How Math Is Sufficient To Explain Small Stock Outperformance

Takeaways from Diversification, Volatility, and Surprising Alpha by Fernholz et al. (Link)


  • It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly simple, systematic investment strategies including equal and random-weighted portfolios.
  • This outperformance is generally attributed to beneficial factor exposures.
  • It turns out this outperformance needn’t invoke factors. It can be explained by stochastic math where correlation and variance play a larger and more predictable roles than returns.
  • Portfolio logreturns can be decomposed into an average growth and an excess growth component. They argue the excess growth component plays the major role in explaining the outperformance of naıve portfolios.

Some basics

Let’s establish some basic definitions.

Stock Returns

There are 3 types of returns commonly used to describe growth rates. But they are not equal.

Arithmetic returns > Geometric Returns > Log Returns

This is important because only logarithmic returns are an unbiased estimate of expected long term returns.  In other words, arithmetic and geometric returns will overestimate expected growths in wealth.

  • Logreturn of an asset= Arithmetic return – .5 * variance
  • .5 * variance is known as the volatility drag or variance drain. I’ve discussed this here in simple terms.

Portfolio Returns

The logreturn of a portfolio can be decomposed into 2 components:

Weighted avg of stock logreturns  + “excess growth rate” (aka EGR)

Understanding the EGR

EGR = (weighted average stock variance – portfolio variance) / 2

The relationship between stock variance and index variance

Now this part is not in the paper, but taking from my index options experience:

Portfolio variance = weighted average stock variance * average cross-correlation of the stocks

This is a common identity used to price index options. It makes intuitive sense.

  • If all components of the portfolio has a correlation of 1 the portfolio variance would be the same as the underlying stocks.
  • If you had a 2 stock portfolio and the correlation were -1 the portfolio variance would be zero. Iimagine a basket comprised of 50% SPY and 50% inverse SPY. It would never move in price (assuming no fees, frictions, etc) regardless of how high SPY variance was.
  • For an average correlation < 1,  the portfolio variance must be less than the average weighted stock variance.

The key insight: the lower the average correlation between the components the wider the spread between the portfolio and weighted average stock variances!

Back to the paper…

Observations about the EGR

Looking at the formula again:

EGR = (weighted average stock variance – portfolio variance) / 2

  • EGR boosts the portfolio returns beyond that of its components since portfolio variance < weighted average stock variance
  • EGR boosts portfolio returns with lower correlations
  • EGR boosts portfolio returns with high stock variance

Relationships Between Market Cap, Logreturns, and Variance

The authors then use a rank based computation to show:

  • Logreturns of individual stocks do not vary by market cap.
  • Variances of individual stocks do vary by market cap. Smaller stocks are more volatile.

This prompts the great reveal:

Small stocks don’t have higher returns but have higher variances which boost EGRs. The volatility and interaction of the stocks is boosting the portfolios that contain them without any need to rely on factors! The increased volatility of the individual stocks did not earn them a risk premium when considered in isolation, but at the portfolio level they contributed to excess growth.


  •  The authors contend that the excess growth component can be estimated relatively easily, since its value depends only on variances, or relative variances, which are not difficult to determine in practice. The average growth component, however, is more difficult to estimate. 
  • Small stocks are riskier and while this might mean higher single period arithmetic returns long term investors care about logreturns. In logreturn space, individual stocks don’t contribute excess returns. This is at odds with conventional wisdom.
  • Instead, the excess returns are coming at the portfolio level via the small stocks’ contribution to “excess growth rates” (EGRs).
  • They tested the expectations of this stochastic portfolio math on 5 commonly employed weighting strategies, some more diversified and some less diversified than the capitalization-weighted portfolio, confirmed these insights. In general, the more diversified portfolios outperform and the single less diversified portfolio underperforms, because the more diversified portfolios have a higher excess growth rate. This arises from the higher variances associated with the smaller stock exposure in these more diversified portfolios, and not because such stocks have inherently higher returns. This higher excess growth rate, in turn, increases the portfolios’ logarithmic return.

My Comments

  • The role of low correlation was not emphasized enough in the paper considering it drives the EGR by setting the gap between portfolio and average weighted stock variances.
  • You should read the paper if you’d like a refresher on computing arithmetic, geometric, and logreturns.
  • You should read the paper to see how they computed rankings since this work established that there was no relationship between logreturns and the market cap of a stock.

Moontower Gift Guide for 2019

It’s the official start of holiday season. I thought this would be a good time to share some of my favorite finds from 2019 including a gift guide in advance of Cyber Monday.

Moontower Gift Guide

Everything in this list is something I have gotten great use out of and found to be reliable. Several of them have been secret Santa gifts I’ve been given. You can’t go wrong buying them for your white elephant exchange or jumping on them if someone else brings it to the party.

1. A car jump starter. I have a VicTsing model (link)

  • Doubles as a high capacity power bank
  • I keep this in my car and have used it many times to give myself a jump without anyone’s help as well as used it to help other motorists. It’s a quicker, easier process than putting 2 cars nose to nose.

2. A 12v/120v power inflator. There are many types to choose from. I have this Bonaire model (link)

  • Set target tire pressure and hit ‘go’. It auto shuts off when the tire is filled. The best part — a digital guage.
  • Comes with regular 120v power cord and the 12v that plugs into your car’s ashtray. I had to use that one on Friday actually.
  • Comes with multiple needles for filling basketballs or bike tires.

3. Asus ZenScreen. My laptop has a 13″ screen but the ZenScreen connect via USB to give you another monitor. (Link)

  • Draws power from your laptop so no need for a separate power supply
  • The included case acts as a stand allowing you position the monitor horizontally or vertically

4. Youtube Premium (Link)

  • I watch more Youtube than I do TV. If you have the same habit, it’s worth the monthly subscription to get rid of the ads. I started paying for it this year and I’m happier for it.

5. Youtube TV (Link)

  • If you are thinking about cutting the cord, I just wanted to share our solution. We switched to Youtube TV in January and we have been happy. It has all the channels we want and there is no lag when you change channel. The lag was always a pet peeve from our Comcast setup. The quality is good and we are saving a little bit of money.

6. Shady Rays Sunglasses (Link)

  • I like their aviator styles. I wrote this summer: I ordered some Aviator style sunglasses from Shady Rays. They are about $60 bucks. If you break or LOSE the glasses up to 2 times they will send you a new pair for the cost of shipping. 

7. A massage gun. I got a Hypervolt (Link)

  • For the person who likes when the masseuse karate chops their back rapidly Street Fighter II- style. They are pricey but if that therapy is your jam, these guns are pretty awesome. Get your kids or partner to do your back. You can do your legs, IT band, etc yourself. There’s 3 intensity settings and it lasts several hours from a single charge.
  • I actually got a knockoff from China for a fraction of the price and it works perfectly. If you are interested and in the Bay Area hit me up and I might be able to help.

8. “Demonslayer” sake (Link)

  • At $50 a bottle this is a treat but it’s very smooth junmai daiginjo. I mentioned a bit about sake from my trip to Japan. (Link)

9. Most practical stocking stuffer

  • White Wizard Spot Remover (Link). We give this to everyone. Takes blood out of white sheets. If you have kids you keep this stuff in a holster. I have no idea if it’s destroying the environment but it’s so good I assume I’m being evil.

10. Games

Boardgames I started playing this year with my 6-year old:

  • Evolution — The Beginning: I wrote about it here
  • Forbidden Island (Link): Simple and fun coop game by the same game designer who brought you Pandemic. The game gets kids to work together and while the replayability for adults is limited there is enough variation in board layout and characters to keep kids engaged. Take about 30 minutes to play and requires no more reading than identifying the names of regions.
  • The Magic Labyrinth (Link): By far the best version of a memory game our household has ever played, No reading required and adults and kids are on equal footing since the game is about trying to remember you and others’ footsteps through a maze with invisible walls. This game was a big hit around here and is one of our favorites to gift since its fun and has few rules to learn.

