# 27 Mathematics

Mathematics is the soul of science as the vowels are the souls of Hebrew letters. The theoretical models of math are similar to the angelic models of our physical world. There are angels representing each nation, each star in the sky, each blade of grass. Mathematical principles of engineering are edifices in the world of Yetzirah; they are tools in the world of Asiyah. We should know that G-d provides us with ideas. He sends them on the ‘wings of angels’ to help us build the world around.

## 27.1 Euler’s Theorem

Euler’s theorem can be used to derive all the trigonometric identities with complex algebra. Basically replace all trigonometric formulas with exponentials having the imaginary j or i in the exponent and solve.

## 27.2 Taylor Series Expansion27.3 Probability

There was an episode of ‘Ed’, a lawyer/bowling alley series, where Danny Davido, a comedian, plays a con artist. He sends out letters to people predicting which baseball team is going to win a particular game. An unfortunate victim who received five letters with correct predictions decides to place a bet with the con-artist’s bookie on the sixth prediction. Needless to say he loses his money. Why, would the trend fail at this point? Danny explains to his lawyer that he sends out 1000 letters with his first recommendation, then 400 letter follow-ups where he was successful, than 100 letter follow-ups, etc. The point is there will be a few people who will get letters where he was right five times in a row.

The point of this story is that one should be very careful about investing on a trend. For example, in mutual fund investing, there may be a handful of funds that seem to beat the averages, but this may be simply because of chance. Human beings have mental functions that are good at picking out trends. When we were hunters and scavengers, this helped us return to good places to find food or water. Information was limited and recognizing a trend was more likely to have reason. Today with massive communication systems, trends of a few are more like the con-artist scheme above. Our minds, has vshalom, overlook that one investment’s success is often by chance because we picked it out of a thousand failures. This is related to the illness of gambling.

The Stock Market resembles gambling, because the holders of individual stocks and mutual funds are distributed to program trading machines. This information is used to manipulate prices for the maximum loss to individuals. For those who wouldn’t gamble in Las Vegas, one must be careful of stock markets. Personal business is a better choice. Corporations have out of proportion salaries for CEOs with stock options that dilute shareholder value. Nevertheless, the CEOs have incentive to grow the stock price in order to sell these holdings, but often this is done on a cyclical basis without a net growth in the assets for the long-term investor.

Text 27-1: Seasonal cycle that sometimes occurs
James Dines: "Stocks: The month of May used to be an inauspicious month for new buying, but it has been improving. Despite many years of rising markets, May has seen the DJI decline in 22 of the last 40 years, 55% of the time, or has proven to be the prelude to significant declines in Junes or Julys - that pave the way for the traditional "Summer Rally." Since 1965, the period 1 May to 30 Jun has been a loser 60% of the time (24 out of 40 years.) But, 13 of the 16 exceptions (81%) occurred in the last 22 years, so the period has been getting less bearish. May is the beginning month for the worst six months (1 May - 31 Oct) of the year, a bullish factor for golds since they tend to move opposite the market. The statistics were based on the Dow's average percentage change for the six-month periods from 1950-2003 as applied to an arbitrary investment portfolio of \$10,000. Specifically, when compounded annually starting in 1950, buying in May and selling by Oct, a \$10,000 investment was reduced to \$9,682 by 2003, reflecting a loss of \$318. On the other hand, buying in Nov and selling by Apr resulted in an appreciated value of \$492,060, reflecting a \$482,060 profit by 2003.
Memorial Week Rally: Investors who nonetheless insist on buying might as well do so before the start of Memorial Day weekend, May 23 to May 27 this year. The DJI rose during that week 12 years in a row (from 1984-1995) - also 1999, 2000, 2003, and 2004. Downers during the Memorial Day holiday include 1996, 1997, 1998, 2001 and 2002. All told, the "memorial week rally" has come true 76% (16 of 21) of the time in the last 21 years.

For the longer term the multi-year business cycle suggests during the market peak, (interest rates climbing) hold money market or drug/medical companies, during the contraction (interest rates have stopped going up), average into the market in dividend instruments. During the trough, (interest rates have started declining), invest in growth stocks, oil, copper, and REITs. During the expansion, (interest rates are flat or have started rising), invest in growth companies.

