How Much Leverage Should Altruists Use?

Cross-posted to my web­site. I have tried to make all the for­mat­ting work on the EA Fo­rum, but if any­thing doesn’t look right, try read­ing on my web­site in­stead.

Last up­dated 2020-05-17.


Philan­thropic in­vestors prob­a­bly have greater risk tol­er­ance than self-in­ter­ested ones. Altru­ists can use lev­er­age—bor­row­ing money to in­vest—to in­crease the ex­pected util­ity of their port­fo­lios. They may wish to lever their port­fo­lios at much higher ra­tios than self-in­ter­ested in­vestors—likely 2:1 to 3:1, and per­haps much higher (prac­ti­cal con­cerns notwith­stand­ing).

Un­like nor­mal in­vestors, al­tru­ists care about re­duc­ing their cor­re­la­tions with other in­vestors, so they should heav­ily tilt their port­fo­lios to­ward un­cor­re­lated as­sets.

This es­say will dis­cuss:

  1. Tra­di­tional vs. al­tru­is­tic investing

  2. Ba­sic ar­gu­ments for us­ing leverage

  3. Ap­pro­pri­ate lev­els of risk for altruists

  4. The im­por­tance of un­cor­re­lated as­sets, and where in­vestors might be able to find them

  5. Po­ten­tial changes for philan­thropic behavior

Dis­claimer: I am not an in­vest­ment ad­vi­sor and this should not be taken as in­vest­ment ad­vice. This con­tent is for in­for­ma­tional pur­poses only. Please do your own re­search or seek pro­fes­sional ad­vice and oth­er­wise take rea­son­able pre­cau­tions be­fore mak­ing any sig­nifi­cant in­vest­ment de­ci­sions. Any given port­fo­lio re­sults are hy­po­thet­i­cal and do not rep­re­sent re­turns achieved by an ac­tual in­vestor.


This es­say uses a num­ber of terms that may ap­pear un­fa­mil­iar. I define most terms when I first use them, but this sec­tion pro­vides an easy refer­ence.

(eta): A nu­meric value for rel­a­tive risk aver­sion that de­ter­mines the shape of an agent’s util­ity func­tion.

  • im­plies a lin­ear util­ity func­tion, that is, no diminish­ing marginal util­ity.

  • im­plies log­a­r­ith­mic util­ity, which is the value most of­ten used in the­o­ret­i­cal mod­els.

  • im­plies sub-log­a­r­ith­mic util­ity, which em­piri­cally de­scribes most in­di­vi­d­u­als’ prefer­ences.

Cash: For our pur­poses, cash does not re­fer to cur­rency, but to an in­vest­ment that’s con­sid­ered safe, such as short-term US Trea­sury bills. See “Risk-free rate.”

Isoe­las­tic util­ity func­tion: A util­ity func­tion that ex­hibits con­stant rel­a­tive risk aver­sion. If some­one has an isoe­las­tic util­ity func­tion, their risk aver­sion does not change based on how much money they have.

The isoe­las­tic util­ity func­tion is defined by

where is con­sump­tion and is a shape pa­ram­e­ter.

Lev­er­age: The pro­cess of bor­row­ing money in or­der to in­vest more than 100% of one’s as­sets.

Marginal util­ity: Utility gained when some­one gains a new good or ser­vice. For our pur­poses, we care about the marginal util­ity of money: how much util­ity a per­son gains from an ex­tra dol­lar.

Rel­a­tive risk aver­sion (RRA): The ex­tent to which an agent fa­vors safe bets over risky ones. See the defi­ni­tion of .

Risk-free rate: The in­ter­est rate in­vestors can earn with­out tak­ing on any risk, such as the rate earned by short-term US Trea­sury bills.

Sa­muel­son share (S): The pro­por­tion of some­one’s port­fo­lio they in­vest in a risky as­set.

  • If , they in­vest their en­tire port­fo­lio in the as­set.

  • If , they in­vest of their port­fo­lio in the as­set and the rest in cash.

  • If , they use lev­er­age to in­vest more than 100% of their port­fo­lio in the as­set.

Utility func­tion: A func­tion that as­signs a real num­ber to how much an agent val­ues each pos­si­ble out­come. In our case, a util­ity func­tion takes in an amount of money and re­turns the util­ity of hav­ing that much money.


What makes al­tru­is­tic in­vest­ing differ­ent from tra­di­tional in­vest­ing?

For the most part, al­tru­ists and tra­di­tional in­vestors have the same in­cen­tives re­gard­ing how they should in­vest—they want to in­vest in the port­fo­lio with the best re­turn for an ac­cept­able level of risk. Altru­ists and self-in­ter­ested in­vestors differ in two key ways:

  • Char­i­ta­ble causes prob­a­bly have slower-diminish­ing marginal util­ity of money than in­di­vi­d­u­als, so al­tru­ists should seek greater risk.

  • Self-in­ter­ested peo­ple only care about how much money they have, while al­tru­ists care about how much money all other (value-al­igned) al­tru­ists have.

And ar­guably a third way (more on this later):

  • Tra­di­tional in­di­vi­d­u­als avoid cer­tain mar­ket-beat­ing fac­tors like value and mo­men­tum for rea­sons that aren’t cap­tured by a stan­dard util­ity func­tion, and al­tru­ists do not share these rea­sons.

Re­gard­ing the first two differ­ences be­tween al­tru­ists and other in­vestors:

  1. The first differ­ence means al­tru­ists should be will­ing to use more lev­er­age than in­di­vi­d­u­als, to the ex­tent that their marginal util­ity diminishes more slowly.

  2. The sec­ond differ­ence im­plies that al­tru­ists who are thought­ful about in­vest­ing should not pro­por­tion­ally in­vest their money in the op­ti­mal port­fo­lio; in­stead, they should at­tempt to push the over­all pool of value-al­igned philan­thropic money in the di­rec­tion of op­ti­mal.

Im­prov­ing on con­ven­tional in­vest­ing wisdom

In Com­mon In­vest­ing Mis­takes in the Effec­tive Altru­ism Com­mu­nity, Ben Todd de­scribes some ways peo­ple can im­prove their in­vest­ment de­ci­sions over the sta­tus quo. Be­fore con­sid­er­ing lev­er­age, in­vestors should un­der­stand and im­ple­ment these sug­ges­tions.

My es­say will as­sume that read­ers agree with Todd’s sec­tion 3, “Not be­ing di­ver­sified enough.” Quot­ing the most rele­vant por­tion:

Many peo­ple I’ve spo­ken to are al­most fully in­vested in US equities. I think the ra­tio­nale for this is that equities have been the best re­turn­ing as­set his­tor­i­cally, so there’s no rea­son to own any­thing else. Another ra­tio­nale is that since you can’t beat the mar­ket, you should put ev­ery­thing into equities.

But US stocks do not equal “the mar­ket”. If you try to tally up all global fi­nan­cial as­sets, you get some­thing like this:

18% US stocks
13% For­eign de­vel­oped stocks
5% For­eign emerg­ing stocks
20% Global cor­po­rate bonds
14% 30 year bonds
14% 10 year for­eign bonds
2% TIPs
5% REITs
5% com­modi­ties
5% gold

This rep­re­sents the truly ag­nos­tic port­fo­lio. If you think you have no abil­ity the beat the mar­ket, then this is the port­fo­lio with the best risk-re­turn. 100% US equities is a huge bet on just one as­set.

From 1973 to 2013, a port­fo­lio like this re­turned 9.9% per year. In com­par­i­son, stocks re­turned 10.2%. So you only gave up a tiny 0.3% to switch to this port­fo­lio.

In re­turn, you had far lower risk. The volatility of the 100% equity port­fo­lio was 15.6%, whereas this di­ver­sified port­fo­lio had a volatility of only 8%. The max­i­mum draw­down was also only −27% com­pared to −51% with equities. The wide di­ver­sifi­ca­tion also makes you less vuln­er­a­ble to un­fore­seen tail risks.

The much lower volatility means you could have lev­ered up 2x and ended up with the same amount of volatility and same draw­downs as equities, but re­turns that were twice as high, at 20% per year.

Ad­di­tion­ally, I would recom­mend read­ers re­view Todd’s sec­tion 6, “Not beat­ing the mar­ket”:

Most peo­ple I speak to are sold on the “ex­pert com­mon sense” view that am­a­teur in­vestors shouldn’t try to beat the mar­ket, and should in­stead in­vest in in­dex funds. I ba­si­cally agree with this view. How­ever, if you’ve got a lit­tle more time to put into in­vest­ing, I think it’s worth con­sid­er­ing the idea of tilt­ing your in­vest­ments to­wards as­sets that have value (are cheap based on met­rics like P/​E and P/​B), high mo­men­tum (have gone up in the last 12 months, are above their 200-day mov­ing av­er­age) and low volatility. If you do this within equities, I think it’s pos­si­ble to beat the mar­ket by a cou­ple of per­centage points.

Most peo­ple are skep­ti­cal when I claim that value and mo­men­tum in­vest­ing beat the mar­ket, as I be­lieve they ought to be. Jus­tify­ing this claim against the effi­cient mar­ket hy­poth­e­sis re­quires a high bur­den of proof. I can­not meet this bur­den in only a few para­graphs, so I will re­fer to some longer writ­ings on the sub­ject for any­one still cu­ri­ous.

As­ness et al. (2013) Value and Mo­men­tum Every­where ex­am­ines eight as­set classes (US stocks, UK stocks, con­ti­nen­tal Euro­pean stocks, Ja­panese stocks, coun­try equity in­dex fu­tures, gov­ern­ment bonds, cur­ren­cies, and com­mod­ity fu­tures) and finds that value and mo­men­tum work in all eight.

Two ad­di­tional pa­pers: Fact, Fic­tion and Mo­men­tum In­vest­ing; and Fact, Fic­tion and Value In­vest­ing. Tar­geted at skep­tics, these ar­ti­cles offer high-level sum­maries and re­fer to other aca­demic pa­pers that provide ev­i­dence for the effi­cacy of mo­men­tum and value.

The books Quan­ti­ta­tive Value and Quan­ti­ta­tive Mo­men­tum, the first by Wesley Gray and To­bias Carlisle, and the sec­ond by Wesley Gray and Jack Vo­gel, dis­cuss value and mo­men­tum, re­spec­tively. Th­ese books dis­cuss why value and mo­men­tum work and how the au­thors be­lieve the strate­gies can best be im­ple­mented. For read­ers speci­fi­cally in­ter­ested in whether or how it is pos­si­ble to beat the mar­ket, I be­lieve Quan­ti­ta­tive Mo­men­tum gives a more con­cise and bet­ter-rea­soned ex­pla­na­tion, par­tic­u­larly in the first two chap­ters.

One could ar­gue that in­di­vi­d­u­als should not avoid value and mo­men­tum and be­have ir­ra­tionally by do­ing so. I mostly agree with this, but in­di­vi­d­u­als do have valid rea­sons for want­ing to avoid value and mo­men­tum—they can un­der­perform the broad mar­ket for long stretches of time, which many peo­ple may find un­de­sir­able for var­i­ous rea­sons:

  • They care about main­tain­ing so­cial sta­tus, and thus are risk averse not only with re­spect to los­ing money, but also with re­spect to los­ing money rel­a­tive to their peers.

  • For pro­fes­sional in­vestors, rel­a­tive (but not ab­solute) un­der­perfor­mance can cause them to lose clients.

Altru­ists should not con­sider these com­pel­ling rea­sons to avoid mar­ket-beat­ing strate­gies and should make efforts to sidestep the be­hav­ioral bi­ases that lead peo­ple to avoid them.

Risk and lev­er­age for tra­di­tional investors

Ac­cord­ing to stan­dard eco­nomic the­ory, in­di­vi­d­u­als have some level of risk tol­er­ance that dic­tates how much risk they are will­ing to take on their in­vest­ments. For our pur­poses, we can as­sume that in­di­vi­d­u­als have some isoe­las­tic util­ity func­tion, which means there ex­ists a con­stant > 0 de­scribing some­one’s rel­a­tive risk aver­sion (RRA), and their util­ity func­tion is given by:

An of 0 cor­re­sponds to lin­ear marginal util­ity of money (that is, no diminish­ing marginal util­ity). = 1 in­di­cates log­a­r­ith­mic util­ity, while > 1 gives sub-log­a­r­ith­mic util­ity. Differ­ent in­di­vi­d­u­als re­port sub­stan­tially differ­ent RRAs, and differ­ent meth­ods of em­piri­cally de­ter­min­ing give differ­ent re­sults; but most peo­ple ap­pear to have val­ues rang­ing from 1 to 4[1].

can be used to de­ter­mine how much risk some­one is will­ing to take on an in­vest­ment. For sim­plic­ity, sup­pose only two in­vest­ments ex­ist: a risk-free in­vest­ment and a risky one. (The risky in­vest­ment might not be a sin­gle as­set, but a di­ver­sified com­bi­na­tion of stocks and bonds.) Let be the rate of re­turn of the risk-free in­vest­ment, and let the risky in­vest­ment fol­low a log-nor­mal dis­tri­bu­tion pa­ram­e­ter­ized by and : that is, the log­a­r­ithm of the dis­tri­bu­tion is a nor­mal dis­tri­bu­tion with mean and stan­dard de­vi­a­tion .