Boardgames for 10 and up:

  • Settlers of Catan (Link): This was the gateway game that got us into European boardgames 11 years ago. Putting it here because we recently re-discovered it when I took a chance and taught it to Zak. Honestly, unless you are used to playing games for hours it might be a reach for age 6 but I’d feel very comfortable teaching it to an 8-year-old. Teaching someone the rules is the most painful part, but I have yet to meet anyone who did not enjoy playing. Even non-gamers. It’s a gateway game for a reason. While its conflict is economic like Monopoly, it feels less punitive and the entire design is one of the most elegant I’ve seen. Dusting it off was like seeing an old friend and wondering why you let it take so long to connect.
  • Decrypto (Link): A party game like Codenames. Both games are great for teams and so many ages. As word games go Codenames and Balderdash are hall of famers but Decrypto is an instant classic we started playing last Christmas.
  • Acquire (Link): I’m putting this game here even though it’s several generations old. It’s a classic game of M&A and stock ownership using the hotel industry as the theme. Many Moontower readers are finance pros so might appreciate the rec if they didn’t know about it already. It’s in a sweet spot of complexity and has clever market-driven dynamics.
  • A general tip: normal people don’t like reading rule books. Learning rules is best done via Youtube videos. Just search for a tutorial of the game you are interested and use the rulebook as a reference. If you need even deeper rule clarifications I’m 99% confident any question you can think of is covered in relevant BGG forum.

Video Game

  • We just discovered Portal : Bridge Constructor (Link). Available on many platforms including iOS and Android. We are playing it via Xbox Game Pass which has 3 months for a $1 intro rate which gives you access to tons of free games. If you liked Lemmings growing up you will dig this game. The whole house is addicted to structural engineering principles now.

11. Finally, a rec I received from Twitter — Storyworth memoirs (Link)

  • Our knowledge of our parents before we are born is very limited. If you have ever asked them to recount their stories, you may have left the conversation wishing it was written down. And that’s just for the stuff they thought of when they were on the spot or during that evening around the table. Storyworth is a service which prompts them every day with a question that they answer and at the end of the year you can get physical and/or digital copies. I’m sure the efficacy of this depends a lot on your parent’s bandwidth, willingness, and literacy but I thought the concept was worth surfacing here. I actually think they should have a version for family recipes!

Sake Tidbits I Learned While in Kyoto

One of the highlights of the past week was spending a few hours with sake apprentice Jorge Navarette. I will do a fuller write up on his awesome story when I compile a guide to Japan based on our trip but I will just say that Jorge is probably the most knowledgable American citizen on the subject of sake. Meeting him was pretty much an accident as we just took a tour where the hotel concierge sent us but he had only been there since July.

But here’s some knowledge you can use today if you like sake or want to get into it.

  • Premium sake is defined by how much the rice is polished down. So a 60% sake has had 40% of its rice polished down before fermentation. Sake must be below 70% to be considered premium.
    • Jumai: “pure rice” typically 60-70% remaining after polish. Large variation in taste.
    • Ginjo: “fragrant” 50-60%
    • Daiginjo: “super fragrant”. Less than 50%. Very smooth, but less variation in taste
  • To give sake shelf life it is pasteurized. In Japan, unpasteurized or “draft” sake is bottled and kept cold. It is known as “nama” sake. While technically you can drink it within 6 months we learned you really want to drink it in less than 30 days. You can find it in the states but it won’t be the same as fresh namasake.
  • Sake is expensive in the US mostly because of taxes and tariffs. A good bottle of sake in Japan need not set you back more than $7.
  • A word on Japanese whiskey. Jorge could trace the boom in popularity and price spike back to a single month in time. ​November 2014. There were 2 reasons at the same time which created a perfect storm:
    1. The rise of international demand for Japanese whiskey in the wake of The Whiskey Bible ranking Yamazaki as number one in the world and ahead of Scottish brands.
    2. A surge in local demand when a documentary about Nikka Distillery’s founder aired in Japan. It told his incredible story full of passion and suffering as he traveled to Scotland to learn the craft during the WW1 era.

Yamazaki 12 went up 6x in price but this was like table whiskey before that! Jorge had been drinking it for over 20 years when it was 25 a bottle.

Guides to Visiting Japan

My wife and I went to Japan in September 2019 to celebrate our 10 year anniversary. We spent 10 days there and it was a trip of a lifetime. Japan can feel like an alternate universe if you are visiting from the US. While many Japanese speak English, they are hesitant to practice it since it will not be perfect. This kind of attitude embodies something I noticed in Japan…something I have almost certainly insensitively labeled ‘samurai culture’. Whatever Japanese people do, they do well. Like shockingly well. You think the best whiskey comes from Scotland. Try again. The best pizza comes from Italy? You haven’t been to Seirinkan in Tokyo. And you have heard of wagyu beef, right?

Shopping, normcore fashion, hipness, history, culture, landscapes. Off the charts. And the counterforces of uniformity, collectiveness, and an aging populace can all be seen as well — the pachinko parlors are the most depressing (and loud) casinos I’ve ever been in.

Some of the the fun oddities were the abundance and tiny size of what feels like a bar or restaurant per capita. I seriously can’t understand how economics works in Japan. How are there so many sushi spots that seat 4 people, or sake bars stacked 6 levels up in what look like office buildings. There are so many nooks and crannies, everywhere you go feels like at least as deep as the Big Apple. Amazing food in the train stations, baked goods I enjoyed as much as Parisian patisseries, and vending machines with Walmart-wide variety (though there is no Diet Coke anywhere — you can find a Coke Zero in McDonald’s if you’re desperate).

As Bill Murray said, “For relaxing times, make it Suntory time.”

Here’s our personal guides to Japan:

  • Japan is expensive but there is no need to tip.
  • You’ll want to carry yen but it’s straightforward to get cash at ATMs and despite warnings from others we had no trouble using credit cards in most places.
  • The Green Japan Rail passes were worth it. The cars were not crowded and comfortable. It’s best to get them in the mail in advance.
  • We rented Ninja wi-fi hotspots and picked up/dropped off at the airport. The experience is simple and the hotspot worked well almost everywhere although a bit patchy on the shinkansen.
  • You can use luggage forwarding services from hotel to hotel even in different cities. Bellhops will organize the transport of your luggage on the trains. Very handy if you want to explore on travel days and have your luggage be at your hotel when you arrive.

Our Itinerary

Friday night to Monday morning: Tokyo. We lucked out and overlapped with one of the great sumo tourneys held each year in Tokyo. Highly recommend sumo, Shibuya, the fish market, Harajuku, and just getting your 20k steps a day all over this expansive city.

Monday: Shinkansen train to Hiroshima where we caught a Carps baseball game then caught the ferry to spend the night on Miyajima Island

Tuesday: Toured the island and the ropeway before heading back to Hiroshima to visit the Peace Museum. Shinkansen to Kyoto at night. We used a guide named Naoko. Reach out if you want her info.

Wed: Shinkansen from Kyoto to Kobe to explore the city. On the way back to Kyoto in the evening made a pit stop in Osaka with a good friend. Hashigo is the Japanese word for basically bar-hopping. We bounced around Osaka, hitting bars and food stops before finding ourselves buying rounds of highballs for a wide array of partygoers.  After an expensive cab ride at 2am we found ourselves back at our hotel in Kyoto

Thursday to Sunday: Exploring Kyoto. We used a great guide named Toby who has lived in Kyoto his whole life, speaks great English, and is very well educated. He was an amazing guide to take us through the shrines and explain the history. If you are going to Kyoto you will want to see my separate post here.

Our friends’ guides who helped us plan the trip:


Hi! This is the info I have from my Japan trip. I went in 2015 so some of it might be dated but I thought the best travel site for any info on Japan (esp the Kyoto and surrounding areas) was this:


It looks like since then the guy has done other cities as well (Tokyo).

This was also a good reference for trip planning:

And for traveling around this was great too:

But Google maps has really gotten better in the past 5 years (I’ve noticed it’s accuracy more when I use it for international travel trips). Might want to download hyperdia as a cross-check/back up though.

