Lesson from coin flip investing

The setup

  • You invest in 2 coins every week for the next 1000 weeks (19.2 yrs)
  • These coins pay a return each week
  • Every 4 weeks, you rebalance wealth equally between the 2 coins
  • Coins have an expected edge of 10%
  • Simulation is run 10,000x
  • Assume no transaction costs

Individual Coin Payouts

Coin Win Payout Loss Payout Expected Annual Return Expected Annual Volatility
A(Low Vol) 2.75% 2.50% 6.70% 18%
B (High Vol) 8.25% 7.50% 21.5% 54%

Results of the 2 Coin Portfolio1

Strategy CAGR Volatility Median Return Max Drawdown
Theoretical  14.1% 28.5% 10%2
Un-rebalanced simulated 17.9% 32% 6% 68%
Rebalanced simulated 13.9% 30% 9% 64%

Observations from many simulations like the one described

  1. The higher the portfolio volatility, the more the mean and median diverge
  2. Rebalancing pushes median returns closer to the theoretical mean
  3. The rebalancing benefit is positively correlated to the difference of volatility between the coins

How much to wager when you have edge? (Hint: median not mean outcomes!)

Link: Rational Decision-Making under Uncertainty: Observed Betting Patterns on a Biased Coin

  • Optimal bet size as a fraction of bankroll is 2p-1 where p is the probability of winning1. You will recognize this as the edge per trial reported as a percent. So a 60% coin has 20% expected return or edge
  • The formula is a solution to a proportional betting system which implicitly assumes the gambler has log utility of wealth

Imagine tossing a 60% coin 100x and starting with a $25 bankroll

Arithmetic Mean Land

The mean of one flip is 20% positive expectancy

Optimal bet size is 20% of bankroll since you have .20 expectancy per toss

Increase in wealth per toss betting a Kelly fraction: 20% of bankroll x .20 expectancy = 4%

Expected (mean) value of game after 100 flips betting 20% of your wealth each time

$25 * (1+.04) ^ 100 = $1,262

Median Land

The median of one flip betting a Kelly fraction is (1.2^.60 * .8^.40 – 1) or 2%

Median value of game after 100 flips betting 20% of your wealth each time

25 * (1.2^60) * (.8^40) = $187.25!

Things to note

  • The median outcome by definition is the increase in utility since Kelly betting implicitly assumes the gambler has log utility
  • After 100 flips, the median outcome is only about 1/10 of the mean outcome! The median outcome gives an idea of how much to discount the mean payoff. If your utility function is not a log function (ie does quadrupling your wealth make you twice as happy) then a different Kelly fraction should be used

Percents Are Tricky

Which saves more fuel?

1. Swapping a 25 mpg car for one that gets 60 mpg
2. Swapping a 10 mpg car for one that gets 20 mpg

[Jeopardy music…]

You know it’s a trap, so the answer must be #2. Here’s why:

If you travel 1,000 miles:

1. A 25mpg car uses 40 gallons. The 60 mpg vehicle uses 16.7 gallons.
2. A 10 mpg car uses 100 gallons. The 20 mpg vehicle uses 50 gallons

Even though you improved the MPG efficiency of car #1 by more than 100%, we save much more fuel by replacing less efficient cars. Go for the low hanging fruit. The illusion suggests we should switch ratings from MPG to GPM or to avoid decimals Gallons Per 1,000 Miles.

Think you got it?

Give “deflategate” a go. The Patriots controversy brought attention to a similar illusion — plays per fumble versus fumbles per play.

If you deal with data analysis you have probably come across the problem of normalizing data by percents and the pitfalls of dividing by small numbers (margins, price returns, etc).

The MPG vs GPM illusion is more clear if you are comfortable with XY plots from 8th grade math recap. Look at the slopes of x/1 versus 1000/x (in this case think of Y=M/G and the recipricol as gallons per mile. I multiplied gallons/mile by a constant 1000 to make the graph scale more legible).