Let Alice be an in­vestor with RRA equal to . Alice can in­vest some pro­por­tion of her port­fo­lio into her risky as­set; Ayres and Nale­buff in their book Life­cy­cle In­vest­ing re­fer to this pro­por­tion as the Sa­muel­son share.

Un­der tra­di­tional model as­sump­tions, the op­ti­mal Sa­muel­son share is given by . If S < 1, Alice should put the rest of her money into the risk-free as­set; if S > 1, she should use lev­er­age to in­vest more than 100% of her port­fo­lio.[2][3]

For most rea­son­able as­sump­tions about fu­ture stock re­turns and for typ­i­cal val­ues of , S < 1. That is, most peo­ple do not want to in­vest more than 100% into stocks.

How­ever, Ayres and Nale­buff ar­gue in Life­cy­cle In­vest­ing that peo­ple should treat their fu­ture in­come as an as­set in their port­fo­lio. Be­cause young peo­ple will have much more money in the fu­ture than they do to­day, they should use lev­er­age to di­ver­sify their in­vest­ments across time. If you ex­pect that 90% of your in­vestable earn­ings lie in the fu­ture and you have all your cur­rent in­vest­ments in stocks, your port­fo­lio is effec­tively 10% stocks/​90% fu­ture-earn­ings. If in­stead you get 2:1 lev­er­age, you now have 20% in stocks and 80% in fu­ture earn­ings, which is more rea­son­able than 10%/​90%. (Ar­guably you should have some­thing like 5:1 lev­er­age so that your port­fo­lio con­tains half stocks and half fu­ture earn­ings; but that much lev­er­age brings ad­di­tional com­pli­ca­tions, so Ayres and Nale­buff only recom­mend go­ing up to 2:1.)

Risk aver­sion for al­tru­is­tic causes

What RRA () should we use for char­i­ta­ble causes and how does it com­pare to in­di­vi­d­u­als’ RRAs? This is a com­plex ques­tion and I can­not offer a com­plete an­swer within the scope of this es­say. Peo­ple of­ten as­sume that al­tru­ists’ util­ity func­tions lie some­where be­tween log­a­r­ith­mic and lin­ear (); e.g., Paul Chris­ti­ano wrote, “I think log­a­r­ith­mic re­turns are fairly con­ser­va­tive for al­tru­is­tic pro­jects [...], and that for most causes the pay­offs are much more risk-neu­tral than that.” I con­sider this a rea­son­able as­sump­tion, but the ques­tion de­serves more in­ves­ti­ga­tion. A full anal­y­sis is out of scope, but I will briefly model how util­ity might diminish for three of the most pop­u­lar cause ar­eas in effec­tive al­tru­ism: global poverty, farm an­i­mal welfare, and ex­is­ten­tial risk.

Global poverty

In the long run, marginal util­ity for philan­thropists can­not diminish any faster than the rate of the self-in­ter­ested in­di­vi­d­ual with the slow­est-diminish­ing util­ity curve. If some per­son has a lower than ev­ery­one else, once ev­ery­one’s welfare has been im­proved to the point where that per­son has the great­est marginal util­ity of money, (which might take a long time, but in the­ory will hap­pen even­tu­ally), philan­thropists can­not do worse than to sim­ply give their money di­rectly to that per­son.

But the per­son with the steep­est util­ity curve might not cur­rently have the high­est marginal util­ity of money, be­cause there prob­a­bly ex­ist many other worse-off peo­ple who need money more. We care about how philan­thropists should in­vest their money right now, which means we want to know what the util­ity curve of global poverty in­ter­ven­tions cur­rently looks like.

GiveWell rates GiveDirectly as one of the top char­i­ties work­ing on global poverty. GiveDirectly sim­ply gives cash di­rectly to the world’s poor­est peo­ple. GiveWell be­lieves cash trans­fers do not rep­re­sent the most cost-effec­tive global poverty in­ter­ven­tion, but that they prob­a­bly aren’t much worse than the best in­ter­ven­tions. And we can an­a­lyze GiveDirectly’s marginal util­ity eas­ily be­cause we can model its value in terms of the util­ity curves of the peo­ple re­ceiv­ing the cash. So for now, let’s as­sume GiveDirectly rep­re­sents the pin­na­cle of global poverty char­i­ties.

For sim­plic­ity, let’s as­sume GiveDirectly has no over­head costs and give the world’s poor­est peo­ple money with­out en­coun­ter­ing any ob­sta­cles. GiveDirectly’s goal is to provide as much util­ity to as many peo­ple as pos­si­ble by in­creas­ing their wealth and thus al­low­ing them to buy more stuff that makes their lives bet­ter—that is, GiveDirectly wants to max­i­mize the ag­gre­gated in­di­vi­d­ual con­sump­tion util­ity func­tion:

where rep­re­sents the per­son, is the con­sump­tion level of the per­son, and gives the util­ity func­tion over con­sump­tion for the per­son. (Note that this ag­gre­gate util­ity func­tion is al­most cer­tainly not isoe­las­tic, which means we can­not as­sign it a rel­a­tive risk aver­sion pa­ram­e­ter .)

GiveDirectly’s math­e­mat­i­cally-op­ti­mal strat­egy—as­sum­ing all in­di­vi­d­u­als have the same value of (which is false, but a com­mon sim­plify­ing as­sump­tion in de­vel­op­ment eco­nomics)—is to give cash to the world’s poor­est per­son un­til they have as much money as the sec­ond-poor­est per­son; then give cash to the bot­tom two poor­est peo­ple un­til they reach the level of the third-poor­est per­son, and so on.

The world’s poor­est per­son and the world’s mil­lionth-poor­est per­son have al­most the same amount of money, so giv­ing a dol­lar to the lat­ter does al­most as much good as giv­ing a dol­lar to the former. The world con­tains many more poor peo­ple than GiveDirectly can af­ford to help, so (given our sim­plify­ing as­sump­tions) cash trans­fers have nearly lin­ear marginal util­ity of money, at least for donors with less than a few hun­dred billion dol­lars[4].

We had to make sev­eral sim­plify­ing as­sump­tions to reach this con­clu­sion. I can­not say with con­fi­dence that the con­clu­sion holds in the real world, but it does seem plau­si­ble that this sim­plified model roughly de­scribes how the global poverty cause op­er­ates in terms of its diminish­ing marginal util­ity:

  • In prac­tice, GiveDirectly can give money only to peo­ple it can reach; but it can reach such a large set of peo­ple that this does not sub­stan­tially af­fect its rate of diminish­ing util­ity.

  • GiveDirectly’s over­head costs scale sub-lin­early with money donated[5], so they don’t cause the util­ity curve to fall off more rapidly.

In­ter­ven­tions that GiveWell con­sid­ers more cost-effec­tive than cash trans­fers, such as de­worm­ing, prob­a­bly ex­pe­rience more rapidly diminish­ing marginal util­ity. Or, as GiveWell might say, these in­ter­ven­tions have less fund­ing ca­pac­ity than cash trans­fers do. Let’s call these “su­pe­rior” in­ter­ven­tions, in the sense that they are su­pe­rior to cash trans­fers. We can de­ter­mine the rate of diminish­ing by an­swer­ing three ques­tions:

  1. How much bet­ter are su­pe­rior global poverty in­ter­ven­tions than cash trans­fers?

  2. How much philan­thropic money has already gone into su­pe­rior global poverty in­ter­ven­tions?

  3. How much money would need to flow into global poverty be­fore all su­pe­rior in­ter­ven­tions be­came fully funded?

I don’t know the an­swers to any of these ques­tions, but let’s make up some on-the-face plau­si­ble an­swers and see what we get.

  1. GiveWell’s cost-effec­tive­ness anal­y­sis gives an an­swer for the first ques­tion: GiveWell es­ti­mates that its top char­i­ties do be­tween 5 and 60 times as much good as GiveDirectly.[6] The 60x char­ity is an out­lier and GiveWell’s es­ti­mate might not be ac­cu­rate, so let’s sup­pose the cur­rent best char­ity is 10x as cost-effec­tive as cash trans­fers.

  2. The USAID bud­get is $40 billion. We can ex­trap­o­late this to per­haps $200 billion per year in global spend­ing, and maybe $10 trillion in the past few decades. Prior to a few decades ago, GDPs were low enough and global poverty spend­ing was un­pop­u­lar enough that, for the sake of this rough es­ti­mate, we can as­sume prior spend­ing was $0. Of this $10 trillion, maybe $1 trillion is spent effec­tively. (One could prob­a­bly come up with a much bet­ter es­ti­mate with a few hours of re­search.)

  3. In 2018, GiveWell moved $111 mil­lion to top char­i­ties other than GiveDirectly (see GiveWell Met­rics Re­port) and moved similar amounts in 2015–2017. GiveWell prob­a­bly can keep mov­ing money for a while, and its top char­i­ties only rep­re­sent a frac­tion of bet­ter-than-cash-trans­fers giv­ing op­por­tu­ni­ties—we should in­clude effec­tively-uti­lized money from other philan­thropic or­ga­ni­za­tions such as the Gates Foun­da­tion. So philan­thropists can spend per­haps $100 billion be­fore run­ning out of such op­por­tu­ni­ties.

As­sign the an­swers to these ques­tions to sym­bols , , and , re­spec­tively. The value of is given by[7]

For the given val­ues of = 10, = $1 trillion, and = $100 billion, we get = 24, which in­di­cates re­mark­ably high risk aver­sion (most in­di­vi­d­u­als have an RRA be­tween 1 and 4[1:1]). This im­plies that the 100 billionth dol­lar spent on global poverty did times as much good as the trillionth dol­lar, which seems highly im­plau­si­ble.

If in­stead we as­sume that global poverty in­ter­ven­tions can ab­sorb a full $1 trillion, we get a rel­a­tive risk aver­sion of 3.3. Per­haps we also be­lieve that few or­ga­ni­za­tions spent their money in­effec­tively, and only $100 billion went to effec­tive causes, which means over 90% of the value of su­pe­rior global poverty in­ter­ven­tions has yet to be cap­tured; giv­ing = 1.

Alter­na­tively, we could ask what pro­por­tion of su­pe­rior giv­ing op­por­tu­ni­ties philan­thropists have funded already. This pro­por­tion equals , so in the first set of num­bers given, this would equal $1 trillion /​ $1.1 trillion = 0.91.

My in­tu­ition sug­gests that philan­thropists have used up most of the su­pe­rior op­por­tu­ni­ties, yet they still need much more fund­ing to ex­haust them—I would guess we are sub­stan­tially less than 90% of the way there. (Some peo­ple who re­viewed this es­say had the op­po­site in­tu­ition—that the ma­jor­ity of su­pe­rior op­por­tu­ni­ties have yet to be filled.) I do not have any par­tic­u­lar ex­per­tise on this is­sue and find a wide range of an­swers plau­si­ble, which means could plau­si­bly take on many val­ues. But it does seem un­likely that cash trans­fers would have a much smaller than self-in­ter­ested peo­ple’s, while other global poverty in­ter­ven­tions would have a much larger .

To view from an­other an­gle, and pump in­tu­itions in a differ­ent way: How much money do we be­lieve su­pe­rior in­ter­ven­tions can ab­sorb rel­a­tive to cash trans­fers? I would guess that su­pe­rior in­ter­ven­tions can ab­sorb close to as much fund­ing as cash trans­fers (to within a fac­tor of two), which sug­gests that the two s can­not differ by much.

It might make more sense to model su­pe­rior giv­ing op­por­tu­ni­ties us­ing a non-isoe­las­tic util­ity curve (i.e., the value of is not fixed). This would vi­o­late many of the as­sump­tions in this sec­tion, re­quiring much more anal­y­sis.

Ul­ti­mately, my weakly-in­formed guess is that su­pe­rior global poverty in­ter­ven­tions ex­pe­rience diminish­ing marginal util­ity at a slightly but not much higher rate than cash trans­fers.

Farm an­i­mal welfare

If we wish to fund in­ter­ven­tions that at­tempt to re­duce the in­ci­dence of fac­tory farm­ing or im­prove farm an­i­mals’ well-be­ing, what diminish­ing marginal util­ity might we ex­pect these in­ter­ven­tions to ex­pe­rience?

We can think about farm an­i­mal welfare as an ex­ten­sion of global poverty in at least two differ­ent ways, each of which re­sults in a differ­ent model and a differ­ent an­swer for .

