Variation of St. Petersburgh Paradox Simulation

Related previous posts: Simulation of St. Petersburg Paradox

In the traditional St. Petersburg Paradox, there is 1/(2^n) chance of an outcome of (2^n), for all positive integers n, summing up to an expected value of infinity, but with realistic outcomes far lower than that. What if the base is changed to an integer other than two? In this exercise, bases of 2, 3, 4, 10, and 100 were used to run similar simulations, each with 100,000 trials. To perform this variation, only two lines of codes needed to be changed from the original Java code. For base 3, for example:

• "while(flip < 0.5)" becomes "while(flip < 1.0/3)"
• "double result =  Math.pow(2,times)" becomes "double result =  Math.pow(3,times)"

After running the simulation, the results and their analysis as follows:

Base	Average/Base	Max/Base	St. Dev	Log(max)/Log(base)
2	9.75	65536	784.25	17
3	7.49	59049	724.34	11
4	6.47	65536	956.40	9
10	5.48	100000	3315.75	6
100	3.46	10000	12282.46	3

Given the different bases, the average result / base value becomes the most significant output to compare. As the value of the base increases, the normalized average decreases, since the probability of getting the lowest value dramatically increases. For base of 100, the result will simply be 100 for 99% of the time. While it's true that the output will also be dramatically greater, the clustering around high-probability, low-value region outweighs the rarity of low-probability, high-value regions.

The max result / base value is a bit harder to decipher. This is the output value were the game started with output of 1, rather than the value of the base, offering a normalized maximum output value. There doesn't seem to be a clear-cut trend of these values as the bases change. The lack of trend is reasonable, given that it only takes one rarity event to record the maximum. The standard deviation drastically increases as the base increases, a more obvious reflection of the greater variance in the output as the base increases. The logarithm of maximum value / logarithm of base is equivalent to the maximum number of times that the game ran for. The decreasing trend is reasonable, given that the probability of extending the game decreases dramatically as the base increases (50% for base 2, 1% for base 100).

The last three trends (or two trends plus the lack of trend for max/base) are reasonable. But the average result / base value is still the most intriguing, since the expected values, regardless of the base, are still all infinity. Instead, as expected, the greater the variance (bigger bases), the further the deviation from the expected value to the realistic output value (smaller normalized results).

Performances of Defensive Stocks from Pre-Crisis to 2011

Defensive stocks are those that perform well during a period of economic slowdown. They generally do not outperform the market during periods of rapid economic growth, and thus serve as low-risk securities. A MSN article from May 2010 lists 14 "safe stocks to stock up on." Of the 14, three are currently listed on the Dow Jones Industrial Average. They include Johnson and Johnson (NYSE: JNJ), which is involved in health care products, Proctor and Gamble (NYSE: PG), which focuses on consumer packaged goods, and Wal-Mart (NYSE: WMT). The historical performances of these stocks from mid-2008 to present were retrieved and analyzed. For comparison, similar data were retrieved for two companies not part of the Dow: Google (NYSE: GOOG) and Caterpillar (NYSE: CAT), and the Dow Jones Index itself. The results are as follows:

Name	High	Low	Decrease	Current	Increase
JNJ	72.22	46.60	35.47%	65.98	41.59%
PG	73.15	44.18	39.60%	66.67	50.91%
WMT	63.17	46.42	26.52%	59.99	29.23%
GOOG	685.33	257.44	62.44%	633.14	145.94%
CAT	115.41	22.17	80.79%	92.25	316.10%
DJI	13058.20	6626.94	49.25%	12294.00	85.52%

Indeed, as the figure shows, during the financial crisis of late 2008 and early 2009, the percentage decrease from the highest closing trades to the lowest closing trades was the greatest for Google and Caterpillar, which are not defensive stocks. Then came the Dow index, which featured a mix of defensive and non-defensive stocks. Finally, the three defensive stocks saw the smallest decrease during those economic downturn times. On the reverse, as the economy has been recovering over the past 2+ years and stock prices have been increasing again, the three defensive stocks saw the smallest percentage gain, while Google and Caterpillar both saw enormous gains, with the Dow motley in the middle.

