the oil sector, are looking a little tired. And having a long-term long position on it to me seems risky (selling puts is a bullish play), but if you elect to go this route, I’d set an alert for any major move below $80 and probably either roll-down the position or exit it altogether.

Well, there you have it–one way to stick it to “the man.” A much better way than siphoning gas–plus, it’s legal. Tomorrow, I’m going to look at other companies in the oil sector to see if there aren’t better (as well as cheaper) candidates for this strategy, so keep your motors running!

PS: Speaking of siphoning gas, if you’re planning on renting a car on your summer vacation, insist on one with a locking gas cap.

*If you’re unfamiliar with options, please refer to my links and as always, don’t trade any strategy until you thoroughly understand it by paper-trading it first.

PMPT Part III: Lognormal Distribution & PMPT Theory Evaluation

Friday, June 6th, 2008

This concluding installment on Post Modern Portfolio Theory (PMPT) will discuss the assertion that investment returns are Lognormally distributed and how that could affect the Modern Portfolio Theory (MPT) assumption that they are Normally distributed.

Are investment returns lognormally distributed?
There is an excellent theoretical basis for the proposition that investment returns are distributed according to a Lognormal distribution.* A random variable is usually thought of as being Lognormally distributed if it can be expressed as the product of other factors. This is exactly the case for compounded investment returns which are computed by expressing the individual period percentage returns as fractions and adding 1.0 to form the relative return expression. For example, a 12% return expressed as a relative return would be given by 1+.12 = 1.12. Thus, a compounded overall return over n number of periods (months, years, etc.) is the product of the relative returns expressed like this: (1+r1)(1+r2)…(1+rn). The data can always be thought of as the result of combining data from shorter intervals, daily data combined to form monthly data, monthly data combined to form annual data, etc.

Further, a Lognormal distribution can never be negative in the same way that with conventional investments the most you can lose is all the money you have invested. Even the vaunted Ibbotson & Associates SBBI Annual reports that the Large Stocks asset class is not exactly Normally distributed.

But let’s see how the distributions for the various asset classes actually look. Using monthly total return data from 1927-2007 for Large Stocks the distribution looks like the following histogram. It contains 81 years x 12 months = 972 data items. On this graph are superimposed the best fit Normal distribution as a red line and the best fit three-parameter Lognormal distribution as a blue line.

As you can see, the Normal distribution in red is completely covered by the Lognormal line indicating that the best fit Lognormal distribution is in actuality a Normal distribution. What this means is that the assumption made by MPT that investment return data is Normally distributed is vindicated by the data and that MPT’s use of standard deviation is an excellent measure of risk even by PMPT standards. At least for the Large Stock asset class anyway. Let’s see if the coincident plots of Normal versus Lognormal distributions continues for the other asset classes. Here is the histogram for the Small Stock asset class.

The two distributions appear coincident on this graph as well. Let’s look at the histogram for Long-Term Corporate Bonds.

Here the two plots diverge only minimally as the red line begins to appear in a few spots. Next is the graph for Long-Term Government Bonds.

Now we begin to see some distribution divergence as the histogram demonstrates a noticeable amount of positive skewness. The assumption of a normal distribution is still quite appropriate however. Lastly, let’s look at the histogram for Intermediate-Term Government Bonds.