Shiller:
The fact is that equities were over-valued for years, making them vulnerable to the kind of brutal, sudden sell-off we've just witnessed. But now that the S&P has declined 40% in 12 months, the question is whether equities are at long last a bargain. The answer is a qualified yes: Stocks aren't exactly cheap, but for the first time in years you can expect decent returns, provided you're patient.
"If you buy now (Dow 8400) and wake up in 10 years, you'll probably get a return around the historic average," said Yale economist Robert Shiller. In the near term, however, Shiller - who correctly predicted the implosion of the stock-market and real-estate bubbles - is more cautious. "There is a substantial risk that with all this economic turmoil, stocks will fall far lower," he warned.
But make no mistake, stocks are now at levels where buying makes sense.
The best measure of stock valuation is Shiller's own index of price-earnings multiples. Shiller uses a 10-year average of inflation-adjusted earnings to calculate an adjusted P/E. The advantage to the Shiller method is that it smoothes out the peaks and valleys in profits.
Example: In the 2003 to 2006 period, earnings soared to historic heights, jumping from a normal 9% of gross domestic product to an extraordinary 12%. The profit bubble made P/Es look artificially low, handing the stock jockeys a logical-sounding reason to claim that equities were a buy, when in fact they were overpriced. Both the "P" and the "E" were in a bubble - the "P" even more than the "E." When the "E" collapsed in the face of the current downturn, the outrageous valuations were rudely exposed.
To see how out of whack P/Es had gotten, let's take a look back. From 1890 to the early 90s, the average Shiller P/E stood at 14.6. It dropped as low at 6 in the early 80s, and never went over 24. Then, in the late 90s, P/Es regularly stood at over 30, and at their peak in 2000 hit 44.
In the bear market that followed, P/Es dropped - but only into the low-20s. Then they took off again, averaging 25 to 28 from 2003 to the beginning of this year. Now they're at 15.7, not far from their pre-bubble average. That decline is tonic for investors. Research by economist and hedge fund manager Cliff Asness shows that buying in at a high Shiller P/E usually leads to poor returns, while grabbing stocks at a low Shiller P/E is a reliable route to riches.[2876]
From today's levels, what can we expect? Stocks' future return is closely related to the inverse of the P/E, also known as the earnings yield. So at a P/E of less than 16, investors should obtain real, or inflation-adjusted, gains of around 6.5%, which is about what Asness found in his research. Add 2.5 points for inflation, and the nominal return comes to a respectable 9%. That's about a point below stocks' long-run return, but it's far better than anything investors could expect for a decade and a half.
The rub is that getting even that 9% return won't be easy. Assuming no escalation of P/Es, stock returns come from a combination of earnings growth and dividend income. Earnings per share grow only at about 2% a year after inflation. (Total earnings grow faster than that, but new issues of stock dilute that growth.) So add in our 2.5% inflation rate to 2% real growth, and you still need a dividend yield of 4.5% to get to that 9% goal. The yield on the S&P 500 is now around 3.3%, versus around 2% earlier this decade. That's better, but not enough.
Finally, remember this: Shiller points out that stocks were cheap in the early 1930s, and investors who bought then eventually made good money. But it took them many years to get there. So if you buy now, stick with strong dividend-paying stocks, and fasten your seatbelts. It will be a bumpy ride.
3/2008 TORONTO — The bottom of the financial-market downturn could still be several months away and will hinge on a bottoming of the U.S. housing market, former Federal Reserve Board Chairman Alan Greenspan said.
Speaking with TD Bank chief economist Don Drummond in a loosely structured discussion in front of a luncheon crowd of about 2,000 in downtown Toronto, Mr. Greenspan said housing values might have another 5 to 10 per cent to decline before they bottom, “sometime in the first half of next year.”