The first way: We can con­sider fac­tory farm­ing in­ter­ven­tions in the same way we ex­am­ined su­pe­rior global poverty in­ter­ven­tions. Some amount of money has been spent on pre­vent­ing fac­tory farm­ing; and some ad­di­tional amount of money can be spent be­fore efforts on farm an­i­mal welfare are no more cost-effec­tive than cash trans­fers. We can an­swer the same three ques­tions we raised in the pre­vi­ous sec­tion and find the value of for fac­tory farm­ing in­ter­ven­tions.

We prob­a­bly be­lieve that the best farm an­i­mal in­ter­ven­tions are more cost-effec­tive than the top global poverty in­ter­ven­tions, mean­ing we as­sign a greater value to . All else equal, a larger re­sults in a larger —that is, greater risk aver­sion.

Push­ing in the other di­rec­tion, prob­a­bly only a small frac­tion of value in the anti-fac­tory-farm­ing space has been cap­tured already, which means the pro­por­tion is small.

Some ex­am­ple val­ues:

  • if and , then

  • if and , then

  • if and , then

  • if and , then

The sec­ond way: When con­sid­er­ing cash trans­fers, we looked at how much money it would take to im­prove the sta­tion of the N worst-off peo­ple so that they be­come as well off as the (N+1)<sup>th</​sup>-worst off per­son. We can ex­tend this model by look­ing at a set of sen­tient be­ings in­clud­ing both hu­mans and fac­tory-farmed an­i­mals (and nat­u­rally we could ex­tend this to in­clude other be­ings as well, but for now let’s just con­sider those two cat­e­gories). As a weird but use­ful and sort-of-cor­rect ab­strac­tion, as­sume fac­tory-farmed an­i­mals con­sume money in the same way that hu­mans do, but that they’re re­ally, re­ally poor, so they have ex­tremely high marginal util­ity of money. Then philan­thropists should give money to help an­i­mals un­til all the an­i­mals are as well off as globally poor hu­mans. Ac­cord­ing to this model, dona­tions to farmed an­i­mal welfare have nearly lin­ear marginal util­ity.

This model seems plau­si­ble in some re­spects, but the first model re­quires un­rea­son­ably small val­ues of and/​or to pro­duce a similar an­swer. Both of these mod­els have sev­eral ob­vi­ous short­com­ings. They are only meant to be a rough start, not a defini­tive an­swer.

Ex­is­ten­tial risk

Work on ex­is­ten­tial risk re­duc­tion of­ten in­volves do­ing re­search: ex­plor­ing ways of neu­tral­iz­ing en­g­ineered su­per-viruses, study­ing how to cre­ate sta­ble AI sys­tems, etc. Other types of work can be done, but I will fo­cus on re­search for now.

Effec­tive re­searchers be­gin by fo­cus­ing on the most promis­ing po­ten­tial di­rec­tions. As fund­ing for a field in­creases, re­searchers be­gin pur­su­ing less promis­ing av­enues. How quickly does the value of re­search diminish?

Owen Cot­ton-Bar­ratt claims in an ar­ti­cle, The Law of Log­a­r­ith­mic Re­turns, that the re­turns to re­search grow log­a­r­ith­mi­cally with re­sources in­vested. He refers to Ni­cholas Rescher’s book Scien­tific Progress: “start­ing with the ob­ser­va­tion that progress (counted in terms of the num­ber of “first rate” dis­cov­er­ies) goes lin­early with time while re­sources in­crease ex­po­nen­tially, [Rescher] de­duces that the un­der­ly­ing be­havi­our is a log­a­r­ith­mic re­turn to re­sources.” Cot­ton-Bar­ratt pre­sents some ad­di­tional em­piri­cal ev­i­dence sug­gest­ing that re­turns to re­search scale log­a­r­ith­mi­cally. His fol­low-up ar­ti­cle, The­ory Be­hind Log­a­r­ith­mic Re­turns, pro­vides the­o­ret­i­cal jus­tifi­ca­tion for this ob­ser­va­tion.

Ad­di­tion­ally, I spoke with some peo­ple in­volved in ex­is­ten­tial risk re­search, and they (in­de­pen­dently) agreed that re­search prob­a­bly pro­duces log­a­r­ith­mic re­turns.

Log­a­r­ith­mic re­turns en­tail . In­vestors with log­a­r­ith­mic util­ity have greater risk tol­er­ance than most in­di­vi­d­ual in­vestors by a fac­tor of per­haps 1.5 to 4, but still much less risk tol­er­ance than some­one with near-lin­ear util­ity (such as a donor to GiveDirectly).

Note that this ap­plies to any re­search-based cause, which could in­clude po­ten­tial ex­is­ten­tial risks such as AI safety, biose­cu­rity, and cli­mate change, but also can in­clude other cause ar­eas such as dis­ease pre­ven­tion, macroe­co­nomic policy, and anti-ag­ing.

Risk aver­sion for altruists

Un­cor­re­lated small donors are nearly risk-neutral

Sup­pose you are a philan­thropist with a rel­a­tively small amount of money, and your in­vest­ments have no cor­re­la­tion with other philan­thropists’. (This sec­ond as­sump­tion rarely holds true in prac­tice, but let’s take it as a given for now.) If you want to donate to a cause that already has much more money than you do, you have nearly lin­ear marginal util­ity of money, which means you should be­have nearly risk-neu­trally.

As a nu­mer­i­cal ex­am­ple, let’s say your preferred cause area has $1 billion in fund­ing and you cur­rently have $1 mil­lion. Fur­ther sup­pose that this cause has log­a­r­ith­mic util­ity of money, which means $1 billion pro­duces log(1 billion) = 20.723 utils (where a util is an ab­stract mea­sure­ment of util­ity). If you add your $1 mil­lion, your cause now gets log(1,001,000,000) = 20.724 utils, giv­ing (ap­prox­i­mately) an ex­tra .001 utils.

If the cause as a whole dou­bles its fund­ing to $2 billion, the util­ity will in­crease from 20.723 to 21.416—so the first $1 billion is worth far more than the sec­ond billion. If you dou­ble your $1 mil­lion to $2 mil­lion, the util­ity of the cause in­creases to 20.725. Your first mil­lion pro­vides .001 utils, and your sec­ond mil­lion pro­vides an ad­di­tional .001 utils. The sec­ond mil­lion does provide less value, but only slightly less—if we use more sig­nifi­cant figures, we can see that the first mil­lion gives .00999 utils and the sec­ond mil­lion gives .00998.

To look at it an­other way, sup­pose you have the choice be­tween get­ting ei­ther (A) $X with cer­tainty, or (B) 50% chance of get­ting $2 mil­lion and 50% of get­ting noth­ing. At what value of X are you in­differ­ent be­tween A and B? Most self-in­ter­ested peo­ple would prob­a­bly say some­thing in the range of $50,000 to $300,000, with some peo­ple go­ing some­what higher or lower.[8] But in this ex­am­ple, your value of X would be $999,500.

How risk-averse are large donors?

While un­cor­re­lated small donors ex­pe­rience nearly lin­ear marginal util­ity of money, large donors do not. For our pur­poses, a large donor is one who can fund a sig­nifi­cant frac­tion of their preferred cause area. A donor who is the sole fun­der of a cause ex­pe­riences diminish­ing marginal util­ity at the same rate as the cause it­self, as dis­cussed in Risk aver­sion for al­tru­is­tic causes.

Cor­re­lated small donors look like large donors

Un­like in­di­vi­d­ual in­vestors, small donors do not care only about how their own port­fo­lios perform. Altru­ists who want their causes to re­ceive more fund­ing also want fel­low donors’ in­vest­ments to perform well.

Quot­ing Paul Chris­ti­ano:

[T]he fact that I am a small piece of the char­i­ta­ble dona­tions to a cause shouldn’t mat­ter. My risk is well-cor­re­lated with the risk of other in­vestors, and if I lose 10% of my money in a year, other in­vestors will also lose 10% of their money, and less money will be available for char­i­ta­ble giv­ing. This holds re­gard­less of whether a cause has a mil­lion donors or just one.

In effect, many small donors with highly cor­re­lated in­vest­ments have similar risk prefer­ences to a sin­gle large in­vestor.

Time diversification

The book Life­cy­cle In­vest­ing ex­plains the con­cept of time di­ver­sifi­ca­tion (an ex­cerpt from the first chap­ter, which in­tro­duces the con­cept, is available on­line). The ba­sic idea: Treat your fu­ture in­come as an as­set in your in­vest­ment port­fo­lio. Sup­pose you have $10,000 in in­vest­ments and ex­pect to in­vest an ad­di­tional $90,000 of your earn­ings over the next 20 years. In that case, you have in­vested only 10% of your life­time sav­ings. If you in­vest in a stan­dard 60% stocks/​40% bonds port­fo­lio, you will have $6,000 in stocks to­day but $60,000 by the time you re­tire (ad­justed for in­fla­tion and mar­ket re­turns). That means you have much more ex­po­sure to mar­ket fluc­tu­a­tions in 2040 than in 2020. You can solve this by di­ver­sify­ing across time: ap­ply­ing lev­er­age to your cur­rent $10,000 port­fo­lio while hold­ing more bonds or cash in 2040.

The prin­ci­ple of time di­ver­sifi­ca­tion ap­plies to philan­thropists as well, in two differ­ent ways:

  1. You should con­sider the dona­tions you can make with your fu­ture in­come.

  2. The fu­ture might con­tain more value-al­igned al­tru­ists who will in­crease the size of the philan­thropic money pool.

Life­cy­cle In­vest­ing pro­vides in­for­ma­tion on how to de­ter­mine op­ti­mal lev­er­age at each point in time given var­i­ous as­sump­tions, par­tic­u­larly in chap­ters 3 and 4.

Ac­cord­ing to Life­cy­cle In­vest­ing, in­vestors should calcu­late their op­ti­mal Sa­muel­son share. Then they should con­sider the dis­counted pre­sent value of their fu­ture in­come as an as­set in their port­fo­lio, and hold stocks in the cor­rect pro­por­tion of their life­time port­fo­lio. For ex­am­ple, an in­vestor with $10,000 in stocks and $90,000 in dis­counted fu­ture earn­ings[9], and with a Sa­muel­son share of 0.6, should ideally hold 60% in stocks. There­fore, they should the­o­ret­i­cally in­vest with 6:1 lev­er­age to get their stock hold­ing up to $60,000. Their over­all port­fo­lio would then look like this:

 $60,000 stocks
-$50,000 debt
 $90,000 fu­ture cash

which can be sim­plified to

 $60,000 stocks
 $40,000 cash

giv­ing the in­vestor their de­sired port­fo­lio.

Similarly, if you are the sole fun­der of a cause, you cur­rently have $10 mil­lion, and you ex­pect the cause’s fund­ing in the fu­ture to in­crease to $30 mil­lion in pre­sent-value dol­lars, then you can treat that ex­tra $20 mil­lion as cash in your port­fo­lio, and in­crease your cur­rent port­fo­lio’s risk by 3x. If you have a Sa­muel­son share of 1, in­crease your risky hold­ings from 100% to 300% (that is, 3:1 lev­er­age); at a Sa­muel­son share of 0.5, in­crease risky hold­ings from 50% to 150%.

For already-pop­u­lar causes such as global poverty, we might ex­pect fu­ture fund­ing to con­tinue in­creas­ing at the his­tor­i­cal rate. If global poverty dona­tions grow with GDP, we should treat this as a bond in the global poverty in­vest­ment port­fo­lio that pays the GDP growth rate in in­ter­est. As­sum­ing $20 billion in an­nual effec­tively-di­rected global poverty dona­tions (from the rough es­ti­mate given above) and 1% real GDP growth, we can treat global poverty dona­tions as a $2 trillion as­set that grows at 1% real per year and has the same risk level as GDP growth. If we look only at the GiveWell money moved of ~$100 mil­lion and as­sume it will grow with GDP (his­tor­i­cally it has grown faster than that), this sug­gests we treat fu­ture GiveWell dona­tions as an as­set worth $10 billion.

Many causes that effec­tive al­tru­ists pri­ori­tize do not see much main­stream in­ter­est, but might ex­pect to get much more in­ter­est in the fu­ture. For ex­am­ple, a few years ago, AI safety had hardly any fund­ing, but re­cently has been grow­ing in pop­u­lar­ity. For such causes, philan­thropic in­vestors may wish to sub­stan­tially lever up their port­fo­lios to cre­ate time di­ver­sifi­ca­tion. But also con­sider that if fu­ture fund­ing looks un­cer­tain, in­vestors may wish to take on less risk.

Ad­just­ing for other al­tru­ists’ in­vest­ing behavior

Sup­pose there ex­ist two risky as­sets A and B. Al­most all in­vestors in­vest in A, and hardly any­one in­vests in B, but both provide similar risk-ad­justed re­turn. An or­di­nary in­vestor should max­i­mally di­ver­sify by split­ting their as­sets into ap­prox­i­mately half A, half B.