Here are some general comments though from memory:

    • SKIP OSAKA. It’s super close to Kyoto too, but not worth a detour. Just do Kyoto, Kobe, and Miyajima.
    • There is so much to see within and right outside Kyoto, like Nara and Fushimi Inari. Kyoto is charming and more authentic and quaint and quiet vs Tokyo.
    • Kobe is an easy day trip. Definitely go for authentic Kobe beef. You can do a lunch there and head back to Kyoto, or stay and check out this beautiful park at the top. All around Japan there are these great parks you need to take cable cars up to. and this is one of them in Kobe (but not a must do):
    • Loved Miyajima. Huge oysters there but don’t make a mistake of waiting to eat there at end of day! Everything closed early evening and we missed out! We ended up getting oysters in Hiroshima before taking the train back to Kyoto. I was very sad. We saw huge ones everywhere for sale at the various food stalls in town. So first thing you should do is eat lunch there, then see the famous shrine at the beach, then go up via cable car to the top of the mountain for a hike. The island is small, you can see it in a day as long as you start early. Not much to see in Hiroshima unless you really want to see the memorial stuff. we just passed thru on the way to Miyajima and back.
    • Love Hakune! we stayed at a nice hotel there. The area is more prone to quakes tho, FYI. So you’ll prob feel tremors here and there while staying there but nothing to be alarmed about
    • There is an outlet near Hakune. Do not go, total waste of time. We only went because we needed to pass thru to catch a bus, so unless you need to do the same, don’t make an effort
    • Mt Fuji is hard to see, it’s mostly cloudy at the top so don’t be disappointed if you dont see it. We did see it eventually but most of the time from what I hear it’s not easy to see entirely naked without clouds. Lake Kawaguchiko is nice if you have time to go around there (you can see Mt Fuji from there) but not a must do
    • The Japan Rail pass works w the airport bullet trains so def get it!

Japan Travel Recommendations


  • we used DK Eyewitness Travel guide for Japan and found it useful. It’s higher level, but covers all you the items you can do probably anyway.
  • Japanese observe the more traditional meal times, meaning lunch is served from like 11:30AM to 3PM, sometimes more limited and dinner usually starts at 5:30 to like 6 PM. In between it is sometimes difficult to find a good place to eat. However, Japan also offers a lot of tea/coffee houses, which serve their traditional sweets. So all day breakfast or lunch, no no.
  • Don’t eat at restaurants, where you see only white people ;-).

Language: It is sometimes difficult to communicate, but they will always make a great effort to get you what you need. A lot of people have voice translation apps on their phone. We ate at restaurants, where nobody really understood English and still found our way around, other customers will help you too.

Train – https://www.jrpass.comOrder your train voucher online and exchange it for the actual pass once in Japan. The pass doesn’t allow you to travel on the fastest train though (fastest means less stops). It’s very convenient and saves you money. Recommend the green pass. It’s a little more comfortable and less crowded during rush hour. It is recommended to make seat reservation head of time if you need to travel during rush hours. Otherwise, just go to the train station to get your seat reservation. Everything is well organized and people stand in line before boarding. Be at the tracks a little ahead of time, the train doesn’t stop for long and sometimes trains come one after one. We once jumped into the wrong train, since it was there already, but 1 min ahead of our scheduled train. It was the fastest train that is not part of the pass. We didn’t have to pay anything, and just got off at the next train station to change to our train.Btw/ the train schedule on google is not reliable, better to use JR’s website or ask concierge.

Luggage ForwardingWhen you travel between cities/hotels, you can take advantage of luggage forwarding, if you want to do day tours on your own and don’t want to log along all your luggage. You just need to plan ahead a bit, since they require you to give them the luggage at least one day before your preferred luggage arrival time. The hotel concierge can help you with this. When they say your luggage will arrive by certain date/time, it will be there, most likely ahead of time. Costs vary depending on the destination but are usually about $15-$30 per luggage. We found this very convenient and used a few time.

Pocket WifiSome people recommend to get a pocket Wifi, so that you get unlimited data in Japan and use the pocket Wifi to connect your phones. We have T-Mobile and an international data plan and never had issues. In the trains they also provide free Wifi. Within the city and to find general direction it is good to have a data plan to make use of Google’s direction. Howard, for example, use a pocket Wifi and was very surprised about the speed and also the convenience to receive and to send it back to the company.

Kabuki TheatreIf you are interested in seeing a traditional Japanese Kabuki theatre, check with your concierge for tickets. The good theatre companies are easily booked out.

Ryokan ( recommend that you try out the traditional Japanese Inn. I probably wouldn’t stay in a Ryokan the whole trip, but two nights its worth it. We stayed at two different one, for one we gave a recommendation below, which was in Arima Onsen, its close to Kobe.


Hotel Hotel Chinzanso TokyoIt’s a little away from the hustle and bustle, but still close to subway and easy to get around. The hotel (former Four Season) has its own huge garden (one could probably say little park) and pagoda. Has a couple good restaurants.

Wiki: “For over six centuries, the land surrounding Hotel Chinzanso Tokyo has been home to wild camellias. It was featured in a wood block print by the ukiyoe artist Ando Hiroshige (1797-1858). The current name of the garden “Chinzanso” was given by Yamagata Aritomo (1838-1922), an influential member of a group of elderly statesmen known as the Genrō in the Meiji (1868-1912) and Taishō (1912–26) eras. In 1918, reflecting the will of Yamagata, the property was assigned to Baron Fujita Heitaro. Though the garden was destroyed during World War II, reconstruction began in 1948 under the direction of Ogawa Eiichi, the founder of the company now known as Fujita Kankō, whose vision of “building a green oasis in Tokyo” included the transportation of more than 10,000 trees.[3]On November 11, 1952, a grand opening ceremony for the new Chinzanso garden and reception center took place. On January 1, 2013, after 20 years of operation as a Four Seasons, the hotel was integrated with Chinzanso reception center and renamed “Hotel Chinzanso Tokyo.”


We were there for cherry blossom so we visited Ueno park. Senso-ji temple and Ginza are staples. Meiji Shrine is the most important Shinto shrine in Tokyo.

Imperial PalaceNaturally one of the must-sees in Tokyo. Get the audio guide on your phone, they have an app, which can also be used at different sites

Kabuki theatre – you are interested in this type of art, highly recommend. We watched a small section. We just finished our city tour and decided to see if they have some availability, so we got a cheap standing tickets. We think that’s enough to get an idea.

Tokyo Cruise Asakusa PierNice tour on the Sumida river and gives somewhat a good overview. At the finish point, we visited the Hama-rikyu gardens.

Tsukiji Fish Market

Formerly the largest fish market in the world. However, the tuna auction has moved to a new location and can only be observed from the second floor through glasses. We didn’t go. If that is the interest, it could still worth a trip.

Whiskey Tour Suntory – close to Tokyo. Very popular. Reserve ahead of time.


We didn’t visit that many restaurants in Tokyo, however, it has a lot of Micheline star rated places, so that makes it easier, the difficult part is to get reservations, since it’s a competitive sport to get a dinner place at good restaurants. Throughout Japan you have also simpler places, even at train stations, which are really good for our standards. Most ramen places have a vending machine, which has the name or picture of each dish. you order and pay there, walk in, give people the ticket, sit down and they deliver the food to you. We are huge noodle/ramen fans. Tried many different ramen places/brands at different cities with a little research online. We were not disappointed. What surprised us is that Japanese eat ramen with rice! That was interesting to us but we can totally understand considering the importance of rice in Japanese diet.

Zoroku Sushi Hiroo, 4 – 2 – 48 Minamiazabu, Minato-ku Tokyo, 03-6721-7255 (+81-3-6721-7255) , Chef Name : Okajima.

SF’s Hashiri Sushi’s restaurant manager, Ken Matsuura, made a reservation for us at this sushi place. Chef Okajima is the brother of the sushi Chef Tokunori at Hashiri. Its only a sushi bar, reservation required. Its in a nicer residential neighborhood. Easy to access via subway. Sushi was excellent, lots of fishes we never or rarely see here. He uses mainly seasonal fishes. We had to ask him to stop “feeding” us, we couldn’t eat anymore. His female assistant sushi chef speaks(very rare to see a woman in this male dominated industry) English, so that helped. But there is really no need to talk much, since there is not really a menu, you just sit down and decide what sake to drink. We recommend not to go just with one bottle, but with different craft recommended by them, this way you can taste different ones throughout the evening. They have a good selection.

