Warren Buffett 90-10 Portfolio and the Kelly Criterion

When Warren Buffett was asked how he would ask his wife to manage their money after he was gone, his advice was simple.

My advice to the trustee could not be more simple: Put 10% of the cash in short-term government bonds and 90% in a very low-cost S&P 500 index fund. (I suggest Vanguard’s.) I believe the trust’s long-term results from this policy will be superior to those attained by most investors…

Can we try and understand this choice through the lens of the Kelly criterion, which we have explored in some depth in previous posts?

Setup

Suppose that returns \(r\) from the stock market obey a Gaussian distribution, with the mean \(\mu = 1.1\) and scale factor \(\sigma = 0.3\). Note that this \[r = 1 + \text{annual rate of return}.\] This choice essentially assumes average annual returns of 10% from the market, and anticipates typical swings (\(2\sigma\)) to lie between -50% and +70%.

We will play with some of these parameters later, but right now, they seem reasonable enough; so lets not fixate on them.

I will also assume that the worst possible return is \(r = 0\), which implies a return of -100%, which means everything is completely wiped out. This has never happened in the past, and it seems like a good bet that this will "never" happen in the future. Similarly, I will assume that the best possible returns are 300%. It seems like a reasonable cap.

So my annual returns are expected to be random variables drawn from a truncated normal distribution. Let's see what this would look like.

Kelly Criterion

In the Kelly criterion, the crucial expression to develop is the mean of the logarithm of the expected return.

In this case, the probability of obtaining a return of \(r\) is \[\mathcal{N}_{T}(r|\mu, \sigma, r_{min}, r_{max}) dr,\] where the subscript "T" reinforces the "truncated" aspect of the normal distribution.

If I wager a fraction \(f\) of my investment pot, then I should expect to end up with, \(1 - f + f r\), where I assume that the uninvested fraction is kept safe in cash, and earns 0%.

Thus I can write the logarithm of the pay off as:
\[\log S = \langle \log R \rangle = \int_{r_{min}}^{r_{max}}  \mathcal{N}_{T}(r) \log(1 + f(r-1)) dr\]  Let's visualize log S for this choice of \(\mathcal{N}_{T}(r)\).

Python codes used to generate these (and subsequent results) are at the end of this post.

The optimal allocation to the S&P portfolio using the Kelly criterion (by maximizing log S) is 82.8%, which is reasonably close to Buffett's suggested portfolio.

Scenarios

Let us consider three scenarios: (i) conservative (\(\mu = 1.05\), \(\sigma = 0.5\)), (ii) moderate (\(\mu = 1.08\), \(\sigma = 0.4\)), (iii) aggressive (\(\mu = 1.12\), \(\sigma = 0.3\)).

The optimal allocations for the conservative, moderate, and aggressive assumptions are approximately, 55%, 83%, and 100%, respectively.

If we assume a \(\sigma = 0.3\), then a 90% stock allocation corresponds to a \(\mu \approx 1.09\), or an expected 9% rate of return from the S&P.

Historically, the standard deviation of S&P returns has been closer to 0.2. I like to use a higher number because the histogram of the actual returns is not perfectly normal, and the tails have been fatter than one would expect from a \(\sigma = 0.2\).

Python Code

Dalio: How the Economic Machine Works

Ray Dalio put together this very nice video to paint the workings of an economy with a broad brush. It presents a compelling lens through which to view the relentless onslaught of "economic" news. For the first time, I think I have a rough idea of what Ben Bernanke is trying to pull off:

Krugman on Freidman

Lately, I have become fascinated by Milton Friedman. I just borrowed a copy of "Capitalism and Freedom" from the library, and look forward to reading it.

I found this interesting piece (2007) by Paul Krugman at the New York Review of Books, entitled "Who was Milton Friedman?"

What’s odd about Friedman’s absolutism on the virtues of markets and the vices of government is that in his work as an economist’s economist he was actually a model of restraint. As I pointed out earlier, he made great contributions to economic theory by emphasizing the role of individual rationality—but unlike some of his colleagues, he knew where to stop. Why didn’t he exhibit the same restraint in his role as a public intellectual?

