SANJIV R. DAS s a professor at Santa Clara Unversty n Santa Clara, CA. srdas@scu.edu Dgtal Portfolos SANJIV R. DAS Dgtal assets are nvestments wth bnary returns: the payoff s ether very large or very small. Dgtal portfolos return dstrbutons are hghly skewed and fat-taled. A venture fund s a good example of such a portfolo. A Bernoull dstrbuton offers a smple representaton of the payoff to a dgtal nvestment: a large payoff for a successful outcome and a very small (almost zero) payoff for a faled one. Dgtal nvestments typcally offer a small probablty of success, around 5% to 25% for new ventures (see Das, Jagannathan, and Sarn [2003]). In a recent paper, Fernandez, Sten, and Lo [2012] advocate the creaton of a bomedcal megafund of dgtal assets. The standard technques used for mean-varance optmzaton aren t useful tools for optmzng dgtal assets. Ths artcle s analyses let us characterze dgtal assets return dstrbutons. The ntutons obtaned from the mean-varance settng may not carry over to portfolos of Bernoull assets. As Bernoull portfolos nvolve hgher moments, the best way to dversfy s by no means obvous. For nstance, s dversfcaton by ncreasng the number of assets n the dgtal portfolo always a good thng? Is t better to nclude assets wth as lttle correlaton as possble, or s there a sweet spot for optmal asset-correlaton levels? Should all the nvestments be of even sze, or s t preferable to take a few large bets and several small ones? Is a mxed portfolo of safe and rsky assets better than one that offers a more unform probablty of success? These are all questons of nterest to nvestors n dgtal type portfolos, ncludng collateralzed debtoblgaton nvestors, venture captalsts, and venture fund nvestors. We use a method based on standard recurson for modelng a Bernoull portfolo s exact return dstrbuton. Andersen, Sdenus, and Basu [2003] frst developed ths method for generatng credt portfolos loss dstrbutons. We examne these portfolos propertes n a stochastc domnance framework to provde gudelnes to dgtal nvestors. These gudelnes are consstent wth prescrptons from expected utlty optmzaton: 1. Holdng all else the same, t s better to have more dgtal nvestments. For example, a venture portfolo should seek to maxmze market share. 2. As wth mean-varance portfolos, lower asset correlaton s better, unless the dgtal nvestor s payoff depends on the upper tal of returns. 3. A strategy of a few large bets and many small ones s nferor to one wth bets of roughly the same sze. 4. A mxed portfolo of assets wth low and hgh probabltes of success s better than one n whch all assets have the same probablty of success. IT IS ILLEGAL TO REPRODUCE THIS ARTICLE IN ANY FORMAT WINTER 2013 THE JOURNAL OF PORTFOLIO MANAGEMENT 41 Copyrght 2013
MODELING DIGITAL PORTFOLIOS Assume that an nvestor has a choce of n nvestments n dgtal assets start-up frms, for nstance. The nvestments are ndexed = 1, 2,, n. Each nvestment has a probablty of success, denoted q. If t succeeds, the payoff s S. Wth probablty (1 q ), the nvestment wll not succeed, the start-up wll fal, and the money wll be entrely lost. Therefore, the payoff (cash flow) s Payo ff wth prob = = S q C 0 wth prob ( 1 q ) Callng ths nvestment a Bernoull tral reflects realty n dgtal portfolos. It mmcs the venture captal busness, for example. Consder two generalzatons. Frst, we mght extend the model and allow S to be random,.e., drawn from a range of values. Ths wll complcate the mathematcs, but not enrch the model s results by much. Second, the falure payoff mght be non-zero: an amount a. That gves us a par of Bernoull payoffs {S, a }. We can decompose these nvestment payoffs nto a project wth constant payoff a plus another project wth payoffs {S a, 0}, the latter beng exactly the orgnal settng wth a falure payoff of zero. The verson of the model we solve n ths artcle, wth zero falure payoffs, s wthout loss of generalty. Unlke stock portfolos n whch the asset choce set s assumed to be multvarate normal, dgtal asset nvestments have a jont Bernoull dstrbuton. Portfolo returns on these nvestments are unlkely to be Gaussan, and hence hgher-order moments are lkely to matter more. To generate a return dstrbuton for a portfolo of dgtal assets, we must account for correlatons across dgtal nvestments. We adopt the followng smple correlaton model. Defne y as the performance proxy for the -th asset. Ths proxy varable s smulated for comparson wth a