Hashiri Ebisu/Daikanyama, 2F Digital Gate Building, 3 – 5 – 7 Ebisuminami, Shibuya-ku, Tokyo, 03-5724-3770 (+81-3-5724-3770), Chef Name : Kokubo & NoharaHashiri has also a Tokyo restaurant, the original, but we had to cancel our reservation there, bec/ our plans changed. Should be pretty good.

Here are a couple more recommendation from Ken Matsuura

Kuroiwa – Kaiseki Ebisu (Tokyo)

Ju Ju – Yakiniku – Nishiazabu (Tokyo)

Imafuku – Sukiyaki Hiroo (Tokyo)Kokoro Mai – Rice Kaiseki – Hiroo (Tokyo)Nihonshu Stand Moto in Shinjuku Great spot for rare and exciting sake and tsumami.Tsunahachi – Tempura restaurant – Shinjuku (Tokyo)Suzu Sushi restaurant – Tameike sannou (Tokyo)Sushi Ono – Sushi – Ebisu (Tokyo)Toritama Shirokane – YakitoriMoe Roppongi – YakitoriChugoku Hanten – Roppongi Great Chinese restaurant for shanghai crab Doteno Iseya – Tempura donburi – Yoshiwara (Tokyo)Isen – Tonkatsu – Akihabara (Tokyo)

Fuji Mountain

A day tour or overnight stay at Fuji Mt is recommended. In town you can stay at a traditional Ryokan, but reservations are really advised. We changed our plans last minute to catch cherry blossom at other parts of the country, so we canceled our reservation there. We had plans for Shuhoukaku Kogetsu (, since one can sit in the hot tub, look over the lake and see the mountain at the other side. Mid-week reservations are more available.


A charming mountain town west of Tokyo. Well preserved Edo period streets and houses, escaped the WWII bombing. Famous for its matsuri festival and Hida wagyu beef. (Hida-gyu is raised for 14 months, almost twice as long as wagyu beef). The wagyu beef is more reasonably priced compared with other major cities and absolutely delicious. Try it with mountain vegetable dishes and local sakes. Also highly recommend the freshly hand cut soba noodles with raw mountain yam puree. It is not common and we certainly have not found it anywhere here.

KanazawaThe birth place of samurai culture. Have a well preserved geisha district, since it escaped WWII bombing. Nagamachi Samurai Quareter retains its earthenwalled streets. Nomura house is one of the most beautiful and best preserved samurai houses. Higashi Pleasure District is full of atmosphere with old-fashioned street lamps and wooden windows.

Kenroku-en garden, another one of the “famous three” gardens.

It is also famous for gold leaves production. It supplies more than 98% of Japan’s needs.

Hakuichi Higashiyama – gold leaf store/ice cream in the Higashi Pleasure District,136.6664916,15z/data=!4m5!3m4!1s0x0:0x70f11d77388c67f!8m2!3d36.5723398!4d136.6664916

The ice cream here is covered with a large sheet of gold leaf. Super decadent. Gold leaf souvenirs sold here are very beautiful, you can also buy eatable gold leaf items, like coffee or tea with tine gold leaves.

Also recommend a sweet shop in that area. Some of the shops are hundreds of years old. Great traditional macha tea with a sweet.

The area is also famous for pottery making. Truly beautiful.



Many things to see and do. Can easily spend a week here.

Kinkaku-ji – the famous golden pavilion

Ryoan-ji Temple

Famous for its rock garden. A composition of white gravels and 15 stones is considered to be the ultimate expression of zen Buddhism. Only 14 of the 15 stones can be seen from any angle.

Higashiyama and Kiyomizu-dera Temple charming and unspoiled district with stone paved roads and lanes. Kiyomizu is a UNESCO World Heritage site and famous for its wood only construction, the entire temple is built on wooden legs on a hill. Traditionally they don’t use nails or screws, but I guess now they do.

Nijo Castle

Known for its ornate interiors and so called nightingale floors.

Sanjusangen-do Temple

The longest wooden structure in the world. Its name derives from the 33 spaces between the building’s pillars.Fushimi Inari Taisha Shrine Senbontorii – shrine with thousands of red torri gates. Its worth to walk all the way up the hill, the first part is full of tourist and further up it gets quieter and still pretty impressive. The round walk is about 1.5 to 2hr

Bamboo walk – very beautiful.

Gion DistrictFormer entertainment district and now has tons of restaurants.

Tea ceremony

Recommend to try it. It does not take long but a great experience. Kyoto is where the tea ceremony was developed.

Kamishichiken Kabuki, pronounced locally as Kamihichiken, is a district of northwest Kyoto, Japan. It is the oldest hanamachi in Kyoto, and is located just east of the Kitano Tenman-gū Shrine. The name Kamishichiken literally means “Seven Upper Houses.” It is the most authentic way to experience it. There are restaurants that provide geisha performance with dinner. But those are more commercial and designed for tourists.

Nishiki Market

The best place to try and shop for food items. We tried and found many interesting and delicious snacks and food items. We bought a variety of dehydrated vegetable and fruit crisps, including cherry tomato (our favorite, perfect texture-crisp but not too dry, with great flavor of tomato still!), shitake mushrooms, bitter melon, red chili pepper, lemon, pineapple, etc. We also tried a fermented black soy bean tea that is supposed to be good for detox and digestion. Never heard about it before. Like it a lot and even brought some home. The quality and variety here are much better than those in Osaka we feel.

RestaurantsKobe Beef Steak Mouriya Gion – didn’t had much luck with getting into any really good restaurants, this one was ok, but one can do probably better. Reservations ahead required. Our hotel had real trouble to make reservations for us.Blowfish restaurant Guenpin – restaurant is right at the entrance of the Gion district. Very reasonable priced, but reservation required.

Kara-Kusa Curry (curry dishes)We went there for lunch, since it was close to our hotel. They only offer a couple different curry dishes. Was really good!

Google Map:,135.7615117,3a,75y,90t/data=!3m8!1e2!3m6!1sAF1QipMWYnvHj54HiZLFoE9f2AvMgqcKdYD4r5xNzdRJ!2e10!3e12!!7i4032!8i3024!4m19!1m13!2m12!1scurry!3m6!1scurry!2zSG90ZWwgTmlra28gUHJpbmNlc3MgS3lvdG8sIO-8lu-8k–8kCBUYWthaGFzaGljaMWNLCBTaGltb2d5xY0ta3UsIEt5xY10by1zaGksIEt5xY10by1mdSA2MDAtODA5NiwgSmFwYW4!3s0x6001089820c1eb3d:0x1f30991ae3944209!4m2!1d135.7607782!2d35.0004694!5m3!5m2!4m1!1i2!3m4!1s0x60010899e750ffff:0xe4c3e4e96a0e904b!8m2!3d35.0024582!4d135.7615117

Nara – close to Oaska 1.5 hr by train

Nara is also known as the city of peace. We did a day tour to Nara and visited the Nara Park, which had free roaming/wild deer’s, which are “messengers of gods”. The park also more than 3000 lantern at the Kasuga Shrine, very unique and picturesque. Most famous temple is probably the Todai-ji Temple – world heritage site.

Arima – A cute Spa/Onsen Town near Kobe 20min inland by train.


Tocen Goshoboh (Ryokan) – night or two is enough to relax in this old spa town. The Inn is one of the oldest in Japan. Excellent food and spa option.


The Osaka castle deserves a visit. Also Dotonburi. It is full of shops and eateries.

Heimeji castle Considered to be the most beautiful castle in Japan by many. We definitely agree.

OkayamaOkayama Castle & Korakuen Garden – example of Japanese Garden with some rare birds. One of “famous three” gardens of Japan.


Has a charming Bikan historial area.



Hiroshima Peace Memorial – good, pretty moving actually bec. of all the personal stories.


