Chapter 4Stock Market Investment, asGambling on a FavorableGameLecture 3 mentioned favorable and unfavorable games before focussing on fairgames. This lecture will soon focus on the question: how do you best exploita setting where the odds are in your favor? This scenario is unrealistic forcasino games but has historically been realistic for the stock market.Novice investors are told to view the stock market as a plac e for long-term investment. This excellent advice is unfortunately rather neglectedin most mathematically oriented discussions, but will be emphas i zed here.Between short-term speculation and long-term investment li es a spectrumof intermediate activites with no clear dividing line, but the ends of thespectrum are quite different to a typ i cal ind iv i d ual .The stock market is really “a market of stocks”, but for most of thislecture I use a conventional shorthand of representing the U.S. market bythe S&P500 index (essentially an actual invest m ent pos si bi l i ty, via an indexfund)..4.1 Unfavor able gamesThere are two things to say about unfavorable games. Mathemati c ian s oftensay the first thing wrong, for instance by sayingGambling against the house at a casino is foolish, because the3536CHAPTER 4. STOCK MARKET INVESTMENT, AS GAMBLING ON A FAVORABLE GAMEodds are against you and in the long run you will lose money.What’s wrong is the because.SayingSpending a day at Di sn ey l and is foolish, because you will leavewith less money than you started withis inane, because people go to Disneyland for entertainment, and know theyhave to pay for entertainment. And the first quote is equally inane. Casinogamblers may have irrational ideas about chance and luck, b ut in the U.S.they typically regard it as entertainment wit h a chance of winning, not as aplan to make money. So it’s worth be in g more carefu l and sayingGambling against th e house at a casino and expecting to makemoney is foolish, because the odds are against you and in thelong run you will lose money.The second thing to say is that buying insurance is mathematically similarto placing an unfavorable bet – your expected gain in negative, because theinsurance company needs to cover its costs and make a profit. But bu yi n ginsurance is oft en sensible on grounds of utility theory (yyy ref discussion inother chapter), which we sh all negl ec t in thi s lectu r e.4.2 Favorable games and the long termNotions of long term versus short term play an important role in investment,so let’s start with a brief discussion. In everyday language, a job which willonly l ast six months is a short term job; someone who has worked for acompany for fifteen years is a long term employee. Joining a softball teamfor a summer is a short term commitment; raising children is a long termcommitment. We judge these matters relative to human lifetime; long termmeans some noticeable fraction of a lifetime.Turning to money matters, consider the differe nc e between simple inter-est and compound interest. The Table compares the value, after increasingnumbers of years, of an initial $1,000 earning 7% interest.year 0 4 8 12 16 20simple interest 1000 1,280 1,560 1,840 2,120 2,400compound interest 1000 1,311 1,718 2,252 2,952 3,870Table 1. Effect of 7% interest, compounded annually.4.3. THE IID MODEL AND THE KELLY CRITERION 37One of several possible noti ons of long term in financial matters is “thetime span over which compoun d in g has a noticeab l e effect”. Rather arbi-trarily interpreting “noticeable effect” as “10% more” and taking the 7%interest rate, this suggests taking 8 years as the cut-off for long term.Beingabout 10% of a human lifetime, this fortuitously matches reasonably well the“noticeable fraction of a lif et i me ” criterion above. And indeed in matte r spertaining to individuals, financi al or otherwise, most writers use a cut-offbetween 5 and 10 years for “long term”.Aside: the one fact from freshman calculus of substantial relevance to yourpersonal life is the inequality1+ρ(ert− 1) >eρrt.This shows the value of unit investment, with interest rate r and tax rate 1 −ρ,isgreater when tax is deferred until the sale time t than if tax i s paid as the interestis earned.The the me of this section is t h e nature of compounding when gains andlosses are unpredictable. The relevant arithmetic is multiplication notaddition: a 20% gain followed by a 20% loss combine to a 4% loss, because1.2 × 0.8=0.96.In class I introduce the topic as follows. Suppose you invest $1,000 todayin the stock market, more precisely in the S&P500 index (v i a a fund wit hvery low expenses). What do you guess the investment will be worth in 10years? In 2011 t h e stu de nt guesses1were$500, $800, $1, 200, $1, 800, $1, 800, $2, 400.I have a deck of cards on which are pasted the annual total returnsof the S&P500 index over each of the 52 years 1956 th rou gh 2007. I say“let’s suppose the annual returns over the next ten years are statisticallylike random years from the past; we can track our hypothetical invest mentvalue over the next ten years by shuffling and dealing ten cards”. Doingthis once i n the 2008 class, the hypothetical investment grew from $1,000 to$1,839.4.3 The IID model and the Kelly crite ri onLet us make explicit the type of model used implicitly above. A “re t ur n ”x =0.2 or x = −0.2 in a year means a 20% gain or a 20% loss.1on September 6, 2011; the index closed at 1,165. In 2008 (onSeptember 17, 2008; the index closed at 1,156) the guesses were$800, $1, 000, $1, 600, $1, 700, $1, 800, $2, 200, $2, 500.38CHAPTER 4. STOCK MARK E T INVESTMENT, AS GAMBLING ON A FAVORABLE GAMEThe IID model. Write Xifor the return in year i. Suppose the (Xi) areIID random variables. Then the value Ynof your investment at the end ofyear n isYn= Y0n�i=1(1 + Xi) (4.1)where Y0is your initial investment.To analyze this model we take logs and divide by n:n−1log Yn= n−1log Y0+ n−1n�i=1log(1 + Xi)and the law of large numbers says that as n →∞the right side converge s toE log(1+X). We want to compare this to an investment with a non-randomreturn of r. For such an investment (interest rate r, compounde d annually)we would have Yn= Y0(1 + r)nand therefore n−1log Yn→ log(1 + r).Matching the two cases gives th e conclu si onIn the IID model, the long term growth rate isexp(E l og(1 + X)) − 1. (4.2)The formula looks strange, because to compare with the IID annual modelwe are working wi t h