Altru­ists care how much money other value-al­igned al­tru­ists have. As dis­cussed above, they wish to re­duce their as­sets’ cor­re­la­tion with other al­tru­ists’, not just with their own. There­fore, the op­ti­mal al­lo­ca­tion would not be 50% A/​50% B, but per­haps 100% B. If most peo­ple heav­ily over­weight as­set A, then an in­vest­ment-minded philan­thropist can move the “al­tru­is­tic port­fo­lio” closer to op­ti­mal by in­vest­ing only in B.

Ad­di­tion­ally, the pool of philan­thropic money has some op­ti­mal Sa­muel­son share. If the ag­gre­gate al­tru­is­tic port­fo­lio does not reach that risk tar­get, then thought­ful in­vestors can com­pen­sate by over-lev­er­ag­ing to make up for oth­ers’ in­suffi­cient lev­er­age. (If most al­tru­ists take on too much risk, one can com­pen­sate by hold­ing ex­tra cash, but this seems un­likely in prac­tice.)

Th­ese con­sid­er­a­tions mat­ter a great deal. If your port­fo­lio only makes up a small part of the al­tru­is­tic port­fo­lio, this sug­gests that you should hold no con­ven­tional as­sets and in­stead in­vest your en­tire port­fo­lio in some­thing weird-look­ing in or­der to max­i­mize un­cor­re­lated re­turn. It also might mean you should take on a huge amount of lev­er­age and in­cur a near-cer­tain prob­a­bil­ity of bankrupt­ing the al­tru­is­tic por­tion of your port­fo­lio in or­der to in­crease the over­all risk and re­turn of the al­tru­is­tic money pool.

Ideally, al­tru­ists would agree about op­ti­mal in­vest­ment choices. Sup­pose Alice in­vests in 100% US equities while Bob goes 100% short on US equities. If they share val­ues, they have a mu­tual in­ter­est in en­sur­ing that the other in­vests op­ti­mally. One of them must be mak­ing a mis­take, so they should come to an agree­ment about how they both should in­vest. In­so­far as it is pos­si­ble, rather than unilat­er­ally at­tempt­ing to shift the al­tru­is­tic money pool, we should co­op­er­ate with other philan­thropists and come to a con­sen­sus on how to in­vest.

Although most al­tru­ists do not di­ver­sify well or take on enough risk, it is plau­si­ble that large philan­thropists—who hold a dis­pro­por­tionate share of the al­tru­is­tic money—in­vest bet­ter. In­vest­ment-minded al­tru­ists should par­tic­u­larly care about how large donors in­vest. If they already in­vest op­ti­mally, that means small donors have less of a rea­son to try to shift the over­all in­vest­ment pool.

What if RRA varies over time?

If a philan­thropist’s util­ity curve does not fol­low the isoe­las­tic util­ity func­tion defined above, they do not have a sin­gle fixed value of . If in­creases or de­creases over time, we can­not sim­ply look at the op­ti­mal amount of in­vest­ment risk to use at a par­tic­u­lar point in time and then main­tain it for­ever. Fur­ther­more, we do not nec­es­sar­ily want to vary lev­er­age over time based on the value of .

First, some sim­plify­ing as­sump­tions:

  1. You will in­vest your money un­til time , at which point you will donate it. Let be your RRA at this time. You will donate all your money at once, rather than donat­ing over time ac­cord­ing to some sched­ule.

  2. You do not have enough money to mean­ingfully shift the util­ity curve, so the value of does not change as a re­sult of your dona­tion.

  3. You know the op­ti­mal time to donate. (In ac­tu­al­ity, the op­ti­mal value of de­pends on the shape of the util­ity curve, so we can­not know it with­out first know­ing .)

Un­der these as­sump­tions, the marginal util­ity of your dona­tion de­pends only on , not on the value of at any other time. de­ter­mines the rel­a­tive risk aver­sion of your dona­tion, which means it tells you how much risk you should take on. The de­sired level of risk is given by the Sa­muel­son share for­mula:

(Re­call that = ex­pected re­turn of in­vest­ment, = stan­dard de­vi­a­tion, and = risk-free rate.)

Donors can de­crease correlation

Be­cause most donors’ in­vest­ments are cor­re­lated, their util­ity curves fall off more rapidly than if they were in­de­pen­dent. But it is not des­tined to be this way.

If cor­re­lated donors ex­pe­rience log­a­r­ith­mic (or worse) marginal util­ity and un­cor­re­lated (small) donors ex­pe­rience nearly lin­ear marginal util­ity, donors should care a lot about find­ing un­cor­re­lated sources of in­come.

What sorts of un­cor­re­lated as­sets can we find, and what rate of re­turn can we ex­pect from them? In this sec­tion, I will sur­vey a few as­set classes worth con­sid­er­ing. This is not meant to be an ex­haus­tive list, nor will I de­scribe in de­tail the pros and cons of each; this list is just meant as a start­ing point. For more on some of these di­ver­sifiers, see AQR (2018), It Was the Worst of Times: Diver­sifi­ca­tion Dur­ing a Cen­tury of Draw­downs.

Note that if all al­tru­ists started in­vest­ing in these di­ver­sify­ing as­set classes, they would no longer provide as sub­stan­tial di­ver­sifi­ca­tion benefits. But I don’t ex­pect that to hap­pen in the near fu­ture.

(In­di­vi­d­ual in­vestors should care just as much about de­creas­ing cor­re­la­tion by di­ver­sify­ing. The differ­ence is that or­di­nary in­vestors can sim­ply hold an op­ti­mal port­fo­lio; but al­tru­ists care about other al­tru­ists’ money, so they may want to over­weight di­ver­sifiers rel­a­tive to what an in­di­vi­d­ual in­vestor would do.)


His­tor­i­cally, bonds have had near-zero cor­re­la­tion to stocks (even anti-cor­re­la­tion at times). Most in­vestors already hold bonds as well as stocks, how­ever, so adding bonds to our port­fo­lio does not do much to re­duce our cor­re­la­tion to other donors.


Fewer donors in­vest in com­modi­ties. Com­modi­ties provide low cor­re­la­tions to stocks (but usu­ally still pos­i­tive), and by care­fully se­lect­ing good in­vest­ment ve­hi­cles, they can provide pos­i­tive (but prob­a­bly not great) re­turn. For more, see Lev­ine et al. (2016), Com­modi­ties for the Long Run.

Man­aged fu­tures (trend­fol­low­ing)

A bet­ter idea than bonds or com­modi­ties might be man­aged fu­tures. Th­ese are ac­tively-man­aged long/​short strate­gies that in­tend to provide un­cor­re­lated re­turns. Man­aged fu­tures funds usu­ally use trend­fol­low­ing: go­ing long on as­sets that have been trend­ing up­ward over a cer­tain time hori­zon, and go­ing short on as­sets that have been trend­ing down­ward. When I dis­cuss man­aged fu­tures in this es­say, I am talk­ing speci­fi­cally about ones that use trend­fol­low­ing strate­gies.

Hurst et al. found promis­ing re­sults in De­mys­tify­ing Man­aged Fu­tures (2013). Ac­cord­ing to a back­test de­scribed in the pa­per, a di­ver­sified man­aged fu­tures strat­egy (in­vest­ing across stocks, bonds, com­modi­ties, and cur­ren­cies) pro­duced a re­turn of 19.4% with a stan­dard de­vi­a­tion of 10.8%, and an alpha over stock, bond, and com­mod­ity in­dexes of 17.4% (figure 2, page 49). That is, the man­aged fu­tures strat­egy pro­duced a 17.4% re­turn that was not ex­plained by the re­turn of stocks, bonds, or com­modi­ties. This sug­gests that man­aged fu­tures provide a strong un­cor­re­lated source of re­turns not cap­tured by most in­vestors.

I highly doubt that ac­tual in­vestors can re­al­ize a risk-ad­justed re­turn as high as what the Hurst pa­per found in its back­test, but I do ex­pect man­aged fu­tures strate­gies to be worth pur­su­ing in some cir­cum­stances. For more on how man­aged fu­tures strate­gies be­have and why they might provide alpha, see the pa­per. And for fur­ther ev­i­dence, see Hurst et al. (2014), A Cen­tury of Ev­i­dence on Trend-Fol­low­ing In­vest­ing, which tests man­aged fu­tures trend­fol­low­ing strate­gies go­ing back to 1880.

Go­ing for­ward, I ex­pect man­aged fu­tures to perform worse for three pri­mary rea­sons:

  1. The 19.4% figure does not in­clude fund fees or trans­ac­tion costs.

  2. Man­aged fu­tures strate­gies de­pend on in­ter­est rates, which are much lower now than they were over the stud­ied pe­riod.

  3. More in­vestors to­day use man­aged fu­tures strate­gies than did in the 80′s, 90′s, or early 00′s. (But note that man­aged fu­tures strate­gies have got­ten less pop­u­lar over the past decade.)

At the same time, Hurst et al. offer some rea­sons to be op­ti­mistic about fu­ture perfor­mance:

  1. Trans­ac­tion costs have been de­creas­ing over time as mar­kets be­come more liquid.

  2. In­vestors have ac­cess to mar­kets that were not pre­vi­ously in­vestable, such as emerg­ing mar­ket equities and emerg­ing mar­ket cur­ren­cies.

  3. Even as­sum­ing muted fu­ture re­turns, trend­fol­low­ing still pro­vides valuable di­ver­sifi­ca­tion.

The pa­per pro­vides some quan­ti­ta­tive anal­y­sis on how man­aged fu­tures benefit a port­fo­lio even if they pro­duce much worse re­turns than they did his­tor­i­cally.

Why do man­aged fu­tures strate­gies provide alpha? Why have they not been ar­bi­traged away? The an­swer is not known for cer­tain, but the most rea­son­able hy­poth­e­sis is that the ma­jor­ity of in­vestors avoid it be­cause it can un­der­perform their bench­mark for long pe­ri­ods of time. I can­not do this hy­poth­e­sis jus­tice, but in brief, con­sider this perfor­mance chart of a man­aged fu­tures mu­tual fund (EQCHX) com­pared against the S&P 500 over the five-year pe­riod from 2014 to 2019 (the dark blue line is EQCHX and the light blue line is the S&P):

<img src=”http://​​mdick­​​as­sets/​​images/​​eqchx-sp500.png″ /​​>

As dis­cussed above, most in­vestors bench­mark their re­turns against an in­dex like the S&P 500, and would be un­will­ing to suffer five years of dra­matic un­der­perfor­mance.

For more on this hy­poth­e­sis, see AlphaAr­chi­tect (2015), The Sus­tain­able Ac­tive In­vest­ing Frame­work: Sim­ple, But Not Easy.

Will man­aged fu­tures con­tinue to work? AQR (2018), Trend Fol­low­ing in Fo­cus, dis­cusses this ques­tion and con­cludes that they prob­a­bly will. In short, man­aged fu­tures strate­gies do not ap­pear over-sub­scribed, and al­though they have performed poorly over the past few years, similar stretches of poor perfor­mance have oc­curred many times in the past.

Even­tu­ally, of course, man­aged fu­tures strate­gies will no longer provide un­cor­re­lated re­turns. (Even if only a small per­centage of in­vestors ever adopts such strate­gies, over time those in­vestors will be­come richer un­til they have enough money to elimi­nate the mar­ket in­effi­ciency.) But un­til then, it ap­pears that small donors with the abil­ity to in­vest in man­aged fu­tures can get nearly lin­ear marginal util­ity of money by do­ing so.

Buy-and-hold long/​short strategies

Altru­ists might con­sider pur­su­ing buy-and-hold long/​short strate­gies (as op­posed to man­aged fu­tures, which are ac­tively-traded long/​shorts). That is, buy some as­set that has mod­er­ate but not perfect cor­re­la­tion to what you be­lieve most philan­thropists own and short-sell the stan­dard philan­thropists’ port­fo­lio. Say you be­lieve all philan­thropists put all their money in the S&P 500[10]. You could buy a global ex­clud­ing-US stock mar­ket in­dex, which has high but not perfect cor­re­la­tion with the S&P 500, and then short the S&P. As a re­sult, you can get a re­turn stream that’s un­cor­re­lated with other al­tru­is­tic in­vestors. For ex­am­ple, if the S&P and the global ex-US mar­ket are cor­re­lated with r=0.8, you could put 500% of your money in the global ex-US mar­ket (that is, 5:1 lev­er­age) and short 400% of the S&P 500, for a net 100% mar­ket ex­po­sure with zero cor­re­la­tion to the S&P. (That is not ac­tu­ally how the math works, but for the pur­poses of this ex­am­ple, it doesn’t mat­ter.)

At first glance, this seems waste­ful: you are short­ing an as­set that other al­tru­ists hold, so your in­vest­ments can­cel each other. But this is still a good idea given the as­sump­tion that the S&P-buy­ers are mak­ing a mis­take. By us­ing a long/​short strat­egy, you can in­crease the com­bined re­turn of all philan­thropic money with­out in­creas­ing risk. In other words, you are mov­ing the pool of philan­thropic money closer to an op­ti­mally-di­ver­sified port­fo­lio, and you are do­ing so more effec­tively than if you just bought the global ex-US in­dex. Of course, it would be even bet­ter to con­vince other al­tru­ists to im­prove their port­fo­lio hold­ings.