The Volatility Drain

I don’t want to torment you this week, but if you trust me play along and you’ll be paid off with some non-obvious lessons.

Imagine the wish you made on your 10-year-old birthday candles comes true. You are magically given $1,000,000. But there’s a catch. You must expose it to either of the following risks:

1) You must put it all on a single spin at the roulette wheel at the Cosmo. You can choose any type of bet you want. Sprinkle the wheel, pick a color, a lucky number, whatever you want.


2) You can put all the money in play on a roulette wheel that has 70% black spaces. Place any bet you want, but you must bet it all. And one more catch…you are required to play this roulette wheel 10x in a row. Your whole bankroll including gains each time.

Think about what you want to do and why. Even if you cannot formalize your reasoning, take note of your intuition. I’ll wait.

Let’s proceed.

First of all, the correct answer for anyone without a private jet is #1. Just spread your million evenly, pay the Cosmo its $52,600 toll and try not to blow the rest of it before you get to McCarran. For many of you who computed the positive expected value of option #2 then you might feel torn.

Welcome to a constrained version of the St. Petersburg paradox.

The expected value of a single spin with a million dollars spread over the favorable blacks is $400,000 (.70 x $1,000,000 – .30 x $1,000,000). A giant 40% return.

But if you are forced to play the game 10x in a row, there is a 97% you will lose all your money (1-.70^10).

What’s going on?

This problem highlights the difference between arithmetic or simple average return vs a compounded return. If you made 100% in an investment over 10 years, the arithmetic average would be 10% per year while the compounded annual return would be 7.2%. I won’t demonstrate the math, but you can always ask me or just Google it. The mechanics are not the point. An understanding of the implications will be, so hang on.

In option #1, you will be in simple return land. In option #2, you are in compounded return land. Compounded returns are not intuitive, but they are much more important to your life. Let’s see why.

Sequencing and the geometric mean

  • Compound returns govern quantities that are sequenced such as your net worth or portfolio. If you earn 10% this year, then lose 10% next year, you are net down 1%., right? While the arithmetic average return was 0% per year, your compound return is -.50% per year (.99^2 – 1).
  • Let’s thicken the plot by increasing the volatility from 10% to 20%. If you win one and lose one, your arithmetic mean is 0, but now your compound return is -2% per year. Interesting.
  • Let’s turn to Breaking The Market  to see what happens when we tilt the odds in our favor and really ramp the vol higher. In his game, a  win earns 50%, while a loss costs you 40%.
    • The expected value of betting $1 on this game is 5%. But this is the arithmetic average. The geometric average is a loss of 5%!
    • If you played his game 20x, your mean outcome is positive but relies on the very unlikely cases in which you have an almost impossible winning streak. You usually lose money.
    • As BTM explains: Repeated games of chance have very different odds of success than single games. The odds of a series of bets – specifically a series of products (multiplication)- are driven by, and trend toward, the GEOMETRIC average. Single bets, or a group of simultaneous bets -specifically a series of sums (addition)-, are driven by the ARITHMETIC average.

The most important insights to remember!

  • Arithmetic means are greater than geometric means; the disparity is a function of the volatility.
  • Mean returns are greater than median and modal returns (Wikipedia pic). In other words, even in positive expected value games, if the volatility is high and you bet the bulk of your bankroll, your most likely outcomes are much worse than the mean. 


Using this in real life

Step 1

Recognize compounded returns when you see them. We have already seen them in the domain of betting and investing. 

Consider these questions.

  • I want to raise the price of my product by 60%, how many customers can I lose while maintaining current revenue?
  • If CA experiences a net population outflow of 20% in the next 20 years, how much would it need to raise taxes on those that stayed behind to make up the shortfall?
  • If muscle burns 2x as much calories at fat and I lose 40% of my muscle mass, how much less calories will I burn while at rest?

