I was talking to a friend over the weekend and he told me a story about a person he knows who made hundreds of millions of dollars of net worth in his career and then lost it all. I asked my friend how that could happen. He said "he made a lot of risky bets and none of them worked out."

I don't get how anyone could do that to be honest. I don't understand how someone gives Madoff all of their money to manage for them. When someone has very little to lose, I totally get betting it all and going for it. But when you have accumulated a nest egg or more, you must be diversified in your investments and assets. You cannot put all of your eggs in one basket.

Last week on MBA Mondays, we talked about Risk and Return. I made the point that risk and return are correlated. If you want to make higher returns, you must take on higher risk. But you can mitigate that risk by diversification. And this post is about that strategy.

One of the things most everyone learns in business school is portfolio theory (that's a wikipedia link if you want to learn more). Portfolio theory says that you can maximize return and minimize risk by building a portfolio of assets whose returns are not correlated with each other.

Let's use some real life examples. Let's say you have a portfolio of stocks and all of them are tech companies. To some degree, they are all correlated. When the tech bubble blew up in March of 2000, every tech stock went down. So if you had that portfolio, your portfolio went down big. Let's say you have a portfolio that has some tech stocks, some oil stocks, some packaged goods stocks, some real estate, some bonds, and some cash in it. When the tech bubble bursts, you get hit, but your portfolio does not "blow up." That is the power of diversification at work.

I have my own tech bubble story that is similar to that example. When the Gotham Gal and I moved back to NYC in the late 90s, we bought a large piece of real estate in lower manhattan from NYU. We sold a big slug of Yahoo stock that we got in the sale of Geocities to fund the purchase. And then we sold another big slug of Yahoo stock to fund a complete renovation of that real estate. Beyond those two sales, we did not get liquid on most of our internet and tech stocks because our funds were locked up on almost everything else.

When the bubble burst, our net worth dropped 80% to 90%. But it could have dropped 100%. That real estate did not drop in price. It actually increased by 2.5x over the eight years we owned it. That is the power of diversification at work.

Of course, we learned our lesson from that experience. We now have a fairly diversified portfolio of assets that includes venture capital investments, real estate investments, hedge funds, and municipal bonds. I am not suggesting that our mix is a good mix. I suspect we could be much more conservative and more "efficient" with our asset allocation if we hired a professional financial planner to do this work for us.

But this post is not really about our portfolio construction or even about asset allocation. It is about the power of diversification as a risk mitigator. 

Let's talk about diversification in venture capital funds. Making "one off" early stage venture capital investments is a bad idea. The chance that you will pick a winner in early stage venture capital is about one in three. I've said many times on this blog that one third of our investments will not work out at all, one third will work but will not be interesting investments. And all of our returns will come from the one third that actually work out. If you are making "one off" early stage investments and make five or six investments over the course of a few years, you do not have enough diversification. You could easily pick five or six investments and not once get to the one third that work.

We put 21 investments into our 2004 fund and I believe we will put between 20 and 25 investments into our 2008 fund. With that number of investments, we have a good chance of finding one investment that will be good enough to return the entire fund. And we have a good chance of finding another four or five investments that will return the fund again. We can handle a complete wipe out on between five and ten investments and still produce excellent returns. That is how diversification helps to manage risk in an early stage venture portfolio.

So if you are building a portfolio of anything, be it financial assets or anything, make sure to fill it with things that are not too similar and not too correlated with each other. To do otherwise is not prudent.

#MBA Mondays

Comments (Archived):

  1. Fernando Gutierrez

    Don’t you think that diversification depends a lot on what your intentions are? if you pretend to keep what you have you definitely have to do it. But if you don’t have enough (and that is subjetive) maybe you need to take bigger risks (risk/return again!).

    1. fredwilson

      Yup. I said that in the post

      1. Fernando Gutierrez

        You’re right, I read too fast and didn’t get the “very little to lose” part.

  2. Harry DeMott

    Now you’re talking my language.The counter argument is the oft quoted Warren Buffet – who says to “put all your eggs in one basket, and watch that basket really closely.”I actually like the theory that Nassim Nicholas Taleb (of The Black Swan fame) has put forth. He says that he is in essence investing in lottery tickets with his way out of the money put strategy – so in order to fund this – he needs a big slug of fixed income investments that he views as 100% safe (basically US Government securities).VC is no different than this. You are buying “lottery tickets” to some degree – in that you are buying into ventures with pretty asymmetric risk reward characteristics. To some degree, you are also long stock markets – as your ultimate exit is likely to come from a publicly traded strategic investor (who will clamp down hard on acquisitions if their stock is in the dumps) or an IPO.So as a VC investor, you really need a lot of ballast – in the form of heavy fixed income allocations.for me, it has always been the same – get you personal debt down as low as you can go – and keep a portfolio of cash or cash equivalents so that a modest return from them will cover the life you want to lead. Everything else goes into growth opportunities.

    1. fredwilson

      That’s a great strategy harry

    2. John Frankel

      Spot on, Harry. Great article, Fred.

    3. Matt A. Myers

      I only buy lottery tickets when they’re $50+ million now – and that’s tax-free in Canada baby!!

      1. JLM

        So when the odds are the longest and your chances of winning are the slimmest, you invest? Just joshing ya, friend.

        1. Matt A. Myers

          That’s why I never invested in you! Too easy of a return!! ;P

      2. pfkrieg

        So a $3 mm windfall is not enough?I calculated the odds of the lottery when I was in high school. At that time, I calculated that the local state lottery was (from memory) about a 1 in 72mm payoff. Then, I NPV’ed the payoff, which is an annual annuity with total payout of $xx mm. So to get $72 mm in NPC, you really need a jackpot greater than $140 mm. And with a payoff that big, you’re talking about multiple winners for that kind of feeding frenzy.End of the day, it’s a a bad investment. I tell my wife, if you are doing it for entertainment, fine. But each dollar is a worse investment than the last.

        1. Tereza

          At the end of the day, it’s a tax on the poor and the dreamy.

          1. Fernando Gutierrez

            Yeah, it’s amazing how a lot of people set all their hopes in the lottery…

          2. Tereza

            You betcha. Hey Fernando i never did a Haiku about you and I feel bad.I would do one now but i’m pretending to get some work done here!

          3. Fernando Gutierrez

            LOL, don’t worry! you (we) better get things done!

          4. PhilipSugar

            The thing that makes them the worst is that enable people to look at them as a plan to get rich versus actually doing the hard work it takes to get there. That being said I agree with having the state make the profits versus anybody else.

          5. Matt A. Myers

            There’s something to be said for letting people dream though.. even if it’s more futile than not.I play for entertainment.P.S. I stopped playing lotteries awhile ago until they introduced LottoMax. $50 million is the top prize, and last draw there were 45 separate prices of $1,000,000 … so the odds are a bit better. πŸ™‚

        2. Druce

          even if the expected value of a $1 ticket is $2, only millionaires or billionaires should play, because with any less money you will go broke before you hit the jackpothttp://r6.ca/blog/20090522T…

    4. Tereza

      Harry how ‘heavy’ is heavy for fixed income allocations for a VC? What % range?

      1. Harry DeMott

        Understanding that there is a good chance that any money you put into a VC investment you will never see again – heavy is the amount necessary to provide you with the income necessary to live your life the way you want if you lost every bit of your VC investments – and just had to work for a living. (you probably want a little more than this in case you lost all of your investments in VC, equities and anything else considered risky, and at the same time lost your ability to work ) For someone like me – older, 2 kids, house, mortgage, etc… it means I need a lot more to start doing VC investments – than someone who is young single etc… I look at the list of typical angel investors – and they tend to be either older and with one or more large successes behind them – or younger with perhaps a more moderate success behind them – but no real meaningful overhead or responsibilities in life.

        1. Tereza

          That’s a good framework. Thanks Harry.Also makes me think that while seeking angel funding it’s higher probability among those on either end of the age spectrum; less likely if they’re just starting a family and buying a house or paying school or college tuition. Even if they are wealthy relative to me. Ramping up on those expenses naturally will compel some to hold back on the riskier investments until they are in the clear.

    5. Dave Pinsen

      “The counter argument is the oft quoted Warren Buffet – who says to “put all your eggs in one basket, and watch that basket really closely.”Warren Buffett’s aphorisms generally aren’t a good guide to how he actually invests. Decades ago, he might have adhered to them a little more closely (e.g., when he piled into American Express during the salad oil scandal). But he’s usually pretty well diversified in reality.What Buffett says and what Buffett does are often not the same thing.

      1. JLM

        This is not only very true but Buffett’s failings have been of the normal and pedestrian variety.I find his story to be intriguing and I applaud his success and do not for a second suggest that he is not a supremely successful investor but he has made some pretty big and tactically flawed bets — Net Jets jumps to mind.He is the poster child for long term compound interest.

  3. Michael R. Bernstein

    Fred, this seems to point out a fundamental misalignment of interest between the VC (who is diversified across many startups) and the entrepreneur (who is ‘all in’ to one). Under these circumstances, the VC would have a much larger appetite for risk than the entrepreneur (especially the first-time entrepreneur).

    1. Matt A. Myers

      The entrepreneur should have many strategies for their project, possible alternates, and multiple possible exits to spread out their own risk, IMHO.. It’ll increase their own chances of success, and of course that makes it safer for the VC.

      1. giffc

        “Diversification to mitigate risk” does not really apply to early-stage entrepreneurs. You need to be aware of possible pivots and exits for your business, but until you are established, you are far better off trying to do one thing at a time and doing it really well. Then you change/pivot as needed. Trying to do many things to see which one sticks… well, that’s a recipe for disaster IMO.

        1. Matt A. Myers

          Right. I wasn’t trying to imply that you spread yourself out in hopes of hitting a market fit; I don’t think any bootstrapping entrepreneurs could be considered fat. πŸ™‚

        2. TC

          Disagree. Look at Scott’s comment below (Buffet).

    2. Fernando Gutierrez

      But that’s true of every investor, not only VCs.And sometimes the misalignment can be to the other side even with VCs: a serial entrepreneur who already has enough money but hasn’t hit it big yet and doesn’t want to be in the middle third of the portfolio that Fred talks about (those that are simply ok).

    3. RacerRick

      The entrepreneur (usually) isn’t putting in $ and (usually) gets paid a salary in VC backed companies. So it’s not the same kind of risk.

      1. Tereza

        It can take a while (if ever) to get to Series A. Typically in these days of bootstrapping until you bring in that large number of users and this hopefully leads to a Series A, until that point the entrepreneur is by definition putting in his/her money in to fund it, and not taking a salary. This is what angel Investors want to see.I’m sure there is variation in this and would like to hear more, but organizations such as Golden Seeds (an angel group for women) like to see $100k of founders capital invested and two customers secured before they even consider you for the process. Personally to me that feels somewhat counter to the objective of “seed” funding, but that’s how they look at it. And I don’t know about the rest of you but $100k is still a lot of money to me!Anyone founding a startup should count on no salary for a year or possibly two. And when it comes it won’t be a rich one because your value is in equity.

      2. JLM

        The entrepreneur is building a business, a company, to produce some good or service.The traditional investment frame of reference in which diversification is an appropriate and important consideration is not the frame of reference when genius and insanity collide.Successful entrepreneurs are not “diversifying”, they are burning the boats and moving out with a singular purpose to succeed or die in the attempt.Resist the temptation to mix our metaphors. The entrepreneur is not a traditional investor. He is a crazy person and in that craziness is his unique genius.

    4. Druce

      The agent-principal problem is maybe the biggest issue in business – how you align incentives which are inherently to some disagree misaligned.http://en.wikipedia.org/wiki/Principal-agent_pr…One reason big companies tend to not take big risks is it’s hard to structure credible compensation arrangements where management’s best strategy isn’t to perform just well enough to keep their jobs.

      1. fredwilson

        This is a great topic. I’ve learned a lot about getting alignment when the deal structure and economics doesn’t create perfect alignment. Building strong relationships with entrepreneurs is a key part of it

      2. Dave Pinsen

        That’s one reason I like investing in small companies where the founder/CEO owns the lion’s share of the stock.