I sus­pect (al­though do not have much ev­i­dence) that philan­thropists un­der-weight quite a few as­set classes, such as global stocks (es­pe­cially emerg­ing-mar­ket stocks), gold, com­modi­ties, and emerg­ing-mar­ket bonds. A philan­thropic in­vestor might want to buy some of these as­sets while short­ing over­weighted ones. An in­vestor might par­tic­u­larly want to do this if they do not be­lieve that man­aged fu­tures perform as well as I sug­gested in the pre­vi­ous sec­tion.

Note that in­di­vi­d­ual in­vestors would never want to use this sort of strat­egy. The only rea­son this type of long/​short might make sense is be­cause you be­lieve some in­vestors’ money brings as much value to your util­ity func­tion as your own money does, but those other in­vestors are un­der-di­ver­sified, so you want to make up for that lack of di­ver­sifi­ca­tion.

Long/​short fac­tor premia

AQR (2018) iden­ti­fied long/​short fac­tor pre­mia as the best di­ver­sifier ac­cord­ing to its back­tests. In brief, a long/​short fac­tors strat­egy in­vests in mar­ket-beat­ing fac­tors (such as value and mo­men­tum, dis­cussed above) by go­ing long on “good” as­sets while go­ing short on “bad” as­sets (ac­cord­ing to the fac­tors), pro­duc­ing a mar­ket neu­tral po­si­tion. For a de­tailed anal­y­sis of this strat­egy, see Il­ma­nen et al. (2019), How Do Fac­tor Premia Vary Over Time? A Cen­tury of Ev­i­dence.

AQR has a Style Premia Alter­na­tive Fund (QSPNX) that ap­pears to at­tempt to fol­low the strate­gies iden­ti­fied in Il­ma­nen et al., al­though I have only briefly read the fund liter­a­ture. How­ever, this fund has a $1 mil­lion in­vest­ment min­i­mum, and I am not aware of any similar funds that re­tail in­vestors can ac­cess. The AQR and Il­ma­nen pa­pers sug­gest that suffi­ciently-wealthy al­tru­is­tic in­vestors may want to in­vest in a long/​short fac­tor fund as a di­ver­sifier, and QSPNX might be a rea­son­able way to do that.


Another low-cor­re­la­tion in­vest­ment op­por­tu­nity, sug­gested by Paul Chris­ti­ano:

[I]f you start or in­vest in a small com­pany, your pay­off will de­pend on that com­pany’s perfor­mance (which is typ­i­cally quite risky but only weakly cor­re­lated with the mar­ket). [...] This spe­cial case is only pos­si­ble be­cause the en­trepreneur or in­vestor is putting in their own effort, and moral haz­ard makes it hard to smooth out all of the risk across a larger pool (though VC funds will in­vest in many star­tups). You shouldn’t ex­pect to find a similar situ­a­tion in in­vest­ments, ex­cept when you are pro­vid­ing in­sight which you trust but the rest of the mar­ket does not (thereby pre­vent­ing you from in­sur­ing against your risk).

Mis­sion hedging

Some al­tru­ists have dis­cussed the con­cept of mis­sion hedg­ing:

How should a foun­da­tion whose only mis­sion is to pre­vent dan­ger­ous cli­mate change in­vest its en­dow­ment? Sur­pris­ingly, in or­der to max­i­mize ex­pected util­ity, it might use ‘mis­sion hedg­ing’ in­vest­ment prin­ci­ples and in­vest in fos­sil fuel stocks. When oil com­pa­nies perform well (which means they are con­tribut­ing more to cli­mate change), the anti-cli­mate change foun­da­tion will have more money. When more fos­sil fuels are burned, fos­sil fuel stocks go up, thus giv­ing the foun­da­tion more money. When fewer fos­sil fuels are burnt and fos­sil fuels stocks go down, the foun­da­tion will have less money, but it does not need the money as much any­more.

A philan­thropist could mis­sion hedge and ob­tain low cor­re­la­tion with other al­tru­is­tic in­vestors by buy­ing a mis­sion-hedge in­vest­ment (such as fos­sil fuel stocks) while si­mul­ta­neously short­ing the broad mar­ket. This would not provide ideal di­ver­sifi­ca­tion be­cause al­tru­ists who in­vest in in­dex funds will still have some ex­po­sure to the same risk fac­tors as the long/​short mis­sion hedge strat­egy, but it will re­duce cor­re­la­tion.

My im­pres­sion is that al­tru­is­tic in­vestors should pri­ori­tize max­i­miz­ing risk-ad­justed re­turn, then fo­cus on re­duc­ing cor­re­la­tion with other al­tru­ists. Mis­sion hedg­ing pro­vides only ter­tiary value and should be avoided in­so­far as it im­pedes the first two goals. But I have not stud­ied mis­sion hedg­ing, and I could be wrong.

What if ev­ery­one did this?

If you in­vest in a par­tic­u­lar source of re­turn that’s un­cor­re­lated to the broad mar­ket, and many other philan­thropists do, too, you have formed a pool of cor­re­lated in­vestors, which means your util­ity curve be­haves as if ev­ery­one in that pool is a sin­gle large donor. Nat­u­rally, you would pre­fer to in­vest in an as­set that’s not cor­re­lated with any other al­tru­ist’s in­vest­ments. But an un­cor­re­lated as­set with, say, $10 mil­lion in philan­thropic money still pro­vides much greater marginal util­ity than the S&P 500, in which or­ders of mag­ni­tude more value-al­igned philan­thropists already in­vest.

I doubt that huge sums of philan­thropic money will flow into un­usual in­vest­ments like man­aged fu­tures in the near fu­ture, which means those peo­ple who do make such in­vest­ments might ex­pe­rience nearly lin­ear marginal util­ity of money. That said, most of the ar­gu­ments in this es­say rely on the as­sump­tion that the over­whelming ma­jor­ity of al­tru­is­tic in­vestors will not change their be­hav­ior. For the most part, the pro­pos­als in this es­say even­tu­ally will stop be­ing a good idea if enough value-al­igned philan­thropists adopt them.

Re­turn expectations

The Sa­muel­son share for­mula re­quires four pa­ram­e­ters: , , , and . We have dis­cussed in some depth, but have not ad­dressed the val­ues of the other three pa­ram­e­ters. What might we ex­pect?

In­vest­ment firm Re­search Affili­ates (RAFI) pub­lishes an es­ti­mate of re­turn ex­pec­ta­tions based on solid method­ol­ogy. RAFI up­dates its pre­dic­tions reg­u­larly based on chang­ing mar­ket con­di­tions, but as of this writ­ing, it pre­dicts a 0.4% re­turn af­ter in­fla­tion with 14.4% stan­dard de­vi­a­tion for US large-cap stocks over the next 10 years (with a 95% con­fi­dence in­ter­val of −3.4% to 4.1% re­turn). But of course, in­vestors should di­ver­sify be­yond US stocks. RAFI es­ti­mates that the op­ti­mally-di­ver­sified port­fo­lio will provide 1.6% real re­turn with 4.0% stan­dard de­vi­a­tion. In­vestors who hold this port­fo­lio may want to use much more lev­er­age than in­vestors in US stocks, if only be­cause it has a much lower stan­dard de­vi­a­tion. Alter­na­tively, RAFI’s op­ti­mal (un-lev­ered) port­fo­lio at the 12% volatility level has 4.5% ex­pected real re­turn; this port­fo­lio has slightly worse re­turn-to-risk ra­tio (0.375 in­stead of 0.4), but re­quires less lev­er­age.

We care about , not just . We sub­tract the risk-free rate be­cause (1) if we hold cash, that cash can earn the risk-free rate; and (2) bor­row­ing money to use lev­er­age re­quires pay­ing the risk-free rate in in­ter­est (at least in the­ory, see Cost of lev­er­age). In­vestors can earn with­out tak­ing on any risk, so in some sense it doesn’t count, and we should sub­tract it out. The cur­rent risk-free rate af­ter in­fla­tion nearly equals zero, so we can as­sume with­out los­ing much pre­ci­sion.

As dis­cussed above, in­vestors prob­a­bly can out­perform a fully di­ver­sified port­fo­lio by tilt­ing to­ward value and mo­men­tum. Most im­ple­men­ta­tions of value and mo­men­tum (such as the Van­guard Value ETF) track a broad mar­ket in­dex and only use a weak value or mo­men­tum tilt—they have low ac­tive share.

What kind of re­turns might we ex­pect from a high-ac­tive-share value/​mo­men­tum fund? In­vest­ment firm AlphaAr­chi­tect main­tains an in­dex it calls the Global Value Mo­men­tum Trend In­dex, in­vestable through the ETF VMOT.[11] This in­dex only in­vests in the top ~5% of stocks match­ing its crite­ria. Com­par­ing AlphaAr­chi­tect’s back­test from 1992 to 2017 against in­dex re­turns from the Ken French data library, VMOT would have re­turned 17.4% with a 13.6% stan­dard de­vi­a­tion (be­fore sub­tract­ing fees and in­fla­tion), ver­sus 9.3%/​15.0% for the US + Europe stock mar­kets (which makes for a rea­son­able bench­mark).

For a longer back­test, I at­tempted to ap­prox­i­mate VMOT’s method­ol­ogy us­ing the Ken French Data Library en­tries on mo­men­tum and value (earn­ings-to-price), which go back to 1952[12]. Th­ese data only in­clude US stocks, and we have the­o­ret­i­cal rea­sons to ex­pect the sim­plified in­vest­ment strat­egy rep­re­sented by these data to (slightly) un­der­perform VMOT[13]. Over the full sam­ple back to 1952, it re­turned 18.3% with a stan­dard de­vi­a­tion of 16.2%—perform­ing bet­ter than in the more re­cent pe­riod. We do not have data on how VMOT might have performed be­fore 1992, but based on re­sults from the Ken French data, it prob­a­bly would have performed about as well or pos­si­bly bet­ter than it did in the 1992 to 2017 back­test.

Go­ing for­ward, we can­not nec­es­sar­ily ex­pect the same re­turns from a strat­egy like VMOT. RAFI pre­dicts an av­er­age 2.8% re­turn for the global stock mar­ket in­dex over the next 10 years. This sug­gests a differ­en­tial of 9.3% − 2.8% = 6.5% be­tween his­tor­i­cal nom­i­nal re­turn and fu­ture real re­turn.[14] If we sub­tract 6.5% from the his­tor­i­cal VMOT re­turn, and sub­tract an ad­di­tional 2% for fees and costs (which is the figure AlphaAr­chi­tect uses), we get an 8.9% ex­pected real re­turn for high-con­vic­tion value and mo­men­tum strate­gies like VMOT. This is a rough es­ti­mate, not an ac­cu­rate figure. Ar­guably we should as­sume value and mo­men­tum will not perform as well in the fu­ture as they have in the past, and ap­ply an ad­di­tional dis­count to this 8.9%. If we dis­count the ex­cess re­turn by half, we get a 5.9% ex­pected re­turn for VMOT. I will not ap­ply this dis­count in my calcu­la­tions, but we should be aware that do­ing so would change our ul­ti­mate es­ti­mate of how much lev­er­age to use.

As some­thing of a cor­rob­o­ra­tion, RAFI pro­vides es­ti­mates of for­ward-look­ing five-year re­turn for var­i­ous long/​short fac­tors. At the time of this writ­ing, it makes the fol­low­ing pre­dic­tions for its long/​short value and mo­men­tum fac­tors (net of trans­ac­tion costs):

  • 5.7% for US large-cap value

  • 1.1% for US large-cap momentum

  • 8.0% for US small-cap value

  • 6.4% for US small-cap momentum

(RAFI’s pro­jec­tions for for­eign de­vel­oped mar­ket fac­tors are similar but gen­er­ally a bit higher.)

A con­cen­trated long-only port­fo­lio on a par­tic­u­lar fac­tor would have ap­prox­i­mately the same ex­pected re­turn as the long/​short fac­tor plus the broad mar­ket (al­though that’s not quite how the math works).

The un­der­ly­ing in­dexes used by VMOT make some im­prove­ments on RAFI’s sim­ple fac­tor model (see Quan­ti­ta­tive Value and Quan­ti­ta­tive Mo­men­tum for de­tails[15]), so it might be rea­son­able to as­sume a higher ex­pected re­turn for VMOT. If we then sub­tract fees, we get some­thing close to the origi­nal es­ti­mate I gave for VMOT (prob­a­bly a bit higher[16]).

RAFI be­lieves the value and mo­men­tum pre­mia will work as well in the fu­ture as they have in the past, and some of the pa­pers I linked above make similar claims. They offer good sup­port for this claim, but in the in­ter­est of con­ser­vatism, we could jus­tifi­ably sub­tract a cou­ple of per­centage points from ex­pected re­turn to ac­count for pre­mium degra­da­tion.