The answer, I suspect, is that he got caught up in an essentially political role. Milton Friedman the great economist could and did acknowledge ambiguity. But Milton Friedman the great champion of free markets was expected to preach the true faith, not give voice to doubts. And he ended up playing the role his followers expected. As a result, over time the refreshing iconoclasm of his early career hardened into a rigid defense of what had become the new orthodoxy. 

In the long run, great men are remembered for their strengths, not their weaknesses, and Milton Friedman was a very great man indeed—a man of intellectual courage who was one of the most important economic thinkers of all time, and possibly the most brilliant communicator of economic ideas to the general public that ever lived.

Too Big to Fail and The Big Short

If I ever had to make a list of the top five world events of my lifetime, the great recession (TGR) of 2008 will probably feature prominently (I hope. I don't think I want to live in "very interesting times").

I recently read two books on TGR: Andrew Ross Sorkin's "Too Big to Fail", and Michael Lewis' "The Big Short". Both books are eminently readable.

Sorkin's book deals with the events leading up to the fall of Lehman Brothers, and reads like a movie screenplay. The "dialogues" of the principal actors in the drama are written in first person. I don't know how, and how accurately, Sorkin managed to do that, but it does make for compelling storytelling.

You get a sense for how chaotic those times were, and how the principals involved had to make important decisions under extreme uncertainty and pressure. And how easy it is for many commentators on the crisis to be Monday night quarterbacks.

The book provides interesting  color on people who have subsequently come to be viewed in the media somewhat uni-dimensionally. For example, you learn how Lehman CEO Dick Fuld, stood up for the weak in an ROTC camp in his college days, before coming to be unanimously reviled as an out-of-touch, and perhaps, criminal operator. You learn how unaware of social niceties former Treasury Secretary Hank Paulson was. I never knew that this Republican, former Goldman Sachs CEO was a Toyota-Prius-driving birdwatcher and environmentalist.

I would strongly recommend Sorkin's book for the scene-by-scene portrayal of some very tumultuous  times, and for the fullness with which it casts some of the most reviled people in America today.

If Too Big to Fail is a view from the inside, The Big Short is a view from the outside.

Michael Lewis' book outlines the stories of a few unlikely characters who foresaw the financial massacre a few years earlier, and smartly bet against it. It follows the trail of social misfits like Michael Burry, a former medical doctor-turned-hedge fund manager, who was among the first to figure it all out, only to be hounded by investors during trying times, and Steve Eisman who managed a fund for Morgan Stanley and lamented that he couldn't short his parent company.

Even if these people knew the whole thing was going to blow up, they did not know when. And even if they bought insurance to bet on the outcome they thought was most likely, they could not be sure that when the house was on fire, the insurer wouldn't go bankrupt. As Warren Buffett put it succinctly “It's not just who you sleep with, it's also who they are sleeping with,”

Here's a commencement speech by Michael Burry, and another one by Michael Lewis.

EconTalk Podcasts and More

I stumbled upon EconTalk podcasts earlier this year, and have been hooked. I find myself listening to these fascinating long-form (~1 hour) discussions on various "economic" topics with intellectual leaders while exercising, driving, doing dishes etc.

The whole discussion, while being in a question-answer format, is not really an "interview" in the Charlie Rose sense. The focus is more on the topic, and less on the person.

Here is the description of the program from the website:

The Library of Economics and Liberty carries a weekly podcast, EconTalk, hosted by Russ Roberts. The talk show features one-on-one discussions with an eclectic mix of authors, professors, Nobel Laureates, entrepreneurs, leaders of charities and businesses, and people on the street. The emphases are on using topical books and the news to illustrate economic principles. Exploring how economics emerges in practice is a primary theme. 

The quality of the discussions in the forum is quite extraordinary.

Milton Friedman videos on YouTube

Milton Friedman may be a controversial economist, but his videos on YouTube reflect why he was such an intellectual juggernaut. Here are a few that I used to while away a perfectly enjoyable afternoon.

