Modern Portfolio Theory: A Case Study on Turnips

And the way having buddies makes the stalk market much more worthwhile.

Turnip is an enchanting addition in Animal Crossing: New Horizons, and arguably what promotes the sport’s social side from a cute distraction to an absolute necessity (that’s, if you wish to play optimally). It’s the de facto inventory market on the abandoned island, the place you attempt to purchase low and promote excessive so as to pocket an honest revenue. Plus, did I point out you can go to your mates’ island for some bonus buying and selling alternatives?

On this publish, I study the “stalk” market by means of the mean-variance optimization lens. Is it value it? When must you promote? To what extent can your mates be of help? As some may count on, the solutions lie within the fashionable portfolio idea.

The Stalk Market Defined

Turnip, although seems to be a typical commodity, has a couple of distinct traits to it –

  • You may solely purchase it from Daisy Mae each Sunday
  • Each morning and afternoon of the next Monday to Saturday, a brand new quote will change into accessible at Nook’s Cranny
  • The entire thing expires the following Sunday

Nevertheless, turnip differs from most monetary property in no less than one significant method. It’s removed from clear whether or not the underlying worth follows some variation of geometric Brownian movement, which is certain to have unintended ramification if we resolve to borrow from the asset pricing equipment. Nevertheless it’s not completely hopeless, as shall be proven shortly – we are able to pin it down numerically.

It’s additionally value noting that within the absence of an optimum promoting technique (remark if you realize higher!), I’m going to make an assumption that makes the issue way more tractable (and fewer path-dependent): let’s solely promote our turnips at one particular time window (out of the entire 12 of them), and such choice shall be made solely based mostly on Sharpe ratio.

Turnip’s Value Dynamics

For a very long time, it has been recognized to the Animal Crossing group that the turnip worth isn’t actually unpredictable. As a substitute, it follows one of many following 4 patterns:

  • random
  • reducing
  • giant spike
  • small spike
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Now due to the extraordinary reverse-engineering work performed by Treeki (see right here), the thriller behind turnip worth has been deciphered in its entirety, to the purpose the place it turns into potential to simulate the costs numerically, enabling Monte-Carlo-style evaluation:

 

Technique: Promote on Wed a.m. and Go Away

Juding from my experiment of 100Ok trials, if we have now to restrict our timing of promoting turnips to one of many 12 accessible slots, Wednesday a.m. appears to be your best option each by way of anticipated return and Sharpe ratio. If we comply with this technique, a return of just about 10% could be anticipated over the 3-day time-frame – considerably outperforming nearly each asset class in the actual world, thoughts you. See appendix for particulars.

 

A Higher Technique: What if You have got Buddies?

To make issues extra fascinating, the sport additionally means that you can purchase (on Sunday) and promote your turnips (within the following week) on a good friend’s island. Profiting from such flexibility, a rational turnip investor will examine as many choices as potential and solely purchase on the minimal and promote on the most throughout all islands.

Analytically, let $P_1$, $P_2$, …, $P_N$ be the turnip costs noticed on $N$ totally different islands. Assuming they’re IID and comply with the cumulative distribution operate of

the utmost worth $Q$ the place $Q = max(P_1, P_2, …, P_N)$ could have the cumulative distribution operate of:

Because of this, the brand new distribution undersamples the smaller values and oversamples the bigger ones on the identical time, successfully shifting the chance mass to the fitting. The extra islands we have now at our disposal, the extra advantageous the state of affairs turns into. A single extra island brings a staggering fivefold enhance in anticipated return and nearly fivefold enhance in Sharpe ratio. As soon as it reaches 6 islands, you possibly can count on a 122% return over the identical 3-day time interval with a Sharpe ratio of 0.98, dwarfing any hedge fund tremendous star who has ever walked this Earth by far.

 

For our closing thought experiment, let’s push the variety of islands to infinity. Folks have already been making the most of worth discovery web sites (equivalent to Turnip Change) to increase their alternative set. Within the excessive case, the return distribution collapses to 1 single quantity because the turnip investor buys on the international minimal and promote on the international most (90 and 660 Bells respectively per my simulation) and is not a random occasion. In different phrases, a assured 633% return for the last word turnip grasp.

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Conclusions

Turnip is a good funding by all means. With a bit of planning, some hardwork and some buddies, it constantly delivers among the finest returns recognized to humankind. Some may argue that the sport’s interface doesn’t actually make it simple for gamers to abuse the system, however that’s one other story for one more day.

In the meantime, to my fellow turnip merchants, completely happy investing. Simply watch out and don’t let the Nooks know.

Appendix

Desk 1: Optimum Promoting Window

TimeAnticipated ReturnVolatilitySharpe Ratio
Monday a.m.-0.08950.2211-0.4049
Monday p.m.-0.11150.2379-0.4687
Tuesday a.m.-0.0850.3118-0.2725
Tuesday p.m.0.04620.69510.0664
Wednesday a.m.0.09530.71690.1329
Wednesday p.m.0.0920.71330.129
Thursday a.m.0.07540.72380.1042
Thursday p.m.0.08290.72290.1147
Friday a.m.0.04370.73280.0596
Friday p.m.-0.03610.7379-0.049
Saturday a.m.-0.20540.3844-0.5343
Saturday p.m.-0.25620.3297-0.7771

Desk 2: Promoting on the Optimum Island

Variety of IslandsAnticipated ReturnVolatilitySharpe Ratio
10.09530.71690.1329
20.47070.89830.524
30.72211.01670.7102
40.921.10970.829
51.08241.18530.9132
61.22341.24950.979
6.3333

Replace on Could 4th, 2020: Corrected previously-inaccurate assertion re: randomness of turnip worth. Because of Chau-Wai Wong.

Replace on Could sixth, 2020: Mentioned the circumstances the place individuals use worth discovery app to pinpoint one of the best worth accessible and what if any individual has infinite variety of buddies. Because of Freya Yi.

That is Half II of my Animal Crossing publish sequence. For Half I, see right here; for Half III, see right here.

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