All this goes to say is that defensive stocks have low risk and variance. In good times, they return less than the others, but in bad times, they don't lose as much as the others. Wal-Mart's low variance is exemplified in an even more profound way by realizing that its post-crisis trough was hit on February 2009; however, that price of 46.24 was actually higher than that during January 2008, and wasn't far off from the prices during much of 2006 and 2007, when the economy was growing strongly. Defensive stocks usually cover common products, used whether the economy is growing or shrinking. Think of the sectors the defensive stocks here cover: healthcare, consumer package, and retail, all of which are rather basic necessities and face relatively inelastic demand. In all, coming together in a group of ideas are: defensive stocks, common goods, necessities, inelastic demand, low risk, low variance, low returns, and low losses. Whether these are better than other stocks will ultimately depend on the state of being of the macroeconomics.

Sources:

• http://articles.moneycentral.msn.com/learn-how-to-invest/14-safe-stocks-to-stock-up-on.aspx
• http://www.google.com/finance?q=NYSE%3AJNJ
• http://www.google.com/finance?q=NYSE%3APG
• http://www.google.com/finance?q=NYSE%3AWMT
• http://www.google.com/finance?q=INDEXDJX%3A.DJI
• http://www.google.com/finance?q=NASDAQ%3AGOOG
• http://www.google.com/finance?q=NYSE%3ACAT

Lockheed Martin Corporation (NYSE: LMT)

Topping the list of top US federal contractors at over $35 billion (followed by Boeing at over $19 billion) in the fiscal year of 2010, Lockheed Martin continues to lead in global aerospace, defense, and security. Headquartered at Bethesda, MD, right outside Washington DC, Lockheed Martin recently secured a Japanese order of 42 F-35 fighter planes. Looking to modernize their fighter fleets, the Japanese will replace the Boeing F-4s, which were last assembled in 1981. It will cost $114 million for each of the first four units.

Stocks for Lockheed Martin currently trade at 80.87. Financial statements of the company include its strong health. The company secured over $500 million of net income in each quarter for the past year. Total equity has hovered around $3 billion during the same time. In terms of cash flow, although there have been a few years recently with negative net cash flows, quarterly data show drastic increase in cash amounts for the first 9 months of 2011. Cash levels hover around 9.4% as of Q3-2011. The current level is around 1.2, indicating that it has more current assets than current liabilities.

Sources:

• http://www.google.com/finance?q=NYSE%3ALMT
• https://www.fpds.gov/fpdsng/index.php/reports
• http://news.businessweek.com/article.asp?documentKey=1377-aXfrkWe.hC7Q-5T53DPIFU42QK5IM2N5TRAHA5B

The Boeing Company (NYSE: BA)

Currently, each share is trading at 72.54. Growth for the company looks mildly healthy. Fitch predicts a stable 2012 outlook for the global aerospace and defense sector. Boeing may be one of the few companies facing short-term gains if further unsettling developments unfold in the Korean Peninsula after the death of Kim Jong Il.

Looking at the financial statements, the numbers look relatively healthy as well. Net income and equity both have been positive and growing for 2011, with net income climbing from $586M for Q1 this year to $1,094M for Q3. At the cash flow statement, there are several negative subtotals. But the cumulative negative cash flow in 2010 mainly stems from investment. For the 9-month period ending this September, cash flow from financing was beyond a billion in the negative, mainly from dividends. A look at the annual cash flow statements from the past 4 years reveals that while dividends are increasing, the growth rate has drastically slowed down.

Sources:

• http://www.google.com/finance?q=NYSE:BA
• http://online.wsj.com/article/BT-CO-20111220-710863.html
• http://www.sfgate.com/cgi-bin/article.cgi?f=/g/a/2011/12/19/investopedia69524.DTL

Response to "First Person: How I Got Myself $80,000 in Debt"

Recently, Yahoo contributor S.L. Carroll wrote an article detailing how she incurred the enormous amount of debt early in her life, and in retrospect seems regretful of her decisions. For two of the categories - student loans and medical bills - most can sympathize with. Rising tuition has been a severe concern in the past few years, and the medical insurance has been a topic of heated political debate. But the other category - credit cards - is less likely to draw sympathy. Although Carroll admits that she "never should have applied for a credit card much less six of them," her reasoning for the prior behavior was what "the free money was addicting."