    5. fredwilson

      Yes that is a great observationIt is one of many reasons why I encourage entrepreneurs to take a little off the table in liquidity offerings when they have that opportunity

  4. Mark Essel

    Every morning I wake up breathing is another opportunity to improve the way I spend my time generating value for my self and my partners.The risk of diversification is not understanding the nuances of many orthogonal markets. What do I know of flux of value in oil, perfumes and and pork bellies? I’m not sure how to judge risk for the variance in value of the dollar.I tend to think of real things keeping their value in times of high market flux. As you mentioned real estate was a solid way for you and GG to diversify after hitting payday with Yahoo stock.What about global diversification?

    1. Dave Pinsen

      There’s definitely logic in investing in faster growing parts of the world, all else equal. I’ve got a bunch of my chips on a little Aussie company levered to growth in China (I’ve paired it with a hedge against a fall off in Chinese growth). But global diversification didn’t offer much protection against systemic risk in ’08. That Aussie stock of mine, for example, dropped 90%. Again, most correlations went to one, as stock markets across the world tanked.

  5. akharris

    Too right, Fred. There’s a real trick, though, in figuring out precisely what assets are uncorrelated to one another from a logical perspective. Without that, you won’t really have a good idea about what will be correlated going forward, you’ll just know what was correlated in the past.That weakness killed a lot of portfolios in 2008. People who thought they had broadly diversified portfolios got smashed when correlations went to 1 between assets that had, historically, been uncorrelated.Past performances is never a good indicator of things to come.

    1. fredwilson

      That’s a good topic for mba mondays. So true

  6. kidmercury

    i hope folks will consider getting into gold as a part of their diversification strategy; gold should be the base currency as it is already the strongest currency in the world and its fundamentals are only getting stronger. to diversify, i recommend silver, gold stocks if you have risk appetite, and oil if you have enough money beyond gold and silver.for those of you who are against gold, i hope you will make your opinions public so i can make fun of you as gold continues to rise. preferably you will get very emotional about this subject as doing so will facilitate my comedic endeavors. thanks.

    1. fredwilson

      I went public with my hate of gold and survived your ‘making fun’ of it

      1. kidmercury

        i regard you as a friend so it is less funny when you dislike gold. maybe you can convince jdawg to hate gold?

      2. kidmercury

        also, we are still in the very early phases of the gold rally. when gold hits 1650, and then 2300, then 3000, then possibly 5,000 or 7,000…..well, at that point, folks will wish they heeded the warning of their good buddy kid mercury. that is also when the kid mercury jokes will really begin — i am just laying the foundation now, as the jokes will be immensely funnier that way.

  7. sigmaalgebra

    Yes, diversification is the wonderful all-time strategy to lower risk. In practice it’s not nearly as good as much better information and tends to need information difficult to get, but, still, it’s nearly magic.The broad idea of ‘diversification’ is to make several investments and, thus, get a more desirable combination of risk and return than possible from any one investment.Note: The referenced Wikipedia article has (there MPT abbreviates ‘modern portfolio theory’):”More technically, MPT models an asset’s return as a normally distributed random variable, …”Sorry, guys, I have to quit reading there: As I show here, don’t need the “normally distributed” (that is, Gaussian) assumption. Yes, at times a Gaussian assumption does enter, and there are ways, essentially based on the central limit theorem, to argue that it is justified, but just for the usual approaches to diversification Gaussian is not needed.With some of the uses of a Gaussian assumption, we can be another Long Term Capital Management (LTCM); it appears that they took Gaussian too seriously and from that believed that their chances of going broke from their high leverage was very small. Taking Gaussian seriously, especially out in the long tails which is essentially what LTCM did, is dangerous.PreliminariesThe mathematical approach is to assume that our investment is part of an ‘experiment’ that we do once but that might be done many times. Much more can be said; this is the ‘modern’ approach with details from A. Kolmogorov — yes, he does a lot in terms of sigma-algebras.A number we get from our one ‘trial’ of our experiment is a ‘random variable’. So, suppose X is a random variable. Then for a number x we can consider the probability that X is <= x and write this as P(X <= x) — this function of x is the ‘cumulative distribution of X’.As can read between the lines in this post, we do believe that a cumulative distribution exists but try to do our work nearly never seeing or using it. Elementary courses can mention a catalog of famous distributions, especially Gaussian, and suggest that in practice we should find distributions and use them directly; here students are being badly misled.What about the ‘expectation’ or ‘average’ of X? Well, that might not exist. Or it might exist and be positive or negative infinity. But the expectation of the random variable the absolute value of X, that is, |X|, will always exist. If E[|X|] is less than infinity, then E[X] will exist. We assume that E[X] exists and is finite.From the cumulative distribution we can calculate E[X]. In practice, instead, we usually estimate E[X] by averaging ‘samples’.The ‘variance’ of X is Var(X) = E[(X – E[X])^2], and the ‘standard deviation’ of X, Std(X), is the square root of the variance. Again, we can calculate Var(X) from the cumulative distribution of X but in practice usually estimate the variance from samples of data.We assume that variance is finite.In investing, the usual measure of ‘return’ is just E[X] or something closely related, and similarly for variance and risk.For random variables X and Y, their ‘covariance’ is Cov(X,Y) = E[(X – E[X])(Y – E[Y]))]. Of course Var(X) = Cov(X,X). It follows, essentially from the Schwarz inequality, that covariance is finite.The ‘correlation’ of X and Y is Cor(X,Y) = Cov(X,Y)/(Std(X)Std(Y)) when we decide that 0/0 = 0 in case we encounter this ratio. It follows that correlation is between -1 and 1, and we can argue that it corresponds to the cosine of an angle that otherwise we don’t define! Right: We are defining an ‘angle’ between two random variables; so, we are implying we have some ‘geometry’ of random variables; yup, we do, but I omit details. Yup, darned near any data in practice can be modeled as random variables, with its geometry, etc.Then Cov(X,Y) = 0 if and only if Cor(X,Y) = 0.If for all real numbers x and y we have that P(X <= x AND Y <= y) = P(X <= x)P(Y <= y), then X and Y are ‘independent’; in that case it follows that Cov(X,Y) = 0.If we know the cumulative distribution of X and that of Y and if knowledge of X does not help us predict the value of Y (beyond what we already know from the cumulative distributions), then X and Y are independent; the proof here is an application of some of the properties of ‘conditional expectation’ that follow from the Radon-Nikodym theorem (with a famous proof by von Neumann); often in practice we can check this criterion from other things we know just intuitively (“Look, Ma, no data!”). Thus in practice the easiest way to conclude Cov(X,Y) = 0 is just to make this intuitive argument that X and Y are independent.Suppose X is the value of stock A in one year and Y is the value of stock B in one year. Suppose we believe that E[X] = E[Y], that Var(X) > Var(Y), and that Cov(X,Y) = 0. So, stock A has more risk than stock B. So should we be willing to pay more for the riskier stock or less?People can differ on this point!Some people will want to pay more today for the riskier stock just because they believe that it may fluctuate more during the next year. So, when the risky stock is up, say, compared with its expected value in one year, sell it. When it’s low in comparison, buy it. This practice is sometimes called ‘pumping’ and can result in higher expected portfolio returns.But, a stock with low risk can justify higher ‘leverage’. So, people can differ.And as we will see, due to covariances, there can be other considerations.So, net, for individual stocks, higher return might not correspond to higher risk; so, risk and return are not necessarily positively correlated.Yes, for a portfolio that we invest in today, hold for, say, one year, and sell, higher return needs higher risk (because in such a simple scenario we do like return and don’t like risk so would never accept higher risk without higher return) so that for such portfolios risk and return will have positive correlation. However, active portfolio managers, even ones following modern portfolio theory, will likely not hold the portfolio for a year but ‘re-balance’ it frequently.Diversification 101We keep this case very simple and illustrate that we can get all the return possible from one investment but, by increasing the number of investments, make risk as small as we please.”Look, Ma, no risk!”.In this case, some people are tempted to apply high ‘leverage’!Suppose for some positive integer n we have n investments. Suppose we wait some interval of time, say, one year, and then sell the investments. Suppose for i = 1, 2, …, n, random variable X(i) is the value of investment i at the end of the year. Suppose, for this simple case, for positive, finite m, and for each i, E[X(i)] = m — that is, all the expectations are the same. Suppose, also for this simple case, for finite a and for each iE[X(i)^2] = a.So, expectations of all the squares are the same. We’re being really careful, here, guys; in particular we are ensuring that we don’t have to subtract one infinity from another, a no-no in such things.Let random variableS = X(1) + X(2) + … + X(n).Then S is the value of the sum of all the investments at the end of the year.Then E[S] = nE[X(1)] = nm.Right: Expectation is a ‘linear operator’ as in Nelson Dunford and Jacob T. Schwartz.So, in expectation, that is, on average, at the end of the year we will have nm.Suppose, again in this simple case, different investments have 0 covariance; this is the key assumption: So, for i, j = 1, 2, …, n with i not j,0 = E[(X(i) – m)(X(j) – m)]= E{X(i)X(j)] – m^2orE{X(i)X(j)] = m^2With these assumptions, what we discover is that the expectation of S grows as n, the variance of S grows as n, and the standard deviation of S grows as the square root of n. So, for large n, the standard deviation of S is as small a fraction of the expectation of S as we please (yes, we assumed that m > 0).Or if we consider S/n, that is, a single, ‘typical’ investment, its standard deviation goes as 1 over the square root of n and, thus, by making n sufficiently large, can be as small as we please.So even if we pick investments where the expected return E[X(1)] = m is really large, with n large enough we can make the risk as small as we please.The derivation is actually easy in full detail:Var(S) = E[(S – E[S])^2]= E[(S – nm)^2]= E[S^2 -2nmS + (nm)^2]= E[S^2] – (nm)^2= E[X(1)S + X(2)S + … + X(n)S] – nn(m^2)= E[X(1)^2] + (n – 1)m^2 +E[X(2)^2] + (n – 1)m^2 + … +E[X(n)^2] + (n – 1)m^2 – nn(m^2)= na + n(n – 1)m^2 – nnm^2= na – nm^2= n(a – m^2)So the expectation is nm while the variance is n(a – m^2). Since standard deviation is the square root of the variance, the standard deviation goes as the square root of n. So, with m > 0, as n grows, the standard deviation of S becomes as small a fraction of the expectation of S as we please. Or, on average each investment has risk as low as we please. Still, the expected return is nm.Note: We have essentially derived the weak law of large numbers.Diversification 102We generalize the diversification above.We assume that for our portfolio we do like return and do not like risk.Suppose for some positive integer n we have n investments to consider. Suppose we want to invest for, say, one year and then sell all the investments. For i = 1, 2, …, n, suppose 1 dollar invested in investment i now will be worth random variable X(i) dollars in one year.We assume that the expectation E[X(i)] exists and that E[X(i)^2] is finite.Suppose the total we invest now is w dollars. Suppose in investment i we invest y(i) dollars. Then, we wanty(1) + y(2) + … + y(n) = wIn one year our investment will be worth random variableS = y(1)X(1) + y(2)X(2) + …, y(n)X(n)For easier notation, we regard y as the n x 1 vector with components (y(1), y(2), …, y(n)). Similarly 1 x n X = (X(1), X(2), …, X(n)).And for the 1 x n transpose of y we write y’.ThenS = y’XWe want return r, that is, E[S] = r.So, we want to find the y to solveminimize Var(S)subject toy(1) + y(2) + … + y(n) = wE[S] = ry(i) >= 0, i = 1, 2, …, nSupposeE[X(i)] = x(i)and regard 1 x n x = (x(1), x(2), …, x(n))The returnr = E[S] = y(1)E[X(1)] + y(2)E[X(2)] + … + + y(n)E[X(n)]= y’xWe recall that matrix multiplication is associative.So,r^2 = (y’x)(y’x)= (y’x)(y’x)’= y'(xx’)yandVar(S) = E[(S – r)^2]= E[S^2] – r^2= E[(y’X)(y’X)] – r^2= E[(y’X)(y’X)’] – r^2= E[y’XX’y] – y'(xx’)y= y'(E[XX’] – xx’]ySo if we let n x nA = E[XX’] – xx’thenVar(S) = y’AySuppose component i, j of A is a(i,j). Thena(i,j) = E[X(i)X(j)] – x(i)x(j)= E[X(i) – x(i))(X(j) – x(j)]which is the covariance of X(i) and X(j).So A is the matrix of covariances.So, we want to find the y to solveminimize Var(S) = y’Aysubject toy(1) + y(2) + … + y(n) = wx’y = ry(i) >= 0, i = 1, 2, …, nOr we will be just as happy withminimize Var(S) = y’Aysubject toy(1) + y(2) + … + y(n) = wx’y >= ry(i) >= 0, i = 1, 2, …, nThen as we increase r, the set of solutions to the constraints shrinks and, then, necessarily, Var(S) = y’Ay will not get smaller and will likely get larger.So, in this case, increasing return for the portfolio likely increases the risk for the portfolio meaning that for the portfolio risk and return have positive correlation (I omit the details of this argument).The set of all such pairs (r, y’Ay) form an ‘efficient frontier’ of all the portfolios we should consider.Easily A’ = A.Next, for any y,y’Ay = y'(E[XX’] – xx’)y= y'(E[(X – x)(X – x)’])y= (E[y'(X – x)y(X – x)’])y= (E[y'(X – x)((X – x)y)’])= E[(y'(X – x))^2] >= 0So A is symmetric, non-negative semi-definite [Halmos] so thaty’Ayis a convex function of y [Fleming]. So, y’Ay is a continuous function of y [Fleming].So, we are trying to minimize a continuous, convex function subject to linear constraints.Easily the set of all y that satisfies the linear constraints is closed and bounded and, thus, compact [Rudin]. Since convex y’Ay is continuous on this set, there exists a solution [Rudin].Considering the Kuhn-Tucker conditions [Mangesarian], a constraint qualification is satisfied at each y that satisfies the constraints. So, the Kuhn-Tucker conditions are necessary and sufficient for a solution.More can be said, but, as here, the basic derivations are simple.Curiously, if we want to make Var(S) large, then the problem is in NP-complete.Here we make use of some of the material in:Paul R. Halmos, ‘Finite-Dimensional Vector Spaces, Second Edition’, D. Van Nostrand Company, Inc., Princeton, New Jersey.Walter Rudin, ‘Principles of Mathematical Analysis, Third Edition’, McGraw-Hill, New York.Wendell H. Fleming, ‘Functions of Several Variables’, Addison-Wesley, Reading, Massachusetts.Olvi L. Mangasarian, ‘Nonlinear Programming’, ISBN 07-039885-2, McGraw-Hill, New York.The first two were at times used as the first two of three main references in Harvard’s Math 55 with a colorful description athttp://www.american.com/arc…If we had been the first to do this derivation, then we would be Harry Markowitz, have a Nobel prize in economics, and be a founder of ‘modern portfolio theory’.Exercise: See how the 102 case solves the 101 case and see what the answer is.Exercise: Take the arguments from the 101 case about reducing risk by increasing n and, with appropriate but meager assumptions, apply them to the 102 case. This exercise may be original research.Easily if we just include some additional investments, then our efficient frontier will be no worse and likely better, that is, give better pairs of risk and reward.We could extend to the capital asset pricing model (CAPM): Here we assume that there is a market with many investors, all with the same information, in particular the stock returns and covariances, and all who like return and do not like risk. Then we get to see what prices the market assigns to the stocks. The idea of an ‘index’ fund, then, is just to take these market prices to the bank.Again, we do not need a Gaussian assumption.

