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stock market investing for individuals

stock market investing for individuals 45
Lecture 2: The Kelly criterion for favorable games: stock market investing for individuals David Aldous January 25, 2016Most adults drive/own a car Few adults work in the auto industry. By analogy Most (middle class) adults will have savings/investments Few adults work in the banking/ nance industries. In my list of 100 contexts where we perceive chance (22) Risk and reward in equity ownership refers to the investor's viewpoint; (81) Shortterm uctuations of equity prices, exchange rates etc refers to the nance professional's viewpoint.Over the last 30 years there has been a huge increase in the use of sophisticated math models/algorithms in nance, and many \mathematical sciences" majors seek to go into careers in nance. There are many introductory textbooks, such as Capinski Zastawniak Mathematics for Finance: An Introduction to Financial Engineering. and college courses such as IEOR221. But these represent rst steps toward a professional career; not really relevant to today's lecture. Today's topic is \the stock market from the typical investor's viewpoint". Best treatment is Malkiel's classic book A Random Walk Down Wall Street. But instead of summarizing that book, I will focus on one aspect, and do a little math.There are many di erent theories/viewpoints about the stock market, none of which is the whole truth. So don't believe that anything you read is the whole truth. A conceptual academic view: nancial markets are mostly about moving risk from those who don't want it to those who are willing to be paid to take over the risk. The rationalist view: today's stock price re ects consensus discounted future pro ts, plus a risk premium. This \explains" randomness mathematically (martingale theory). Most math theory starts by assuming some oversimpli ed random model without wondering how randomness arises. Many \psychological" theories say stock prices can stay out of alignment with \true value" for many years cycles of sectors becoming fashionable/unfashionable. \Fundamental analysis" see the Decal course Introduction to Fundamental Investing seeks to assess \true value" better than the market. The ecient market hypothesis says this is not practical. The \just a casino" view emphasizes the fact than most trading is from one owner of existing shares to another, rather than raising new capital for a business to start or grow.Our starting point in this lecture . . . . . . The future behavior of the stock market will be statistically similar to the past behavior in some respects but will be di erent in other respects and we can't tell which. This is true but not helpful So go in one of two directions. 1 Devise your investment strategy under the assumption that \the future will be statistically similar to the past", recognizing this isn't exactly true. 2 Decide (by yourself or advice from others) to believe that the future will be di erent from what the market consensus implies in certain speci c ways, and base your strategy on that belief. Wall St makes money mostly by (2), selling advice or speculating with their own money.I'm not going to discuss whether you should invest in the stock market at all. If you choose to do so, here's the academic viewpoint. The default choice is (something like) a SP index fund, available with very low expenses. As a matter of logic, because most investments are made via professional managers, their average gross return must be about the same as the market average, so the actual return to an individual investor must be on average be less than the market average, because managers charge fees and expenses. There's overwhelming empirical evidence (next slide, and course project: survey the literature) that individual investors on average do even worse, typically by going in and out of the market, or switching investments.In comparing our default (index fund) with more sensible possible alternatives, the issue is measuring risk and reward in the stock market. Here is the \anchor" data for this lecture. show IFA page Implicit in the gure is that we measure reward = longterm growth rate risk = SD of percentage change each year.Here is a rst issue that arises. it is very hard to pin down a credible and useful number for the historical longterm average growth rate of stock market investments. Over the 51 years 19652015 the total return (including dividends) from the SP500 index rose at (geometric) average rate 9.74 from IFA let's check another source. Aside from the (rather minor) point that we are using a particular index to represent the market what could possibly be wrong with using this gure Well, it ignores expenses it is sensitive to choice of start and end dates; starting in 1950 would make the gure noticeably higher, whereas ending in 2009 would make it noticeably lower. to interpret the gure we need to compare it to some alternative investment, by convention some \riskfree" investment. it ignores in ation it ignores taxes.Here are two graphs which give very di erent impressions of longterm stock market performance. show SP index click on max show in ationadjusted SPIn comparing our default (index fund) with more sensible possible alternatives, the issue is measuring risk and reward in the stock market. Regarding