threshold performance level to determne whether the asset yelded a success or falure. It s defned by the followng functon, wdely used n the correlated default-modelng lterature. See Andersen, Sdenus, and Basu [2003] for an example. (1) y + 1 ρ 2 Z, = 1... n (2) common factor drves the correlatons between the portfolo s dgtal assets. We assume that Z N(0, 1) and Corr (, Z ) = 0,. The correlaton between assets and j s gven by ρ ρ j. The mean and varance of y are: E(y ) = 0, Var (y ) = 1,. Condtonal on, the values of y are all ndependent, as Corr(Z, Z j ) = 0. We can formalze the probablty model governng the dgtal nvestment s success or falure. We defne a varable x, wth dstrbuton functon F( ), such that F(x ) = q, the dgtal nvestment s probablty of success. Condtonal on a fxed value of, the probablty of success of the th nvestment s defned as p P [ y < x ] (3) Assumng F to be the normal dstrbuton functon, we have p 2 + 1 x Z x = Pr Z < 2 1 ρ 1 F ( ) ρ = Φ 1 ρ where Φ(.) s the cumulatve normal densty functon. Gven the level of the common factor, asset correlaton ρ, and the uncondtonal success probabltes q, we obtan the condtonal success probablty for each asset p. As vares, so does p. For the numercal examples n ths artcle we choose the functon F(x ) to the cumulatve normal probablty functon. We use a fast technque for buldng up dstrbutons for sums of Bernoull random varables. In fnance, Andersen, Sdenus, and Basu [2003] ntroduced ths recurson technque n the credt portfolo-modelng lterature. We call a dgtal nvestment successful f t acheves ts hgh payoff S. The portfolo s cash flow s a random n C = 1 varable C. The portfolo s maxmum possble cash flow s the sum of all dgtal asset cash flows, because each and every outcome s a success: (4) where ρ [0, 1] s a coeffcent that correlates threshold y wth a normalzed common factor N(0, 1). The C max n S = 1 (5) 42 DIGITAL PORTFOLIOS WINTER 2013
To keep matters smple, we assume that each S s an nteger and we round off amounts to the nearest sgnfcant dgt. If the smallest unt we care about s a mllon dollars, then each S wll be n nteger unts of mllons. Recall that, condtonal on a value of, dgtal asset s probablty of success s gven as p. The recurson technque wll let us generate the portfolo cash flow probablty dstrbuton for each level of. We then compose these condtonal (on ) dstrbutons usng the margnal dstrbuton for, denoted g(), nto the uncondtonal dstrbuton for the entre portfolo. We defne the probablty of total cash flow from the portfolo, condtonal on, as f(c ). Then, the portfolo s uncondtonal cash f low dstrbuton becomes fc ) = f ( C g ( ) d (6) x Compute the dstrbuton f(c ) numercally as follows: We ndex the assets wth = 1 n. Cash flow from all assets taken together wll range from zero to C max. Break ths range nto nteger buckets, resultng n a total number of buckets N B, each one contanng an ncreasng level of total cash flow. We ndex these buckets by j = 1 N B, wth the cash f low n each bucket equal to B j. B j represents the total cash flow from all assets (some pay off and some do not), and the buckets comprse the dscrete support for the entre dstrbuton of total portfolo cash flow. For example, suppose we had 10 assets, each wth a payoff of C = 3. Then C max = 30. A plausble set of buckets comprsng the cash flow dstrbuton s support would be: {0, 3, 6, 9, 12, 15, 18, 21, 24, 27, C max }. Defne P(k, B j ) as the probablty of bucket j s cash flow level B j f we account for the frst k assets. For example, f we had just three assets, wth payoffs of value 1, 3, and 2 respectvely, then we would have seven buckets,.e., B j = {0, 1, 2, 3, 4, 5, 6}. After accountng for the frst asset, the only possble buckets wth postve probablty would be B j = 0, 1, and after the frst two assets, the buckets wth postve probablty would be B j = 0, 1, 3, 4. We begn wth the frst asset, then the second and so on n order, and compute the probablty of seeng the returns n each bucket. Each probablty s gven by the followng recurson: P( k, B P ( k B p j p j )[ P(k (, B k j S k + 1 k,, n 1 k + k+ 1, (7) The probablty of a total cash flow of B j after consderng the frst (k + 1) frms s equal to the sum of two probablty terms. Frst, the probablty of the same cash flow B j from the frst k frms, gven that frm (k + 1) dd not succeed. Second, the probablty of a cash flow of B j S k+1 from the frst k frms and the (k + 1)-st frm does succeed. We start