1. No free lunch:

2. Young Michael Moore challenging Friedman: He was so much thinner then. (Edit: Apparently not the real Michael Moore)

3. An older black and white video:

etc.

More Physics Envy

On the heels of a recent related link, comes this talk, in which Andrew Lo deliberates on what physics envy may have done to the dismal science (H/T Farnam Street). From the video:

In physics it takes 3 laws to explain 99% of the data. In finance it takes more than 99 laws to explain about 3%.

Interesting video, especially for those with some knowledge of both fields. Also check out the link (from the Farnam Street blog) to a paper (PDF) by the same title. 

Barry Schwartz and Efficiency

Psychologist Barry Schwartz (or the Paradox of Choice fame) pens in an interesting opinion on how to think about economics as a competition between efficiency and friction (and offers a quasi-defense of Mitt Romney's role at Bain Capital in the process).

On efficiency:

It may seem heartless to worship efficiency at any cost, including lost jobs and decimated communities, but it is important to understand that increased efficiency is the only way a society’s standard of living will improve. If your company raises your pay without becoming more efficient, it will have to raise its prices in order to pay you. This is true of all companies. And if all companies raise their prices to allow for higher wages, you will end up just running in place, with your higher wages exactly matched by the higher prices of the things you buy. It is only if your company and others find a way to pay you more without charging more that your living standard goes up.

On friction:

ALL these examples tell us that increased efficiency is good, and that removing friction increases efficiency. But the financial crisis, along with the activities of the Occupy movement and the criticism being leveled at Mr. Romney, suggests that maybe there can be too much of a good thing. If loans weren’t securitized, bankers might have taken the time to assess the creditworthiness of each applicant. If homeowners had to apply for loans to improve their houses or buy new cars, instead of writing checks against home equity, they might have thought harder before making weighty financial commitments. If people actually had to go into a bank and stand in line to withdraw cash, they might spend a little less and save a little more. If credit card companies weren’t allowed to charge outrageous interest, perhaps not everyone with a pulse would be offered credit cards. And if people had to pay with cash, rather than plastic, they might keep their hands in their pockets just a little bit longer.

Life is not as predictable as driving. We don’t always know where we’re going. We’re not always in control. Black ice is everywhere. A little something to slow us down in the uncertain world we inhabit may be a lifesaver.

Howard Marks on Taxing the Rich

I like Howard Mark's writing because it cuts the rhetoric and attempts to look at an issue from multiple angles. In a recent memo, he frets about the increasing use "the rich should pay their fair share" in political circles. "Fairness" may really be a eye-of-the-beholder thing, he argues quite convincingly.

As as example he says about tax deductions:

The drafters called them deductions: provisions that reduce the net income on which  taxes are levied. Critics call them loopholes, suggesting there’s something underhanded  about those provisions. And politicians use the laudatory-sounding term tax incentives to describe tax code provisions that reduce tax revenues in order to encourage certain behavior. It all depends on your point of view.

And later:

As I’ve written before, I was very impressed when, as a young man, I heard an interesting explanation for America’s economic progress relative to Great Britain: “When the worker in Britain sees the boss drive out of the factory in his Rolls Royce, he says ‘I’d like to put a bomb under that car.’ When the worker in America sees the boss drive out of the factory in his Cadillac, he says ‘I’d like to have a car like that someday.’ ”

In recent weeks, much has been made of nation-wide polls which say something like "75% of the people support increasing taxes on the rich" etc. or something similar.

That is such a stupid question to ask/poll.

Sure, as rational self-interested individuals, why wouldn't they? It is a classic case of tyranny of the majority. Despite the superficial difference, it is not completely unlike popular support for banning the hijab in France (which much of the media saw as somewhat bigoted).

PS: I am a card-carrying member of the 99% :). I think the memo is a great read, simply for the nuanced views it presents dispassionately.