reward, as said above it is very hard to pin down a credible and useful number for the historical longterm average growth rate of stock market investments. But a typical conclusion is Over the very long run, the stock market has had an in ationadjusted annualized return rate of between six and seven percent. It is not obvious how to measure assign a numerical value to the risk of an investment strategy. If I lend you 100 today and you promise to pay me back 200 in 10 years then my \reward" is 7.0 growth rate but my \risk" is that you don't pay me back and I don't know that probability. A convention in the stock market context is to interpret risk as variability, and measure it as SD of annual percentage returns.IFA and similar sites start out by trying to assess the individual's subjective risk tolerance using a questionnaire. The site then suggests one of a range of 21 portfolios, represented on the slightly curved line in the gure. The horizontal axis shows standard deviation of annual return, (3 to 16), and the vertical axis shows mean annual return (6 to 13). Of course this must be historical data, in this case over the last 50 years. Notwithstanding the standard \past performance does not guarantee future results" legal disclaimer, the intended implication is that it is reasonable to expect similar performance in future. Questions: (a) How does this relate to any theory (b) Should you believe this predicts the actual future if you invested (c) Does the reward/risk curve continue upwards further Answers: (a) The Kelly criterion says something like this curve must happen for di erent \good" investment strategies. (b) Even if future is statistically similar to past, any algorithm will \over t" and be less accurate at predicting the future than the past. (c) No.Expectation and gambling. Recalling some basic mathematical setup, write P() for probability and E for expectation. Regarding gambling, any bet has (to the gambler) some random pro t X (a loss being a negative pro t), and we say that an available bet is (to the gambler) favorable ifEX 0 unfavorable ifEX 0 and fair ifEX = 0. Note the word fair here has a speci c meaning. In everyday language, the rules of team sports are fair in the sense of being the same for both teams, so the better team is more likely to win. For 1 unit bet on team B, that is a bet where you gain some amount b units if B wins but lose the 1 unit if B loses, Epro t = bp (1 p); p =P(B wins) and so to make the bet is fair we must have b = (1 p)=p. (Confusingly, mathematicians sometimes say \fair game" to mean each player has chance 1=2 to win, but this is sloppy language).Several issues hidden beneath this terminology should be noted. Outside of games we usually don't know probabilities, so we may not know whether a bet is favorable, aside from the common sense principle that most bets o ered to us will be unfavorable to us. One of these days in your travels, a guy is going to show you a brandnew deck of cards on which the seal is not yet broken. Then this guy is going to o er to bet you that he can make the jack of spades jump out of this brandnew deck of cards and squirt cider in your ear. But, son, do not accept this bet, because as sure as you stand there, you're going to wind up with an ear full of cider. spoken by Sky Masterson in Guys and Dolls, 1955.The terminology (fair, favorable, unfavorable) comes from the law of large numbers fact that if one could repeat the same bet with the same stake independently, then in the long run one would make money on a favorable bet but lose money on an unfavorable bet. Such \long run" arguments ignore the issues of (rational or irrational) risk aversion and utility theory, which will be discussed later. In essence, we are imagining settings where your possible gains or losses are small, in your own perception.Unfavorable bets. Roughly speaking, there are two contexts in which we often encounter unfavorable bets. One concerns most activities we call gambling, e.g. at a casino, and the other concerns insurance. Regarding the former, mathematicians often say ridiculous things such as Gambling against the house at a casino is foolish, because the odds are against you and in the long run you will lose money. What's wrong is the because. Saying Spending a day at Disneyland is foolish, because you will leave with less money than you started with is ridiculous, because people go to Disneyland for entertainment, and know they have to pay for entertainment. And the rst quote is equally ridiculous. Casino gamblers may have irrational ideas about chance and luck, but in the U.S. they typically regard it as entertainment with a chance of winning, not as a plan to make money. So it's worth being more careful and sayingGambling against the house at a casino and expecting to make money is foolish, because the odds are against you and in the long run you will lose money. The second context is that buying insurance is mathematically similar to placing an unfavorable bet your expected gain in negative, because the insurance company wants to cover its costs and make a pro t. But the whole point of buying insurance is risk aversion, so this needs to be treated in the setting of utility theory and psychology of probability (a later Lecture).So where can I nd a favorable bet The wiseacre answer \start your own casino or insurance company" is