    1. JLM

      OK, so why do economists and mathmeticians live in such ratty houses?

      1. Tereza

        Not the ones that live a town over from me. Oh, wait! They all run hedge funds!

      2. sigmaalgebra

        I will give a straight answer:If want to avoid incompetent mistakes, such as in the Wikipedia piece that need a Gaussian assumption, then about have to work through the details about as I did. There’s no real alternative.For your question, James Simons, Andrew Viterbi, Robert Bixby, Edward O. Thorp, or John von Neumann (long gave nice parties at Princeton). For Rudin, he and his mathematician wife lived in a Frank Lloyd Wright house. Might also consider David Luenberger although I like his ‘Optimization by Vector Space Methods’ better than his ‘Investment Science’.Likely some students of Avellaneda at Courant, Cinlar at Princeton, Karatzas at Columbia, Shreve at CMU, Breiman at Berkeley, and Doob at Illinois made enough money on Wall Street to live in nice houses. Also consider Shiryayev and students. Some of those people actually know what a ‘current of sigma-algebras’ is, along with Brownian motion, a Wiener process, the strong law of large numbers, the central limit theorem, martingales, Ito integration, the Brownian motion solution of the Dirichlet problem, and the martingale convergence theorem, if only because often they have to just to read the first few paragraphs of some relevant research papers.Uh, the Markowitz and Sharpe work has been taken VERY seriously on Wall Street including by a lot of people in Stamford and Greenwich who live in nice houses: E.g., investment advisors and portfolio managers have learned in very clear terms NOT to pay traders based only on return without also considering risk.You have a valid, broad point: Mathematics has a lot of tools that now can buy low and sell high. It’s like bread is $100 a loaf, ovens are cheap, and flour is free. Computer science knows ‘how’ to program but, beyond just what is intuitive, has poor means to know what to program. Some topics in mathematics can provide some powerful means to know what to program.

        1. JLM

          OK, so the Domino’s guy and the Pappa John’s guy — did they make their billions in pizza or math or investing? They built companies around pedestrian and mundane products and earned their billions the old fashioned way by meeting a perceived market demand.Anybody who shorted the market a year before Obamamania did pretty damn well if they closed their positions out promptly. A red or black bet at the end of the day, no?Hedging, wedging, edging, shorting financial instruments does not produce a product and is not really “business”. Derivatives and phony insurance wrecked our economy.That’s all just about redistribution of wealth and yes, some folks have done very, very, very well but they have not produced anything of lasting value to the economy. And they wrecked the economy in the process.The Long Term Capital chaps — the smartest wizards in the world — went broke like a 2-bit whore.The world might have been better served if the internet service of some of those wizards had been terminated.

          1. sigmaalgebra

            For making money, there are lots of ways. I wish I’d done Dominoes: Heck, I was in grad school, ordering pizza carryout, and should have seen the opportunity. But at the time I just didn’t have the ‘entrepreneurial’ insight.Also I wish I’d gotten what I was promised at one startup: I’d be worth about $500 million now. The stock was promised “in two weeks”; 18 months later I’d done my part, saved the company twice, still had no stock, was neglecting my marriage, so went for a Ph.D.My career is to make money; I’m trying. I want it “mostly legal” and …. Well, mostly legal. My project is to exploit some applied math, some original with me, the Internet, and Moore’s law to provide some new information a billion people might like. Actually this project is fully legal, squeaky clean legal. Okay, that and a dime might cover a ten cent cup of coffee.I wanted a career on Wall Street, deliberately studied much of just the right stuff in grad school (from a star student of Cinlar long running the financial engineering program at Princeton), got recruited as I was finishing my Ph.D., but was too busy taking care of my ill wife and didn’t go. I wanted to go and very much wish I had.For what happens on Wall Street, yup, there has been a lot of nonsense. And, yup, one of the worst things that can happen to the world is Wall Street nonsense.That Wall Street doesn’t make anything doesn’t bother me: In a sense, if they do their job ‘right’, maybe can conclude that they work very hard establishing the ‘correct’ prices of things so that no one pays too much. I don’t really believe this, but could argue it.To me, Wall Street is a strange, challenging, and important phenomenon for society and an opportunity for some people. I do wish I was one of the people taking the opportunity.For just how to make the world safe from Wall Street, I don’t really know and don’t have the time now to work on it. Besides, I believe I have better paths to a nice house, yacht, collection of cars, chartered plane to take a pretty woman to ‘La Traviata’, etc.! Besides, instead of trying to fix Wall Street, I’d rather write music and do mathematical physics.I enjoyed writing out the math in such simple terms for diversification and the Markowitz model, in the Markowitz model got some results from the Kuhn-Tucker conditions that look nice (that I omitted), but now need to get back to my project.

          2. Tereza

            OK Sigma, a human angle….now you’re talkin’ a language I understand! (and I have an MBA too) :-)thanks for sharing.

          3. ShanaC

            When you come back, pretty please reccomend a textbook so I can learn stochastic processes. I’ve always wanted to do that. How to I build up????It’s like knowing a matrix is an array in computer programming, and I managed to figure that out without any help. So how do I figure out the next steps???

          4. sigmaalgebra

            My Ph.D. research was in stochastic optimal control.Once I got a nice letter back from Fisher Black (right, Black-Scholes and Goldman Sachs) saying that there were no applications of my background at Goldman Sachs. Guess I caught him on a bad day. Or maybe the market was down. But he was not seriously wrong: Goldman Sachs has made plenty of money without me.For the subject of stochastic processes, especially as needed by (the math sometimes used by) Wall Street, there are some severe problems.The first problem is the list of prerequisites. Basically need a nicely solid undergraduate major in pure mathematics. E.g.,Bernard R. Gelbaum and John M. H. Olmsted, ‘Counterexamples in Analysis’, Holden-Day, San Francisco.needs to be fun reading, where you turn pages faster than eat popcorn at a good movie, and where nearly all the examples are close to obvious.I had those prerequisites and did a lot of applied work in stochastic processes, especially for military problems; that work was important, but I wouldn’t know an easy way for others to do the same.Then need measure theory and functional analysis as in a pure math Master’s program. For this material, likely the standard is the first half, the ‘real’ half, of ‘Papa Rudin’:Walter Rudin, ‘Real and Complex Analysis’, ISBN 07-054232-5, McGraw-Hill, New York.I like this book, but there have been some people at Courant who regard it as too severe. An unguided tour of this book will make an unanesthetized root canal procedure feel good in comparison.ThenJohn C. Oxtoby, ‘Measure and Category: A Survey of the Analogies between Topological and Measure Spaces’, ISBN 3-540-05349-2, Springer-Verlag, Berlin.has to be more fun than eating an elegant dessert to some Tchaikovsky ballet music.The second problem is learning stochastic processes based on that pure math material.Actually, before stochastic processes, need to learn probability based on measure theory as introduced by Kolmogorov. Good authors here are Breiman, Neveu, Chung, and Loeve. This material is NOT commonly taught in US pure math departments. An unguided tour of this material is NOT promising. E.g., in a course I took, the primary reference wasJacques Neveu, ‘Mathematical Foundations of the Calculus of Probability’, Holden-Day, San Francisco.I very much like it: It’s succinct, as math elegant and just gorgeous, but it will be widely regarded as impenetrable. Loeve was long at Berkeley, and Neveu was a Loeve student. Neveu has long been back in France; he’s a bright guy.Of course, Loeve’s main book is hisM. Loeve, ‘Probability Theory, I and II, 4th Edition’, Springer-Verlag, New York.It’s huge. There’s a LOT in it, including on stochastic processes, e.g., the second order stationary material long popular in parts of electronic engineering for military problems. Neveu has the core, plus some, on many fewer pages.Loeve writes English like it was French; I find it just different, but some people find it too difficult.The pure math material is widely available in the US; the applications to probability and stochastic processes are not common in the US and where they are available usually are NOT in a math department; the number of non-math departments where the material is available is TINY. It’s rock solid math but just is not popular in US pure math departments and more popular in France, Russia, and maybe Japan.Uh, trying to get through this material without full attention to fully detailed theorems and proofs will bog down in total nonsense after just a few steps. So, some departments that try to play with this material, maybe in economics or electronic engineering, without the theorems and proofs will make a mess and, then, start to make silly mistakes such as the Gaussian assumption in the Wikipedia article. Uh, in such math, the theorems and proofs are about all there is that is solid for you to make progress and about the only quality control on what you’re reading: Without all the details, serious mistakes are too common.Would also be good to have a solid course in mathematical statistics: That material is even more difficult to get because it is rarely taught in pure math departments, and the number of advanced statistics departments is tiny. Generally statistics as a field needs a LOT of work.It’s important to work with E[X], E[(X – E[X])^2], E[Y|X], etc., but in an application when actually need numerical values usually have to estimate those from data, and that is ‘statistical estimation’. E.g., a severe problem directly applying the Markowitz material is getting good enough estimates for the components of the covariance matrix. The difficulty is so severe that making direct, large scale application of the Markowitz model is nearly a joke. Or the Markowitz model is what we would do if we had a LOT of data we don’t have and, also, don’t have a lot more data we might have.The field of statistics of stochastic processes has long been regarded as wide open for research; this likely means that there isn’t much even on the shelves of the research libraries and still less in courses.Sadly, I listed a major fraction of all the profs in the US should study with: Breiman, Karatzas, Avellaneda, Cinlar. Should also include Varadhan, also at Courant and Chung at Stanford. Dynkin, a Kolmogorov student, long at Cornell, is likely retired. Doob and Halmos both recently died.I can’t recommend that anyone strike out to learn stochastic processes. You would be putting the solution ahead of the problem, i.e., have a solution looking for a problem. I suspect that there are easier ways to make money then learning and then applying stochastic processes. Also, you’d have nearly no one to talk to about what you were doing! And if Fisher Black were still alive, he might say that there were no applications at Goldman Sachs.That doesn’t mean that I think that knowledge of stochastic processes should be ignored: My view is that it stands to be a good part of another Moore’s law for much of the rest of this century. Still, easy it ain’t.

