Bernoulli’s Prisoner’s Dilemma: A Goals-Based Perspective
In 1738, the Swiss mathematician and physicist Daniel Bernoulli proposed a simple thought experiment:
“A rich prisoner who possesses two thousand ducats but needs two thousand ducats more to repurchase his freedom, will place a higher value on a gain of two thousand ducats than does another man with less money than he.”
Let’s continue to play this out and place Bernoulli’s prisoner within the context of modern markets and ask him to evaluate various investments. What becomes immediately clear is that his ducats are dedicated to one objective: getting the heck out of prison!
Our prisoner has a goal for his money, just like we do.
Our prisoner can invest his ducats as he sees fit, and because he wants to maximize his chances of release, we can describe his use for various investments with goals-based portfolio theory.
We don’t need to bother too much with the details right now, but clearly our prisoner will evaluate both the expected returns and expected volatility of a given security over time through the prism of achieving his freedom. His willingness to trade off return and volatility is presented in the following graphic. The line is the minimum return he requires for any given level of volatility. As volatility, or the X axis, increases, our prisoner requires ever-higher levels of return, as depicted by the Y axis. This is hardly a revelation: It is exactly what traditional theory would expect.
The Prisoner’s Dilemma: Return and Volatility
But what if we build a stock exchange in our prison and let our wealthy prisoners trade shares among themselves? This is where things get interesting.
In the second graphic, we plot three different prisoners, A, B, and C, each of whom has different starting wealth, required ending wealth, and time horizon. For the sake of simplicity, we’ll suppose each has the exact same view of a security’s future volatility and return, which are labeled as s and m in the figure.
Three Prisoners’ Dilemma: Return and Volatility
Here’s the thing: Each investor is willing to accept completely different returns for the same security!
Moreover, if the security’s price is simply the inverse of return — 1/m, a simple but not unreasonable model — then each investor is willing to pay a completely different price for the exact same security!!
There is no difference of opinion about the characteristics of the security driving differing acceptable prices, but rather a difference in investor needs.
When we place these three prisoners in the marketplace, we would expect Prisoner A and Prisoner B to sell their shares to Prisoner C at the price of 1/c until Prisoner C exhausts his liquidity or Prisoner A and Prisoner B exhaust their inventory. Then, the price drops to 1/b, and Prisoner A continues to sell to Prisoner B. From there, the price drops to 1/a, and Prisoner A would buy, but no one would be willing to sell.
Prisoner C is an enigma. Traditional utility models would not expect anyone to accept lower returns in response to higher volatility. But goals-based investors can be variance-seeking when their initial wealth is low enough. Behavioral finance characterizes their goals as “aspirational.” This is why people buy lottery tickets and gamble: Increasing the volatility of outcomes is the only way of increasing their chance of achieving life-changing wealth.
Of course, all this is more than a simple thought experiment: It reveals some critical lessons about markets.
First, when setting capital market expectations or target prices for stocks, analysts would do well to assess the marketplace of buyers and sellers to determine how their needs and liquidity will influence the coming price. This is more complicated than our example, of course, because in addition to different needs, everyone also has a different outlook for a given security.
This is no surprise to practitioners. Markets dominated by institutional buyers look vastly different than those dominated by aspirational investors and “YOLO” traders.
A very present example is our current regime of ongoing quantitative easing (QE) from central banks around the world. For investors befuddled by sky-high stock valuations, the difference between Prisoner A and Prisoner B is illuminating. They are exactly the same except for one thing: Prisoner B is wealthier today.
In general, then, this means that adding cash to financial markets creates investors who are willing to pay more for the exact same security. Conversely, when excess liquidity is drained from markets, prices should drop, all else equal, because investors with less cash today require higher returns. Thus line B moves back to line A.
Second, and most striking: There is no “correct” market price. No security has a “fair value” or “fundamental value.” Rather, price emerges from a security’s characteristics interacting with the needs of the investors in the marketplace.
Another key component of price: each investor’s relative liquidity in the marketplace. If enough aspirational investors, or Prisoner Cs, deploy their cash into a security market, prices can remain elevated or spike until their liquidity is exhausted. Sound familiar, GameStop?
This may seem obvious, but it is not the traditional perspective on markets. The efficient market hypothesis asserts that securities always trade at their fair value and that market timing cannot work. Of course, predicting the development of the fundamentals of a security is a difficult task. But that is only half the equation. As our hypothetical prison stock market demonstrates, understanding the marketplace of investors and their behavior can yield insights that are just as valuable.
What’s even crazier: Every investor in the market is acting rationally. Prisoner C is offering a perfectly rational price for the security even if it is the highest bid in the marketplace! Prisoner A is acting just as sanely despite having the lowest buy price.
And this is some of the promise goals-based portfolio theory offers. Behavioral finance would describe the price action of our prison market as irrational albeit predictable investor behavior, and traditional theory would dismiss it as nonexistent. But goals-based investors can more clearly see what is really happening.