not so practical, but a variant of the latter is. For those who can, following the advice increase your insurance deductibles to the maximum you can comfortably a ord to lose is a favorable bet, likely to save you money over a lifetime. In this lecture we consider investing in the stock market as mathematically similar to making a sequence of favorable bets (and letting your winnings ride). Exactly why one could consider this a favorable bet could be debated endlessly standard economic theory asserts that investors need to be rewarded for taking risk rather than using alternative riskfree investments, while empiricists observe that, in countries without anticapitalist revolutions, the historical performance of stock markets actually has been better than those alternatives.long term versus short term. In everyday language, a job which will only last six months is a short term job; someone who has worked for a company for fteen years is a long term employee. Joining a softball team for a summer is a short term commitment; raising children is a long term commitment. We judge these matters relative to human lifetime; long term means some noticeable fraction of a lifetime. Table: E ect of 7 interest, compounded annually. year 0 4 8 12 16 20 simple interest 1000 1,280 1,560 1,840 2,120 2,400 compound interest 1000 1,311 1,718 2,252 2,952 3,870 One of several possible notions of long term in nancial matters is \the time span over which compounding has a noticeable e ect". Rather arbitrarily interpreting \noticeable e ect" as \10 more" and taking the 7 interest rate, this suggests taking 8 years as the cuto for long term. Being about 10 of a human lifetime, this fortuitously matches reasonably well the \noticeable fraction of a lifetime" criterion above. And indeed in matters pertaining to individuals, nancial or otherwise, most writers use a cuto between 5 and 10 years for \long term".The mathematical theme of this lecture is the nature of compounding when gains and losses are unpredictable. The relevant arithmetic is multiplication not addition: a 20 gain followed by a 20 loss combine to a 4 loss, because 1:2 0:8 = 0:96. Let's move on to some mathematics . . . . . .First let us make explicit the type of model used implicitly above. A \return" x = 0:2 or x =0:2 in a year means a 20 gain or a 20 loss. The IID model. Write X for the return in year i. Suppose the (X ) are IID random i i variables. Then the value Y of your investment at the end of year n is n n Y Y = Y (1 + X ) (1) n 0 i i=1 where Y is your initial investment. 0 Conceptually, we are assuming the future is statistically the same as the past, and assuming independence over di erent time periods.To analyze this model we take logs and divide by n: n X 1 1 1 n log Y = n log Y + n log(1 + X ) n 0 i i=1 and the law of large numbers says that as n1 the right side converges toElog(1 + X ). We want to compare this to an investment with a nonrandom return of r. For such an investment (interest rate r, n compounded annually) we would have Y = Y (1 + r) and therefore n 0 1 n log Y log(1 + r). Matching the two cases gives the conclusion n In the IID model, the long term growth rate is exp(Elog(1 + X )) 1: The formula looks strange, because to compare with the IID annual model we are working with the equivalent \compounded annually" interest rate. It is mathematically nicer to use instead the \compounded instantaneously" interest rate, which becomes just Elog(1 + X ).The main impact of this result is that what matters \in the long term" about the random return X is not precisely the meanEX , but rather its \multiplicative" analog Elog(1 + X ). Let us note, but set aside for a while, the points Is the model realistic for stock market investing The phrase long term here refers to the applicability of the law of averages as an approximation to nite time behavior this is a third meaning of the phrase, logically quite distinct from the two previous meanings. Instead we focus on the conceptual point is that there are many investment possibilities, that is ways to allocate money to di erent risky or safe assets. Write for a portfolio, that is a way of investing given proportions of your fortune in di erent assets. We can now jump to our rst key mathematical point.The Kelly Criterion. Suppose you have a range of possible investment portfolios , which will produce return X , with known distribution. Then (assuming the IID model) the portfolio that maximizes the long term growth rate is the portfolio that maximizes Elog(1 + X ). So choose that portfolio. The math here is just STAT134, but the implications are rather subtle, as we will see from playing with some hypothetical, very simple models. Mathematics of the Kelly criterion: one risky and one safe asset. Suppose there is both a \risky" (random return) asset (a \stock", more realistically a SP500 index fund) and a riskfree alternative investment (a \bond") that pays a xed interest rate r.Suppose we choose some number 0 p 1 and at the start of each year we invest a proportion p of our total \investment portfolio" in the stock market, and the remaining proportion 1 p in the bond. In this case our return in a year is  X = pX + (1 p)r where X is the return on the stock. The long term growth rate is now a function of p: in the continuous setting growth(p) =Elog(1 + pX + (1 p)r): (2) The Kelly