          5. ShanaC

            Ok- So Happily I got about half of the “how to do matrix algrebra” post on my own (with my decent understanding of calculus, no less)It did provide the essential hint to a thought problem which I understood: yet now I understand much better- why arrays in multiple work the way they do, and that there are much better ways to work through them by collapsing them through multiplication and addition.I just like the social sciences, and recently realized that there is a misapplication of math underneath them, so I better go study more math. Which means building up from freshman calculus (which is never taught well, stop letting grad students teach it) into some of the fun stuff.My gut feeling is that from what little I know of stochastic- you would be better able to draw down the proper linear regressions. I keep wondering if we don’t add enough noise to the system- for we are a little random.How do I build up? (And I am totally passing on these textbooks to a friend of mine-Engineer on wallstreet who won’t give up his textbook from college on stochastic processes, it is among his favorite books)

          6. sigmaalgebra

            How do you “build up” and pursue social science?Uh, for how to get past freshman calculus, there are university courses for that!For a book, consider authors Protter and Morrey or Thomas. When I taught calculus, I used Protter and Morrey: It’s a very well written, well balanced, although relatively easy, college calculus book. Yes, the books I learned from were more difficult!I’m big on independent study: E.g., I never really took freshman calculus. As a freshman I got plopped into some math beneath me, had a girl friend who went to class and told me when the tests were, showed up for the tests, made an A, and had the poor prof state that I was “the best math student I’ve ever had”. He wasn’t a very good math prof!So, with the time I saved in his course, I got a calculus book and dug in. As a sophomore I transfered to a much better school, with a surprisingly good math department, was a math major, and took the class on sophomore calculus and did fine.”Look, Ma, skipped freshman calculus!”.I took a reading course in topology, where I gave 1-2 lectures a week, and wrote an honors paper.Nearly all the stuff I learned — sometimes even from when I was in a course — was from ‘independent’ study.When you get far enough, you are supposed to lose patience with courses, and the best grad departments encourage that. And in research and, really, nearly anything in significant applications, you just have to work independently or nearly so.Still, really independent study is not good: Some expert guidance can be from good up to crucial. E.g., when I get back to mathematical physics, I intend to go to research seminars in physics departments.My wife’s Ph.D. was in essentially ‘mathematical sociology’. She learned from two of the guys, both presidents of the ASA, Rossi and Coleman who tried to do such things. There is:”Boldness becomes rarer, the higher the rank.”-Karl Von ClausewitzFor her dissertation she took some Rossi survey data where the people giving answers varied on “rank” and what they did and tried to test this claim.Past freshman calculus, the usual steps are sophomore calculus, abstract algebra, linear algebra, ‘Baby Rudin’, and then a wide range from algebra, analysis, topology, applications, etc.Abstract algebra: Go back and do arithmetic again but do everything as theorems and proofs. Learn about equivalence relations — the data base people like relations, and the social networking people might like equivalence relations since they generate partitions.Generalize simple arithmetic to groups, rings, fields, Galois theory, vector spaces, likely some linear algebra and matrix theory, and maybe more — adjoints, reflexivity. Maybe do some category theory (some computer science people think that they like it).Learn about countable and uncountable infinity. For the natural numbers, justify ‘proof by induction’ (the main technique in proving programs correct). Construct the rational numbers and show that they are countably infinite. Construct the real numbers, show that they are uncountably infinite, and prove that they are ‘complete’ — “calculus is the elementary consequences of the completeness property of the real number system”. Maybe construct the ordinals and do transfinite induction and the axiom of choice. Maybe get a lecture on model theory.A major point of such a course is to get some real progress in creating and writing correct proofs. The subject is simple enough that the proofs are not very difficult.Linear Algebra: Yes, do the axioms of vector spaces (where we do a lot of our work in applications), study linearity and maybe kernels and quotients, of course do elementary row operations and Gauss elimination, the Hamilton-Cayley theorem, eigenvalues/vectors and inner products, the Schwarz inequality, the Gram-Schmidt process, and orthogonal, unitary, symmetric, and Hermitian matrices, spectral theory, and the polar decomposition.The classic text remains Halmos.Maybe do some numerical linear algebra and maybe the simplex algorithm.Baby Rudin. That is,Walter Rudin, ‘Principles of Mathematical Analysis’, McGraw-Hill, New York.So do calculus again and this time prove everything. Cover metric spaces, separable, second countable, compactness, and continuity and show that the last two are enough for the Riemann integral to exist and also for the continuous functions under the max norm to be complete and, thus, a Banach space. Show that the Riemann integral exists if and only if the function is continuous everywhere except on a set of measure zero.Do some exterior algebra of differential forms, e.g., in case any student wants to do modern versions of relativity theory.Between Halmos, Baby Rudin, and exterior algebra, at least at one time will have covered essentially all of Harvard’s Math 55 as in the URL I gave.Try to get in both classic and modern versions of Stokes and related theorems. Side reading: R. Buck’s ‘Advanced Calculus’.Maybe the prof will also do partial derivatives, gradients, convexity, and at least something on Lagrange multipliers and/or the Kuhn-Tucker conditions, e.g., as for the Markowitz model and, really, quite a lot of now classic mathematical economics.Apply what you’ve learned to some of ordinary differential equations, at least for something simple like planetary motion or A/C circuit theory (e.g., how people used to build electronic oscillators).Take a first cut at Fourier theory (will do it again and better in Papa Rudin using measure theory). In the real world, Fourier theory happens, whether want it to or not, whether Fourier had done it or not. Can’t get away from Fourier effects — spectral lines in physics, X-ray crystallography, holography, overtones of organ notes, tone controls in music systems, digital filtering and the fast Fourier transform, effects of time invariant linear systems, ringing in mechanical and electronic systems, JPG compression, the core of Shannon’s information theory on the maximum about of data can send over a band-limited channel (why you can’t watch movies over a phone line), solving the heat equation, and more.Hopefully do some ‘applied advanced calculus’: The classic is a text by F. Hildebrand at MIT. But for some of the most important material, by far the best source I know is the old, FIRST edition of Apostol’s book — he gives a NICE presentation of the classic stuff on Stokes theorem, etc. in about 20 pages. If don’t mind setting aside getting all wound up in the more modern stuff of manifold theory and exterior algebra, are happy in just three dimensions, and are willing to draw pictures, then it’s NICE and STILL what nearly everyone in physical science and engineering will use instead of the modern stuff.My view is that mathematical sociology and mathematical social science more generally are wide open fields for good work. I’m not saying that the good work is easy, just that it’s needed and essentially fully missing. The first main problem is that the social scientists just do NOT know NEARLY enough math to build a mathematical science.My guess is that a crucial pillar of any such success will be a mature approach to random variables, stochastic processes, and related topics. So, dig into Papa Rudin, Neveu, Breiman, etc. Learn the classic limit theorems, the Poisson process, Markov processes, martingales, ergodic processes, etc.Then hold your nose at the stench and take a course in mathematical statistics. The usual mud hole is Mood, Graybill, Boas.Then take a course in analysis of variance, experimental design. Take a fairly practical course since by then the usual math that material uses will be trivial for you.Uh, understand some of the special power of simple, old cross tabulation — that is, connect it with the L^2 optimality of conditional expectation!Might see what the EE people do, first, with linear systems and, then, non-linear systems. Or, maybe regard a social system you are studying as a ‘black box’ and try to identify what mechanisms are in it. That work might help you avoid reinventing some wheels and see some of where there are some possibilities and some of where there are not.Then take some ‘social theory’ and see if can actually get something mathematical and solid. My guess would be to try to get a grip on ‘causality’ — SUPER tough to do but, with my guess, just crucial for real progress. Yup, somewhere in there you will be enticed by principle components and/or log-linear — don’t be!If you do get some results, then publish about three papers. Then go to grad school for maybe a year, submit your papers as your dissertation, get your Ph.D., and then get back to research, if that’s what you want to do. What grad school? Maybe the one of the prof who was the editor in chief of the journal of your best paper! Uh, some VCs make a big deal of ‘introductions’: In academics there’s a really simple, powerful form of ‘introduction’ — a good published paper!If you can actually do something mathematical and solid for a ‘mathematical social science’ (and I’d want something much better than mathematical economics), then you have a shot at being an academic star. There won’t be many in your field who can read your papers!To do this, likely you will have to be ‘creative’ but also rock solid. Here’s my technique: Start with a free running, fairly wide ranging, guessing machine! Can also use intuition and, when develop some good intuition, should. Then evaluate the guesses with ‘thought experiments’; e.g., check some simple cases and some extreme cases. Begin to see what likely is solidly true, what is likely solidly false, and then the rest. In math, can usually do some little derivations to evaluate some of the guesses. Then you are beginning to see what the possibilities are for something new, powerful, and solid — the usual publication criteria are “new, correct, and significant”.Sometimes follow a hint like pulling on a thread: Sometimes the hints are subtle but, if looked at carefully, are incongruous and, thus, indicative of maybe something important. So, follow the thread CAREFULLY, handling the subtle parts with great care so as not to miss any of the incongruity. At the end of the thread, maybe explain the incongruity and, thus, maybe have discovered something.For math, in the end, you will have a theorem and a proof, and those are solid. If they are also new and significant, they you have some progress in research and maybe more.Even in math, sometimes people have enough good intuition to be quite sure something is true long before anyone has a proof.Exercise: List some famous cases!Exercise: Find and read the A. Wiles description of how he wanders in a dark room, bumps into the furniture, finds where things are, and eventually finds a light switch.Uh, there is a theme in mathematical theories: Set up and use theoretical constructs that can’t actually see with real data! So, don’t insist that all your constructs be tangible in simple senses. Such constructs can be just intermediate in the work but still crucial. For a simple case, given 10 random variables, they have a cumulative distribution as a surface in 11 dimensions, but without some really severe assumptions have no chance of making it tangible; it’s only intermediate, but still it’s crucial.Another theme: When things are a total mess, e.g,, getting an algebraic solution is hopeless, maybe go to some limiting cases; here things can be MUCH simpler and maybe powerful enough for your purposes. For a really simple example, in the diversification discussion, I gave a proof of the weak law of large numbers: I said next to nothing about the distribution of the random variables but, still, in the limit of large numbers of stocks, got a very simple and useful result. Here, too, the distribution of the random variables was an important intermediate construct; I used it; but I never got any data on it and left it intangible; still it was crucial.A theme in mathematical physics: Assume an ideal case, e.g., a perfect sphere, a frictionless ball, a capacitor with infinitely large plates, etc. That is, pick a version of the problem that is not too far from reality and that lets you get some simple results. Then argue that the actual cases of interest in reality are likely close to the ideal cases. E.g., for a lot — of course not everything — that might want to do with the physics of the earth, assuming it is a perfect sphere is fine and MUCH easier than accounting for every nook and cranny all over the planet. For a simple example, in the first approach I took to diversification, I took a really simple, ideal case and got an important result — portfolio standard deviation, that is, risk, goes to zero. Then we are prepared to believe that something quite similar will happen for a realistic case that is close to the ideal one.One broad approach, maybe not super powerful but should be used more, is to find and exploit symmetry. In a sense, sometimes we are given too much and fail to see it’s all just simple symmetry transformations of something simple. So, at times, take out the redundant symmetry and get back to something simpler.Make money with such things? Uh, just now, and likely for a decade or so, a big deal on the Internet is ‘the social Web’, but here nearly everything is just intuitive. If could get some relevant, solid social science, then could blow away the intuitive stuff like Newton’s planetary motion blew away the astrologers, chemistry blew away the alchemists, steam replaced muscle power, etc. Yes, just now there’s money to be made with just some good intuitive stuff — what you want to do about this fact is up to you.Uh, just now advertising is on the way to getting MUCH more ROI from the Internet and the ad targeting that can be done there, but this situation is still clumsy, won’t remain forever, and will be replaced by some much more powerful approach to the basic economic function. Nearly everything serious in ads and their targeting cries out for some solid social science.Uh, I actually never wanted to be a college prof, was only for a short time to help my ill wife recover, and certainly am not now. “Those that can, do; those that can’t teach; ….”. I got to get back to doing.