After groping around with those you may have found the general formula:  X / (1-X)

Credit: emathhelp.net

If you lose 20%, you need to recover 25% to get back to even. Lose 50%, and you need 100% to get back to even. 100% volatility and you are certain to go broke. Look at the slope of that sucker as you pass 2/3.

In other words, negative volatility is a death spiral. Let the brutality of the math sink in.

Why has nearly every real estate developer you know went bust at some point? Because they are in the most cyclical business in the world and love leverage. Leverage amplifies the volatility of their returns by multiples. Compounded returns are negatively skewed. Mercifully for them, zero (aka bankruptcy) is an absorbing barrier.

Step 2

Protect Yourself

  • Diversify your bets. In the earlier casino example, if you could divide your million dollars into 10 100k bets you would now have a basket of uncorrelated bets. If you could bet 1/10th of your bankroll on 10 such wheels you’d expect to make 400k in profit (7 wins out of 10 spins). With a standard deviation of 1.45 you now have a 95% chance of getting at least 5 heads and breaking even on the bet instead of a 97% to go bust in the version where you bet everything serially.
  • When a bet is very volatile, reduce your bet size. If you put 100% of your net worth into a 20% down payment on a home you lose half your net worth if housing prices ease 10%. In investing applications, variations of Kelly criterion are good starting points for bet sizing.
  • Remember that for parallel bets to not be exposed to disastrous volatility, your investments must not be highly correlated. Having a lot of investment in the stock market and high beta SF real estate simultaneously is an illusion of diversification. Likewise, if you own 10 businesses, you will likely want them in separate LLCs. For those in finance, you will immediately recognize the divergence in interests between a portfolio manager of a multi strat fund and the gp of the fund. Izzy Englander wants his strategies to diversify each other while he gets paid on the assets, while the individual PM wants to take maximum risk. Izzy risks his net worth, the PM just her job. If you take one thing away from this paragraph: a basket of options is worth more than an option on a basket.
  • Insurance is by necessity a negative expected value purchase. You buy it because it ensures financial survival. In arithmetic return land it’s a bad deal, but if the insurance avoids ruin, it may have a profoundly positive effect on compounded returns which is what we actually care about.
  • Finally, the power of portfolio rebalancing. If you hold several uncorrelated assets, by rebalancing periodically you narrow the gap between the median and mean expected returns. This is more apparent if there is wide differences in the volatilities of your assets.
    • I ran a bunch of Monte Carlo sims on “coin flip assets” with positive drift. Some takeaways were a bit surprising.
      • If the volatility of your portfolio is about 9% per year, median returns are about 90% of the mean returns. At this level of volatility, rebalancing has little effect.
      • If the volatility of your portfolio is about 15% per year, median returns are about 50% of the mean returns if you rebalance.
      • Rebalancing actually lowers your mean returns when the volatility of the portfolio is high even though it raises the median. My intuition is by taking profits in the higher volatility assets it truncates the chance of compounding at insane rates, but it also cuts the volatility by so much that it provides a much more stable compounded return. The higher the volatility the more of the mean return is driven by highly unlikely right upside moves.
      • The impact of high volatility is stark. It is extremely destructive to compounded returns.
For finance folk and the curious
  • Compounded returns are negatively skewed. Black-Scholes option models use a lognormal distribution to incorporate that insight. The higher the volatility, the greater the distance between the mean and mode of the investment. Example pic from Quora.
    • A recollection from the dot com bubble. Market watchers like to say the market was inefficient. The options market would disagree. Stock prices and volatilities were extremely high reflecting the fact that nobody understood the ramifications of the internet. Had you looked at the option-implied distributions is was not uncommon to see that a $250 stock had a modal implied price of $50. To be hand-wavey about it, the market was saying something like “AMZN has a 10% of being $2050 and a 90% chance of being worth $50.” In other words, if you bought AMZN there was a 90% chance you were going to lose 80% of your money. If you are itching to get technical on the topic Corey Hoffstein’s paper explores how risk-neutral probabilities relate to real-world probabilities.
    • For option wonks, (assuming no carry costs) you’ll recall the concept of variance drain. The median expected stock price is S – .5 * variance. The mode is S – 1.5*variance. The higher the variance, the lower the median and mode! The distribution gets “squished to the left” as the probability the stock declines increases in exchange for a longer right tail like we saw during the dotcom days.
    • The expensive skew embedded in SPX option prices reflects 2 realities. First, the average stock in the index will see its volatility increase but more critically the cross-correlation of the basket will increase. Since index option variance is average stock variance x correlation, there is a multiplicative effect of increasing either parameter. The extra rocket fuel comes from the parameters themselves being positively correlated to each other!