Again, here we see a similar amount of divergence between the Normal and Lognormal distribution fits as a visually perceptible amount of positive skewness appears. The Normal distribution is still an excellent approximation to the Lognormal for this asset class however.
No histogram is presented for the U.S. Treasury Bills asset class as that data in monthly total return fractional form is available to only four places to the right of the decimal point. The lack of sufficient data resolution results in an almost random uniform distribution rendering it useless for this analysis.
The previous three graphs above for the various Bond asset classes exhibit an excess of “peakiness” (known as kurtosis) relative to the histograms for the Large and Small Stock asset classes. The high kurtosis of these distributions suggests that neither Normal nor Lognormal distributions completely explain the behavior of these debt based asset classes. A high kurtosis in a distribution usually means more of the variation is due to infrequent larger deviations as opposed to frequent smaller sized deviations. This sounds very much like the behavior of bonds where a fixed interest rate provides much of the return with interest rate market fluctuations providing a smaller relative capital appreciation/loss portion.
Conclusion & PMPT Evaluation
We have shown that the MPT assumption that investment returns are Normally distributed is quite workable and serves to confirm the validity of the portfolio allocations presented in earlier installments of this article.
To summarize this evaluation of PMPT:
1. PMPT’s criticisms of MPT are basically theoretical and have been shown here to have little practical validity.
2. The tools that PMPT provides run contrary to the basic tenets of MPT, those being efficient risk minimization through broad diversification over many asset classes and the employment of Index funds to represent those asset classes as asserted by the Efficient Market Hypotheses (EMH).
3. Investment returns appear to very closely follow a Normal distribution particularly the stock asset classes in contradiction to the assertions of PMPT. The bond asset classes do exhibit some skewness but a Normal distribution nevertheless provides a very close fit. As such, the standard deviation, derided by PMPT as a measure of risk, is in fact an excellent objective measure.
4. PMPT is silent on correlations of asset class behavior and is thus useless for determining optimum portfolio allocations among various asset classes.
I hope you have enjoyed my series on MPT and PMPT and I am grateful to Dr. Kris for allowing me the space to present it.
*A lognormal distribution is the probability distribution of a random variable whose logarithm is normally distributed.
Posted by Prof. Pat

PMPT Part II: Mathematical Tools–Sortino Ratio & Volatility Skewness

Thursday, June 5th, 2008

Today we continue with our investigation of Post Modern Portfolio Theory (PMPT) and look at a couple of its mathematical tools–the Sortino Ratio and the concept of volatility skewness.

The Sortino Ratio
The Sortino ratio measures the risk-adjusted return an of investment asset, portfolio or strategy. It is a modification of the Sharpe Ratio but penalizes only those returns falling below a user-specified target, or required rate of return, while the Sharpe ratio penalizes both upside and downside risk equally. It is thus a measure of risk-adjusted returns that some people find to be more relevant than the Sharpe. The ratio is calculated as follows:

S = (r – t) / d

where,
r is the annual rate of return for the investment,
t is the Required Rate of return,
d is the downside risk as computed above.

The Sortino Ratio is touted as providing a more realistic risk assessment tool than the Sharpe Ratio as it it based on the semi-variance notion of downside risk and the asserted enhancement of using a Log-normal model rather than a Normal model of investment performance data. The same criticisms PMPT makes against MPT are made against use of the Sharpe Ratio in favor of the Sortino Ratio. That is, rankings of investment funds within the same asset class made using the Sharpe Ratio can potentially be distorted because it ignores the preference of investors to avoid volatility on the downside while tolerating it of the upside. The Sortino Ratio is designed to specifically address this investor concern.

Volatility Skewness
Volatility skewness
is another concept promoted by PMPT enthusiasts. It is the ratio of the total variance of a log-normal distribution that occurs above its mean to that which occurs below its mean. A volatility skewness greater than 1.0 indicates positive skewness, i.e. a tailing off of the distribution toward positive returns to the right. A volatility skewness of less than 1.0 indicates a tailing off of the distribution toward negative returns to the left. There is already a formal statistical definition of skewness for a probability distribution known as the third standardized moment so it is questionable why such a new parameter is necessary. Perhaps volatility skewness is simply a more easily and intuitively understood manner in which to provide arguments in favor of PMPT for those situations in which the assumption of an unskewed Normal distribution and therefore MPT and the Sharpe Ratio can be credibly subjected to criticism.

PMPT short-comings
That is essentially the crux of PMPT. The volatility skewness and Sortino ratio are useful only as tools for fund evaluation and ranking, but there’s nothing in PMPT that addresses the asset class portfolio optimization problem or computes cross-correlation among asset classes.

PMPT is contradictory to the MPT tenet that asset classes be represented by Index funds as suggested by the Efficient Market Hypothesis (EMH). The EMH was first proposed by French mathematician Louis Bachelier and developed by Dr. Eugene Fama. It contends that markets are informationally efficient in their pricing of assets and that any known information about them is already incorporated in their price. It therefore follows that it is impossible to consistently or reliably outperform market averages except through random chance. As such MPT recommends the holding of widely diversified market indices as individual asset classes represented in an overall portfolio. This concept is an anathema to PMPT.