          7. ShanaC

            Thank you so much. Hmm.email shana dot carp at gmail Over time, I may have questions. And forsome reason, you are right, math is lonely. I don’t always like that aboutmath…

          8. sigmaalgebra

            Actually you asked for some texts in stochastic processes. Okay.I will draw mostly just from my own bookshelf; for those books I keep the text as markup for the mathematical typesetting language TeX by D. Knuth.For nearly all of these books and for the subject itself, (A) the prerequisites for meaningful understanding are severe, (B) in US universities good courses are rare, (C) for a beginner an unguided tour is not promising, and (D) it is not entirely clear just what hiring managers on Wall Street would think of this material.(1) A nice start on the classic limit theorems, martingales, and diffusion is in:Leo Breiman, {it Probability,/} ISBN 0-89871-296-3, SIAM, Philadelphia.I like it in part because it discusses ‘regular conditional probabilities’ that I used in my dissertation. Breiman has long been at Berkeley.Read this along with Neveu.(2) Likely the most important early text in stochastic processes in English is:J. L. Doob, {it Stochastic Processes,/} John Wiley and Sons, New York.This is an early book in Doob’s career and tries to be easy to read and is in places but not in total.(3) Likely the last book in Doob’s career, and not on my bookshelf, is:Joseph L. Doob. ‘Classical potential theory and its probabilistic counterpart’, ISBN-13 978-3540412069, Springer-Verlag, New York.So, in a region with a boundary with values at the boundary, release a Brownian motion particle, and at the point where it first strikes the boundary make the money in the value at that point. That is the same problem as the classic Dirichlet problem in potential theory in physics but now solved via Brownian motion. Yes, part of the connection with investing is that hitting the boundary is the ‘exit’.(4) Two of the best places in the US for stochastic processes have long been Stanford and Cornell, and from there are two books that may have the best elementary introduction:Samuel Karlin and Howard M. Taylor, {it A First Course in Stochastic Processes, Second Edition,/} ISBN 0-12-398552-8, Academic Press, New York.Samuel Karlin and Howard M. Taylor, {it A Second Course in Stochastic Processes,/} ISBN 0-12-398650-8, Academic Press, New York.(5) Also trying to be elementary is:Erhan c Cinlar, {it Introduction to Stochastic Processes,/} ISBN 0-13-498089-1, Prentice-Hall, Englewood Cliffs, NJ.The book has a lot of “implicit monotone class arguments” (a standard argument in the measure theory approach to probability and crucial for it) that are likely confusing to beginning students, that is, frequent places where a student will ask “How did he get from that line to the next one?” with no good answer.(6) Likely the most comprehensive and authoritative single stack of books is the three volumes:I. I. Gihman and A. V. Skorohod, {it The Theory of Stochastic Processes/} ISBN 0-387-06573-3, Springer-Verlag, New York.Yup, the palace of capitalism draws its math from the old Soviet Union!(7) Maybe in second place are the two volumes of:R. S. Lipster and A. N. Shiryayev, {it Statistics of Random Processes,/} ISBN 0-387-90226-0, Springer-Verlag, New York.(8) Likely the best single text for a good background in the mathematics for ‘continuous time finance’ is:Ioannis Karatzas and Steven E. Shreve, {it Brownian Motion and Stochastic Calculus, Second Edition,/} ISBN 0-387-97655-8, Springer-Verlag, New York.Karatzas is at Columbia and Shreve is at CMU and has been running a Master’s program in ‘computational finance’ or some such.(9) To read along with Karatzas and Shreve would beK. L. Chung and R. J. Williams, {it Introduction to Stochastic Integration, Second Edition,/} ISBN 0-8176-3386-3, Birkha”user, Boston.Chung has long been a pillar of stochastic processes at Stanford.(10) Of course there isAlbert N. Shiryaev, {it Essentials of Stochastic Finance: Facts, Models, Theory,/} ISBN 981-02-3605-0, World Scientific Publishing, New Jersey.(11) Classic, and in part from Courant, is:K. It^o and H. P. McKean, Jr., {it Diffusion Processes and their Sample Paths, Second Printing, Corrected,/} ISBN 0-387-03302-5, Springer-Verlag, New York.Ito integration is one of the main topics in stochastic differential equations.(12) Central topics include Markov processes, Brownian motion, and diffusion, and for these three considerKai Lai Chung, {it Lectures from Markov Processes to Brownian Motion,/} ISBN 0-387-90618-5, Springer-Verlag, New York.David Freedman, {it Brownian Motion and Diffusion,/} ISBN 0-387-90805-6, Springer-Verlag, New York.Freedman has long been doing statistics at Berkeley and recently died.(13) It may be that one of the best ways to make money on Wall Street is to write better automatic trading software. Likely the first approach people will consider is second order stationary processes as in, say,Yu. A. Rozanov, {it Stationary Random Processes,/} Holden-Day, San Francisco.(14) For connections with the statistics of such processes, consider:David R. Brillinger, {it Time Series Analysis: Data Analysis and Theory, Expanded Edition,/} ISBN 0-8162-1150-7, Holden-Day, San Francisco.He was a Tukey student at Princeton and has long been at Berkeley.Making MoneyGetting good with stochastic processes as in this list of texts could take a major effort in life, but good courses from experts could considerably shorten the effort. Mostly the people with such expertise have pursued focused Ph.D. degrees at likely Courant, Columbia, Princeton, Cornell, Stanford, or Berkeley and at the time intended to pursue careers as research professors in academics.My guess is that few people on Wall Street hiring would have such backgrounds or know how to evaluate someone with such a background.So, if hired, the situation would be a technical expert working for a much less technical person, and commonly this situation is not good for the technical expert. Supposedly lawyers responded to this situation with a point of professional ethics that a lawyer must work as a lawyer only for a lawyer. Also what physicians do about this problem might help. Here, however, maybe the technical person could ‘learn the business’ and then start their own business, say, a hedge fund.One of the mathematicians who has made the most money on Wall Street has been James Simons. As I recall, he at least once remarked that he didn’t really use mathematics in making money.Maybe there is an auto mechanic who, working late one night, gets visited by an Auto Fairy who gives him a marvelous tool box with tools (that he must keep secret!) that repair cars unaided. Nice. But, at least first cut, the business is still auto mechanics.Maybe knowledge of stochastic processes can help for making money on Wall Street or elsewhere, might in some circumstances even be a magic toolbox, but to make money likely still have to be in business. Also likely no one else will have any faith in the mathematics and, instead, judge based just on results. Yes, such judgment can also long give high praise to Madoff, but that’s just part of the reality in business.There is an old recipe for rabbit stew that starts out, “First catch a rabbit.”. Well to make money with applied mathematics, likely somewhere need to find an application.A movie starts out:”The early history of America is a tale of great first times.”Making a lot of money on Wall Street where, say, study of Karatzas and Shreve was important, might be such a first time.For such “first times”, typically there won’t be many people going your way or even praising your effort. The lesson in part goes back just to kindergarten and the Mother Goose “Little Red Hen”.That nearly no one else ‘believes’ until the loaf is out of the oven is a problem, the flip side of which is some of the opportunity: If many people understood stochastic processes, then making money with it would be more difficult.But, here’s a point: This mathematics comes complete with rock solid theorems and proofs. That’s most of the most solid information in civilization. Some of the material in the texts I’ve listed has to be counted as among the top, center crown jewels of civilization.Mathematics doesn’t give ‘truth’; instead, if you bring assumptions, then maybe mathematics can use those as raw material and deliver conclusions. So, if you are going to try to get from A to B in something, e.g., business, then maybe with some assumptions that from all you can tell DO look reasonable, you can get from A to D, maybe mathematics gets you to E, and maybe E is really close to B. So you have made part of your trip faster and more reliable. Everyone else is at best floundering around with “Do I need a Gaussian assumption”, and you are way, WAY past that. What I’m describing should be among the more powerful advantages.Example: Various unpredictable, exogenous things happened that ruined some of my planning so that for a while I was supporting myself and my wife through our Ph.D. degrees by doing some military applied math and computing. The Navy wanted to know what would happen to the SSBN fleet in a special, controversial scenario of global nuclear war limited to sea and wanted the answer in two weeks. Fine: My wife had a vacation planned for us starting the day after the due date! People floundered around. I had some ideas, but no one had much confidence so pursued some other approaches in parallel. Uh, there’s an old WWII Koopman’s report OEG-56 on looking for submarines that has some slightly crude but cute math. I borrowed from that and got a finite state space, continuous time Markov process, wrote some corresponding software, and delivered some solid software and some results on-time. Then people believed.I didn’t believe: What I did was crude. But it was many steps ahead of everything else.So, one way to have success is not to be perfect or even very good but just much better than anything else: I did it in that SSBN case with some applied math — actually I make good use of Cinlar. The applied math was a crucial, powerful advantage, really, just blew away all the other efforts on the problem.Net, for making money knowledge of stochastic processes, and applicable mathematics more generally, can be an advantage, maybe a very valuable, even crucial, advantage, but is likely not the whole story.

          9. PhilipSugar

            This is the point I was trying to make when I said trying to make money without a purpose causes large distortions.Again making money is not a bad thing what so ever.

    2. ShanaC

      Now that was brilliant! I just need an overview of how to do matrices and how it relates to statistics (actually I need to get past single variable calculus, however this makes sense)This is just which areas under a curve being multiplied with each other in very specific ways in very specific formulations (because of the matrices) Brilliant! Now how do I do that?And what is up with the notation, why do the matrices not look square- that took me a while to understand….