Levered ETF/ETN tool

Use this tool to estimate how much a levered fund would need to buy or sell to maintain its mandated levered exposure. You should make a copy of the sheet for your own use.

A few points to consider:

  • AUM changes faster than the position size by the amount of the leverage factor
  • Inverse funds require 2x the adjustment of their long counterparts! So a levered inverse SPY fund would require 2x the adjustment of a levered long SPY fund.
  • For more detailed explanation of why funds must adjust their positions see my explanation of shorting.

Preview below:

The difficulty with shorting and inverse positions

Shorting is hard

Shorting assets is intrinsically difficult because
  1. while your position goes against you it gets bigger
  2. and when you win your position is getting smaller
Consider the impact of a $1mm fund that is designed to mimic a $1mm short in stock X.
  • X down 50% scenario

    • The fund earns 50% return. So now the fund has $1.5mm aum and the short is only $500k. For the fund to match the return of X going forward the fund must now triple its position.
      • Note this requires selling into a declining market (negative gamma)
      • The fund must keep its initial Position/AUM ratio constant. So initially this was 1:1 but then became 1/3 which is why it needs to triple the position
  • X up 50% scenario

    • If the stock increased 50% the fund loses 50%. Its AUM is $500k and the short is $1.5mm. The fund must cover much of the short.
      • The fund must buy in a rallying market (negative gamma).
      • The new position/AUM ratio is 3:1 so the fund must buy back $1mm or 2/3 of its position so that its AUM is $500k and its position is $500k. In this case the fund is insolvent.

Inverse ETFs and ETNs

The above dynamic is also how an inverse ETF or ETN work. The ETN must match the inverse return of a reference asset. So if all the AUM is exposed to the asset then we calculate the fund PositionSize/AUM.
  • NAV = AUM / Shares Outstanding
  • The down case

    • As the reference asset moves lower the fund must sell more of it to maintain the PositionNotional/AUM ratio. In this case, as the reference asset moved lower, the fund AUM increased due to profits while its position size decreased as the price of the reference asset declined. The fund must sell enough of the asset to rebalance the initial PositionNotional/AUM ratio. Selling into a declining market. This ensures the ensuing percentage move in the reference asset corresponds to the percentage change in NAV.
    • Redemptions are stabilizing as they require the position rebalance to be smaller as the AUM declines and the reference asset is purchased
  • The up case

    • As the reference asset rallies the fund must cover its notionally increasing short. PositionSize is increasing while AUM declines, so the reference asset must be purchased to reduce the position size and again normalize the notional/AUM ratio.
    • In this case, redemptions are de-stabilizing as they reduce AUM which further moves away from its initial value and the redemption also prompts an in-kind purchase of the already appreciated reference asset.

In sum:

  • For inverse etfs to maintain a constant exposure in return space to their reference asset they must rebalance such that the dollar size of the underlying position is a fixed ratio to the AUM.
  • The inverse nature means that the AUM and position size are always moving in opposite directions requiring constant rebalance (negative gamma). This creates a downward drift to the product NAVs.
  • As the reference asset rallies, position size gets bigger and AUM drops due to losses. As reference asset falls, position size shrinks while AUM increase due to profits.
  • Redemptions can stabilize rebalance requirements in declines and exacerbate rebalance quantities in rallies as redemptions reduce shares outstanding and in turn AUM while in both cases triggering the fund’s need to buy the reference asset which again is stabilizing after declines but not after rallies. In other words, profit-taking is stabilizing while puking is de-stabilizing.
  • I extend this explanation to levered funds here.