Economics: A love affair

Bill Gross of Pimco uses the metaphor of love, marriage, and divorce to describe disturbing financial events in Europe, United States, and the rest of the world. An entertaining take on a sobering state of affairs.

Oh those feisty Europeans! Always fighting like a dating couple and then finally resolving their differences by saying “I do” sometime in the 1950s with the creation of the Common Market and the European Economic Community (EEC). In doing so, France and Germany said “never again,” and even though they didn’t like each other (read “hate”) they decided to make economic lurv in the hopes that they wouldn’t destroy the continent again. It later turned into a formal union, a European Community (EC), where they invited lots of witnesses to the ceremony and created instant family members, if that’s metaphorically possible. Twenty-seven of them, including Italy, Spain and the U.K. were now relatives despite some liking pasta and others preferring horrid cuisines featuring Shepherd’s Pie or fish and chips. The marriage progressed to the point of a smaller monetary union sometime in 1999, but critically, without a common budget. Husband and Wife – Germany and Greece – decided to have a joint bank account, but with separate allowances and no oversight. Greece could issue bonds at nearly the same yield as could its Northern hard-working neighbors, but were free to spend it any way they chose. This was an economic version of an open marriage where one party gets to have all the fun and the other worked nine-to-five and came home too exhausted for whoopee.

Gray and Dirty Swans!

My opinion of Nicholas Nassim Taleb's "The Black Swan" is not particularly charitable. But you know, this guy put his philosophy into practice by running a hedge fund and making billions of dollars for himself and his clients. What do you say about that, huh?

I did not know this while I was reading the book, but it turns out that the magnitude of his exploits was  "somewhat" misleading.

NNTs investment strategy is very crudely speaking the opposite of the insurance business. That is, you continuously bleed money, with the expectation of making monster sized gains when an extremely low probability event (a Black Swan) occurs. The rationale behind the strategy is that the probabilities "we" assign to low-probability events (could be zero) are usually lower than warranted. That is, there is an  asymmetry between expected and empirical probabilities.

As Janet Tavakoli puts it:

The black swan fund's strategy is purportedly to buy out-of-the-money put options on stocks and broad market indices and hedge tail risk for clients. The strategy may produce long periods of mediocre--or even negative--returns followed by a large gain and vice versa. No one can tell you for certain exactly when (or for how long) large gains are possible.

She reports here that:

Taleb’s Empirica Kurtosis “black swan” fund had negative returns in 2001, the year of the 9/11 black swan event. Taleb later claimed he only called it a hedge fund “in May-Oct 2001.” Perhaps he meant something else, because Empirica Kurtosis wound up at the beginning of 2005 with lackluster returns , and performance specifics are not public, but it may have been a stranded swan.

She also discovered that when he was quoted as saying the following (in the aftermath of the present crisis), in an article in GQ fawning over his intellectual prowess:

I went for the jugular--we went for the max. I was interested in screwing these people--I'm not interested in money, but I wanted to teach them a lesson, and the only way you can do it is by trying to take it away from them. We didn't short the banks--there's not much to be gained there, these were all these complex instruments, options and so forth. We'd been building our positions for a while...when they went to the wall we made $20 bln for our clients, half a billion for the Black Swan fund.

he was actually being borderline deceitful. Upon being pressed NNT admitted that he really had made about 250-500 million dollars, and not the 20 billion which was his notional exposure.

Sure, whats a factor of 40 or 80 between friends?

But then, Jim Rogers, caught him on that as well. Making 0.25-0.5B on a 20B exposure is a 1-2% return, not the super-sized return you would expect for all the waiting and bleeding you've been doing.

In the linked articles, Janet Tavakoli also attacks his claim of having been one of the first to foresee the present crisis. What does she get in return? This hilarious stuff! He puts a big yellow post it note on the GQ story as posted on his webpage:

Note that NUMBERS are wrong. This is not a business/finance, but a philosophy article written by Will Self. So read the article for its ideas. Janet Tavakoli used the errors as a platform for her (failed) smear campaign.

I have very, very stupid enemies.

As Tavakoli says "he plays the victim and resorts to unwarranted name calling when asked legitimate questions."