Instead, asset allocation optimization among asset classes in PMPT is left to opaque and proprietary products from professional investment advisory houses who typically recommend expensive commissions and load-laden managed funds. That of course does nothing to rightly address the concerns of astute investors about what exactly it is that is being optimized by these products.

As such, there is little to combat the charge that PMPT is merely a contrivance invented for the singular purpose of rehabilitating the proposition that commission or fee-based professional active asset management is necessary and essential and that individual investors cannot competently nor advantageously see to their own investment management affairs as MPT would permit them to do.**

The next installment will discuss the PMPT assertion that investment returns are Log-normally distributed and present some actual distribution data to let us see for ourselves.

*See Part V of the MPT series (May 7th blog)

**Note from Dr. Kris: I do not share Prof. Pat’s view that markets are efficient because there is overwhelming evidence that they aren’t. However, that is not to discount MPT as a valuable and useful tool for investment management.

Posted by Dr. Pat

PMPT Part I: Overview of Post-Modern Portfolio Theory

Wednesday, June 4th, 2008

In Part IV of this article I made a closing reference to the evolution of Modern Portfolio Theory (MPT) into Post Modern Portfolio Theory (PMPT). In fact, the use of the term “evolution” in this context is actually similar to speculating on the evolution of humans into apes. PMPT is like the heckler in the audience of a cosmology symposium given by Stephen Hawking. In fact, PMPT’s most striking accomplishment is successfully co-opting the name of MPT and including it as part of its own. All right, I’m not being completely fair. PMPT does make some valid criticisms of MPT but, as we shall see, falls far short of being in the same league when it comes to providing a comprehensive toolbox for decision making regarding portfolio allocation.

Major Criticisms of MPT
The criticisms of MPT laid out by PMPT are basically twofold:
1. MPT equates portfolio risk with standard deviation. This makes no allowance for the investor’s greater aversion to variances in portfolio returns on the downside. Excessive variance on the upside, that is, greater gains than anticipated, are not negatively considered when risk is evaluated by the investor. Harry Markowitz, the father of MPT himself, made reference to a concept such as semi-variance that he deems would have been preferable.
2. The symmetry of the standard deviation is a result of the assumption that investment returns follow a Gaussian or Normal distribution. PMPT asserts that a three-parameter log-normal distribution is more appropriate. In this distribution, the logarithms of the investment returns follow a Normal distribution. This type of distribution is characterized by the allowance for skewness of investment returns with a tail in one direction.
Today’s discussion will focus on the first criticism while the second one will be addressed in Part III on Friday.
Downside risk defined
PMPT provides a replacement for the standard deviation as a measure of risk. It is known as downside risk and is given by the following formula:

where,
d is the downside risk as a percentage,
t is the annual Required Return as a percentage,
r is the random variable for the range of investment returns,
f(r) is the best three-parameter log-normal distribution that fits the data.
The integration in this formula is performed from negative infinity up to the target Required Return. In this way it measures only the probability of returns that the investor would consider to be on the downside. To obtain a more accurate value for the downside risk, PMPT requires the use of annualized monthly asset class performance data rather than more limited annual data to provide sufficient confidence for the computation of the estimates of the three parameters of the log-normal distribution.
Note that the above formula is in integral form which means that it’s what is mathematically known as a continuous function. PMPT prefers this continuous function over the commonly used discrete version which simply uses the raw time series data for asset class performance evaluation over time. For comparison purposes, the discrete version is given by:

where,
d is again the downside risk as a percentage,
3.464 is the monthly to annual conversion factor, the square root of 12,
E is the Expected Value operator used to obtain the average return,
t is again the annual Required Return as a percentage,
r is again the random variable for the range of investment returns,
n is the total number of monthly returns in the data.
Why PMPT prefers a continuous function
The preference for the continuous version of the expression for downside risk is a consequence of the aforementioned requirement of using monthly data to quantify returns. This can increase the level of perceived risk by imposing more frequent periodic performance goals. Annualizing returns with a continuous formula tends to smooth out any monthly fluctuations and present the investor with the impression of lower risk, or so the argument goes. Another reason PMPT prefers a continuous function is that it permits forward-looking predictions as well as back testing to be conducted by creating a prediction model from the data.
Conclusion
In summary, what PMPT does is to fit a three parameter log-normal distribution to monthly data, annualize the numbers, then apply the above formula to assess the downside risk.
The next installment of this article will describe additional aspects of PMPT.