      1. sigmaalgebra

        Sorry I omitted matrix algebra. I was trying to be short!Here’s matrix algebra:In high school you likely sawax + by = ucx + dy = vSo you were given numerical values for everything but x and y and wanted to find x and y. Depending on the given numbers, there will be none, one, or infinitely many solutions.So, on the left, just rip out the x and y and get the 2 x 2 ‘matrix’a bc dCall this matrix A.Then also consider the 2 x 1 (2 rows, 1 column) matricesxycall it wanduvcall it z.Now DEFINE ‘matrix product’ Aw so that Aw = z is just the same as the two equations in two unknowns.So, so far, all there is is just notation. It’s just shorter notation for the two equations in two unknowns.Next, for positive integers m, n, and p, consider m x n matrix A and n x p matrix B. So, A has m rows and n columns, and B has n rows and p columns. Typically we write A = [a(i,j)] where i = 1, 2, …, n and j = 1, 2, …, m and the number in row i and column j of A is just a(i,j). Typically we write the i and j as subscripts but in computer software we return to what I wrote! Similarly we write B = [b(i,j)] where here i = 1, 2, …, n and j = 1, 2, …, p.Then we want to be clear on the matrix product AB. So, this will be m x p C = [c(i,j)] = AB. How to do that? For column j of C, take column j of B and multiply it by A just as in our 2 x 2 example. Or pick up column j of B, rotate it 90 degrees counter clockwise, overlay it on row i of A, multiply the pairs that are on top of each other, and add the products, and that’s the number in row i and column j of the product. Or in full detail:c(i,j) = a(i,1)b(1,j) + a(i,2)b(2,j) + … + a(i,n)b(n,j)So, are multiplying pairs of numbers and adding the result. This arithmetic is called an ‘inner product’, has a lot of profound and powerful properties, and is so common in scientific and engineering computing that it’s one of the first enhancements in a ‘super computer’ ‘vector’ instruction set.”Look, Ma, just one instruction!”.For a short explanation, that should be clear enough on matrix product.Suppose x is n x 1. The set of all such x is called the n-dimensional vector space R^n. Here we think of R as the set of real numbers, that is, the numbers on the line and used in single variable calculus. Yup, we live in R^3 (at least locally).If matrix A is m x n, x is n x 1, y is m x 1, and Ax = y, then we can think of this as a function from R^n to R^m. It is the most important such function. Or if we want to call the function f, we would have f(x) = Ax = y.Or we can think of Ax = y as m equations in the n unknowns x.Turns out that the two most important pillars of ‘mathematical analysis’, e.g., calculus, etc., are linearity and continuity. Continuity is the easiest way to know that the integral of single variable calculus exists. The best example of linearity is just Ax = y.Why is this thing ‘linear’? Because easily enough for numbers s and t and for n x 1 u and v, we haveA(su + tv) = s(Au) + t(Av)Right: How to add these things? The obvious way, just add corresponding components. Now you know how to add vectors and matrices.Uh, what is su? Simplest possible: Take each of the n components of u and multiply it by the number s. Now you know how to multiply a vector by a number.Now you know matrix algebra and what linearity is.Why is linearity important? A huge fraction of physical systems are linear in this sense, and the algebra of linearity, that is, that the two sides ofA(su + tv) = s(Au) + t(Av)are the same, often lets us get good things for free.Also for things that are not linear, the standard first-cut approximation (e.g., as in Taylor series in calculus) is just linear.If we have matrix products A(BC) then necessarilyA(BC) = (AB)CSo matrix product is ‘associative’. Often cute; it cleaned out lots of thrashing around in my Markowitz derivation.If have a function f: R^n –> R^m, that is, f(x) = y where x is in R^n and y is in R^m, then want to define the ‘derivative’ of f as an m x n matrix, say, A. Then the number in row i of column j of A is the partial derivative of component i of f with respect to variable j of x.So, letting D abbreviate derivative, might write Df = A. Now you know multi-variable differential calculus. Yup, with meager assumptions, A is the local linear approximation to f at x. Also D is linear. So is the integral of calculus.Yup, in particular, nicely enough D(Ax) = A.The classic text on matrix theory is Halmos: He got his Ph.D. from Doob (long the best guy in the US in stochastic processes now important on Wall Street) and wrote it in about 1942 when he was an assistant to von Neumann; the book is a baby version of a favorite von Neumann topic Hilbert space generally regarded as crucial for quantum mechanics. Yes, R^n is a Hilbert space although not nearly the most interesting one.For more details on Df(x) = A, see Fleming, long at Brown’s Division of Applied Math. Fleming also considers the slightly stronger version called the Frechet derivative.You asked about matrix algebra and statistics:Suppose we interview m people and for each person ask them n questions and one more question. Maybe we ask each of m = 200 people n = 10 questions each and then one more question. The usual social science case is to ask about age, number of years in school, and gender and the one more question is income.Suppose each answer is a real number.So, put the data in m x n A and m x 1 y where for person i = 1, 2, …, m and question j = 1, 2, …, n, a(i,j) has the answer to question j for person i and y(i) has the answer to the one additional question for person i.Then suppose we want to find n x 1 b so thaty = Abat least approximately.Once we have b, we can try to use it on more people: We ask them the n questions and predict their answer to the one additional question.Well for our approximation, we might try to minimize the errors y – Ab. To do that we might minimize (y – Ab)'(y – Ab) where (y – Ab)’ is the ‘transpose’ of (y – Ab): So, (y – Ab) is m x 1: Rotate it 90 degrees counter clockwise and get 1 x m (y – Ab)’. Now you know what transpose is.So we seek b to make(y – Ab)'(y – Ab)small. Yup, this is the sum of squared errors. Yup, it’s a lot like a variance.Yup, for the second grade version, we regard the answers as values of random variables.For how to find b, one approach is to take the partial derivatives with respect to b, set them all to zero to get where we are at a minimum (several points of contact with the Markowitz derivations), and solve. There are also easier ways, e.g., based on a cute separation result in Fleming. Whatever, end up with n x n matrix A’A which, as in the Markowitz work, is symmetric, non-negative semi-definite. The quantum mechanics people call an A’A a self-adjoint operator.When we find b, the Ab is in R^m. It is in the ‘space’ of the n columns of A and is the ‘projection’ of y onto that space and, thus, is the point in that space closest to y. The projection is perpendicular, and the Pythagorean theorem applies which in this part of statistics istotal sum of squares = regression sum of squares + error sum of squaresThis is the main connection between matrix algebra and applied multivariate statistics. Close here is the A’A which is close to ‘factor analysis’ which is likely the key to eHarmony.For an application, once we have b, can go to anyone, ask them three questions and predict their income, etc. Or, more generally, find a pattern and then apply it. Or, in the pattern, take what we do know and use it to find more that we want to know. Such information might be valuable; at times it has been.Justification? That’s deeper!Social scientists love this applied math; also do marketing research people, and this material was long standard in MBA programs.Starts include:N. R. Draper and H. Smith, ‘Applied Regression Analysis’, John Wiley and Sons, New York.Donald F. Morrison, ‘Multivariate Statistical Methods: Second Edition’, ISBN 0-07-043186-8, McGraw-Hill, New York.More advanced is:C. Radhakrishna Rao, ‘Linear Statistical Inference and Its Applications: Second Edition’, ISBN 0-471-70823-2, John Wiley and Sons, New York.

  8. Dave Pinsen

    “Portfolio theory says that you can maximize return and minimize risk by building a portfolio of assets whose returns are not correlated with each other.”In reality, though, when the crap hits the fan, nearly all correlations go to one. Think of 2008, for example: it didn’t matter if you owned 10 stocks or 100 — chances are your portfolio plummeted. The reason is that diversification among stocks protects against idiosyncratic risk (the risk of something going wrong with a particular company), but not against market, or systemic risk*. One way to protect against market risk is to hedge (preferably, before the crap hits the fan; i.e., buying umbrellas when it’s sunny out)**. Another way to protect against systemic risk is to build a market-neutral portfolio (or invest in a market-neutral fund). In a market-neutral portfolio, instead of just owning the tech stock you think is best, you’d own half as much of it, and short an equivalent dollar amount of a tech stock you think has awful prospects. You’d do the same with other stocks, pairing a short position with a long position. Had you done that in 2000, for example, you would have lost money on your long tech stock position, but made money on your short tech stock position. You would have essentially canceled out market and industry risk. This year, when I’ve been unsure of which way the market was going, I’ve tried my hand at pairs trades (Here’s one example).*Diversifying among different asset classes in addition to stocks was also problematic in 2008, as the correlations between most asset classes also approached one (with the exception of Treasuries, which rallied). **Incidentally, one tech entrepreneur who was savvy enough to be hedged during both the crashes of 2000 and 2008 is Mark Cuban.

    1. fredwilson

      The comments are going to be required reading in all mba mondays. You guys finish the class so well.Thanks!!

      1. Mark Essel

        I like to think of Dave as the master of contingency plans. I have much to learn from him in this area.

  9. Etc

    yes twitter, foursquare, zynga etc in one portfolio is a true example of diversification

    1. fredwilson

      We are not trying to create a diversified portfolio by sector. We are a sector bet. Our LPs will get diversification across sectors by investing in a bunch of different fundsBut we do try to build a large enough portfolio to diversify specific deal risk

      1. Dave Pinsen

        Good, concise explanation.

  10. Druce

    The saying goes that diversification is the only free lunch in investing – reduces risk without reducing return.Of course, a pessimist would say it differently – since smart investors diversify, the market sets prices so you don’t get increased reward for diversifiable risk, therefore the investor who doesn’t diversify is taking risk that he is not getting compensated for.So if anyone says they offer uncorrelated returns, ie the risks the market doesn’t compensate you for, in most cases run! Unless, of course, you are able to judge what risks they are actually taking and what the price of that risk is. In most cases they exaggerate, or don’t know what they’re talking about, or are rip-off artists.

  11. RichardF

    OT – Is 8 pages (2570 words) for a Disqus comment a World Record ?

    1. fredwilson

      That is an essay, not a blog comment. It should be on a blog, not a comment thread. But it is a great read nonetheless and I appreciate that it was left here as value add to the discussion

      1. RichardF

        undoubtedly a great intellect behind the comment. It’s why MBA Mondays (or even the whole site) needs a wiki and a resources page with additional links.It’s also why from a UI perspective Disqus needs to limit the display of a comment to say (arbitrary) 1000 words with a disq.us link to the whole comment imo.

  12. JLM

    Theories of diversification are worthy of discussion and it is difficult to find fault with any comment which argues for a bit of safety amongst one’s investments whether through diversification or othewise.I think the real issue is not so much “diversification” amongst individual securities but rather diversification amongst categories of assets and whether they are lifestyle investments or financial investments.A general note of caution — never, ever, ever comingle or mix your business risks/investments with your personal risks/investments/stuff. Make damn sure that you have a Chinese wall and lots of Chinamen between what you do at the office and what you do at home.It is perfectly reasonable to compartmentalize your business risks in such a manner that you are using OPM in your business endeavors (even if you co-invest w/ your own funds after you’ve made a few trips to the pay window yourself) and keeping your own net worth guarded from business risks.In looking at one’s personal balance sheet there is an opportunity to “fully fund” specific elements of your life and to then take the inherent risk of a setback changing your own personal lifestyle for the worse out of the equation.To have no debt on your house in a state in which homestead laws may protect you from business creditors is such an example. To own your basic “stuff” free and clear is another example.I also argue for “paying yourself first” in every instance in which money changes hands (including payday if you have such a wonderful ocassion in your life regularly). Don’t wait until you are 65 to start thinking about how you will pay for the rest of your life. Start SAVING at 20 and keep saving for the rest of your life. Put those funds in the safest possible type of financial investments.I must make a general plea to consider real estate as a critical part of your personal long term investment strategy.It is tangible. It rarely drops to zero value. It can generate meaningful cash flow while capturing the impact of inflation. It is easily leveraged and borrowings are not a taxable event. It can be increased in value by sweat equity and you can inject sweat equity at your own schedule.

    1. PhilipSugar

      I certainly agree with the concept of paying off your house and not buying other toys unless you can do so in cash.Many people take the view if you don’t have a mortgage that’s foolish but it really is freedom….certainly for Fred it worked out very well.Regarding investing in additional real-estate what is your strategy…..I’ve thought about it but if you’re renting it out (like a house) it really seems like that’s a full time job, and if you don’t have enough units you have quite a bit of risk (getting the one tenant that screws you over or dealing with the complete unknown like you do with vacation rentals). Land doesn’t require work, but doesn’t produce enough income to offset taxes.

      1. JLM

        Real estate requires attention but it is still working while you are sleeping and it sops up inflation like a Shamwow!Everything in real estate gets down to underwriting your tenants and ensuring they have the ability and propensity for paying the landlord first.Having been in institutional real estate — high rise office buildings, thousands of apartments, warehouses, shopping centers and land — I am highest on, of all things, STORAGE!If you are just starting out, then rent houses are the way to go.I particularly like rehabs wherein you can inject sweat equity to leverage your investment.The key thing is to get into the game because inflation is coming.

        1. Tereza

          I am with you on storage, JLM. I hadn’t looked at it from an investor’s side, but should.It is a racket!Cheap locations, cheap buildings.Boy do they have customers’ nuts in a vice. Once you’re in it’s hard to get out. And all paid, in a recurring payment, on a credit card.

          1. JLM

            Plus no elevators to maintain, no HVAC, no windows to clean and yes, the credit card payments are quite nice.Did I mention that the typical deal yields 9-12% cash on total cost when stabilized including all void funding which with 25% equity and a long term 5% loan makes for a nice Christmas pudding, no?Calculate the ROE and see what else can match it just now? Of course it is an operating business as you have to service the tenants.

          2. Tereza

            JLM you should create a storage fund and we’ll all invest in it.

          3. Matt A. Myers

            I’ll put my lotto winnings into it!

          4. Tereza

            Dahlink, for JLM to get started, he needs money you do have, not money you may (but probably will not) have.However, CAN$52 is a great place to start, to live a piece of the JLM dream!;-)

          5. Matt A. Myers

            How about I put $42 into it + I spend $10 on lottery? πŸ˜› Diversification!

          6. Tereza

            BTW my DH (that’s code for Darling Husband) buys a lottery each week, too, and I tell him the same thing. So I’m not pickin’ on you.I tell him he should be focusing his dream mojo toward something he can effect and has a great chance of achieving.