Debt is not free money. It is a liability, an obligation of reimbursement. Applying for credit cards wasn't the problem. Taking on debt gives access to more assets, which could be a beneficial short-term decision for an individual or a firm, particularly when new ventures are sought or begun. But it also adds onto liabilities, and requires the responsibility to recognize that obligation, at the bare minimum. Debt is not free money. Recognize the obligation to get the reimbursement under control, or one is stuck under a heap of burden. For a corporation, it could mean the difference between survival and bankruptcy. For an individual, it could mean the difference between proactive financial planning and reactive financial recovery.

It is difficult to grasp that concept when the current government debt is over $15 trillion, or over $48K per person, in the United States. But fiscal responsibility can begin at the individual level. Better yet, fiscal responsibility needs to begin at the individual level, where one can and does control his or her own destiny. In this case, it's not about reaching bipartisan agreements or arguing over political values. Instead, it's crystal clear as recognizing that debt is not free cash; taking on loans means incurring the responsibility and obligation to repay the amount.

Sources:

• http://finance.yahoo.com/news/first-person-got-myself-80-000-debt-163400779.html
• http://www.brillig.com/debt_clock/

Simulation of St. Petersburg Paradox

St. Petersburg Paradox exemplifies a situation where the expected value of an outcome doesn't reflect the realistic outcomes. The game can be played by flipping a coin until heads comes up. Let the outcome be 2 to the power of the number of coin flips. The derivation won't be shown here in detail (but is available on the link below), but essentially, there is 1/(2^n) chance of an outcome of (2^n), for all positive integers n. The expected payoff for each integer n is therefore 1, and the sum of the expected payoffs, the expected value of this game, is therefore infinity.

Realistic outcomes are far from this expected value. One can briefly visualize this by realizing that there's 50% chance of an outcome of 2, 25% chance of an outcome of 4, 12.5% chance of an outcome of 8, etc. Simply put, the chances of very high outcomes are very small. Let's use programming to simulate numerous rounds of this game. The Java code for the simulation method is as follows:

public static int simulation()
 {
     int times = 0;
     double flip = 0;
     while(flip < 0.5) //reflecting the 50% chance of getting heads
     {
         flip = Math.random();
         times++;
     }
     double result =  Math.pow(2,times);
     return (int)result;
 }

Writing a for loop code, this method was called 1,000,000 times, with the output values exported onto Microsoft Excel. The values were sorted and counted, and the results are as follows, along with the expected frequency of the different outcomes.

n	Outcome (2^n)	Frequency	Expected Freq	Deviation
1	2	500,592	500,000.00	0.118%
2	4	249,969	250,000.00	-0.012%
3	8	124,870	125,000.00	-0.104%
4	16	62,262	62,500.00	-0.381%
5	32	30,951	31,250.00	-0.957%
6	64	15,741	15,625.00	0.742%
7	128	7,843	7,812.50	0.390%
8	256	3,883	3,906.25	-0.595%
9	512	1,900	1,953.13	-2.720%
10	1,024	993	976.56	1.683%
11	2,048	484	488.28	-0.877%
12	4,096	233	244.14	-4.563%
13	8,192	144	122.07	17.965%
14	16,384	71	61.04	16.326%
15	32,768	34	30.52	11.411%
16	65,536	12	15.26	-21.357%
17	131,072	11	7.63	44.179%
18	262,144	3	3.81	-21.357%
19	524,288	4	1.91	109.715%

Even with 1 million trials, the greatest output value was only 524,288, which reflects the rare instance of 18 straight flips of tail before finally getting a heads. This only happened 4 times out of the 1 million trials, and was even considered strong positive deviation from the expected frequency of 1.91. The expected value of this game (expected average value) is still infinity, as the math demonstrates.

In this simulation of 1,000,000 trials, the average output value was 20.496084. The results are, as expected, very skewed to the right. If we let the output value measure monetary amount, it's interesting to note that the top 996 trials (0.0996%) contains 51.30% of the values. The top 0.7772% contains 65.86%. The bottom 96.8644% contains only 24.33%.

Sources:

• http://plato.stanford.edu/entries/paradox-stpetersburg/

Public Ownership of Sports Franchises

One may not initially associate sports franchises with issuing stocks, but Green Bay Packers are doing exactly that to finance the addition of seats at Lambeau Field. A publicly owned corporation since 1923, the Packers had over 110,000 shareholders (larger than the population of Green Bay) owning 4.75 million shares before the most recent sale on Tuesday. Although deficient of many advantages associated with traditional stocks, the Packers stocks hold more important sentimental values for the stakeholders.