Warren Buffett on Gold

Following up on gold, here are a couple of interesting perspectives on gold from the world's most famous investor.

Gold gets dug out of the ground in Africa, or someplace. Then we melt it down, dig another hole, bury it again and pay people to stand around guarding it. It has no utility. Anyone watching from Mars would be scratching their head.

And another:

You could take all the gold that’s ever been mined, and it would fill a cube 67 feet in each direction. For what that’s worth at current gold prices, you could buy all — not some — all of the farmland in the United States. Plus, you could buy 10 Exxon Mobils, plus have $1 trillion of walking-around money. Or you could have a big cube of metal. Which would you take? Which is going to produce more value?”

Is gold a good investment at these prices?

Here (pdf) is a balanced viewpoint by Howard Marks of Oaktree. You can suspect a bona fide attempt to look at all sides of the argument, when he starts with:

In 1952, Noah S. “Soggy” Sweat, Jr., a member of the Texas House of Representatives, was asked about his position on whiskey.  Here’s how he answered: 

If you mean whiskey, the devil’s brew, the poison scourge, the bloody monster that defiles innocence, dethrones reason, destroys the home, creates misery and poverty, yea, literally takes the bread from the mouths of little children; if you mean that evil drink that topples Christian men and women from the pinnacles of righteous and gracious living into the bottomless pit of degradation, shame, despair, helplessness, and hopelessness, then, my friend, I am opposed to it with every fiber of my being.  

However, if by whiskey you mean the oil of conversation, the philosophic wine, the elixir of life, the ale that is consumed when good fellows get together, that puts a song in their hearts and the warm glow of contentment in their eyes; if you mean Christmas cheer, the stimulating sip that puts a little spring in the step of an elderly gentleman on a frosty morning; if you mean that drink that enables man to magnify his joy, and to forget life’s great tragedies and heartbreaks and sorrow; if you mean that drink the sale of which pours into Texas treasuries untold millions of dollars each year, that provides tender care for our little crippled children, our blind, our deaf, our dumb, our pitifully aged and infirm, to build the finest highways, hospitals, universities, and community colleges in this nation, then my friend, I am absolutely, unequivocally in favor of it.  

This is my position, and as always, I refuse to compromise on matters of principle. 

Sweat’s response shows, depending on how you look at it, either how views can diverge on a given subject or how differently a tale can be spun.  Thus it serves well to introduce the topic of this memo: gold.  

And a little further on:

My view is simple and starts with the observation that gold is a lot like religion.  No one can prove that God exists . . . or that God doesn’t exist.  The believer can’t convince the atheist, and the atheist can’t convince the believer.  It’s incredibly simple: either you believe in God or you don’t.  Well, that’s exactly the way I think it is with gold. Either you’re a believer or you're not.

 It is a great read. Check it out.

John Hussman Commentary

I find John Hussman's weekly commentaries on financial markets very informative, mostly because they are very different from noise that important looking "experts" with (faulty?) crystal balls spew on CNBC. Sometime back, I linked to this page, which clearly demonstrates that forecasters, as a group, merely extrapolate the past into the future.

This is a great disservice, because most of the numbers that these experts divine, pretty useless from a practical standpoint. In addition, the mask of conviction with which these experts conduct themselves instills a false sense of confidence in people who like to listen to such talking heads.

I like John Hussman's balanced commentaries because of the how heavily he uses concepts of probability, and Bayesian inference in his analysis. I wish they taught this stuff more widely. For example, in his most recent commentary linked above, he comments about the Gulf of Mexico oil-spill:

With regard to oil spills, however low one might have believed P(we'll have an oil spill) to be, prior to the recent accident, the "prior" probability estimate should change given that we've now observed one of the worst oil spills in history. Even if the oil industry previously argued that the probability of an oil spill was one in a million, it's hard to hold onto that assessment after the oil spill occurs, unless your faith in the soundness of the technology is entirely unmoved in the face of new information.