Posted by Prof. Pat

MPT Part VII: Revised Asset Allocation Tables Using Monthly Data

Tuesday, June 3rd, 2008

Yesterday we compared asset allocations based on a simple average versus a compounded average. Today we’ll be comparing allocation tables derived from monthly versus annual data. In previous MPT segments, my asset allocation calculations were based on annual data for the past 81 years. My goal has always been to do these calculations based on monthly data but this involved manually inputting almost 6000 pieces of data which took me a while, especially since I rechecked every input for accuracy. I hope you’ll forgive me for not doing this in the first place! In my defense, the allocation tables derived from annual data are by no means invalid; they’re just not quite as accurate as the ones I’m about to show you. The fact is that the more data points you have, the more accurately will your results reflect reality.

The Revised Allocation Table Based on an Arithmetic Average
So without further apologies, let’s look the newly derived monthly tables. The first one is based on arithmetic averages (compare it to the annual chart shown in yesterday’s blog):

The toughest part of annualizing returns based on monthly data is in the conversion of monthly standard deviations. Remember that the standard deviation is another term for portfolio risk. [For you academicians who revel in hairy mathematical equations, the conversion formula is given at the end of this article.*] The standard deviations in these tables generally reflect the smaller variations that occur over a greater number of shorter time intervals, at least for the more conservative portfolio allocations.

Note that this table begins at the lower Required Return level of 3.8%. In general, the optimum allocations based on monthly data compared with allocations derived from annual data reveal that the percentage of total portfolio funds allocated to Large Stocks should be slightly decreased in favor of increasing funds to Small Stocks and Long-Term Corporate Bonds.

The Revised Allocation Table Based on a Compounded Average
The table below represents the optimum allocations using monthly data with compounded averages. Compare these results with the table given in yesterday’s blog (and also in MPT Part II).

As we saw yesterday, the allocations for the same risk level (standard deviation) are only the same at the low end of the Required Annual Return. Comparing the above chart with the annual chart, we again see that Large Stocks are also slightly deemphasized until about the 11% level of Required Return in favor of Small Stocks and Long-Term Corporate Bonds.

Revised Asset Class Correlation Table
You’ll recall from MPT Part IV (April 24th blog) that the more that asset classes are uncorrelated with each other, the less risky is your overall portfolio. The asset allocation table given previously was based on annual data, and if you compare it with the one below that was derived using monthly data, you’ll see that in most cases there is less correlation in the monthly table which is what we want.

Conclusion
This concludes my discussion of Modern Portfolio Theory, at least for the time being. Tomorrow I’ll be introducing some concepts of Post-Modern Portfolio as well as my opinions on the subject.

*Calculation of Annualized Standard Deviation
The annualized standard deviation values were derived using the exact form of the period conversion formula shown below rather than the common approximation,



Posted by Prof. Pat

MPT Part VI: Arithmetic vs Compounded Returns

Monday, June 2nd, 2008

I’m in Wisconsin this week chowing down on BBQed brats and watching the Brewers. My guest columnist, Dr. Pat, has graciously agreed to write this week’s blogs. He’ll be sharing his profound knowledge on Modern Portfolio Theory, revising his previous asset allocation charts to incorporate monthly daily instead of yearly data. He also compares portfolio allocations based on arithmetic averaging to compounded averaging of returns. Which approach an investor should use depends on her portfolio goals and investment horizon.The week will be closed out with a discussion on Post-Modern Portfolio Theory. Professor Pat casts a jaundiced eye on the claims of the theory, noting its good points, bad points, and points of sheer absurdity.

Professor Pat’s discourse is mathematically rooted and although it may take an academician to grasp all the nuances, even the average investor can easily employ his charts to determine the optimum asset allocation for his portfolio. It is with tremendous gratitude and deep respect for Prof. Pat’s knowledge and time that I give my blog over to him for the next five days.