Visit Itsukushima Shrine and the Floating Torii. A UNESCO World Heritage site. Recommend to be there to see both high tide and low tide. You can climb the mountain in between.

Restaurant Oyster Ship Kanawa – about oysters, very good.

Hi – below is a pretty comprehensive list. Some of the best restaurant and food experiences are not very kid friendly but you can always try and of course, if you want you can do a date night. In bold are my top recs below.

These restaurants will all allow kids. Highly recommend Tofuya Ukai and Tanya

These places may not be as kid friendly – but you can try. MUST DO Sushi Kanesaka AND Tempura Ginya. If you don’t do Sushi Kanesaka, do Sushi Masuda above.

Lunch (from above listSushi Masuda and Tofu Ukai we have done for lunch as well). Recommend Zauo as pretty fun/unique, Okonomiyaki and a ramen experience anywhere. Also great are ANY bakery in Tokyo – they are all over and best breads of all kinds.

Bars: Highly recommend a Magic Bar, The Whisky Library Bar (and surrounding are) and also the bar in the Ritz Roppongi or Park Hyatt (further out) for views)

Magic Bar Ginza Jyuniji –[] we actually take the older kids there, they love it

Bar High Five in Ginza – cool innovative cocktails

The Whisky Library Bar

Park Hyatt

Jus de Peche in Roppongi for elegant drinks in a relaxed setting. (Eoyang)

Bar Zingaro

Trump Room

Gen Yamamoto

to do: MUST DO Harajuku w Meiji shrine

Super fun to do Oedo Onsen with the kids

Daikanyama is great for boutique shopping, cafes, restaurants

Harajuku – a must for kids – great people watching and there is a squishy store called Mushi which kids our age were totally into. You can do that and Meiji shrine as they are next to each other and then walk around Omotesando for shopping

Tsukiji market

Meiji jingu/harajuku




Cup Noodle Museum/Yokohama

Thomas Land for Tai indoor amusement park for K

Tokyo Edo Museum

Kyoto: In Kyoto, I really liked Wakuden**, although it’s also fun to just go eat at one of the amazing izakaya on Nijo-doori. Check out the Hakuhodo honten (make up brushes, not the ad agency!).

Omen noodles for lunch

Nanzen-Ji – this is the one that does the tea ceremony tea (abridged and just mostly serving you the tea)

Ryoan-Ji – famous zen garden


Ginkaku-ji – beautiful views and temple

IF you have time Kinkaku-ji (golden temple)

Nijo-Jo – samurai castle

And a side trip to Nara just to see Todai-Ji and the big buddha – spectacular

Shopping street (leading up to Kiyomizu dera and also the other famous one)

Maps and more…

These Google maps were very helpful when we were in Japan. They divide attractions by category (“ramen”, “bars”, etc). They correspond to many of the recommendations in the above guides.

The places we went on our anniversary trip (Map)

TF’s amazing guide to Tokyo (Map)

The map we compiled based on everyone’s recs when we planned the trip (Map)

A 37-page guide to Japan incl maps, attractions, and specific instructions (pdf)

Our own experience in Kyoto (Link)

Your Portfolio Intuition Is Poor

Summary and takeaways from Bridge Alternatives’ Portfolio Intuition (Link)

Intuition Test


  • Your current portfolio has 5% return and 15% volatility for a Sharpe ratio of .33
  • You want to allocate 10% of your portfolio to a prospective asset
  • You want to maximize the Sharpe ratio of the resulting portfolio

Choose between A1 and A2

A1 A2
Return 4.00% 4.00%
Volatility 7.96% 46.04%
Correlation -.20 -.20

Unsurprisingly, most people prefer A1 since it has the same attributes as A2 with 1/6 the risk.

Now let’s run the numbers 

Expected return of the new portfolio is the same whether we choose A1 or A2:

Volatility of the new portfolio if we choose A1:

Sharpe ratio of original portfolio = .33

Sharpe ratio when we add A1 = .049/.13363 or .3667

The Sharpe ratio improved by about 10%

Now what is the Sharpe ratio if we add A2 instead of A1.

First, we must compute the volatility. Go ahead, plug and chug…

That’s right, the volatility is the same!

The volatility of the new portfolio is the same whether we add A1 or A2 which means the new combined portfolio has the same improvement to Sharpe whether we add A1 or A2. This is true despite A2 having a far worse Sharpe than A1! It is counterintuitive because portfolio math and the role of correlation is not intuitive.

To see why, look at the formula for portfolio volatility:

Let’s zoom in on the last 2 terms which come from adding the second asset:

Plot of change in overall portfolio volatility vs volatility of prospective asset (A1 or A2)

As we increase the asset’s risk, the first term grows exponentially, and the second term shrinks linearly (remember, the correlation is negative). It turns out that, at least temporarily, the shrinking effect from the negative correlation outweighs the exponential term.

There are 2 observations to note once you are done reeling from the bizarre impact of correlation.

  1. When adding a negatively correlated asset to a portfolio its risk must be incredibly high before it starts to degrade the Sharpe ratio of the final portfolio.
  2. Notice how, at least until we hit the vertex, if we move from left to right, representing an increase in risk, we’re actually reducing return. Put differently, if we added risk and didn’t reduce return we’d deliver more than a 10% improvement; risk has a positive payoff here, which is very cool. There is a significant range where we are reducing the prospective assets’ Sharpe and actually reducing the volatility of the new portfolio.

More Preference Tests

B1 B2 C1 C2 D1 D2
Return 10.54% 3.57% 9.33% 6.50% 6.43% -2.64%
Volatility 20.00% 20.00% 27.50% 12.50% 10.00% 40.00%
Correlation .80 -.20 .40 .40 .50 -.60

Most people agree:

  • B1 was slightly preferred to B2. For the same risk, B1 delivers much more return, though B2’s correlation is better.
  • C2 was preferred. It’s Sharpe is higher (about 0.52 versus about 0.34).
  • D1 was preferred to D2. D1’s Sharpe ratio is much higher. D2’s return is negative

The punchline, of course, is that every one of these assets improves the Sharpe of the portfolio by the same 10%. Your intuition would tell you would prefer a portfolio in the upper left green box since those assets have the best Sharpe (risk/reward), so it is probably uncomfortable to learn that the final portfolio is mathematically indifferent to all of these assets.

Correlation Is The Key

Here’s the same plot relating these equivalent portfolios by their respective correlations

As the correlation drops (corresponding to lines of “cooler” coloring), less return is required to deliver the same 10% improvement!

While Sharpe ratios are “mentally portable”, they are shockingly incomplete without being tied to correlation. To create a compact formula which links Sharpe ratios with correlation, it is helpful to view indifference curves.

Indifference Curves

RRR= Sharp Ratio of prospective asset
RRRb = Sharp Ratio of original portfolio

If Relative RRR > 1 the Sharpe of the prospective asset is greater than the Sharpe ratio of the original portfolio

The indifference curve represents an equivalent tradeoff between Sharpe ratio and correlation for various mixing weights. For example, the light green line assumes you will allocate 20% of the original portfolio to the prospective asset.


  • As the weight allocated to the asset increases (the lines move upward, from green to purple), the asset must be more performant in order to do no harm; it must be better relative to the portfolio. Put differently, as the role played by the asset increases, more is required of it, and that sounds about right.
  • A less performant asset, ie one with a worse Sharpe ratio than the original portfolio can compensate with low or negative correlations

Getting Practical

The investor’s natural question when evaluating a new asset or investment is:

“What is required from an asset (in terms of return, risk and correlation) in order to add value to my portfolio?”

With math that can be verified in the paper’s appendix we find a very handy identity:

This equality describes what’s required, in an absolute bare-minimum mathematical sense, of a prospective asset in order to do no harm. 

How to use it

For a given prospective Sharpe ratio, you very simply compute the maximum correlation the new asset can have to be accretive to the portfolio. For example, if the prospective asset has a Sharpe ratio of .10 and the original portfolio has a Sharp ratio of .40 then the prospective asset requires correlation no greater than .25 (ie .10/.40).

For a given correlation, you can compute the minimum required Sharpe ratio of the new asset to improve the portfolio. If the correlation is .80 and the original portfolio has a Sharpe ratio of .70 then the prospective asset must have a Sharpe ratio of at least .56 (ie .80 x .70).