Adam Robinson’s Game Theory Approach to Markets

Distilled from his interview with Shane Parrish on the Knowledge Project

Markets are smart

When people or in disagreement with prices or confused they are in denial or are missing something from their model

Dunking on fundamental value investing

  • Relies on Ben Graham’s undefined notion of “intrinsic value”
  • It is defined by “the value justified by the facts”. This is a meaningless definition. Like “gravity is when things go down”.
  • Thinking fundamental investing works is hubris. You must believe:
    1. There is a true value
    2. You can ascertain it
    3. Others will come around to your view in a reasonable timeframe
  • What about Buffet and Munger?
    • They hold things forever.
    • They are geniuses.
    • It is a stretch to attribute their success to this idea of fundamental investing.

Dunking on technical analysis

  • Exercise in confirmation bias and data mining

Adam’s approach: Game theory

  • He doesn’t try to predict market prices. He follows the smart money
  • The market is a predatory ecosystem. Books like Peter Lynch “One Up on Wall Street” give retail the illusion they can win in what is a ‘gladiatorial pit’
  • Keynes who was also a great investor described investing: “How do we anticipate the anticipation of others?”
  • What pattern of behavior have you seen that correlates with a different future?
    • People placing bets are wagers on a view of the future
    • His favorite investing book is not an investing book: 1962’s Everett Roger’s “The Diffusion of Innovation”
      • A trend at its core is the spread of ideas
      • Roger’s decomposes the lifecycle of an idea. Early adopters are ridiculed, the masses begin to come around, the idea is enshrined and seen as ‘self-evident’
  • His ordering of traders and how they express their views. Traders near the top of the order will be “right” on a lagged basis. The giant caveat is that these orderings may not have applied as strongly before the 2000s because he claims the world was different (different investment flows, presence of EU, etc). But he makes the case they still held. He looks for strongly divergent views between asset classes to make probabilistic bets on the future. He prefers this because it is the expression of bets vs say using economic statistics. You don’t trade statistics, you trade assets.
    1. Metals traders sentiment is proxied by the copper/gold ratio. They are the “Forrest Gumps” of the investing world —simplistic. They are the closest to economic activity. They are very far-sighted because of mine timelines. They have never been wrong in the past 18 years on the direction of interest rates. In September 2018, during this conversation, the copper/gold ratio implied that interest rates should be at 1-year lows instead they were at 1-year highs. He thinks the metal traders will again be right, they are just early. (9 months later as I write this, interest rates have gone back to 1-year lows!)
    2. Bond traders sentiment proxied by the ratio of LQD/IEF. Basically, credit spreads
      • When they disagree with equity traders, the bond traders tend to be right and early
    3. Equity traders
    4. Oil traders sentiment reflected in XLE vs SP500 spread. The price of oil is less reliable because of sovereign intervention
    5. FX traders sentiment reflected in commodity currency crosses
    6. Economists: Always wrong as a group
    7. Central Bankers: not in touch with the real economy; rely on models only. And economists
  • 3 Ways a Trend Can Form
    1. A stock very sharply reverses a long-standing trend. The trend needs to have been in place for a long time (long is ambiguous; he says ‘months’ or ‘years’). The stock will retrace after its sharp move but if it runs out of gas then the early adopter of the new direction are starting to win converts
    2. Parabolic moves precede a change in direction and a new trend in the opposite direction (reminds me of dynamics of a squeeze)
    3. An asset in a long-established, tight range starts to break out. The less patient hands have been transferring their position to hands that have more conviction evidenced by them willing to wade into a dead name.