Goals-based portfolio theory may, in fact, be a helpful bridge between normative and descriptive theories.
Like the prisoner in Bernoulli’s thought experiment, we have specific objectives to achieve with our money. And like the prisoner, we interact with public markets with those objectives in mind.
Those objectives influence prices in ways that traditional theory might not expect. And while behavioral finance offers some models to predict irrationality, goals-based theory suggests that people may be more rational than initially thought.
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All posts are the opinion of the author. As such, they should not be construed as investment advice, nor do the opinions expressed necessarily reflect the views of CFA Institute or the author’s employer.
Image credit: ©Getty Images / erlobrown
"Kama ilivyoelezwa katika kitabu cha Bernoulli's Prisoner's Dilemma, ""Mfungwa tajiri ambaye ana ducats elfu mbili lakini anahitaji zaidi ya ducats elfu mbili ili kununua uhuru wake, ataweka thamani ya zaidi ya faida ya ducats elfu mbili kuliko mtu mwingine mwenye pesa kidogo kuliko yeye.""" Kile kinachokuwa wazi mara moja ni kwamba ducats zake zimejitolea kwa lengo moja: kupata heck nje ya gereza! Mfungwa wetu ana lengo kwa ajili ya fedha zake, kama vile sisi kufanya. Mfungwa wetu anaweza kuwekeza ducats yake kama anaona inafaa, na kwa sababu anataka kuongeza nafasi zake za kuachiliwa, tunaweza kuelezea matumizi yake kwa uwekezaji mbalimbali na nadharia ya mkoba wa malengo. Hatuna haja ya kusumbuliwa sana na maelezo sasa hivi, lakini ni wazi kwamba mfungwa wetu atathmini mapato yanayotarajiwa na kutofautiana kwa muda wa usalama fulani kupitia prism ya kufikia uhuru wake. Utayari wake wa biashara mbali kurudi na kutofautiana ni iliyotolewa katika grafu ifuatayo. Mstari ni mapato ya chini anahitaji kwa kiwango chochote cha kutofautiana. Kama volatility, au X-mhimili, huongezeka, mfungwa wetu inahitaji ngazi ya juu zaidi ya kurudi, kama ilivyoonyeshwa na Y-mhimili. Hilo si jambo la kufunua: Ni kile hasa ambacho nadharia ya kitamaduni ingetarajia. "Kama tunavyoona, ""Dilemma ya Mfungwa: Kurudi na Volatility"" ni kama ""kujenga soko la hisa katika gereza letu na kuruhusu wafungwa wetu matajiri biashara hisa kati yao wenyewe.""" Hapa ndipo mambo yanapokuwa yenye kupendeza. Katika picha ya pili, tunaonyesha wafungwa watatu tofauti, A, B, na C, kila mmoja wao ana utajiri tofauti wa kuanza, utajiri wa mwisho unaohitajika, na upeo wa wakati. Kwa ajili ya unyenyekevu, sisi itabidi kudhani kila mmoja ana mtazamo sawa kabisa ya usalama wa baadaye kutofautiana na kurudi, ambayo ni labelled kama s na m katika takwimu. Three Prisoners' Dilemma: Return and Volatility Hapa ni jambo: Kila mwekezaji ni tayari kukubali kurudi tofauti kabisa kwa usalama sawa! Zaidi ya hayo, kama bei ya usalama ni tu inverse ya kurudi <unk> 1 m, rahisi lakini si unreasonable mfano <unk> basi kila mwekezaji ni tayari kulipa bei tofauti kabisa kwa ajili ya usalama sawa!! Hakuna tofauti ya maoni kuhusu sifa za usalama kuendesha tofauti ya bei ya kukubalika, lakini badala tofauti katika mahitaji ya wawekezaji. "Kama tunaweka wafungwa hawa watatu katika soko, tunatarajia wafungwa A na B kuuza hisa zao kwa wafungwa C kwa bei ya ""1 c"" hadi wafungwa C kumaliza fedha zake au wafungwa A na B kumaliza hifadhi yao." Kwa hiyo, bei ya bidhaa hiyo inashuka hadi 1 kwa kila mfungwa, na mfungwa A anaendelea kuuza kwa mfungwa B. Kwa hiyo, bei ya bidhaa hiyo inashuka hadi 1A, na mfungwa A ananunua, lakini hakuna mtu anayetaka kuuza. Mfungwa C ni fumbo. Mifano ya jadi ya matumizi haitatarajia mtu yeyote kukubali mapato ya chini kwa kukabiliana na kutofautiana kwa juu. Lakini wawekezaji wenye malengo wanaweza kuwa wakitafuta tofauti wakati utajiri wao wa awali ni mdogo vya kutosha. "Mfano wa ""behavioral finance"" ni