          7. Matt A. Myers

            Buying lottery tickets doesn’t break the bank. A habit I broke, but I keep the rules pretty tight. Though I realize $10 every month is $120 a year, is $1,200 every 10 years not including compounded interest.. I hope to be making more than that in the next 3-5 years though where $10 in a month isn’t so detrimental; If something happens for that not to happen, I’ll change my behaviour – maybe I’ll start buying $20 a month in tickets. πŸ˜›

    2. Fred T

      Well said, JLM. Firstly, putting away 10% for yourself (and probably giving back 10%) has been an age-old mantra. The power of compounding interest still works wonders. Real estate has always been one of the best financial leverages in diversifying one’s portfolio. Gone are the golden days of prevalent creative real estate investing such as short sales and taking over the title and monthly payments while the mortgage note is still under the homeowner’s name – thanks to an SB761 initially passed in Maryland. All I’m saying is investing is a laser-focused entrepreneur’s strategy; only spread out in many forms. Warren may have been a proponent of long. But short is also good if you are passionate with technical analysis and fundamental economic patterns. Some book by Hopkins called “The Art of Selling” said it well. It is a numbers game. The more you go out and take calculated risks, the bigger chance a good percentage will fly off well. Of course, you gotta have liquid assets always ready just in case.

  13. Mike

    “I’ve said many times on this blog that one third of our investments will not work out at all, one third will work but will not be interesting investments.”I have to ask, have any of the 2/3 of your investments that either did not “work out at all” or were not “interesting” contributed in a non-financial way to the success of Union Square? Either through an opportunity to work w/ a specific entrepreneur, learn an entirely new line of business, what NOT to do in the future, etc…?

    1. fredwilson

      yes, all the timethat is why we don’t walk away from them

  14. Vikas Desai

    Diversification is one of the oldest strategies out there, but greed often takes over and this strategy gets lost amongst all the inflated returns from the stock market. Its just unfortunate but its true. In order t understand the importance of diversification you need to be burnt once but not that badly that you are not able to re-invest. When I started investing, I had no debts and was responsibility free so I was investing with a lot of risk. Now I have a mortgage, a kid and currently am looking for a job. So now I have taken a step back and learnt to keep my investment in cash as well which by far is the safest, considering the current situation of government loans.But coming back to your topic, I am with you and have always tried to educate all my friends about the same. I just hope I am able to follow the same path in the future.

  15. Alan Warms

    Fred -Great post. I think one topic that naturally flows from this is angel investing….but I don’t personally think the answer is do a ton of angel investments to get at the portfolio effect. The one difference between your VC investing and other investing is you and your partners add tremendous value and have a huge impact on the investment itself — you are not passive but are active investors. So your 1/3 1/3 1/3 math is a result of personal experience on your investing criteria with you as active investor. As opposed to just investing in 30 deals passively. Why one of the things I am trying to do now is invest in deals where I can be active, like a Yidio.com.Al

  16. andyswan

    It’s all relative.”When someone has very little to lose, I totally get betting it all and going for it. But when you have accumulated a nest egg or more, you must be diversified in your investments and assets.”So, what’s “little to lose” and what’s a comfortable nest egg? To the guy born in the mud in Africa, $30k/year with a car, cable TV and a cell phone would be extremely comfortable. Here that’s what we call “getting started”. I’m sure your $100m-man example just wanted to make his next move to a billion.I have given up on diversification. I hate it. I can see it as a fine strategy for the content, but it absolutely bores me. It’s not about the cash, it’s about the win.I don’t see is as risky to bet it all, because my most valuable assets are my mind’s capability and my heart’s passion.

    1. Evan

      Totally a legit argument. You like to gamble. You aren’t trying to build wealth slowly and specifically.Lots of entrepreneurs are like this: a little bit crazy. Which is great for us, consumers, and great for VCs like Fred.

    2. thewalrus

      i’m down with andy. its all relative….and the only real hedge is yourself.my personal approach is to try to own my home outright. then to invest/take risk in situations i have maximum direct influence over. start with your passions, shape them into the ideal work and lifestyle, then use all your creative and financial resources to make it happen.its a zero risk approach cause you can’t take it with you, and you’re only as good as the experiences you enjoy and share with others while you’re around πŸ™‚

      1. fredwilson

        I totally agree with the ‘you can’t take it with you’ mindset. But if you have capital, you can fund other people’s hopes and dreams

        1. thewalrus

          i think your belief in helping make others hopes and dreams happen is why you have such strong respect from entrepreneurs and frednation. you’re not playing a zero sum game.i am cash poor (in relation to investing), so i give my time and experiences…..maybe that equation might shift a bit at some point in my life….either way…. 1+1=3

          1. JLM

            Time and experience always play trump to Benjamins.

    3. Matt A. Myers

      I think there are other elements that can come into play with the right diversification.

    4. L. Marie Joseph

      “To the guy born in the mud in Africa” There are poor people all over not just in Africa

      1. andyswan

        I never suggested otherwise.

  17. steveplace

    It’s all fun and games until correlations run to 1.Global markets have been on a giant liquidity (risk) conveyor belt since 2009, so it’s tough to find true diversification– at least from a retail perspective.

    1. Dave Pinsen

      FYI, for those who haven’t seen it yet, Steve offers an interesting hedging strategy on his blog today, “SPX Butterfly Hedging Strategy”.

  18. ShanaC

    You might want to put sigmaalgebra’s comment in plain english (or How Markowitz’s work relates back to diversification, since we did that last week.)

    1. sigmaalgebra

      For the connectiion between diversification and the Markowitz model, I listed that as an exercise!

  19. Marc Brandsma

    VC portfolios are hardly diversified.Some large funds pretend to diversify by investing in different sectors (IT, Life Science, Energy) in different stages (early to pre-IPO). But due to the high growth/high tech nature of their investees, their portfolio is more or less aligned around the same beta.Smaller funds, like USV (no offense intended), tend to concentrate/specialize. This is not bad if fund managers master the art of selecting/nurturing/exiting companies in that particuliar segment. If not, it can be disastrous.Diversification is good from a market perspective. When it come to asset class level, it doesn’t work.

    1. Evan

      There are many different kinds of risk.For example, if I choose one fund that is extremely well-diversified across every asset class known to man, am I diversified?Yes and no. Obviously I am diversified across assets very well. But I have very specific investment manager risk. If my manager is getting a divorce, or was simply lucky before and never really had alpha, or leaves the fund, etc, etc etc, I may be very undiversified.So Marc, you are talking about not being diversified on a very specific risk, but that doesn’t limit the point that diversification is fundamental when you want to mitigate risk.

  20. David Repas

    The key is to reduce risk and at the same time increase potential returns. Diversification is often pretty good at reducing both – which, depending on your financial position, isn’t always a bad thing.Capital preservation can also be a great strategy, unless of course you don’t have very much to preserve in the first place. Here’s a quote for the latter group :o)”Diversification is something that stock brokers came up with to protect themselves, so they wouldn’t get sued [for making bad investment choices for clients]. Henry Ford never diversified, Bill Gates didn’t diversify. The way to get rich is to put your eggs in one basket, but watch that basket very carefully. And make sure you have the right basket. You can go broke diversifying. Ask anyone who’s diversified in the last three years. They’ve lost money.” — Jim Rogers / April 2009

    1. TC

      I didn’t lose money diversifiying.

  21. bramcohen

    A point about peoples’s personal finances: Making investments when you have debt isn’t diversification. It’s leverage. Anyone who has any debt should pay it off before making any investments whatsoever. In principle if someone has an extremely low interest loan, like some student loans, it’s profitable to invest elsewhere, but it turns out that that’s not worth the risk in the vast majority of cases, and even when it isn’t the time taken doing the investment and the risk you’ll screw it up aren’t even vaguely worth the small amount of return.While one in three investments pay off, an even smaller fraction pay off in a big way. I’d much rather be diversified in about 3x the number of investments you are, although beyond that I think further diversification wouldn’t pay off much, and there’s a serious question of how many investments you can do proper due diligence on in practice.

    1. JLM

      Hmmm, debt and equity are just different prices for money. Investments of all types require money. The question is do they work w/ debt or equity or some combination?The real question is can you personally “manage” debt? It is a skill. Because, of course, equity does not require managment as it never calls up and says — please pay the vig!Today, interestingly enough, debt is so damn cheap that it is literally like peanut butter but the real issue is can you actually access debt? The answer is — not bloody likely! The banks are frozen. Totally FROZEN.If you can chin the debt, get it by the dump truck load today and hold hard assets until inflation kicks in as we are going to be having Argentina type gaucho inflation when the economy begins to recover. You will be paying back dollars w/ dimes.

  22. Charlie Wang

    Fred – Thanks for the great blog posts, I found it today via your NY Times Q&A. Just have some brief points on diversification:1) Diversification is all well and good, until you really need it to work. “Uncorrelated” assets become very correlated in financial panics and crises, as we have seen all too often in the past.2) There is a difference between being a passive investor in investments (typical investments such as stocks, bonds, mutual funds) and being an owner/operator (VC and PE firms). The owner/operator can have a lot of influence over the direction and success of their investments.3) To achieve greater reward, one does not necessarily have to take on greater risk. This is an often overlooked aspect of investing. It is possible to achieve higher return with lower risk, but these situations are rare. Value and special situation investors have been doing this for years.Cheers,Charlie Wang

    1. fredwilson

      welcome charlie!great comment right off the bat.you are going to be popular in this community if you keep that up

  23. Matt Smith

    Here’s an interesting thought to consider: While diversification can protect you against big losses, it is also true that diversification dilutes superior investment results.

    1. Evan

      Not true. If 100 investments all have similar alphas, then it will not dilute your results. It will protect against systematic risk.

      1. Matt Smith

        Evan- I thought alpha was an investment’s performance either above or below that of the index that it is benchmarked against. Is that right? I guess I’m not sure how you would select 100 investments with similar alphas, except in hindsight. I’d love to understand your point better.The situation that I am describing is one noted by investors like Warren Buffet and Charlie Munger. It is a situation where an investor has an investment opportunity in an area that they understand very well and where if they invest a larger chunk of their assets in that opportunity then their performance will be higher than if they put a lower percentage of their assets to work in that particular investment and diversify their investments in other areas where they don’t have a comparative advantage. That looks a lot more like the real world to me than a hypothetical situation where you have 100 equally good investments to choose from and that’s where diversifying your portfolio would actually dilute your otherwise superior investment results.

        1. Evan

          Matt, you should google CAPM or Portfolio Theory. Or maybe take the CFA πŸ˜‰

          1. Matt Smith

            I’m familiar with portfolio theory but I’m somewhat doubtful about some of the assumptions it makes, particularly that markets are efficient and that people behave rationally.I’ll stick with Warren Buffett on this one. If you have a deep understanding of a business and recognize that it is undervalued by the market you should concentrate your investment activity in that area. In that circumstance, diversification would dilute your investment results. (See the comment from Scott Semple above to see more of Buffett’s thoughts on this issue.)VCs do this professionally. They invest their funds in places where they believe they have a comparative advantage – typically all in the same asset class and without diversification. They may try to get exposure to different segments but they typically invest in the same stage in similar companies. Also, when they feel that they have found a winner in their portfolio they put more of their fund to work in that company if they have the opportunity.

          2. Evan

            you don’t understand what alpha is, so it’s pretty difficult that you understand capm.

  24. ShanaC

    Actually, I also really want someone to go into risk-return since apparently there is some stuff floating around how you can make your money back much more easily now in blue chips that are diversified.Are we also saying that diversify your risk?

  25. Kevin Dorren

    Fred, love your blog – make it a read every few days.Just a point of clarification in your comment stream – Although Warren Buffet said “Put all your eggs in one basket, and watch that basket really closely” he was actually quoting Andrew Carnegie – who at the end of his business career invested in improving the education of millions of people through investment in libraries, colleges and educationPerhaps some comment on how to include Philanthropic investments in your portfolio maybe useful?

    1. fredwilson

      that’s a good topici’ll find a way to talk about it

  26. Eric Friedman

    This begs the question – can you be professionally diversified by working at one company? Many folks I know and am friends with build “side” projects all the time for this very reason. The “rocket ships” they join do have the potential to go to space – but they also have the ever present potential to explode. I think these side projects keep people sane – and allow for some professional diversification.