criterion says: choose p to maximize growth(p). Let's see two examples. In the rst X is large, and we end up with p small; in the second X will be small, and we end up with large p. In these two examples we take the time unit to be 1 day instead of 1 year (which doesn't a ect math formulas).Example: pure gambling. Imagine a hypothetical bet which is slightly favorable. Suppose each day we can place a bet of any size s; we will either gain s (with probability 0:5 +) or lose s (with probability 0:5), independently for di erent days (here  is assumed small). Take r = 0 for the moment. What proportion p of our portfolio do we want to bet each daygrowth rate 2 2 p 2 4 Here, for small , 1 1 Elog(1 + pX ) = ( +) log(1 + p) + ( ) log(1 p) 2 2 2 2 1 1  ( +)(p p =2) + ( )(p p =2) 2 2 2 = 2p p =2: Thus the asymptotic growth rate is approximately the quadratic function of p 2 G (p) = 2p p =2 (3) shown in the Figure. The Kelly criterion says to choose p 2 and then 2 your long term growth rate will be 2 .Now recall that we simpli ed by taking r = 0; when r 0, the fact that a proportion 1 p 1 of the portfolio not put at risk each day can earn 2 2 interest, brings up the optimal growth rate to r + 2 ; the quantity 2 represents the extra growth one can get by exploiting the favorable gambling opportunity. To give a more concrete mental picture, suppose  = 1. The model matches either of the two following hypothetical scenarios. (a) To attract customers, a casino o ers (once a day) an opportunity to make a roulettetype bet with a 51 chance of winning. (b) You have done a statistical analysis of daytoday correlations in some corner of the stock market and have convinced yourself that a certain strategy (buying a portfolio at the start of a day, and selling it at the end) replicates the kind of favorable bet in (a).2 In either scenario, the quantity 2 = 2=10; 000 is the \extra" long term growth rate available by taking advantage of the risky opportunity. Note this growth rate is much smaller than 2 \expected gain" on one bet. On the other hand we are working \per day", and in the stock market case there are about 250 days in a year, so the growth rate becomes about 5 per year; recalling this is \5 above the riskfree interest rate", it seems a rewarding outcome. But if  were instead 0:5 then the 1 extra growth rate becomes 1 , and (taking into account transaction 4 costs and our work) the strategy hardly seems worth the e ort. Implicit in the Figure (back 2 pages) is a fact that at rst strikes everyone as counterintuitive. The curve goes negative when p increases above approximately 4. So even though it is a favorable game, if you are too greedy then you will lose in the long run This setting was arti cially simple; here is a rst step towards a more realistic setting.Example: what proportion of your portfolio to put into the \stock" As before, suppose there is both a \risky" (random return) asset (a \stock", more realistically a SP500 index fund) and a riskfree alternative investment (a \bond") that pays a xed interest rate r. Now we imagine typical values of the return X as0:2., on a yearly timescale. As before we choose some number 0 p 1 and at the start of each year we invest a proportion p of our total \investment portfolio" in the stock market, and the remaining proportion 1 p in the bond. We know that the long term growth rate is a function of p: growth(p) =Elog(1 + pX + (1 p)r): There is a nicer algebraic way of dealing with the interest rate r. Set  X = r + (1 + r)X  and interpret X = (X r)=(1 + r) as \return relative to interest rate". Then a couple of lines of algebra let us rewrite the formula as  growth(p) = (1 + r) exp(Elog(1 + pX )) 1 (4) and the optimization problem now doesn't involve any r.If we imagine the stock market on a daily timescale and suppose  2 changes X are small, with mean  and variance  , then we can use the series approximation   1  2 log(1 + pX ) pX (pX ) 2 to calculate  1 2 2 2 1 2 2 Elog(1 + pX ) p p ( + ) p p  2 2 2 2 (the latter because  and  are in practice of the same order, so  is of 2 smaller order than  ). So the Kelly criterion says: choose p to maximize 1 2 2 p p  , that is choose 2 2 p == : (5) This is another remarkable formula, and let us discuss some of its mathematical implications.1. The formula is (as it should be) timescale free. That is, writing 2 2  ; ; ; for the means and variances over a day and a day year day year Nday year, then (because compounding has negligible e ect over a year) 2 2 2   N and   N , so we get the same value for = year day year day whether we work in days or years. 