ths recurson from the frst asset, after whch the N B buckets are all of probablty zero, except for the bucket wth zero cash flow (the frst bucket) and the one wth S 1 cash flow: P p (8) 1 P(, S ) = p (9) 1 1 All the other buckets wll have probablty zero P(1, B j {0,S 1 } = 0. Wth these startng values, we can run the system from the frst asset to the n-th one by repeatedly applyng Equaton (7). The entre dstrbuton P(n, B j ) s condtonal on a gven value of. We compose all the dstrbutons that are condtonal on nto a sngle cash-flow dstrbuton usng Equaton (6), numercally ntegratng over all values of. PORTFOLIO CHARACTERISTICS Armed wth ths establshed machnery, a dgtalportfolo nvestor (a venture captalst, for nstance) may pose several questons: Frst, s there an optmal number of assets,.e., ceters parbus, are more assets better than fewer assets, assumng no span of control ssues? Second, are Bernoull portfolos dfferent from mean-varance portfolos, n that s t always better to have less asset correlaton than more asset correlaton? Thrd, s t better to have an even nvestment weghtng across assets, or mght t be better to take a few large bets among many smaller ones? Fourth, s a hgh dsperson of probablty of success better than a low dsperson? These questons are very dfferent from the ones facng nvestors n tradtonal mean-varance portfolos. We shall examne each of these questons n turn. WINTER 2013 THE JOURNAL OF PORTFOLIO MANAGEMENT 43
E HIBIT 1 Dstrbuton Functons for Returns from Bernoull Investments as the Number of Investments (n) Increases Notes: Usng the recurson technque we compute the probablty dstrbuton of the portfolo payoff for four values of n = {25,50,75,100}. The dstrbuton functon s plotted n the left panel. There are four plots, one for each n, and f we look at the bottom left of the plot, the leftmost lne s for n = 100. The u next lne to the rght s for n = 75, and so on. The rght panel plots the value of [ ( x) ( x)] dx for all u (0, 1), and confrms that t s always 0 100 25 negatve. The correlaton parameter s ρ = 0.25. E HIBIT 2 Expected Utlty for Bernoull Portfolos as the Number of Investments (n) Increases Notes: The exhbt reports the portfolo statstcs for n = {25,50,75,100}. Expected utlty s gven n the last column. The correlaton parameter s ρ = 0.25. The utlty functon s U(C) = (0.1+C) 1 γ /(1 γ), γ = 3. How Many Assets? Wth mean-varance portfolos, n whch the portfolo s mean return s fxed, t s better to have more securtes n the portfolo, because dversfcaton reduces portfolo varance. In mean-varance portfolos, hgherorder moments do not matter. But wth portfolos of Bernoull assets, ncreasng the number of assets mght exacerbate hgher-order moments, even though t wll reduce varance. Beyond a pont, t may not be worthwhle to ncrease the number of assets (n). To assess ths ssue we conduct an experment. We nvest n n assets, each wth payoff of 1/n. If all assets succeed, the total (normalzed) payoff s 1. Ths normalzaton s only to make the results comparable across dfferent n, and s wthout loss of generalty. We assume that the correlaton parameter s ρ = 0.25, for all. To make results easy to nterpret, we assume that assets are dentcal, wth a success probablty of q = 0.05 for all. Usng the recurson technque, we compute the probablty dstrbuton of the portfolo payoff for four values of n = {25, 50, 75, 100}. The dstrbuton functon s plotted n Exhbt 1, left panel. There are four plots, one for each n. At the bottom left of the plot, the leftmost lne s for n = 100. The next lne to the rght s for n = 75, and so on. 44 DIGITAL PORTFOLIOS WINTER 2013
One way to determne whether greater n s better for a dgtal portfolo s to nvestgate whether a portfolo of n assets stochastcally domnates one wth fewer than n assets. On examnng the shapes of the dstrbuton functons for dfferent n, we see that t s lkely that as n ncreases, we obtan portfolos that exhbt second-order stochastc domnance (SSD) over portfolos wth smaller n. The return dstrbuton when n = 100 (denoted G 100 ) domnates that for n = 25 (denoted G 25 ) n the SSD sense, f xdg x xdg ( x ), and u 0 x ( x)] dx 0 25 [ ( x) ( x)] dx 100 for all u (0,1). That s, G 25 has a mean-preservng spread over G 100, or G 100 has the same mean as G 25 but lower varance, mplyng superor mean-varance effcency. To show ths we plotted the u 0 ntegral [ ( x) ( x)] dx 100 25 and checked the SSD condton. We found that ths condton s satsfed (see Exhbt 1). As others have verfed, SSD mples meanvarance effcency as well. We also examne whether hgher n portfolos are better for a power utlty nvestor wth utlty func- C ton, U( C ) ( + = ) 1 γ.1, where C s the Bernoull portfolo s normalzed total payoff. Expected utlty s gven 1 γ by Σ C U(C) f(c). We set the rsk averson coeffcent to γ = 3, whch s n the standard range n the assetprcng lterature. Exhbt 2 reports the results. The x 25 expected utlty ncreases monotoncally wth n. For a power utlty nvestor, havng more assets s better than havng fewer, keepng the portfolo s mean return constant. Economcally, and specfcally for venture captalsts, ths hghlghts the goal of capturng a larger share of the number of avalable ventures. The results from the SSD analyss are consstent wth those of expected power utlty. We have abstracted away from ssues around the span of nvestor management. Investors actvely play a role n ther nvested dgtal portfolos assets, and ncreasng n beyond a pont may become costly, as modeled n Kannanen and Keuschngg [2003]. The Impact of Correlaton As wth mean-varance portfolos, we expect that ncreases n payoff correlaton for Bernoull assets wll adversely affect portfolos. In order to verfy ths ntuton we analyze portfolos, keepng all other varables the same, but changng correlaton. In the prevous subsecton, we set the correlaton parameter to be ρ = 0.25. Here, we examne four levels of the correlaton parameter: ρ = {0.09, 0.25, 0.49, 0.81}. For each correlaton level, we compute the normalzed total payoff dstrbuton. The number of assets s fxed at n = 25 and E HIBIT 3 Dstrbuton Functons for Returns from Bernoull Investments as the Correlaton Parameter (ρ 2 ) Increases Notes: Usng the recurson technque we computed the dstrbuton of the portfolo payoff for four values of ρ = {0.09, 0.25, 0.49, 0.81} ordered from top u to bottom on the left plot, respectvely. The dstrbuton functon s plotted n the left panel. The rght panel plots the value of [ ( x ) ( x)] dx 0 ρ 0.09 09 ρ= 0.81 for all u (0,1), and confrms that t s always negatve. WINTER 2013 THE JOURNAL OF PORTFOLIO MANAGEMENT 45
E HIBIT 4 Expected Utlty for Bernoull Portfolos as the Correlaton (ρ) Increases Notes: The exhbt reports the portfolo statstcs for ρ = {0.09, 0.25, 0.49, 0.81}. The last column shows expected utlty. The utlty functon s U(C) = (0.1 + C) 1 γ /(1 γ), γ = 3. the results, whch confrm that as wth mean- varance portfolos, Bernoull portfolos also mprove f ther assets have low correlaton. Dgtal nvestors should optmally attempt to dversfy ther portfolos. Insurance companes are a good example: they dversfy rsk across geographcal and other demographc dvsons. Uneven Bets? the probablty of success of each dgtal asset s 0.05 as before. Exhbt 3 shows the probablty dstrbuton functon of payoffs for all four correlaton levels. The SSD condton s met: lower-correlaton portfolos stochastcally domnate (n the SSD sense) hgher-correlaton portfolos. We also examne changng correlaton n the context of a power utlty nvestor wth the same utlty functon as n the prevous subsecton. Exhbt 4 shows Dgtal-asset nvestors are often faced wth the queston of whether to bet even amounts across dgtal nvestments, or to nvest wth dfferent weghts. We explore ths queston by consderng two types of Bernoull portfolos. Both have n = 25 assets, each wth a success probablty of q = 0.05. The frst has equal payoffs: 1/25 each. The second portfolo has payoffs that monotoncally ncrease,.e., the payoffs are equal to j/325, j = 1, 2,, 25. The sum of the payoffs n both E HIBIT 5 Expected Utlty for Bernoull Portfolos when the Portfolo Comprses Balanced Investng n Assets vs. Imbalanced Weghts Notes: Both the balanced and mbalanced portfolo have n = 25 assets, each wth a success probablty of q = 0.05. The frst has equal payoffs, 1/25 each. The second portfolo has payoffs that monotoncally ncrease,.e., the payoffs are equal to j/325, j = 1, 2,, 25. The sum of the payoffs n both cases s 1. The correlaton parameter s ρ = 0.55. The utlty functon s U(C) = (0.1 + C) 1 γ /(1 γ), γ = 3. E HIBIT 6 Expected Utlty for Bernoull Portfolos when the Portfolo Comprses Balanced Investng n Assets wth Identcal Success Probabltes vs. Investng n Assets wth Mxed Success Probabltes Notes: Both the unform and mxed portfolos have n = 26 assets. In the frst portfolo, all the assets have a success probablty equal to q = 0.10. In the second portfolo, half the frms have a success probablty of 0.05 and the other half have a probablty of 0.15. All nvestments have a payoff of 1/26. The correlaton parameter s ρ = 0.55. The utlty functon s U(C) = (0.1 + C) 1 γ /(1 γ), γ = 3. 46 DIGITAL PORTFOLIOS WINTER 2013