kwamba watu wanapata faida kwa kununua tiketi za bahati nasibu na kucheza kamari, na kuongeza kutofautiana kwa matokeo ni njia pekee ya kuongeza nafasi yao ya kufikia utajiri wa maisha." Bila shaka, yote haya ni zaidi ya majaribio ya mawazo rahisi: Inafunua baadhi ya masomo muhimu kuhusu masoko. Kwanza, wakati wa kuweka matarajio ya soko la mtaji au bei ya lengo kwa hisa, wachambuzi watafanya vizuri kutathmini soko la wanunuzi na wauzaji ili kuamua jinsi mahitaji yao na uwiano wa fedha utaathiri bei inayokuja. Hii ni ngumu zaidi kuliko mfano wetu, bila shaka, kwa sababu pamoja na mahitaji tofauti, kila mtu pia ana mtazamo tofauti kwa usalama fulani. Hilo halishangazi watendaji. Masoko yanayodhibitiwa na wanunuzi wa taasisi yanaonekana tofauti sana na yale yanayodhibitiwa na wawekezaji wenye tamaa na wafanyabiashara wa "YOLO". Mfano wa sasa ni mfumo wetu wa sasa wa kurahisisha kiasi (QE) unaoendelea kutoka kwa benki kuu ulimwenguni kote. Kwa wawekezaji waliochanganyikiwa na thamani ya hisa ya juu sana, tofauti kati ya Mfungwa A na Mfungwa B ni ya kuangaza. Wao ni sawa kabisa isipokuwa kwa jambo moja: Mfungwa B ni tajiri zaidi leo. Kwa ujumla, hii inamaanisha kwamba kuongeza fedha kwenye masoko ya kifedha huunda wawekezaji ambao wako tayari kulipa zaidi kwa usalama sawa. Kwa upande mwingine, wakati fedha za ziada zinapoondolewa kwenye soko, bei zinapaswa kushuka, kila kitu kingine kikiwa sawa, kwa sababu wawekezaji wenye fedha kidogo leo wanahitaji mapato ya juu. Hivyo mstari B unarudi kwenye mstari A. Pili, na ya kushangaza zaidi: Hakuna bei ya soko "sahihi". Hakuna usalama ina "thamani ya haki" au "thamani ya msingi". Badala yake, bei hutokea kutoka kwa sifa za usalama zinazohusiana na mahitaji ya wawekezaji sokoni. Sehemu nyingine muhimu ya bei: kila mwekezaji uwiano wa fedha katika soko. Ikiwa wawekezaji wa kutosha, au wafungwa wa C, watatumia pesa zao katika soko la usalama, bei zinaweza kubaki juu au kuongezeka hadi usawa wao utakapokwisha. Je, sauti ya ukoo, GameStop? Hii inaweza kuonekana dhahiri, lakini si mtazamo wa jadi juu ya masoko. Nadharia ya soko la ufanisi inasema kwamba dhamana daima biashara kwa thamani yao ya haki na kwamba wakati wa soko haiwezi kufanya kazi. Bila shaka, kutabiri maendeleo ya misingi ya usalama ni kazi ngumu. Lakini hiyo ni nusu tu ya usawa. Kama soko letu la hisa la gereza linavyoonyesha, kuelewa soko la wawekezaji na tabia yao inaweza kutoa ufahamu ambao ni muhimu kama vile. Nini ni hata wazimu: Kila mwekezaji katika soko ni vitendo kiakili. Mfungwa C anatoa bei ya busara kabisa kwa ajili ya usalama hata kama ni zabuni ya juu zaidi katika soko! Mfungwa A anafanya mambo kwa busara licha ya kuwa na bei ya chini zaidi ya kununua. Hii ni moja ya sababu za msingi wa mkoba wa ahadi. "Mfumo wa fedha wa tabia unaelezea mwenendo wa bei ya soko la gereza kama ""mwenendo usio na busara, ingawa unaoweza kutabiriwa wa mwekezaji,"" na nadharia ya jadi inaifuta kama haipo." Lakini wawekezaji wenye malengo wanaweza kuona waziwazi kile kinachotokea kweli. Lengo-msingi mkoba nadharia inaweza, kwa kweli, kuwa daraja muhimu kati ya nadharia ya kawaida na maelezo. Kama mfungwa katika jaribio la mawazo la Bernoulli, tuna malengo maalum ya kufikia kwa pesa zetu. Kama mfungwa, sisi pia tunashirikiana na masoko ya umma kwa kuzingatia malengo hayo. Malengo hayo huathiri bei kwa njia ambazo nadharia ya kawaida haiwezi kutarajia. Ingawa fedha za tabia hutoa mifano ya kutabiri kutokuwa na busara, nadharia ya malengo inadokeza kwamba watu wanaweza kuwa na busara zaidi kuliko walivyofikiri awali. Kama wewe kama chapisho hili, usisahau kujiunga na Investor Enterprising. Machapisho yote ni maoni ya mwandishi. Kama vile, haipaswi kueleweka kama ushauri wa uwekezaji, wala maoni yaliyoelezwa hayana lazima kutafakari maoni ya CFA Institute au mwajiri wa mwandishi. Image credit: Getty Images