Insights and Caveats

  • Correlation is best understood as a sort of performance hurdle. For assets exhibiting low correlation, less is required of their standalone performance (i.e. return over risk), all else equal.
  • Prospective assets with a Sharpe ratio greater than the original portfolio are always additive.
  • If you happen to find a truly zero-correlation asset it will be additive as long as it has positive returns. And as we saw with asset D2, a negative Sharpe Ratio asset can be additive if it has a negative correlation!
  • This cannot be used to somehow rank prospective assets. It can only serve as a binary filter: yes or no. This might feel like a real limitation. Sharpe ratios are absolutely rankable. They are measurements of the same unit (risk). But as we’ve shown in this paper, those rankings are not indicative of their true value within the context of a portfolio. Making decisions based only on return and risk is like ranking runners based on their times without asking how far they ran. It doesn’t make sense. If you take away one thing from this paper, this should be it!

My Own Conclusions

  • Correlations make portfolio math extremely unintuitive.
  • Negative and low correlations can make poor or losing stand-alone investments great additions to a portfolio. The implications for the diversifying power of low or negative-yielding assets are significant. Bonds, cash, commodities, gold.
  • Highly volatile assets with a negative correlation are tamed and even subtractive to the total risk of a portfolio.
  • While the importance of low or negatively correlated assets is well known it’s possible it remains underappreciated.

Further reading

Breaking The Market’s outstanding post Optimal Portfolios For Two Assets

You will learn:

  • How to mix assets by comparing their geometric returns.
  • Correlation’s effect on portfolio construction is not linear.
    • The closer correlations are to 1 the more they impact the recommended mix.
    • Negative correlations are deeply valuable in portfolio construction, adding to the long term return. Positive correlations are harmful, limiting the benefit of diversification.
    • The mixing range for the geometric returns is the combination of each asset’s variance, expanded or contracted based on the correlation between the two assets.
    • Negative correlation is wonderful.


You can save your own copy here

You can also play with the numbers directly below

Notes on Statistics Done Wrong

Statistics Done Wrong
Alex Reinhart

P Values

  • Measure of ‘surprise’. The smaller the p the larger the ‘surprise’. P values work by assuming that there is no difference between the 2 samples. If you want to show a drug works you counterintuitively show that “the data is inconsistent with the drug not working”
  • P values say nothing about the magnitude of the effect. A small p can reveal a massive effect or a tiny effect with great certainty (say if you collected massive data). Statistical significance does not mean practical significance! Similarly, statistical insignificance does not mean zero and is simply the best evidence-based on the trial data you studied.
  • Neyman-Pearson uses p-values in a conceptually different way. They estimate an acceptable false positive rate called ‘alpha’. Statistical significance allows us to reject the null hypothesis for the established alpha or false positive rate. This rate is informed by the experimenter’s understanding of the procedure.

Confidence Intervals

  • These are preferable to p-values since they provide a point estimate and a measure of uncertainty and if they can be supplied instead of just p-values they should.
  • They are less common in the literature perhaps b/c they can be very wide

Statistical Power

  • The probability that a study of a given amount of data is capable of showing statistical significance. For example, how many coin flips do you need to be 95% sure that your experiment can reveal a biased coin that is 60% weighted towards heads? Statistical power is a function of :
    • The size of the effect; a smaller effect requires more data
    • Sample size; more data means the study has higher statistical power
    • Measurement error; more subjective measurements have less power
  • Many studies are underpowered because there is not enough data. This often occurs because it is expensive/risky (ie drug studies) or unethical (studies on animals)
  • While an antidote for multiple comparison problems can be to require lower p values, the trade-off is that studies will become underpowered
  • The concept of power is often forgotten because it is not taught in intro stats and is not readily intuitive. Again, confidence intervals which are wide can reveal a lack of statistical power again supporting their use over p values.
  • Truth inflation or type M error (‘magnitude’) is the effect of there being many experimenters ‘competing’ to publish extreme results.
  • Small samples have more variance; be careful to draw conclusions from them since they are more likely to be underpowered
    • Rural states have counties with both the lowest AND highest kidney cancer rates; this is likely due to small populations, not a real effect. The same is true for test scores in smaller schools; we may interpret them to be ‘better’ based on test scores but this is because their average extremes are higher than the average extremes of bigger schools!
    • Remedies for this include shrinkage. Weighting the average from a small sample with a weighted average from a larger population (ie weighting a small county with a higher weighted national average). This will, unfortunately, bias truly abnormal cases too much towards normal. The best remedy is to try to find a larger sample (ie use congressional districts instead of counties). Shrinkage is a good technique in measuring average product reviews (products with few reviews are shrunk towards a generic version of the product).


  • Using additional measurements that are highly dependent on highly correlated to previous data. This form of replication doesn’t allow you to generalize inferences.
  • If you cannot eliminate hidden sources of correlation between variables you must try to statistically adjust for confounding factors.

Base rate fallacy

  • A low p-value is often touted as evidence of significance but significance also depends on the base rate. Consider Bayesian examples like mammogram testing. If mammograms have a false positive rate of 5% and a 90% chance of accurately identifying cancer then if you test 1000 people and 50 of them test positive then it is still quite unlikely that most of those people have cancer. Why? Because the base rate is a mere 1%. Only 10 people in that sample have cancer and we expect 9 of them to be accurately identified but more than 50 will test positive! When testing for conditions with very low base rates false-positive rates will swamp true positive rates.
  • An extreme example of these false discovery rates you are looking for an effect which definitely does not exist, no matter how low you set your p threshold we know your so-called significant results are still false positives, and you are bound to record significance results with a large enough sample.
  • Combatting false discovery rates with multiple comparisons is challenging but important since you expect many false discoveries. Tips include:
    • Remember p < .05 doesn’t mean there’s a 5% chance your result is false
    • When making multiple comparisons using a procedure such as Bonferroni or Benjamin-Hochberg will make your required p values much more conservative by accounting for the number of tests
    • Be aware of stat techniques specific to your field for testing data
    • Have an idea of base rates to estimate how prevalent false positives are likely to be

Confounding variables

  1. Correlation is not causation
  • Because you do not know if there is a confounding variable. If you create a model that predicts heart attack rates based on weight, exercise, and diet it’s tempting to say that if you change one of them x% that the heart attack rate will change by y%. However, that is not what you tested. You didn’t change the variables in a real experiment and measure the outcomes. It is not clear that a confounding variable is actually influencing the heart attack rate.
  • Also, to say a variable changes all else equal is a fantasy. In reality, it is unusual for single variables to change in a vacuum.
  1. Simpsons Paradox
  • When a trend in the data disappears when the data is divided into natural groups. It tends to occur in observational studies with biased samples thus obscuring a confounding variable.
  • Examples:
    • Berkeley admission bias against women in 1973. In aggregate, women looked discriminated against but at the department level, the opposite was true. The bias occurred bc women applied in higher numbers to competitive, underfunded departments. The bias happened earlier in the process: women were systematically pushed towards these fields
    • Penicillin appeared to improve outcomes for meningitis cases in the UK. At a closer look, the sample was biased since it was only administered to children who were not rushed to the hospital so they were the milder cases. Isolating the sample to those who visited a general practitioner first we find that penicillin, in fact, seemed to correlate with worse outcomes (there are theories about the breakdown of the contagion causing shock but there aren’t experiments for testing if penicillin actually causes meningitis patients to die)
    • Looking at aggregate data United flights are delayed more frequently than Continental. But at individual airports, the trend reverses. The aggregate data doesn’t account for the fact that United flys out of more airports with bad weather.