Note that RAFI’s es­ti­mates use fac­tor timing—at­tempt­ing to guess how well fac­tors will perform based on the cur­rent mar­ket en­vi­ron­ment, rather than just look­ing at his­tor­i­cal be­hav­ior. This prac­tice is not widely ac­cepted; for ex­am­ple, see As­ness et al.’s Fac­tor Timing is De­cep­tively Difficult (2017). Also note that these num­bers only give ex­pected mean re­turn. Even if these es­ti­mates are ac­cu­rate, we could still see much higher or lower re­turns due to mar­ket volatility.

Un­cor­re­lated returns

As dis­cussed above, al­tru­ists have a few op­tions for seek­ing un­cor­re­lated (or weakly-cor­re­lated) re­turns. Of the (in­vestable) op­tions dis­cussed, man­aged fu­tures ap­pear best.

The pre­vi­ously-dis­cussed pa­per A Cen­tury of Ev­i­dence on Trend-Fol­low­ing In­vest­ing found that a di­ver­sified man­aged fu­tures strat­egy re­turned 11.2% (nom­i­nal) with 9.7% stan­dard de­vi­a­tion over the pe­riod 1880 to 2013, af­ter ad­just­ing for es­ti­mated fees and trans­ac­tion costs. As I said above, I do not ex­pect in­vestors to re­al­ize re­turns this high in prac­tice; I would ex­pect a man­aged fu­tures fund to un­der­perform VMOT af­ter fees and costs, but prob­a­bly out­perform the US stock mar­ket on a risk-ad­justed ba­sis over the next 10 years. I don’t know much about man­aged fu­tures—I’ve read a few pa­pers and done some anal­y­sis based on AQR’s pub­lished data, and that’s my guess based on what I know.

Ch­e­sa­peake Cap­i­tal’s man­aged fu­tures fund (EQCHX), dis­cussed pre­vi­ously, has pub­lished perfor­mance data (net of fees) go­ing back to 1988. From 1988 to 2020, it re­turned 9.8% (nom­i­nal) with a 19.2% stan­dard de­vi­a­tion. If we sub­tract in­fla­tion and de-lev­er­age to match the risk level of the AQR man­aged fu­tures strat­egy, we come up with about a 4% real re­turn with 11% stan­dard de­vi­a­tion.

As dis­cussed in Trend Fol­low­ing in Fo­cus (refer­enced pre­vi­ously), we have rea­son to ex­pect man­aged fu­tures to perform about as well go­ing for­ward as they did in the past. We don’t know if the 31-year sam­ple for EQCHX is rep­re­sen­ta­tive of more long-term perfor­mance, but it gives a more con­ser­va­tive es­ti­mate than the AQR data, so we can use this as a rough pro­jec­tion of fu­ture perfor­mance. Perfor­mance could look dra­mat­i­cally differ­ent in the fu­ture, but this es­ti­mate makes about as much sense as any.

Long-run mar­ket return

Mar­ket-beat­ing strate­gies such as value and mo­men­tum, and low-cor­re­la­tion, pos­i­tive-re­turn strate­gies such as man­aged fu­tures, al­most cer­tainly will perform worse as time goes on. At pre­sent, it ap­pears that only a small per­centage of in­vestors has high enough tol­er­ance for track­ing er­ror to in­vest in these sorts of strate­gies. But over time, these in­vestors will be­come richer and even­tu­ally elimi­nate the mar­ket in­effi­ciency.

The Ram­sey equa­tion gives the long-run rate of re­turn in an effi­cient mar­ket:


  • = in­vest­ment rate of return

  • = pure time preference

  • = rel­a­tive risk aversion

  • = eco­nomic growth rate

This equa­tion holds be­cause in­di­vi­d­u­als dis­count fu­ture money at rate , so they will re­quire that in­vest­ments re­turn at least this much; and in­vest­ments that re­turn more than this will ex­pe­rience in­flows un­til the ex­pected re­turn drops to .

(If his­tor­i­cal mar­ket re­turns roughly con­tinue, we can prob­a­bly ex­pect equities mar­kets to re­turn 3-5% af­ter in­fla­tion in the long run.)

An in­vestor max­i­mizes long-run ex­pected util­ity by, at each point in time, max­i­miz­ing ex­pected util­ity for the next point in time[17]; and max­i­mizes ex­pected util­ity at each time step by in­vest­ing at the level given by the Sa­muel­son share for­mula based on the ex­pected re­turn and stan­dard de­vi­a­tion at each time . There­fore, the long-run ex­pected re­turn does not af­fect what risk level in­vestors should adopt to­day.


Mar­kets do not be­have the way stan­dard the­o­ret­i­cal mod­els sug­gest they do. This sec­tion dis­cusses some de­vi­a­tions from the­ory and how that af­fects lev­er­age.

Be­hav­ior of lev­er­aged in­vest­ments in practice

Be­fore us­ing lev­er­age, in­vestors should un­der­stand how differ­ent im­ple­men­ta­tions of lev­er­age can be­have in prac­tice. Brian To­masik’s Should Altru­ists Lev­er­age In­vest­ments? dis­cusses this in de­tail, with ex­ten­sive simu­la­tions; Colby Davis’s The Line Between Ag­gres­sive and Crazy pro­vides a more con­cise dis­cus­sion of the most im­por­tant points. Note that Davis’s ar­ti­cle, al­though it does not say so ex­plic­itly, as­sumes in­vestors have a log­a­r­ith­mic util­ity func­tion; so some of its con­clu­sions do not ap­ply in the same way for other util­ity func­tions. For ex­am­ple, the for­mula he gives for op­ti­mal lev­er­age is a spe­cial case of the Sa­muel­son share for­mula with .

Mean reversion

If as­set prices fol­low a ran­dom walk, in­creas­ing lev­er­age pro­por­tion­ally in­creases ex­pected value. Un­for­tu­nately, prices prob­a­bly do not (en­tirely) fol­low a ran­dom walk.

In the short run (on the timescale of days to weeks), stock prices tend to over­re­act and then mean re­vert[18]. This es­sen­tially means that me­dian out­comes hap­pen more of­ten, while high-var­i­ance out­comes (where the mar­ket goes up and then up again, or down and then down again) oc­cur less of­ten.

As ex­plained by Brian To­masik, when as­set re­turns are in­de­pen­dent across time, lev­er­age pro­por­tion­ally in­creases mean re­turn, but less-than-pro­por­tion­ally in­creases (and pos­si­bly de­creases) me­dian re­turn. Be­cause short-term mean re­ver­sion in­creases the prob­a­bil­ity of me­dian out­comes, it makes lev­er­age look less ap­peal­ing.

Ad­di­tion­ally, mar­kets ex­hibit long-run mean re­ver­sion over the timescale of years: as­sets that have gone up (or down) a lot over the past three to five years tend to go down (or up) over the fol­low­ing year. For­tu­nately, we can adapt to this phe­nomenon by ad­just­ing our re­turn ex­pec­ta­tions based on as­set val­u­a­tions, which Re­search Affili­ates does in its es­ti­mates that I quoted pre­vi­ously. (Per­haps we could adapt to short-term mean re­ver­sion as well, but it would re­quire up­dat­ing re­turn ex­pec­ta­tions on a fre­quent, e.g., daily, ba­sis.)

As­sets ex­hibit medium-term mo­men­tum: price trends over the past 6-12 months tend to con­tinue over the next 1-3 months. This works in fa­vor of lev­er­age by in­creas­ing the prob­a­bil­ity of high-var­i­ance out­comes while de­creas­ing the fre­quency of me­dian out­comes.

In sum­mary, there ex­ist three well-es­tab­lished types of price trend: short-term mean re­ver­sion, medium-term mo­men­tum, and long-term mean re­ver­sion. The lat­ter two can work to our ad­van­tage (or at least they don’t hurt us), while short-term mean re­ver­sion makes lev­er­age look worse. How this af­fects the value of lev­er­age over­all war­rants fur­ther in­ves­ti­ga­tion.

Left skew of in­vest­ment returns

So far, I have as­sumed that as­set re­turns fol­low a log-nor­mal dis­tri­bu­tion. As­set pric­ing mod­els tra­di­tion­ally as­sume log-nor­mal re­turns, so this as­sump­tion has well-es­tab­lished prece­dent. Un­for­tu­nately, it is false.

In prac­tice, as­set re­turns skew to the left: highly nega­tive re­turns oc­cur more fre­quently than a log-nor­mal model pre­dicts. Any in­vestor with a con­cave util­ity func­tion (i.e., any rea­son­able in­vestor) dis­likes left skew: peo­ple dis­pre­fer bad out­comes more strongly than they pre­fer good out­comes. That means in­vestors should take on less risk than log-nor­mal mod­els sug­gest.

Un­pre­dictabil­ity of fu­ture return

The Sa­muel­son share for­mula as­sumes we know the ex­act val­ues of for­ward-look­ing ex­pected re­turn and volatility. But ob­vi­ously we don’t. Get­ting too much lev­er­age gen­er­ally hurts more than not get­ting enough, so the more un­cer­tain we are about these pa­ram­e­ter val­ues, the less lev­er­age we should want to use (to avoid risk­ing ac­ci­den­tally tak­ing too much risk).

We could ac­count for this by treat­ing mean re­turn and stan­dard de­vi­a­tion as dis­tri­bu­tions rather than point es­ti­mates, and calcu­lat­ing util­ity-max­i­miz­ing lev­er­age across the dis­tri­bu­tion in­stead of at a sin­gle point. This raises a fur­ther con­cern that we don’t even know what dis­tri­bu­tion the mean and stan­dard de­vi­a­tion have, but at least this gets us closer to an ac­cu­rate model.

For more, see Cho­pra, V. K. and W. T. Ziemba (1993). “The effect of er­rors in mean, var­i­ance and co-var­i­ance es­ti­mates on op­ti­mal port­fo­lio choice.”

Cost of leverage

In the­ory, bor­row­ing money for lev­er­age costs only the risk-free rate. In prac­tice, lev­er­age costs more than that un­der most cir­cum­stances.

Life­cy­cle In­vest­ing recom­mends buy­ing deep-in-the-money call op­tions. Such op­tions on highly liquid se­cu­ri­ties tend to cost about the risk-free rate, but op­tions on less-traded se­cu­ri­ties will carry much higher im­plicit in­ter­est rates.

In­vest­ing with mar­gin al­lows for more flex­i­bil­ity. In­ter­ac­tive Bro­kers charges the risk free rate plus be­tween 0.3% and 1.5%, de­pend­ing on the amount of mar­gin used, with higher mar­gin cost­ing lower rates. In­vestors can use mar­gin to buy any in­vestable as­set. Large in­sti­tu­tions can po­ten­tially ne­go­ti­ate cheaper rates.

Brian To­masik gives a more thor­ough sur­vey of types of lev­er­age, along with their ad­van­tages and dis­ad­van­tages.

Higher costs make lev­er­age look some­what less at­trac­tive. We can roughly ap­prox­i­mate the im­pact of a 1% cost over the risk-free rate by sub­tract­ing 1% from the ex­pected re­turn of an in­vest­ment.[19]


In­vestors who hold their money in tax­able ac­counts should pay spe­cial at­ten­tion to min­i­miz­ing taxes. A port­fo­lio that makes fre­quent trades will in­cur sub­stan­tial perfor­mance drag due to taxes. (This ap­plies to any in­vest­ment, not just in­vest­ment with lev­er­age.)

Taxes re­duce ex­pected re­turn, but they also re­duce volatility by the same amount. If in­vestors can bor­row money for free, they can in­crease lev­er­age in pro­por­tion to the tax rate, which will in­crease the ex­pected re­turn and stan­dard de­vi­a­tion of their as­sets by ex­actly enough to can­cel out the tax bur­den. Un­for­tu­nately, in­vestors can­not get free lev­er­age, so taxes will hurt their risk-ad­justed re­turn.

Ideally, al­tru­ists should in­vest in tax-sheltered ac­counts. Foun­da­tions can in­vest how­ever they want, but small donors have more limited op­tions. Donor-ad­vised funds (DAFs) typ­i­cally only al­low a small set of in­vest­ments. But some funds offer more flex­i­bil­ity in some cases: for ex­am­ple, Fidelity Char­i­ta­ble al­lows donors with at least $5 mil­lion to man­age their own in­vest­ments. It also per­mits donors with $250,000 to nom­i­nate an in­vest­ment ad­vi­sor to man­age their money. DAFs typ­i­cally charge 0.6% per year, which sub­stan­tially hurts your re­turn if you keep money in them for a long time, but this might be prefer­able to pay­ing taxes ev­ery year.

Tax-sen­si­tive in­vestors should pre­fer ETFs over mu­tual funds be­cause they do not dis­tribute tax­able gains as of­ten. In the long run, an ETF in a tax­able ac­count may in­cur lower costs than a DAF.