Among the disadvantages, these stocks receive zero dividend and can only be transferred to family members. As a result, there is no re-sale value and doesn't even help the shareholders gain an advantage on the long waiting list for season tickets. NFL rules also prohibit the shareholders from betting on games. The only practical perks for the shareholders are voting rights and the rights to attend annual stockholder meeting in the summer.

Yet despite these restrictions, Packers fans rushed to own part of the team when the offering began on Tuesday. During the first 11 minutes of the offering, 1600 shares were sold. Packers hope to raise $22 worth of stocks before the offers closes next February. The greatest value is the claim of partially owning the franchise. It's this sentimental value, not monetary incentives, that motivates the fans to "invest" money into their team, and the benefits in the form of larger stadium, will help to benefit the fans themselves.

Green Bay is unique among all North American professional sports franchises. With a population of only around 300,000 in the metropolitan area, it by far has the smallest local fan-base. Nevertheless, Green Bay Packers have remained loyal to the Wisconsin city, over 100 miles north of Milwaukee, since its founding. Public ownership has helped that trend, but certainly the success of the Packers has allowed such passion. The Packers have won more Super Bowls than any other franchises, most recently in 2011. With a perfect 12-0 record heading to this week, it is aiming to go back where they were.

Similar small-town professional franchises have not fared as well. NBA's Sacramento Kings recently considered moving to Anaheim to provide more financial stability for the struggling team. While ideas were never proposed, it seems unlikely that Kings, which has fared poorly for most of the latter half of the past decade, could've generated much revenue from stock offerings.

Certainly, America continues to embrace its professional sports franchises, and most teams have very loyal fan-base in their hometown. Even in Cleveland, where teams haven't fared well in recent years, loyal fans bought their personalized bricks at the Heritage Park in Progressive Field, to be part of the history of the Cleveland Indians. In the turn of events this year that saw labor strikes in NFL and NBA, teams could consider selling part of the ownership to the devoted fan-base. It would strengthen the stability and presence of the team in the area while sustaining financial growth.

Avid sports fans flock to stadiums or ESPN to follow their passion, but at the end of the day, money and sports can't be separated. Franchises operate to maintain financial growth. Fans pay to see their teams succeed. Offering partial ownership of the teams to the public could be the link between the two sides.

Sources:

• http://online.wsj.com/article/SB10001424052970204903804577082524238902912.html
• http://www.csmonitor.com/USA/Latest-News-Wires/2011/1206/Green-Bay-Packers-stock-goes-on-sale-receives-mixed-feelings
• http://blogs.sacbee.com/sports/kings/archives/2011/05/kings-announce-1.html

Response to "Some Asians' college strategy: Don't check 'Asian'"

By now, I'm sure most have read the article that was posted on Yahoo and circulated massively on Facebook. For those who haven't, a article published yesterday details a strategy used by half-Asians on the college admission process to not reveal the Asian background of the candidate. The argument stems from the notion that Asians are held to a higher standard in the process, and thus, the chances of enrolling is higher without the Asian tag. Supporters of this claim have the numbers on their side. In the most vivid example, UC Berkeley is more than 40% Asian, but before a law was passed to forbid considering race in the admission process, the same count was only about 20%.

As a value judgement, there may be no consensus on the practice of affirmative action. Proponents cite the increased diversity; critics cite the indirect discrimination. Similarly, as a value judgement, there may be no consensus on the strategy of checking only one ethnicity for a student of multiple backgrounds, simply to boost the chances of admission. It isn't completely incorrect that the half-Asian students cited in the story are "white," but nor is it completely correct that they are only "white."

Yet here is something that hopefully can draw some consensus. Which college one ultimately gets admitted into and attends isn't all that matters. It is true that graduates of more prestigious schools have higher starting and mid-career salaries, on average. They may have more access to resources and opportunities, on average. However, that says nothing about how a particular student will exactly perform. Those who work hard and take advantage of available, despite scarce, resources will likely perform better in the future than those who put little devotion into anything and don't take advantage of multitudes of available resources.