MPT: Arithmetic vs Compounded Returns
Previous installments of this series on Modern Portfolio Theory have presented optimum portfolio allocations for various Required Returns. These returns have been based on the arithmetic average of the history of returns for each asset class. This installment will present the optimum portfolio allocation results based on compounded annual returns. Remember that a return is said to be compounded if the investment amount is adjusted based on the previous return. For example, if you have $10,000 invested in the Small Stocks asset class and it gains 10% in a year, then you’ll be adding that gain to next year’s investment amount for a total of $11,000. (And the reverse process goes for losses.)
The arithmetic average annual return for an asset class is simply the sum of the percent returns divided by the number of years. For example, if the annual returns over a three year period were +8%, -5% , and +12%, the arithmetic average return would be (8 – 5 + 12) / 3 = 5.00%.
The compounded average annual return is more complicated. For this same example it is:

To further illustrate the differences between arithmetic average return and compound average return let’s consider another example. The Small Stock asset class is the highest returning asset class over the 81 years from 1927-2007, but the difference between the arithmetic return is significantly different from the compounded return. For this class the arithmetic average annual return is about 17.3% while the compound average annual return is approximately 12.6%. The arithmetic average is higher because it weights percentage gains equally with percentage losses.
If one asset class experiences a 50% loss in any one year then a 50% gain the next year will not entirely recoup the earlier loss but will in fact recoup only one half that loss (resulting in 75% of the value of the original investment). The arithmetic average return of the two years is zero but there is still an overall loss over the two years as shown by the compound average return which is -13.4%. It actually takes a 100% gain to recover from a 50% loss. In this way, a compounded average return actually numerically weights losses more heavily than gains. In general, the arithmetic average will be greater than or equal to the compounded average with the difference between the two averages positively related to the standard deviation of the data.
An arithmetic average analysis is better at predicting what is more likely to occur in a single time period such as a month or a year. A retiree who needs to withdraw portfolio gains on a periodic basis would use the arithmetic return tables. In contrast, the compounded average analysis is backward looking but is more representative for what is likely to occur over multiple time periods. Investors who don’t need to make portfolio withdrawals and are interested in the long-term accumulation of wealth would elect to use the compounded return tables.
Comparison of Asset Classes in a Basic Portfolio
For the six asset classes considered by Ibbotson & Associates in their Stocks, Bonds, Bills, and Inflation (SBBI) yearbook, the results of configuring the asset allocation optimization algorithm, also known as the mean-variance optimizer, to obtain the best asset class allocations for the range of possible compounded annual Required Returns is shown in the table below. This table ranges to 12.6% because that is the highest possible compound average annual return that can be obtained. That is accomplished with a 100% allocation to the highest returning, but riskiest, asset class–Small Stocks.

As before, all of my calculations below are based on annual return data from 1927-2007 for all of the asset classes described earlier. (See April 21-24 plus April 27 blogs.)

This can be compared to the results using an arithmetic average presented in Part II of this article and reproduced below for convenience.

Comparison of Asset Classes in an Extended Portfolio
For the expanded list of asset classes discussed in Part III where REIT’s, International Stocks, and International Bonds are added to the mix the optimum allocations based on compound Required Returns are shown in the following table.

This can be compared to the results using an arithmetic average presented in MPT Part III and reproduced below for convenience.

In general the differences in the tables can be summarized by noting that a greater allocation toward Large Stocks relative to Small Stocks is preferred when compounded returns are the objective rather than arithmetic average returns. Note that the allocations for the same standard deviation (measure of portfolo risk) are only the same at the low extreme of Required Return. They are not the same elsewhere.

Conclusion
Usage of these tables can be best explained by recommending that the allocations derived using the arithmetic average Required Returns are preferred in the short-term. This is applicable to situations where a particular return is needed for income and any portfolio gains are withdrawn is they are generated. The allocations based on compounded returns are best for long-term financial planning and wealth accumulation.

Tomorrow I’ll be comparing the asset allocations based on annual data with monthly data.

Posted by Prof. Pat