This sub­ject de­serves much more dis­cus­sion, and much has been writ­ten el­se­where on the sub­ject. Altru­ists should con­sider other ways to min­i­mize taxes or, if pos­si­ble, to in­vest in tax-sheltered ac­counts.

Ad­di­tional caveats

Ed­ward Thorp, The Kelly Cri­te­rion in Black­jack Sports Bet­ting, and the Stock Mar­ket (p. 32), lists some ad­di­tional false as­sump­tions of this model:

Stock prices do not change con­tin­u­ously; port­fo­lios can’t be ad­justed con­tin­u­ously; trans­ac­tions are not costless; the bor­row­ing rate is greater than the T-bill rate; the af­ter tax re­turn, if differ­ent, needs to be used[.]

How should this change philan­thropists’ be­hav­ior?

Rea­son­able lev­er­age ra­tios (un­der the the­o­ret­i­cal model)

This sec­tion ig­nores all the caveats pre­sented above.

So far, we have ar­rived at the fol­low­ing con­clu­sions about al­tru­ists’ risk prefer­ences:

  • Un­cor­re­lated small donors ex­pe­rience nearly lin­ear marginal util­ity of money, re­gard­less of their preferred cause.

  • In­ter­ven­tions that straight­for­wardly benefit their re­cip­i­ents, in­clud­ing typ­i­cal global poverty and farm an­i­mal welfare in­ter­ven­tions, prob­a­bly have nearly lin­ear marginal util­ity; but marginal util­ity might diminish more rapidly if the best in­ter­ven­tions dry up quickly.

  • Re­search-based cause ar­eas, in­clud­ing ex­is­ten­tial risk, prob­a­bly ex­pe­rience roughly log­a­r­ith­mic marginal util­ity.

Philan­thropists with near-zero risk aver­sion the­o­ret­i­cally max­i­mize the util­ity of their in­vest­ments by tak­ing on as much lev­er­age as they fea­si­bly can.

For philan­thropists with log­a­r­ith­mic util­ity—likely in­clud­ing large donors and cor­re­lated small donors—the op­ti­mal Sa­muel­son share de­pends on in­vest­ment re­turn ex­pec­ta­tions. This table gives Sa­muel­son shares for var­i­ous in­vest­ment port­fo­lios, given the re­turn ex­pec­ta­tions laid out pre­vi­ously. The fourth column gives de­sired lev­er­age ac­cord­ing to the Sa­muel­son share for­mula, and the fifth column gives ex­pected re­turn af­ter ap­ply­ing lev­er­age. This as­sumes the real risk-free rate equals 0% (which is ap­prox­i­mately true as of this writ­ing).[20]

Be aware that these es­ti­mates de­pend on as­sump­tions about re­turn ex­pec­ta­tions that may be false. Fur­ther­more, the es­ti­mated op­ti­mal lev­er­age only holds un­der the the­o­ret­i­cal model de­scribed pre­vi­ously. Although this model is com­monly used to ap­prox­i­mate in­vest­ment be­hav­ior, it does not ac­cu­rately re­flect real-world trad­ing, for rea­sons listed in Caveats as well as other rea­sons.

di­ver­sified global in­dex, low-vol 1.6% 4% 10.2 16%
di­ver­sified global in­dex, high-vol 4.5% 12% 3.4 15%
value/​mo­men­tum/​trend (VMOT) 9% 13% 6.1 55%
man­aged fu­tures (MF) 4% 11% 3.5 14%
VMOT + MF 7% 10% 7.8 54%

Th­ese es­ti­mates as­sume log­a­r­ith­mic util­ity. If al­tru­ists have higher or lower , they should pre­fer lower or higher amounts of lev­er­age (re­spec­tively).

De­vi­a­tions from the the­o­ret­i­cal model

On the other hand, a num­ber of caveats re­duce how much lev­er­age al­tru­ists may want to use. Th­ese in­clude mar­ket be­hav­iors such as mean re­ver­sion, left skew of in­vest­ment re­turns, and the un­pre­dictabil­ity of fu­ture re­turn; as well as prac­ti­cal con­sid­er­a­tions such as the cost of lev­er­age and taxes.

After ac­count­ing for these caveats, large philan­thropists or cor­re­lated small donors should prob­a­bly take on more lev­er­age than most in­di­vi­d­ual in­vestors, but sub­stan­tially less than the num­bers given in the pre­vi­ous sec­tion. Some have pro­posed us­ing “half Kelly” in­stead of the Kelly crite­rion,[21] where for our pur­poses, the Kelly crite­rion cor­re­sponds to and half Kelly means . This might roughly ac­count for the var­i­ous fac­tors that make lev­er­age look less ap­peal­ing. Or we could look at how his­tor­i­cally suc­cess­ful in­vestors have cal­ibrated their risk. For ex­am­ple, War­ren Buffett ap­pears to use some­thing similar to the full Kelly crite­rion, as do many other suc­cess­ful in­vestors.[22]

Dou­bling re­sults in halv­ing ac­cord­ing to the Sa­muel­son share for­mula. Us­ing the same re­turn es­ti­mates as be­fore, this gives for the di­ver­sified global in­dex (high-vol) and for VMOT + MF.

Push­ing in the other di­rec­tion, re­call that we made two ar­gu­ments for in­creas­ing lev­er­age even fur­ther:

  1. Most al­tru­ists prob­a­bly do not take on enough risk, so in­vest­ment-minded philan­thropists should in­crease lev­er­age to com­pen­sate.

  2. Ac­cord­ing to the prin­ci­ple of time di­ver­sifi­ca­tion, philan­thropists should in­crease lev­er­age now if they ex­pect their preferred cause area(s) to have greater ac­cess to fund­ing in the fu­ture.

The first fac­tor weighs over­whelm­ingly in fa­vor of us­ing lev­er­age for small donors, al­though not nec­es­sar­ily for large donors (be­cause large donors’ choices more heav­ily af­fect the over­all al­tru­is­tic in­vest­ment pool). The sig­nifi­cance of the sec­ond fac­tor de­pends on one’s ex­pec­ta­tions about fu­ture move­ment growth.

We can never defini­tively de­ter­mine the cor­rect amount of lev­er­age to take, and any at­tempt to model the ex­pected util­ity of an in­vest­ment will have many short­com­ings. But in­vestors still need to pick a num­ber, even if that num­ber is 1:1 (i.e., no lev­er­age). I do not claim that the lev­er­age ra­tios offered in this sec­tion are op­ti­mal—they’re just rea­son­able guesses based on what I laid out in the pre­vi­ous sec­tions of this es­say. Tak­ing too much lev­er­age is worse than not tak­ing enough, so I per­son­ally would prob­a­bly not use this much lev­er­age in prac­tice.

Good Ven­tures /​ Open Philan­thropy Project

Ac­cord­ing to many effec­tive al­tru­ists’ value sys­tems, most value-al­igned money re­sides with Good Ven­tures, the foun­da­tion that funds the Open Philan­thropy Pro­ject. Donors should care about how Good Ven­tures in­vests its money and about how to adapt their own in­vest­ments to di­ver­sify against Good Ven­tures. Un­for­tu­nately, I do not know how Good Ven­tures in­vests, so I have noth­ing to say about this other than that it mat­ters a lot.

If Good Ven­tures in­vests a large por­tion of its as­sets in Face­book and does not hedge this in­vest­ment (which it may or may not do, I don’t know), other philan­thropists should pay par­tic­u­lar at­ten­tion to di­ver­sify­ing against Face­book.

Im­ple­men­ta­tion details

I am not an in­vest­ment pro­fes­sional and do not have suffi­cient ex­per­tise to make a recom­men­da­tion about how to in­vest. For in­for­ma­tional pur­poses, I will pro­pose what I be­lieve to be a rea­son­able and achiev­able in­vest­ment plan based on the rea­son­ing laid out in this es­say, but with the caveat that a thought­ful in­vestor could likely find a bet­ter im­ple­men­ta­tion. This should be con­sid­ered a best guess, not an en­dorse­ment.

First, in­vest with In­ter­ac­tive Bro­kers be­cause it offers the cheap­est mar­gin rates (at least in the United States[23]). Create an al­tru­is­tic in­vest­ment ac­count that’s sep­a­rate from your per­sonal ac­count, be­cause per­sonal ac­counts should take much less risk.

Se­cond, in­vest all your al­tru­is­tic funds into a man­aged fu­tures fund. As dis­cussed above, philan­thropists should care greatly about re­duc­ing cor­re­la­tion with other in­vestors; and man­aged fu­tures look like a par­tic­u­larly promis­ing way of do­ing that.

I spoke to an in­vest­ment ad­vi­sor with knowl­edge of man­aged fu­tures, and he sug­gested AQR Man­aged Fu­tures Strat­egy HV Fund (QMHIX) as a good choice. This fund tar­gets high (15%) volatility, which has the dual benefit that (1) in­vestors with high risk tol­er­ance can get higher re­turn with less lev­er­age and (2) the fund pro­duces greater ex­pected re­turn rel­a­tive to fees. QMHIX is not available to all in­vestors; as an al­ter­na­tive, AXS Ch­e­sa­peake Strat­egy Fund (EQCHX)[11:1] has similar method­ol­ogy, as well as rel­a­tively low fees for a man­aged fu­tures fund.

Third, use as much mar­gin as In­ter­ac­tive Bro­kers will al­low with­out sub­stan­tially risk­ing a mar­gin call. In­vestors us­ing Reg T mar­gin can­not get any­where close to as much lev­er­age as is the­o­ret­i­cally op­ti­mal ac­cord­ing to the anal­y­sis above, so they should get as much as they can. (At least small donors should, to com­pen­sate for other donors’ lack of lev­er­age; large philan­thropists may want to take less than 2:1 lev­er­age (but per­haps should take more than 2:1; I am un­cer­tain about this).)

Alter­na­tive in­vest­ment strategies

Another similar in­vest­ment idea: In­stead of buy­ing a man­aged fu­tures fund, buy value and mo­men­tum funds while short­ing the broad mar­ket to pro­duce net zero stock ex­po­sure, and then ap­ply lots of lev­er­age. This prob­a­bly re­quires port­fo­lio mar­gin rather than Reg T mar­gin, which has cer­tain qual­ifi­ca­tion re­quire­ments, so not as many in­vestors will be able to im­ple­ment this strat­egy. Some ex­am­ples of high-con­vic­tion value ETFs: IVAL, QVAL, GVAL, SYLD, FYLD, EYLD. And some mo­men­tum ETFs: IMOM, QMOM, GMOM.

As a third op­tion, philan­thropists could in­vest in more liquid as­sets (which would have worse risk-ad­justed re­turn) and then use op­tions to get much more lev­er­age. I am in­clined to be­lieve that this does not make as much sense. For philan­thropists with log­a­r­ith­mic risk aver­sion, this strat­egy may re­sult in lower ex­pected util­ity. Even for those with near-lin­ear marginal util­ity, us­ing high lev­er­age (say, 10:1 or higher) may im­pose suffi­ciently high costs that the re­sult­ing in­vest­ment port­fo­lio will have nega­tive ex­pected re­turn.

In­vestor psychology

Philan­thropists with the­o­ret­i­cally high risk tol­er­ance should con­sider their psy­cholog­i­cal re­ac­tion to pur­su­ing risky and un­cor­re­lated strate­gies. How will you re­act if:

  1. you in­vest your philan­thropic money in a risky strat­egy and lose 90% or more of your money?

  2. you in­vest in an un­cor­re­lated strat­egy, and slowly lose money over the course of 5-10 years while most peo­ple you know are mak­ing money?

Whether a strat­egy is op­ti­mal in the­ory doesn’t mat­ter if in­vestors can’t fol­low through with it in prac­tice. I per­son­ally do not in­vest in the man­ner I de­scribed in the pre­vi­ous sec­tion; I use some­thing like a com­bi­na­tion of that strat­egy and a more tra­di­tional in­vest­ment port­fo­lio.

Large investors

Suffi­ciently large in­vestors can non-triv­ially shift the al­tru­is­tic money pool by them­selves. Rather than in­vest­ing ex­clu­sively in as­sets with low cor­re­la­tion to other al­tru­ists, they should build a well-di­ver­sified port­fo­lio with a tilt to­ward un­der-weighted as­set classes.

Large in­sti­tu­tions or ul­tra high-net-worth in­di­vi­d­u­als most likely can im­ple­ment a port­fo­lio more effec­tively than in­di­vi­d­ual in­vestors. They might be able to:

  1. ac­cess some types of in­vest­ments that in­di­vi­d­ual in­vestors can­not (such as long/​short fac­tor funds);

  2. hire in­vest­ment firms to cre­ate cus­tomized prod­ucts;

  3. ne­go­ti­ate bet­ter mar­gin rates;

  4. use higher lev­er­age ra­tios.

Wealthy philan­thropists or in­sti­tu­tions with rel­a­tively long time hori­zons should con­sider how best to use these ad­van­tages.