Why do people go to college in the first place? More likely than anything else, students choose to go to college for better employment options for the rest of their lives. On that note, or even by other reasons, there are thousands of successful graduates from "non-prestigious" schools and thousands of unsuccessful graduates from "prestigious" schools each year. There isn't any magic or rocket science in the results. While it's understandable that high-school students try all types of strategies to get into the college of their dream, it's even more crucial to keep in mind that which college one gains acceptance into, isn't as crucial as how one might perform there. It's better to go to a "mediocre" college and get the most out of it, rather than go to a "top-notch" school and get little out of it. It's an individual decision and outcome, but an important remark in a society obsessed with the admission process.

Sources:

• http://news.yahoo.com/asians-college-strategy-dont-check-asian-174442977.html
• http://www.payscale.com/best-colleges/top-us-colleges-graduate-salary-statistics.asp

Response to Siri Controversy

Last weekend, I was trying out Siri for the first time and asked the question of "how do I get from Cleveland to New York?" I already had a plane ticket booked for the next day, but I wanted to see how the voice software would respond. To my dismay, I was told "sorry, I can't help you find flights." First of all, where came the assumption that I automatically was looking for a plane ticket? And furthermore, while a quick Google search would bring numerous results, why couldn't Siri find an answer for me?

Recently, it has hit the headlines that Siri can't help users find abortion clinics. This issue seems to have so much salience for abortion groups, that American Civil Liberties Union even launched an online petition saying that "if Siri can tell us about Viagra, it should not provide bad or no information about contraceptives or abortion care. Send a message to Apple: Fix Siri." In response, Apple claims that the omission is not intentional. An Apple spokeswoman stated the omissions aren't meant to offend anyone, but rather "that as we bring Siri from beta to a final product, we find places where we can do better and we will in the coming weeks."

Even if Siri moves away from its beta stage, it's highly doubtful that it will always find what users want, given the limitation of artificial intelligence. From finding restaurants to voice calling, Siri could be quite helpful. But it's not meant to do everything. To balance its usefulness and convenience, it needs, like all other software, a limitation. So far the public knows little of Apple's development behind Siri. Why can't it find flights, when it's easily done on a Google search, just like a restaurant search? Such function must not have been part of Siri's predefined capabilities. But for airline companies to claim that Siri and Apple are against those industries far bends the argument toward exclusive, self-motivated interests.

On a more practical note, those wishing to find abortion clinics could simply Google search for abortion clinics on the iPhone. It would take around a few more seconds to type out the search. It's not that those angry at Siri are unhappy at the inability to find their solutions. It's their discontent at the technology to not include their interests in the limitation of capabilities of the software. Software is as useful as its capabilities. Just like how a spell-check cannot detect all flaws, Siri is not meant to do everything. Only when can people understand that technology has definite limitations, will new software truly be appreciated for what it can offer.

Sources:

• http://www.cnn.com/2011/12/01/tech/mobile/abortion-clinic-siri-iphone/index.html?hpt=hp_bn6
• http://news.yahoo.com/apples-siri-irks-abortion-rights-advocates-231159078.html

Stock Market Recap

On Wednesday, the Dow increased by 490 points, the largest increase since March 2009, when the Dow hit its low point during the financial crisis. When the index closing at 12045.68 for the month, it showed an 0.8% increase in the month of November, and up 4% for the year. Three key developments led to the tremendous growth observed.

First, banks around the globe announced a coordinated plan to make US dollar funding cheaper for European banks. The announcement was well received in European markets, where the German index DAX jumped nearly 5%. However, this move "did not address the fundamental financial problems threatening the survival of the European currency union." Instead, it seemed more of a temporary move extend more time to Europe to plan its efforts on the preservation of the euro. Governor of Japan's central bank was quoted in saying that "the European sovereign debt problem will not be solved only with liquidity." Instead, what actions taken as crisis response will be crucial.

In other developments that contributed to the rise in markets on Wednesday, China indicated that it would loosen monetary policy by lowering the reserve ratio by 0.5% for banks, thereby enabling the increases in banks' lending activities. Lastly, private-business hiring increased by over 200,000 in November, well ahead of economic forecasts in the largest monthly gain this year.

Sources:

• http://online.wsj.com/article/SB10001424052970203833104577069782912816156.html?mod=WSJ_Markets_LEFTTopStories
• http://www.nytimes.com/2011/12/01/business/central-banks-move-together-to-ease-debt-crisis.html