His ranking model jives with how I think about trading

  • The science part of trading is the constant measuring of market prices and implied parameters.
    1. Rank which markets are the most efficient
    2. Find the parameters which are in conflict with one another
    3. The parameters in the less efficient markets that conflict with more efficient markets represent an opportunity set
  • The art part is to then investigate why those parameters are priced “inefficiently”.
    • Flow-based? Who’s the sucker? Who’s better capitalized?
    • Behavioral? Confirmation, anchoring, recency biases? Others?
    • Is there an aspect of the inefficient market that is unaccounted for and therefore not normalized for in the comparison to the efficient market?

Are car leases confusing?

We leased a Toyota Highlander this year and found leases a bit trickier than meets the eye. Being in finance this is embarrassing to admit. With the risk of sounding of being obvious let’s have a look.

Think of a lease payment as having 2 components:
  1. Depreciation. For a 36 month lease, you will use ‘consume’ the vehicle
  2. A loan. During the 36 months, you are borrowing the amount of vehicle that has not been depreciated. A bank actually owns the vehicle and you are borrowing the yet “unused” portion of it. This is the portion you pay interest on. Lease lingo calls it a “money factor”. To convert a money factor to APR just multiply by 2400.
So how do you evaluate the cost of a lease?

There are 3 levers (sales price, residual and money factor) the salesperson can play with and the interaction of the levers is what makes shopping leases complicated. The “residual value” is the buy-out price of the lease at the end of the term, and is represented as a percentage of the purchase price.

Let’s pretend you are looking at a $50,000 car for a 36-month lease. Let’s say the residual is $30,000 or 60% and the money factor is .00001 (ie 2.4% APR). Your lease payment has a depreciation component of $20,000, the amount of car you will consume, divided over 36 months plus a financing charge of 2.4% divided by 12 months times the amount of car remaining. These numbers average out to $555 for the depreciation and $80 for the financing charge for a total payment of $635 per month.

It’s typical to want a higher residual which translates into less depreciation but here’s the catch — the higher the residual, the more car you are “borrowing”. So a high residual AND a high money factor will lead to smaller depreciation expenses but HIGHER financing charges making the lease more expensive than a lower residual lease.

You may find that if you buy the car outright you get quoted a different price. It may be the case that you could buy the car and sell it after 3 years, effectively creating your own lease, with more favorable economics but remember that the lease is an option to buy or as I prefer to call it — an option to sell (it back to the dealer). For the finance inclined, it’s actually a put struck at the residual value. If the car is worth more in the secondary market in 3 years than the residual, you will buy the car at the residual and flip it. If not, you will simply ‘put’ it back to the dealer.

Here’s my spreadsheet allowing you to compare leasing, buying, and what we like to do — a one pay lease, where you make all the payments up front in exchange for a lower money factor. I put a field for “savings account rate” in it to be complete about your opportunity costs when borrowing less money. The calculator assumes no money down and no taxes. Taxes are state specific, and if you do put money down then whether it improves or detracts from the economics depends on whether your loan amount is at a higher or lower rate than your savings account.

This table shows the interaction between money factor and residual. For a given APR you always want a higher residual. It gets tricker when comparing across both axes.

Links to learn about private markets

Canon (IMO)
http://reactionwheel.net/  (Jerry Neumann, angel and professor)
http://www.paulgraham.com/articles.html  (Paul Graham, founder of Y Combinator)
https://continuations.com/ (Albert Wenger of USQ Ventures)
https://stratechery.com/ (Ben Thompson)
The smartest writing I’ve seen after the canon
http://kwokchain.com/ (Kevin Kwok)
https://www.ben-evans.com/ (Benedict Evans)
https://sivers.org/ (Derek Sivers)
https://startupboy.com/ (Naval Ravikant)
A listen:
Non-Venture Business
Amazing resources
And this long-form Breaking Smart from Venkat Rao (here’s my notes)
More blogs that are worthy to follow:
https://superlp.com/  (Chris Douvos)
https://avc.com/   (Fred Wilson)
https://a16z.com/ (Andreesen Horowitz team)