    1. fredwilson

      i think you can do this eric, but not if you are the founder or key member of the founding team. then you need to be “all in”

  27. Donna Brewington White

    completely, completely off-topicJust came across your “Doubling Down” post from August 2009 and once again thought, it would be great to have some of your posts compiled into some sort of book or something!…and maybe a few choice comments from each. (I know that part would be hard– how do you choose?)BTW, if anyone is interested…here is the link to above referenced…a GREAT post.http://www.avc.com/a_vc/200

  28. scottsemple

    The strategy we’ve adopted precludes our following standard diversification dogma. Many pundits would therefore say the strategy must be riskier than that employed by more conventional investors. We disagree. We believe that a policy of portfolio concentration may well decrease risk if it raises, as it should, both the intensity with which an investor thinks about a business and the comfort-level he must feel with its economic characteristics before buying into it. In stating this opinion, we define risk, using dictionary terms, as “the possibility of loss or injury.”[A] situation requiring wide diversification occurs when an investor who does not understand the economics of specific businesses nevertheless believes it in his interest to be a long-term owner of American industry. That investor should both own a large number of equities and space out his purchases. By periodically investing in an index fund, for example, the know-nothing investor can actually out-perform most investment professionals. Paradoxically, when “dumb” money acknowledges its limitations, it ceases to be dumb.– Warren Buffett, 1993 Chairman’s Letter (http://www.berkshirehathaway.com/letters/1993.html)And comments from a speech to MBA students in the following video between 1min and 3min: http://www.youtube.com/watch?v=P-PobeU4Ox0

  29. Tim Johnson

    Pardon for asking a newbie question. So what strategies do you use to find that top 1/3 or 2/3 in early stage? Or is that the Holy Grail of VC? How much community vetting do you use now that you didn’t/couldn’t 10 years ago?

    1. fredwilson

      gut instinct, pattern recognition (but the right patterns), and people skills

  30. Kyle Healey

    I love reading your blog, you are on another level then the people in my industry.

  31. paulcub

    Tech stock was out of whack in the late 90s. The subprime market went whack in the middle of the last decade. When you put them in different markets, it takes a lot of risks.portable media player

  32. Tim Lind

    I’m interested to know more about the diversification strategies within your tech investment portfolio.I think that this is a very important aspect of the tech funding world that isn’t talked about at all.My concern is what effect investors diversification strategy has on the tech industry as a whole.Policy on competition is also something that I think plays an important role in the big picture of tech industry success, or at least how an entrepreneur should address the funding.

  33. stanleyyork

    The only slight argument someone might have (in the form of a case study) is Warren Buffett. I could be mistaken, but he’s argued against diversification in some instances – not all though (if I remember correctly from his biography – ‘The Snowball’).

    1. stanleyyork

      No response necessary – i see the thread below re: buffet.

  34. wfjackson3

    Is investing in 21 companies instead of 6 really diversification? Isn’t that just the power of statistics at work? Is that really a good metaphor for diverse investments which do not carry the same general statistical information?

  35. Kwillcox

    If a larger company does a stock buyback, it is to allow investors to profit, without having to pay taxes on dividends. Between that and a DRIP program, it is possible for a savvy investor to maximize any portfolio strategy by limiting both taxes and commissions. Nothing stupid about that, just no money for the broker and deferred money for Uncle Sam.

  36. Kwillcox

    In the beginning, there were many railroads, now only a few. There were many car companies, now only a handful. There were many airplane companies, now only a few. And I remember when there were 8 major computer companies, including RCA, Xerox, and GE.So, my point is that there is often a big cull of companies in a field, and we see only the survivors. In fact, almost all the investments in the losers are destroyed. So, a large company that buys back its stock is actually making a statement that it wants to be a survivor. It can live without the cash, and it can benefit its investors.

  37. Akira Hirai

    Diversification is great for investors, but deadly for startup entrepreneurs.At the startup stage, diversification equals lack of focus, and lack of focus equals failure.Of course, by “focus” I don’t mean blindly placing one giant bet. It simply means resisting the urge to solve too many problems or serve too many markets all at once.You’ll have plenty of time to diversify your offerings once you have decent cash flow. Until then, get the whole team focused on one discrete area.

  38. JLM

    Been there, done that.Of course, entrpreneurs are not paid only in financial assets. The real currency is that satisfaction of saying — yeah, I did that! Next?Every person who goes to the pay window because of their own entrepreneurial performance should the day of the closing engage a professional money manager to diversify their holdings, hedge their risk and to manage their money.The same lazer like focus which makes entrepreneurs successful is their blind spot when it comes to investing.

  39. Mark Essel

    Charlie you know how to pick ’em huh ;)Pain heals, chicks dig scars.Glory, lasts forever.http://www.youtube.com/watc

  40. PhilipSugar

    It could be worse….you could have held onto common stock in a heavily leveraged company with VC money that went belly up. I’ve done that. I love JLM line that Good judgement comes from experience, experience comes from bad judgement.

  41. fredwilson

    such good advice charlietoo bad we all have to learn this the hard way

  42. kidmercury

    which run on gold are you talking about? are you talking about the bull market that started in 2000? the one where the “gold is a bubble” crowd started calling a bubble when it was at 400/oz? and then again at 700….900….1000…1200 has to be a bubble…….

  43. Tereza

    I’m personally more interested in the bracelets, earrings, and other fine jewelery to be made from Kid’s stock.But I love gold as a continuing thread in the convo. Always spurs good comments.

  44. Dave Pinsen

    Maybe the derivative of that comment is less than 3 paragraphs.

  45. sigmaalgebra

    Read it or not: If you want an MBA view of diversification, then you need at least what I wrote (yup, I used to be an applied math, MBA program prof — some of what I taught was optimization as in the Markowitz work).What I wrote gives solid definitions for nearly everything, proves nearly everything, and for this much actually is quite short. I take you from next to zero in probability through a relatively careful treatment of the Markowitz work.Also I show in nearly full detail just how a lot of uncorrelated investments can get risk as small as you please; then with leverage, can get return still higher.I also explain why sometimes people prefer a stock with higher risk.Also I avoid incompetent nonsense such as the Gaussian assumption in the Wikipedia article.

  46. JLM

    While I am not an emotional great fan of gold, I must admit that my instincts tell me that there is a 2-3 bagger left in gold not because it is such a sound fundamental investment but rather that it is the safe harbor of last resort and I think that things are going to get very, very, very, very much worse.Oh, yeah, and because my favorite RIA is touting it — Glenn Beck. Of course, Glenn Beck — like or abhor him — is a very smart and successful guy. Kind of like Suzanne Sommers, no? LOL

  47. ShanaC

    I’d short kidbucks.

  48. kidmercury

    now we’re talking! πŸ˜€

  49. kidmercury

    kidbucks (or mercbucks as i like to call them) operate in virtual financial markets that are honest and thus do not allow shorting — short selling is for the corrupt real world markets that many fredland citizens have chosen to be psychologically enslaved to. but even if shorting mercbucks was allowed, mercbucks will be pegged to a gold/silver composite, in accordance with the US constitution….as gold and silver continue to embarrass “real world” currencies, shorting them would be a grave mistake, whose only redemption would come in the form of mocking jokes delivered by kid mercury.

  50. RichardF

    That’s a lot of very’s JLM… what’s your thinking behind it?

  51. Tereza

    ….or TruthBucks.Kid could you create a Truth portfolio, possibly with Pinsen’s help, where you long Truthy assets and short non-Truthy ones? Or create a truth-neutral portfolio where you buy half of long truthy assets and short half of truthy ones?Somepin’ like that…

  52. JLM

    Well, basically nothing in the US is really working. I have become so pessimistic as to be politically agnostic — unable to gather the energy to even articulate my opposition anymore. I have simply resigned myself to an inevitability which borders on surrender preferring to rebuild from the smoldering ashes rather than attempting — even intellectually — to put out the fires.I am afraid that the current administration is simply incompetent, hopelessly over their head and swimming in the deep end with no plan — absolutely no plan — nor any confidence that they can catch their breath long enough to make a plan.Even worse, they have begun to realize it themselves and what passed for competence — bravado — is taking a hike also. I mean this on every front but would use as an example Iran where we have been made to look so foolish that even our allies are seemingly supporting the enemy.We are approaching a financial tipping point — old fashioned running out of money — as it relates to mandated spending, new programs, the accumulation of debt and the inability to fire up the economy — as well as some old fashioned bad luck.The oil spill, the actual economic impact and the knee jerk reaction to ban drilling will suck the life out of the Gulf Coast. A region of the Nation wherein the economy had been a bit sounder than others. All but the knee jerk reaction to ban drilling not being the Administration’s fault in any manner.I find that incompetent folks are generally also unlucky but this is way beyond unlucky, this is snakebit.Once those deep water rigs leave the Gulf, as they have begun to do, it takes 2-3 years to turn them around as they will undertake 24 month contracts at their new locations.As a Nation, we are getting tired.

  53. kidmercury

    yup….gold is the ultimate in truth, the US dollar is non-truth (i.e. lie, deception, tyranny, etc)….long gold/short the US dollar….simple truth portfolio that’s worked in recent history, and is only going to do even better in the years to come….toss in some silver, and that’s basically all you need….truly the truth that sets you free

  54. Dave Pinsen

    LOL @ “truth-neutral portfolio”.

  55. RichardF

    unfortunately the UK fairs no better but I was hoping there was greater optimism in the US.

  56. fredwilson

    sadly i agree about most of thisi am not as pessimistic about obama but the basic observations are sound

  57. kagilandam

    “As a Nation, we are getting tired ” … that is one of the scariest statement i have read in this blog.

  58. ShanaC

    Short to death based on what kid wrote- how do you exit the money into othergoods and services- especially when you want to do the semi-legal (i thinkall money to be practical, should have that space, even if you don’tpractice the semi legal…)

  59. Fernando Gutierrez

    Yeah, if someone as JLM is so pesimistic we should start worring. I’m inSpain and the economy is going to hell, but I didn’t expect the same feelingin the US. I was already thinking on selling my business and going toanother country, but maybe I should just sell and hide in a cave!

  60. Dave Pinsen

    “the UK fairs no better” could be the understatement of the day. My impression (from reading the FT, etc.) is that you guys are in a bad way fiscally; you became overdependent economically on the City and North Sea oil; you’ve let foreigners buy some of your blue chip companies and hollow out your industrial base; etc.

  61. RichardF

    Last time I was down in Mojacar I think there were some nicely priced caves for sale.

  62. JLM

    I don’t mean to come across too pessimistic but I guess what really has me spooked is the sheer incompetence in the face of an increasing level of motionless activity — all arm movement and no foot movement.Some of the stuff is almost purely symbolic — the inability to get Gitmo closed down as an example and no real accountability — but in symbolism we often find huge core truths. If you can’t execute on your own playbook, I worry that you can thrust and parry when the unexpected shows up. If you don’t discipline and hold accountable persons on your own staff — granted they took Greg Craig’s top knot — who are you going to hold accountable?What makes it worse is this sense of faux toughness coupled with what is an obvious attempt to do stuff behind the curtain — an example being the attempted offering of jobs to prospective same party candidates and then not being able to close the deal. If you are going to kill the King, remember to kill the King.The amateurish handling of l’affaire Sen Specter — why bring a bum into the fold? While turning your back on an otherwise good candidate? It all seems so amateurish.Last observation — the banks are essentially closed down for business. I am as nimble a borrower of money as I would ever hope to be — having borrowed or raised over $1B I certainly should have the requisite experience and track record — and my history counts for nothing, absolutely nothing. I have worked through the problem but it has taken every bit of my talent to make the sale where once upon a time I just made a phone call and signed the docs.In the midst of this everybody is going along thinking there is actually going to be some increased level of income to tax?

  63. Fernando Gutierrez

    LOL! Too hot for me down there, but thanks for the tip! πŸ™‚

  64. Tereza

    ….but HEY! Cheer up!!!Before long someone’ll write a really sing-songy musical about it!;-)

  65. RichardF

    All true Dave, our industrial base went years ago and no doubt the fiscal mess we are in is a big concern.As you intimated in another comment Australia is probably the best bet for a western economy. I have a friend who has just moved over there and tells me that there is no recession there at all. The downside is that at the rate the Chinese are buying their assets they are set to become a Chinese colony.

  66. PhilipSugar

    As usual well said. My biggest frustration is this line has been turned on its head…And so, my fellow Americans: ask not what your country can do for you – ask what you can do for your country.

  67. ShanaC

    When was the last time you heard of marketmakers participating in an outcry market? Our markets are virtual?Why not just use normal derivatives like everyone else if I want to deal with gold and silver. Or how about palladium?

  68. kidmercury

    wall st is more fraud than honesty, but scams cannot last forever and wallst scams are coming to an end soon enough. that is why there will be a shortsqueeze in the gold and silver market, upon which the spot price ofgold/silver will soar while futures will be worth nothing.