I would sug­gest that in­vestors who want ad­di­tional guidance speak to AlphaAr­chi­tect. It is an in­vest­ment man­age­ment firm that pro­duces high-qual­ity re­search and in­vest­ment prod­ucts (hence why I have cited it re­peat­edly), and offers cus­tom solu­tions for large in­vestors. Per­haps more im­por­tantly, I be­lieve the own­ers of the firm gen­uinely want to help peo­ple in­vest well—an all-too-rare trait among in­vest­ment man­agers. The peo­ple at AlphaAr­chi­tect know far more than I do about in­vest­ing, and likely have bet­ter ideas about how to in­vest in un­usual situ­a­tions.



I will con­clude by restat­ing the main claims of this es­say, with cor­re­spond­ing con­fi­dence lev­els.

On risk and leverage

Highly likely: More al­tru­ists should use lev­er­age.

Highly likely: Altru­ists should con­sider their own psy­chol­ogy when de­cid­ing how to in­vest.

Likely: Most al­tru­ists should have higher risk tol­er­ance than typ­i­cal self-in­ter­ested in­vestors.

Likely: The best way to get lev­er­age for most in­vestors is with an In­ter­ac­tive Bro­kers mar­gin ac­count.

Pos­si­ble: Many al­tru­ists should use as much lev­er­age as they can rea­son­ably man­age.

Pos­si­ble: For most al­tru­ists, .

On as­set allocation

Highly likely: More al­tru­ists should tilt their in­vest­ments to­ward value and mo­men­tum.

Highly likely: Altru­ists should pay par­tic­u­lar at­ten­tion to re­duc­ing cor­re­la­tion with other al­tru­ists’ in­vest­ments.

Highly likely: There ex­ists a bet­ter in­vest­ment strat­egy than the one I pro­posed in Im­ple­men­ta­tion de­tails.

Likely: Altru­ists should over­whelm­ingly fa­vor as­sets with low cor­re­la­tion.

Likely: Altru­ists should speci­fi­cally in­vest in man­aged fu­tures as a way of re­duc­ing cor­re­la­tion.

Pos­si­ble: Altru­ists should in­vest their en­tire (al­tru­is­tic) port­fo­lios into man­aged fu­tures.

Pos­si­ble: Altru­ists should speci­fi­cally in­vest in any of the funds that I named.

Ques­tions for fu­ture consideration

  • Can we build bet­ter mod­els for the util­ity func­tions of var­i­ous causes?

  • What if we use a util­ity func­tion with non-con­stant RRA?

  • How do effec­tive al­tru­ists as a whole cur­rently al­lo­cate their in­vest­ments across as­set classes? Which as­set classes do al­tru­ists un­der-weight most severely?

  • How ex­actly do de­vi­a­tions from the as­set pric­ing model (in­clud­ing left skew and short-term mean re­ver­sion) af­fect de­sired lev­er­age?

  • How should one in­vest if their in­vest­ment cor­re­la­tion to other al­tru­ists lies be­tween 0 and 1?

  • How should fu­ture dona­tions be treated with re­spect to time di­ver­sifi­ca­tion? How does the an­swer vary by cause area?

  • Should his­tor­i­cal dona­tions fac­tor into the util­ity func­tion? It seems to me that we should count his­tor­i­cal good done as part of to­tal util­ity, but the liter­a­ture I have read on util­ity of con­sump­tion gen­er­ally does not do this. Why not?


Thanks to Brian To­masik, Jake McKin­non, Kit Har­ris, Linda Neavel Dick­ens, and Michael St. Jules for pro­vid­ing me with valuable feed­back. Re­view­ers do not nec­es­sar­ily en­dorse any of the claims made in this es­say.



  • Up­date man­aged fu­tures re­turn ex­pec­ta­tions to in­cor­po­rate ac­tual his­tor­i­cal perfor­mance data from man­aged fu­tures fund EQCHX.

  • Add a back­test go­ing back to 1952 of a strat­egy similar to VMOT.


  • Add ad­di­tional dis­claimers.


  • Provide more de­tails and refer­ences on man­aged fu­tures.

  • Add sec­tion on long/​short fac­tor strate­gies.

  • Mis­cel­la­neous small im­prove­ments.


  • Add more refer­ences to pa­pers on value and mo­men­tum that provide fur­ther ev­i­dence that they work.


  • gives the amount of start­ing cap­i­tal.

    We wish to max­i­mize ex­pected util­ity with re­spect to :

    is nor­mally defined as . Utility func­tions are equiv­a­lent up to af­fine trans­for­ma­tions, so we can sim­plify this to (or in the case that = 1). For the same rea­son, we can omit the term. Fur­ther­more, this means the max­i­mum of does not de­pend on past re­turn, only on the ex­pected dis­tri­bu­tion of fu­ture re­turn.

    First, con­sider the case with only two time steps. Call the lev­er­age-ad­justed re­turn in each time step and .

    Thus we have:

    Us­ing the rule de­ter­mined in the case, as well as the fact that when and are in­de­pen­dent, we can mod­ify the ex­pected value for­mula as fol­lows:

    This product is max­i­mized with re­spect to by max­i­miz­ing each term of the product.

    In the case, . By lin­ear­ity of ex­pected value, this equals . This sum is max­i­mized with re­spect to by max­i­miz­ing each in­te­gral.

    Thus, for any value of , we max­i­mize long-run ex­pected util­ity by in­de­pen­dently max­i­miz­ing ex­pected util­ity at each time step.

    Thanks to Michael St. Jules for pro­vid­ing feed­back on how to sub­stan­tially sim­plify this proof.

  1. For a short sum­mary of em­piri­cal re­search on peo­ple’s risk prefer­ences, see Es­ti­mat­ing the Coeffi­cient of Rel­a­tive Risk Aver­sion for Con­sump­tion. ↩︎ ↩︎

  2. Do­mian, Racine, and Wil­son (2003). Lev­er­aged Stock Port­fo­lios over Long Hold­ing Pe­ri­ods: A Con­tin­u­ous Time Model. ↩︎

  3. Mer­ton (1969). Life­time Port­fo­lio Selec­tion Un­der Uncer­tainty: The Con­tin­u­ous-Time Case. ↩︎

  4. Aside: I at­tempted to write a Python pro­gram to de­ter­mine the marginal util­ity of cash trans­fers and at what point risk aver­sion starts to mean­ingfully in­crease. But the pro­gram didn’t work very well be­cause, given the way I wrote it, NumPy had to strug­gle to com­pute triple-nested in­te­grals far out on the tails of fat-tailed prob­a­bil­ity dis­tri­bu­tions. But I only spent a cou­ple of hours on it, so I think such a pro­gram should be writable given a more care­ful ap­proach. ↩︎

  5. I did not look deeply into this, but GiveDirectly’s staff has not ex­panded as rapidly as its in­crease in dona­tions over the past few years, so this state­ment ap­pears to be true. ↩︎

  6. GiveWell gave sub­stan­tially lower num­bers in 2018, rang­ing from 4.2 to 12.2. I have not in­ves­ti­gated the rea­sons for the differ­ence. ↩︎

  7. Deriva­tion: Marginal util­ity of the dol­lar is times greater than the marginal util­ity of the dol­lar. The stan­dard isoe­las­tic util­ity func­tion is defined as

    and its deriva­tive is . The ra­tio of marginal util­ities be­tween and should equal :

    Solv­ing this equa­tion for gives the de­sired re­sult. ↩︎

  8. I es­ti­mated this based on a range of typ­i­cal net worths and rel­a­tive risk aver­sions. ↩︎

  9. This as­sumes that in­di­vi­d­u­als can treat their fu­ture earn­ings as risk-free. This might not be the case, de­pend­ing how much volatility one ex­pects in their fu­ture ca­reer. Life­cy­cle In­vest­ing dis­cusses this more, as well as Milevsky (2012), Are You a Stock or a Bond? ↩︎

  10. Philan­thropists are over-rep­re­sented in the US, and peo­ple in ev­ery coun­try over-rep­re­sent their own coun­try, so it is al­most cer­tainly the case that philan­thropists over-rep­re­sent the S&P 500, al­though ob­vi­ously they don’t all put all their money in it.

    I did a small in­for­mal Face­book poll that sup­ports this claim. ↩︎

  11. Dis­claimer: At the time of this writ­ing, I have money in­vested in this fund. ↩︎ ↩︎

  12. Method­ol­ogy:

    1. Take the top decile by mo­men­tum and the top decile by E/​P.

    2. Create a port­fo­lio weighted 5050 be­tween these two, re­bal­anced monthly.

    3. Use a com­bi­na­tion of the 12-month sim­ple mov­ing av­er­age and 12-month time se­ries mo­men­tum to risk man­age the port­fo­lio, as de­scribed by AlphaAr­chi­tect.

  13. For com­par­i­son, the strat­egy I tested re­turned 14.4% with stan­dard de­vi­a­tion 16.0% from 1992 to 2017, hy­po­thet­i­cally un­der­perform­ing VMOT by 5 per­centage points on a risk-ad­justed ba­sis. VMOT makes a num­ber of method­olog­i­cal im­prove­ments over the ba­sic strat­egy I tested, the biggest differ­ences be­ing (1) di­ver­sify­ing in­ter­na­tion­ally, (2) us­ing a value met­ric that bet­ter cap­tures com­pany val­u­a­tion, and (3) us­ing a more ro­bust risk man­age­ment method­ol­ogy. ↩︎

  14. Does it make sense to treat mar­ket fac­tors as ad­di­tive like this? Yes. The aca­demic liter­a­ture on fi­nance gen­er­ally treats fac­tors as ad­di­tive: see Fama and French (1992), The Cross-Sec­tion of Ex­pected Stock Re­turns. ↩︎

  15. Any de­vi­a­tions from a well-es­tab­lished in­vest­ing strat­egy should raise con­cerns about data min­ing. The two books ex­plain why they be­lieve their im­prove­ments are not data-mined. They also dis­cuss and re­ject some al­ter­na­tive po­ten­tial im­prove­ments be­cause they sus­pect the strate­gies are data-mined. ↩︎

  16. Based on my coarse es­ti­mate that VMOT has two per­centage points higher ex­pected re­turn than the av­er­age of RAFI’s eight US+de­vel­oped large+small value+mo­men­tum fac­tors plus mar­ket beta, and sub­tract­ing VMOT’s fee, we get a 10-11% ex­pected real re­turn. ↩︎

  17. Proof:

    • Char­ac­ter­ize in­vest­ing as a dis­crete se­ries of time steps from 0 to , af­ter which the in­vestor stops in­vest­ing.

    • Each time step has an in­vest­ment re­turn ran­dom vari­able fol­low­ing a nor­mal dis­tri­bu­tion pa­ram­e­ter­ized by and .

    • is the isoe­las­tic util­ity func­tion, pa­ram­e­ter­ized by some con­stant .

    • gives Sa­muel­son share at time (>1 in­di­cates lev­er­age).

  18. De Bondt and Thaler (1985). Does the Stock Mar­ket Over­re­act? ↩︎

  19. This un­der­es­ti­mates the cost of lev­er­age be­cause on a left-skewed dis­tri­bu­tion, sub­tract­ing a con­stant across mul­ti­ple time pe­ri­ods re­duces the mean by more than that con­stant. And it over­es­ti­mates the cost of lev­er­age be­cause fees re­duce the volatility of an in­vest­ment. Whether it un­der- or over-es­ti­mates on net de­pends on the spe­cific prop­er­ties of the in­vest­ment. ↩︎

  20. Cor­rec­tion 2020-01-20: Origi­nally, I calcu­lated as , but tech­ni­cally this is not cor­rect.

    Let and be the mean and stan­dard de­vi­a­tion of the as­set, and let and be the pa­ram­e­ters used in the Sa­muel­son share for­mula.

    The pa­ram­e­ter in the Sa­muel­son share for­mula should be the log of the mean, and the pa­ram­e­ter should be the stan­dard de­vi­a­tion of the log of the re­turn (where the log of the re­turn is nor­mally dis­tributed). That is:

    and app­prox­i­mately equal and , so sim­ply us­ing and is not a bad ap­prox­i­ma­tion, but it’s not quite ac­cu­rate.

    Thanks to Gor­don Ir­lam for pro­vid­ing this cor­rec­tion. ↩︎

  21. MacLean, Thorp, and Ziemba (2010). Good and bad prop­er­ties of the Kelly crite­rion. ↩︎

  22. MacLean, Thorp, and Ziemba (2010). The Kelly Cap­i­tal Growth In­vest­ment Cri­te­rion: The­ory and Prac­tice. Part VI: Ev­i­dence of the Use of Kelly Type Strate­gies by the Great In­vestors and Others. ↩︎

  23. In­ter­ac­tive Bro­kers op­er­ates in other coun­tries, but I do not know if other coun­tries have bet­ter bro­ker­age firms. ↩︎