The Economic Mess and Financial Disaster that Obama Will Inherit
Nouriel Roubini | Nov 6, 2008
The good news is that America has just elected a president with leadership, vision and great intelligence. President Obama will also choose a first rate economic team: individuals such as Larry Summers and Tim Geithner would be excellent choices for the position of Treasury Secretary. Obama and his team are fully aware of the very difficult economic and financial challenges that the country is facing and will work hard to resolve them.
However, Obama will inherit and economic and financial mess worse than anything the U.S. has faced in decades: the most severe recession in 50 years; the worst financial and banking crisis since the Great Depression; a ballooning fiscal deficit that may be as high as a trillion dollar in 2009 and 2010; a huge current account deficit; a financial system that is in a severe crisis and where deleveraging is still occurring at a very rapid pace, thus causing a worsening of the credit crunch; a household sector where millions of households are insolvent, into negative equity territory and on the verge of losing their homes; a serious risk of deflation as the slack in goods, labor and commodity markets becomes deeper; the risk that we will end in a deflationary liquidity trap as the Fed is fast approaching the zero-bound constraint for the Fed Funds rate; the risk of a severe debt deflation as the real value of nominal liabilities will rise given price deflation while the value of financial assets is still plunging. This is the bitter gift that the Bush administration has bequeathed to Obama and the Democrats.
Given this dismal background, let us consider next in more detail the macro outlook for the U.S. and global economy and its implications for financial markets...
The latest U.S. macro news have been worse than awful: collapsing retail sales and consumption, free fall in capex spending by the corporate sector, sharply falling industrial production, sharply falling employment, housing still in free fall and home prices bound to fall 40% from the peak, collapsing auto sales, forward looking indicators of business (ISM) and consumer confidence dropping to multi-decade lows, sharp surge in corporate defaults, a wrecked banking system and financial system that will have to be partially nationalized. This is the most daunting set of economic and financial challenges that any president has had to face since FDR during the Great Depression. And in the meanwhile in the rest of the world things are as bad: a severe recession in Europe, Japan and other advanced economies; the risk of a hard landing in many emerging markets including China; an almost certain global recession; a severe global financial crisis.
So let us not delude each other: the U.S. and global recession train has left the station; the financial and banking crisis train has left the station. This will be a long and severe and protracted two-year recession regardless of the best intentions and good policies of the new U.S. administration. It will take a lot of hard work and sound policies to clean up this mess and reduce the length and severity of this economic contraction.
And in the meanwhile the brief bear market sucker’s rally in the equity market has lost its steam and U.S. and global equities are starting to plunge again. As I argued for the last few weeks this was a bear market rally and markets could not defy the laws of gravity: a slew of ugly and worse than expected macro news, earnings news and financial news was bound to take a toll on equities and other risky assets. And now, after a brief rally markets are starting to plunge again. For 2009 the consensus estimates for earnings are delusional: current consensus estimates are that S&P 500 earnings per share (EPS) will be \$90 in 2009 up 15% from 2008. Such estimates are outright silly and delusional. If EPS fall – as most likely – to a level of \$60 then with a multiple (P/E ratio) of 12 the S&P500 index could fall to 720, i.e. 20% below current levels; if the P/E falls to 10 – as possible in a severe recession, the S&P could be down to 600 or 35% below current levels. And in a very severe recession one cannot exclude that the EPS could fall as low as \$50 in 2009 dragging the S&P500 index to as low as 500. So, even based on fundamentals and valuations, there are significant downside risks to U.S. equities.
So the brief sucker’s rally is over and a reality check is now dawning on markets and investors. Expect this financial crisis and economic recession to get much worse in the next 12 months before it gets any better. We are nowhere near a bottom for housing, the U.S, economy, the global economy and financial markets. The worst is ahead of us rather than behind us.