(John Maynard Keynes would have paraphrased it as: "When the facts change, I change my mind. What do you do sir?")

Hussman goes on to do a back-of-the-envelope calculation which suggests that since the number of deep sea oil rigs has increased dramatically, the chances of seeing catastrophic oil spills are actually quite significant.

Behavioral Finance Lectures

I have found behavioral finance to be an interesting connection between psychology, decision-making and economics. Wikipedia defines it as:

a separate branch of economic and financial analysis which applies scientific research on human and social, cognitive and emotional factors to better understand economic decisions by consumers, borrowers, investors, and how they affect market prices, returns and the allocation of resources.

Here are a couple of lecture series on this subject:

1. Russel James at the University of Georgia (via Farnam Street, and Simolean Sense)
2. Sanjay Bakshi's "Behavioral Finance and Business Valuation" class at MDI.

They are both quite fascinating, and present a great synthesis which make them worth a look even if you are familiar with many of the ideas.

Interesting Economics/Finance Links

Three interesting links for the weekend.

1. A fascinating article in Vanity Fair on the state of Harvard's endowment (Rich Harvard, Poor Harvard). It's hard for me to really feel sorry, although I know it hurts a lot of innocent bystanders. From the article:

Only a year ago, Harvard had a $36.9 billion endowment, the largest in academia. Now that endowment has imploded, and the university faces the worst financial crisis in its 373-year history. Could the same lethal mix of uncurbed expansion, colossal debt, arrogance, and mismanagement that ravaged Wall Street bring down America’s most famous university?

2. This NYT article (free sign up required) recounts how the governor of India's Reserve Bank, Y. V. Reddy, played it tough during the bubble years, and saved the country from a financial crisis. He seems like the anti-thesis of former Fed-chairman Alan Greenspan, both in action and in popularity. From the article:

Unlike Alan Greenspan, who didn’t believe it was his job to even point out bubbles, much less try to deflate them, Mr. Reddy saw his job as making sure Indian banks did not get too caught up in the bubble mentality. About two years ago, he started sensing that real estate, in particular, had entered bubble territory. One of the first moves he made was to ban the use of bank loans for the purchase of raw land, which was skyrocketing. Only when the developer was about to commence building could the bank get involved — and then only to make construction loans. (Guess who wound up financing the land purchases? United States private equity and hedge funds, of course!)

Seeing inflation on the horizon, Mr. Reddy pushed interest rates up to more than 20 percent, which of course dampened the housing frenzy. He increased risk weightings on commercial buildings and shopping mall construction, doubling the amount of capital banks were required to hold in reserve in case things went awry. He made banks put aside extra capital for every loan they made. In effect, Mr. Reddy was creating liquidity even before there was a global liquidity crisis.

3. An interesting email conversation  (pdf) between Buffett and Raikes, regarding Microsoft and Berkshire (via Reflections on Value Investing).

Taking Stock

These days I find myself browsing a lot of financial websites, hoping to catch the next big slide. No, my intention is not really to avoid it, since I have a full-time job and don't believe in market timing anyway. But really, to do some bottom fishing once the party starts. I thought I'd bookmark some of the sites I visit often, in no particular order.

Yahoo Finance: I started using this a long time ago, when I bought my first few shares on the stock market in 2001. I retain only Priceline.com and was foolish not to take a significant position. Anyway I was (am?) still learning.

Seeking Alpha: Really a anthology of random blogs. It is funny these days I turn to blogs and the comments section of mainstream media articles for more nuanced information (financial or political).

Investopedia: Lots of fundas, written legibly. It also has a nice collection of articles on recent happenings, but like I said - it is best for knowledge, not information.

wikinvest: Wikipedia's little brother. I like the relatively clean interface (not as advertizementless as wikipedia), and contains fairly good information on liquid stocks.

gurufocus: Good place to see what investors I admire are up to.

Motley Fool: Used to be a vastly superior site. Now it reeks of commercialism. They want to sell you something every step of the way. But still, CAPS is a good filter for good ideas.

Other value investing blogs I like are here, here and here.