Notes on How Not to Be Wrong: The Power of Mathematical Thinking

How Not to Be Wrong: The Power of Mathematical Thinking
by Jordan Ellenburg

  • Math gives you tools to extend your common sense of reasoning and logic. He uses the analogy of Ironman’s suit.
  • Competitions including war are often decided by small edges. Being 5% better at x or y can decide the outcome over a long enough game. (Similar to my experience with boardgames. A more efficient engine in a game of 7 Wonders or Settlers can save you actions; like reducing the cost of capturing winning victory points.)
  • Zooming in on points of a curve they look and can be approximated by lines. Trajectories are curves influenced by gravity but objects appear to move in straight lines. Using lines to approximate curves is the basis of calculus and even the derivation of pi as the area of a circle (Archimedes did this by iteratively computing the edges of a circumscribed circle as a polygon. Imagine an octagon, then a polygon with 64 sides, and so forth until it looks like a circle. You can then use trigonometry to compute the area of the triangles you keep creating until the sum of all the area approximates the area of the circle)
    • Critics of this method like Zeno highlighted the uncomfortable paradox of constantly halving something until you get nowhere. Comically, the skeptic Diogenes countered Zeno by simply walking across the room to make the point that motion is indeed possible!
  • The law of large numbers explains why South Dakota can have the highest rate of brain cancer and North Dakota the smallest. They both have small populations. Be careful when comparing quantities from 2 very different sample sizes. Small samples are more volatile. If you flip 10 coins, your odds of getting 8 heads is unlikely but real. But flip 1000 coins it’s nearly impossible to get 800 heads.

Data mining 

“The more chances you give yourself to be surprised the higher your threshold for surprise had better be.”

“A significance test is a scientific instrument and like any other instrument, it has a certain degree of precision. If you make the test more sensitive by increasing the size of the studies population, for example, you enable yourself to see ever-smaller effects. That’s the power of the method but also the danger”

  • An underpowered study has the opposite problem. You dismiss an effect that your method was too weak to see. A good example is the original 1985 hot hand studies. They rejected the idea of a hot hand but it turns out the methods they used rejected a hot hand even on data sets that were generated by simulations that deliberately baked in a hot hand! In fact, their methods failed to notice even the effects of good vs bad defenses which we know influences offensive shooting percentages.
    • The final verdict is there may be some hot hand effect but it is too difficult to detect because if it exists it is very small. In fact, players who think they are hot take harder shots and perform worse so it’s best for them to not believe in the effect since it will be more than offset by an unjustifiably confident shot selection.

The Bayesian examples in the book are great.

  • In a Bayesian framework how much you believe something after you see the evidence depends not just on what the evidence shows but much you believed it to begin with. Posterior probabilities still depend on the strength of your priors.
  • On conspiracy theories: “If you do happen to find yourself partially believing a crazy theory, don’t worry — probably the evidence you encounter will be inconsistent with it, driving down your degree of belief in the craziness until your beliefs come in line with everyone else’s. Unless, that is, the crazy theory is designed to survive the winnowing process. That’s how conspiracy theories work”.

Tradeoffs and cost of perfection

  • Stigler type arguments that optimal decisions often leave a margin for error. Getting to the airport early enough to have a 100% chance of making the flight is probably so conservative it’s wasteful (depends on your utility curve but almost certainly wasteful to be 100% certain vs say 95%). When you read a story about social security overpaying people bc they were actually dead, it turns out that mistake represents less than 1 basis point of payments. In other words, they do a great job not making this mistake and the cost of being 100% compliant may simply not be cost-effective to be worthwhile.

St Petersburg and the role of expected utility

  • Fran Lebowitz utility curve of money: she would drive a cab each month until she could eat and pay rent. Afterwards, she would write. In other words, she had a linear utility curve which flattened abruptly. If you raise her taxes she works more as opposed to someone with a logarithmic curve who is at the point of indifference between work and leisure
  • Ellsburg Paradox highlights the limitations of utility theories. It highlights the difference between what Rumsfeld called “known unknowns” or what mathematics refers to as risk vs “unkown unknowns” or uncertainty. Utility theory may help with uncertainty but formal math is less useful.

Regression to the mean explains many phenomena that are usually attributed to another reason.

  • Examples, best-performing companies (competition attribution), musician/writer sophomore slump, RB after signing a big contract, dietary fiber speeding or slowing digestion, Scared Straight juvenile detention program, diet effects when people are at their peak weights. When something is at an extreme we should expect reversion simply bc of math and therefore be very careful of attributing to an intervention.

Correlations between variables reduce the information content of the variable.

  • You try to identify criminals by foot and hand size you are choosing highly correlated variables.
  • Strong correlations lie behind how we compress images and music files. A green pixel is probably next to a green pixel.

Notes From Lessons of History

The Lessons of History
by Will and Ariel Durant


  • As technology evolves, the influence of geography is diminished
  • The prosperity of civilizations is closely tied to geography notably rivers and coasts
  • Ultimately, humans create culture, not the earth

Biological lessons of history

  • Life is competition
  • We rely on the protection of our tribes and may cooperate within its confines but this is adaptive behavior in service of a wider competitive landscape. Until a group is as large as a state it will “continue to act like individuals and families in the hunting stage”
  • Life is selection
  • Nature knows nothing of our egalitarian ideals or Bill of Rights. “Freedom and equality are sworn enemies”. If people are free, relative advantages will grow nearly geometrically. Inequality is inborn, we can only aspire to ideals such as equal access to education and justice.
  • Natural selection benefits from the diversity and range of natural ability as it is the basis of evolution.
  • Life must breed; birth rates shape history
  • Nature cares only about the species, not the individual. Diversity and large “litters” are the fertile grounds that natural selection needs as fuel. Nature’s check on overpopulation is famine, pestilence, war.
  • Higher birth rates can give rise to stronger tribal powers while progression into a higher standard of livings and industry appears to slow the birth rates as the tribe advances.
  • To track the future of ideas, it may be instructive to watch birth rates; weakened ties to ethnicity or tribalism can leave the incumbent group vulnerable (Caesar and Augustus were aware of this and sought to penalize birth control; Italy’s ethnicity diluted over time making the empire vulnerable to its neighbors)
  • On race: “A knowledge of history may teach us that civilization is a cooperative product, that nearly all peoples have contributed to it; it is it common heritage and debt; and the civilized soul will reveal itself in treating every man or woman, however lonely, as a representative of one of these creative and contributory groups.”


  • Monarchy has been the historical norm while democracies have been ‘hectic interludes’
    • Roman democracy crumbled under class wars giving way to Pax Romana, a 200 year succession of benevolent dictators until the murder of Caesar. This period was followed by disgraceful monarchs including Caligula before giving way to the greatest succession of monarchs ever, the last being Marcus Aurelius. Some of these monarchs had no heirs and promoted by merit until Aurelius died. Afterward his son Commodus ruled when no heir was named.
    • Monarchy has a mixed record, typically at its worst is when it is determined by blood and accompanying incompetence
  • Most modern, complex governments  are oligarchies — ruled by a minority
    • Aristocracy by birth
    • Theocracy by religion
    • Democracy by wealth
  • Plato reduced the cycle to monarchy->aristocracy->democracy->dictatorship. Repeat. This was based on Greece and repeated with the Romans. Democracies are overthrown or conquered as envy amongst the majority tires of the ‘sham’ of having a vote and uses the state to seize wealth. Other oligarchies fall when they wield their great power to narrowly or incompetently.
  • The US democracy started from a wider base and in unity against British rule. Rural land-owning enhanced the sense and dedication to freedom, while geographic isolation created a national sense of freedom. These conditions have given way as land as sparse and cities have grown “Every advance in the complexity of the economy puts an added premium upon superior ability and intensifies the concentration of wealth and [power]”
  • Democracy is a difficult form of gov’t since effective mob rule requires widespread intelligence to avoid manipulation by ‘the forces that mold public opinion’.
  • Democracy, however, has done the most good and it can maintain its promise only if it affords equal opportunity for education [my own thought is this will always be uneven, and the unevenness grows with population size. For a small population, it is reasonable that the average access to education can be even]
  • “If a race or class war divides us into hostile camps, changing political argument into blind hate, one side…may overturn the hustings with the rule of the sword. If our economy of freedom fails to distribute wealth as ably as it has created it, the road to dictatorship will be open to any man who can persuasively promise security to all”

History and War

  • Competition via war is often for the same reasons as competition amongst individuals however while individuals are restrained by law and morals a state ‘acknowledges no substantial restraint either because it is strong enough to defy any interference…or because there is no superstate to offer its basic protection, and no international law or moral code wielding effective force’
  • Religious wars in the 16th century and the wars of the French Revolution were battles between aristocracies leaving the masses to maintain mutual respect for their foreign counterparts while wars of the 20th century in accordance with technology and ‘means of indoctrination’ made war all-consuming ‘struggles between people’ and completely destroying centuries of labor and property.
  • He presents the general case for the inevitability of war — competitive human nature, envy, beliefs so fundamentally different that cannot be settled by negotiation. Perhaps and only if we were united against aliens could we imagine our species not warring with each other. The opposing view is that the imperative to avoid large scale war because of the destructive power of current technology will prevail over any disagreements over ways to organize our lives and economies.