Indeed, as I put in in a note in mid-October:
So risks and vulnerabilities remain and the downside risks to financial markets (worse than expected macro news, earnings news and developments in systemically important parts of the global financial system) will dominate over the next few months the positive news (G7 policies to avoid a systemic meltdown, and other policies that – in due time – may reduce interbank spreads and credit spreads). So beware of those who tell you that we reached a bottom for risky financial assets. The same optimists told you that we reached a bottom and the worst was behind us after the rescue of the creditors of Bear Stearns in March, after the announcement of the possible bailout of Fannie and Freddie in July, after the actual bailout of Fannie and Freddie in September, after the bailout of AIG in mid September, after the TARP legislation was presented, after the latest G7 and EU action. In each case the optimists argued that the latest crisis and rescue policy response was “THE CATHARTIC” event that signaled the bottom of the crisis and the recovery of markets. They were wrong literally at least six times in a row as the crisis- as I consistently predicted here over the last year – became worse and worse.
So enough of the excessive optimism that has been proven wrong at least six times in the last eight months alone. A reality check is needed to assess the proper risks and take the appropriate actions. And reality tells us that we barely literally avoided only a week ago a total systemic financial meltdown; that the policy actions are now finally more aggressive and systematic and more appropriate; that it will take a long while for interbank markets and credit markets to mend; that further important policy actions are needed to avoid the meltdown and an even more severe recession; that central banks instead of being the lenders of last resort will be for now the lenders of first and only resort; that even if we avoid a meltdown we will experience a severe US, advanced economy and most likely global recession, the worst in decades; that we are in the middle of a severe global financial and banking crisis, the worst since the Great Depression; and that the flow of macro, earnings and financial news will significantly surprise (as this past week) on the downside with significant further risks to financial markets.

## 27.4 Work and savings

One should choose a field that will give one time to study and practice Torah. The sages would recite mishnayot while practicing manual labor. One should pay his workers at the end of the day for their service. One should treat ones supervisor lightly because they are under a burden.

## 27.5 Investments

### 27.5.1 Rule of 72

72 divided by the interest rate is the number of months to double ones money. For example, at 7% interest, it would take a little over 10 years to double.

### 27.5.2 Division of investments

1. Bava Metzia 42a Rabbi Yitzchok: A person should invest 1/3 of his money in business, another third in real estate and the last third in cash.
2. Another rule is to give after all business expenses 10% to Jewish education or charity which can include Jewish businesses.

This refers to ones personal business or other types of business. The 1/3 is the maximum value when one is young and just starting out.

The recommendation is to use Vanguard target retirement funds which scale down the business ratio and increase the cash ratio overtime. For example, VTINX is Vanguards retirement fund after the age of 65. Another example is VTTVX for expected retirement in 2025 and VTHRX is for expected retirement in 2030. The way to determine the target retirement year is:

(65 – current age) + current year = Target retirement year.

For example at the age of 49 this gives (65-49)+2011 = 2027. In this case ones investment would be 60% in VTTVX and 40% in VTHRX.

#### 27.5.2.2 Real Estate (1/3)

This would be a personal house and perhaps a rental. Real estate investment trusts (REITs) would occur here as well.

#### 27.5.2.3 Cash (1/3)

Basically saved liquid assets. This might include bonds. Put 30% in VFIDX is the intermediate bond fund. Put 50% in VWEAX is the long-term industrial bond fund. Put 10% in a municipal bond fund preferably for ones state of residence. Leave 10% in short term treasuries or money markets.

## 27.6 Taxes

Joint tenancy can be subjected to the gift tax if it is a gift. It also doesn’t update the cost basis on property.

2013 Defense budget à \$518B.

2014 Defense budget \$498B à \$520.46B with the OCO (overseas contingency operations) à \$80B supporting troops and maintainence of overseas deployed equipment. Non-defense discretionary spending \$469.39B à \$491.77B.[2877] Total would be \$1,012.3B. \$498B à \$486.9B with OCO à \$85.2B à \$92B – \$572.6B is \$20B less than requested. Of the \$486.9B: \$72.9B à \$63B for new defense R&D. \$5.1B for science research. \$29.9B à \$29.9B for NIH.

2015 Defense budget \$512.05B à \$521.27B. Non-defense discretionary spending \$483.13B à \$492.36B.

## 27.7 Philosophy

I include philosophy in the area of mathematics although most would prefer the reverse. Philosophy at its best is mathematical. Overall, it is an area of study that takes ones mind out of conventional thought and offers advantages here is well. Finally, if studied occasionally as opposed to becoming a religion, it can refresh ones own identity with confidence and new ideas. In this regard I recommend, “The Philosophy Book, Big Ideas Simply Explained,” a compilation of most philosophies in the world for browsing and peruse.