2. Even though we introduced the setup by stating that 0 p 1, the model and its analysis make sense outside that range. Economic theory and experience both say that the case  0 doesn't happen (investors are risk averse and so would buy no stock; this would cause the current 2 price of stock to drop), but if it did then the formula p == 0 says that not only should be invest 100 of our wealth in the bond, but also we should \sell short" (i.e. borrow) stock and invest the proceeds in the bond. 3. More interesting is the case p 1. Typical values given for the SP500 index (as noted earlier, stating meaningful historical values is much harder than one might think) are (interestrateadjusted)  = 5:6 and  = 20, in which case the Kelly criterion says to invest a proportion p = 140 of your wealth in the stock market, i.e. to borrow money (at xed interest rate r) and invest your own and the borrowed money in the stock market.What about the notsolong term We started with the multiplicative model, which assumes that returns in di erent time periods are IID. This is not too realistic, but the general idea behind the Kelly criterion works without any such assumption, as we now explain. Going back to basics, the idea to invest successfully in the stock market, you need to know whether the market is going to go up or go down is just wrong. Theory says you just need to know the probability distribution of a future return. So suppose (a very big SUPPOSE, in practice) at the beginning of each year you could correctly assess the probability distribution of the stock return over the coming year, then you can use the Kelly criterion (2) to make your asset allocation. The fact that the distribution, and hence your asset allocation, would be di erent in di erent years doesn't make any di erence this strategy is still optimal for longterm growth.The second key insight from mathematics The numbers for growth rates that come out of the formula of course depend on the distributions of each next year's returns, but there's one aspect which is \universal". In any situation where there are sensible risky investments, following the Kelly strategy means that you accept a shortterm risk which is always of the same format: 40 chance that at some time your wealth will drop to only 40 of what you started with. The magical feature of this formula is that the percents always match: so there is a 10 chance that at some time your wealth will drop to only 10 of what you started with. The math here is from the theory of di usions or stochastic calculus; a little too hard to explain here.x chance that at some time your wealth will drop to only x of what you started with. For an individual investor, it is perfectly OK to be uncomfortable with this level of mediumterm risk and to be be less aggressive (in investment jargon) by using a partial Kelly strategy, that is using some smaller value of p = proportion of your assets invested in stocks than given by the Kelly criterion. Theory predicts you will thereby get slower longterm growth but with less shortterm volatility. A memorable quote The Kelly strategy marks the boundary between aggressive and insane investing.How one might expect this theory (based on assuming known true probability distributions for the future, and on seeking to optimize longterm growth rate) to relate to the actual stock market is not obvious, but one can certainly look at what the actual percentages have been. The next gure shows the historical distribution (based on hypothetical purchase of SP500 index on rst day of each year 19502009 and on subsequent monthly closing data) of the minimum future value of a 100 investment.0 50 60 70 80 90 100 This is obviously very di erent from the \Kelly" prediction of a at histogram over 0; 100. This data is not adjusted for in ation or for comparison with a riskfree investment, and such adjustments (course project) would make the histogram atter, but still not close to the Kelly prediction. We mentioned before that over the historical long term it has been more pro table to borrow to invest more than 100 of your assets in the market. Both observations re ect the fact that the stock market uctuates less than would the fortune of a Kellyoptimizing speculator.Wrap up The math I've shown is still a long way away from explaining the IFA graphic. They have many di erent \asset classes", each represented by an index fund show IFA. We want to construct a portfolio by weighting over each asset class (note we still avoid individual stocks). Any possible portfolio has (historical data) some average and some SD of annual return. We just choose a range of portfolios that maximize average for given SD (this exploits past pattern of correlations as well as averages and SDs). Note that WolframAlpha will do this kind of thing for you show. The standard textbook math is Modern portfolio theory which is essentially the shortterm Normal approximation for price uctuations. The reason I emphasize Kelly instead is that (in principle) you could model \Black Swans" (rare severe shocks) or at least use actual historic annual return distributions rather than assuming logNormal.I'm not going to discuss whether you should invest in the stock market at all. If you choose to do so, here's the academic viewpoint. The default choice is (something like) a SP index fund, available with very low expenses. If you really want to do better then there are three vaguely plausible ways none of which I recommend. Borrow money, if you can nd a low interest rate. Diversi ed Kelleyoptinmized portfolio, like IFA. Long term market timing such as the Shiller PE ratio showFurther reading The two books most related to our \Kelly criterion" are William Poundstone Fortune's Formula: a history of precisely this topic Aaron Brown RedBlooded Risk: The Secret History of Wall Street: his experience of 1980s risk management in nance. A memorable quote from Brown: Kelly enables you to get rich exponentially slowly. Any book by Robert Schiller. For a little more math see the paper Good and bad properties of the Kelly criterion.
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