  • Are we the ‘same trousered apes’ merely armed with increased tech, locomotion, and knowledge but weighed down by the persistent features which moor us to our primitive ancestors? Progress is defined not by happiness or similar measure but to the degree in which we may control our environment. By this standard, as evidenced by our increased adaptability and lifespans we have made tremendous progress. While subject to lapses and regressions, the avergae human has been the beneficary of an accumulated history and knowledge which has been succeeds in being recorded and transmitted to subsequent generations so that they may continue to build on this heritage from its most advanced point.

Moontower Thanksgiving 2019

This letter started with a discussion of cosmic luck. There are 7 billion people in the world. You have about a 1 in 6 chance of being born into the “developed” world. My gratitude starts there.

I didn’t quite catch the heads required to end up in the crib of a median American family but I got something more valuable. Parents who love me and cared about my future.

They met here in the US but they are both from Cairo and they were not in the US long before meeting one another and having me in 1978. As all immigrants must, they navigated a strange land and a strange language. One of my favorite bits of family lore is how weird they thought the spaghetti sauce was in the US. They thought Americans’ taste buds were broken. Turned out my grandpa was buying ketchup.

In my spoiled existence today it’s hard to imagine what they went through and with so little. And yet despite the odds, I was given great opportunities and have had amazing luck:

  •  I’m 6’1
  • My parents are hard-working and my mom was especially demanding. She worked in treasury ops at banks when I was growing up. I was sorting fixed income traders’ mail when I was 16 thanks to her. I commuted from Hazlet, NJ to Manhattan at 5am in my summers to work. I got paid $16/hr in the 90s.
  • I have thus far dodged serious illness. I survived 2 serious car crashes before age 21 (at age 13 I had the experience of seeing and touching my right shinbone before I had even realized what happened). My apartment building burned down my junior year of college. Everyone has crazy experiences. That you are still around to have them has nothing to do with merit. Blind luck.

Speaking of college one of the most influential moments in my life was my choice to go to Cornell. My mother had always pushed me and spoke of this place Cornell before I was even in HS. When I visited the campus, I immediately loved it and applied early decision to maximize my chance of getting in. When I did get in, she and I were over the moon. But in a moment of major doubt and looking at the cost, I hesitated. You see Rutgers would have been a full academic scholarship. Cornell was gonna cost close to 30k per year. I didn’t want to saddle my mother with any more burden. She had already shouldered the bills of a private Catholic all-boys HS (CBA in Lincroft). And when she couldn’t afford it, she begged the school to let me attend without paying. She never gave up and it worked. They graciously made an exception for part of my tuition.

She had a one-tracked mind. Give her son a chance.

In the end, she convinced me with a plan I could bear. Cornell offered 1/3 of the cost in a need-based grant that she wrote many letters for every year. She and I would split the remaining cost equally. I’d pay my share with loans and my summer job earnings. She believed Cornell was going to mean more opportunities.

She had a one-tracked mind. Give her son a chance.

If you simulate my life 100x over I probably don’t end up with the professional luck I’ve actually enjoyed in 90-95 of those sims. I punched above my intrinsic gifts. But I have respectable outcomes in most of those sims because of who my mom is. Sure I had to meet her part of the way with effort but ultimately she gave me the best chance.

I’m thankful for being born in America and that my family fought their way to get here. I’m fortunate to grow up in a home that taught me how to fish even though it didn’t have much fish to give me. I’m not very interested in politics but if I had to say, I’m a lot more liberal than my parents. But something I would agree with them on…when I watch politicians fan the flames against the rich on what I feel are extremely exaggerated claims about what income inequality really is I recoil.

The prosperity of the United States and the possibility to push yourself or your kids into a better life has been calling to foreigners for over a hundred years. Anecdotes are not data. But having seen enough people from humble roots succeed beyond their expectations, makes me thankful I live where those anecdotes are not outliers. Billionaires are outliers. Kids from Hazlet or Newman, CA (Yinh’s hometown) who end up in our shoes are very much part of the real distribution.

if there is a devil he wants you to compare yourself to your neighbor. It’s that simple. It’s his most effective trick. And it’s subtle. Don’t pay attention to his whispers.

This Thanksgiving I’m grateful for all the luck it took for me to be able to sit in a warm house, with wifi and the time to write this. To have my children fed and asleep in bed (they’re in my bed tonite the rascals) and for my wife to be peacefully passed out while watching TV. There’s a lot of heads that had to come up to be here but it all started with a chance to play.

Enjoy your loved ones this Thanksgiving and much love from me. As always, thanks for reading.

Longitudinal Studies > Snapshot Studies

Longitudinal studies described by Roberts are very difficult while snapshot comparisons are relatively much easier to implement. But even the longitudinal studies are done on wide demographic cohorts. You may be interested in a much narrower subset of people. For example the children of decamilionaires, the class Kylie was born into.

The way to do a study like that would probably be a Bayesian experimental design. For example you’d take the base rates from the broader longitudinal studies and adjust them for the specific reference class you are studying. A statement that sounds like “how did children born in the 1980s do given they had access to a PC or given they grew up in Beverly Hills?” The conditions after the word “given” comprise the reference class.

In Kylie’s case, what was the right reference class? What’s the relevant condition? Being a decamillionaire to start? What about her IG followers? Not all decamillionaires enjoyed her fame. What about the fact that she became a billionaire so quickly?

The critical observation: experimental design has an enormous impact on the results of studies that show up in our feeds that then spread like truth at happy hour. A poorly constructed design or reference class has downstream effects that can swamp the conclusions.

A recurring Moontower theme is the world is messy, be critical of tidy takes. Alas, don’t despair, here are some guides and concepts to keep in mind.

Making Better Comparisons

One of my favorite researchers Michael Mauboussin wrote a guide to making better comparisons. Since all non-split second decisions require conscious comparison we can all do better by getting meta about the process. Mauboussin decomposes the steps and highlights pitfalls that await along the way. The link to the pdf and my summary notes can be found here.

Simpson’s Paradox

We learned how easy it can be to overreach for conclusions from snapshot data. Russ Roberts pointed out the danger of averaging averages — Simpson’s Paradox. It states that the change in the average is not the same as the average of the changes. When I interviewed at Susquehanna coming out of college, one of the questions was a perfect example of the paradox, even if I didn’t know the term for it back then.

The question: If batter A has a better batting average than batter B for the first half of the season AND batter A has a better batting average for the second half of the season, is it possible for batter B to have a better batting average for the whole season?

The answer.

Life expectancy example

In 1900, life expectancy was about 47 for a US male. Does that mean you were middle-aged by the time you left college? Of course not. If you made it to 22 years old there was a good chance you’d live well into your 60s or older. Life expectancy is extremely sensitive to infant mortality rates. In 1900, infant mortality was around 20%. Today it’s closer to .5%.

Society 1:

Infant mortality = 30%
Survivors who make it past infancy live to 80
Life expectancy = .20 x 0 + .80 x 80 = 64 years old

Society 2:

Infant Mortality = .5%
Survivors who makes it past infancy live to 80
Life expectancy = .005 x 0 + .995 x 80 = 79.6 years old

Without tracking the lives of people, the snapshot of life expectancy can make people jump to all types of silly conclusions about the 2 periods. When we study longitudinally we would have seen that someone who makes it to 32 is still far from middle-aged.