cases s 1. Exhbt 5 shows the nvestor utlty where the utlty functon s the same as n the prevous sectons. The utlty for the balanced portfolo s hgher than that for the mbalanced one. The balanced portfolo evdences SSD over the mbalanced portfolo, but the return dstrbuton has fatter tals when portfolo nvestments are mbalanced. Investors seekng to dstngush themselves by takng greater rsk n ther early careers may be better off wth mbalanced portfolos. Mxng Safe and Rsky Assets Is t better to have assets wth a wde varaton n probablty of success or wth smlar probabltes of success? To examne ths, we consder two portfolos of n = 26 assets. In the frst portfolo, all the assets have a success probablty equal to q = 0.10. In the second portfolo, half the frms have a success probablty of 0.05 and the other half have a probablty of 0.15. All nvestments have a payoff of 1/26. The probablty dstrbuton of payoffs and the expected utlty for the same power utlty nvestor (wth γ = 3) are gven n Exhbt 6. Mxng the portfolo between nvestments wth both hgh and low probabltes of success results n hgher expected utlty than does choosng nvestments wth smlar probabltes of success. Imbalanced success probablty portfolos also evdence SSD over portfolos wth smlar success rate nvestments. Ths result does not have a natural analog n the mean-varance world wth non-dgtal assets. For emprcal evdence on the effcacy of varous dversfcaton approaches, see Lossen [2006]. CONCLUSIONS Dgtal asset portfolos are dfferent from meanvarance ones because ther asset returns are Bernoull, wth small success probabltes. We used a recurson technque borrowed from the credt-portfolo lterature to construct the payoff dstrbutons for Bernoull portfolos. We fnd that many ntutons for these portfolos are smlar to those for mean-varance portfolos: dversfcaton by addng assets s useful, low correlatons between nvestments s good. However, we also fnd that nvestors prefer unform bet szes over a varety of small and large bets. Instead of constructng portfolos wth assets havng unform success probabltes, t s better to have a mx of assets, some wth low success probabltes and others wth hgh success probabltes Venture funds are just one possble applcaton. These nsghts augment the standard understandng obtaned from mean-varance portfolo optmzaton. Usng ths approach s smple. Investors need the expected payoffs of the assets C, success probabltes q, and the average correlaton between assets, gven by a parameter ρ. Broad statstcs on these nputs are avalable, for venture nvestments from papers such as Das, Jagannathan, and Sarn [2003]. Fernandez, Sten and Lo s [2012] botechnology fund, on the other hand, requres scentsts to help determne nputs. Wth that data, t s easy to optmze a dgtal asset fund s portfolo. Investors can easly extend the techncal approach to features that nclude the cost of nvestor effort as the number of projects grows (Kannanen and Keuschngg [2003]), syndcaton, and so on. The marketplace shows a growng number of portfolos wth dgtal assets, and the results of ths artcle provde useful nsghts and ntuton for asset managers. The modelng secton shows just one way n whch to model jont success probabltes usng a common factor. There are other ways, too, such as modelng jont probabltes drectly, makng sure that they are consstent wth each other, whch may be mathematcally trcky. The results may dffer from the ones developed here for some dfferent system of jont success probabltes. The system we adopt here wth a sngle common factor may also be extended to more than one common factor, an approach often taken n the default lterature. We leave these nterestng extensons for future work. ENDNOTE I am grateful to George Chacko, Bob Hendershott, and Hoje Jo for helpful comments on the paper. REFERENCES Andersen, L., J. Sdenus, and S. Basu. All Your Hedges n One Basket. Rsk, November 2003, pp. 67-72. Das, S., M. Jagannathan, and A. Sarn. Prvate Equty Returns: An Emprcal Examnaton of the Ext of Asset- Backed Companes. Journal of Investment Management, Vol. 1, No. 1 (2003), pp. 152-177. WINTER 2013 THE JOURNAL OF PORTFOLIO MANAGEMENT 47
Fernandez, J.M., R. Sten, and A. Lo. Commercalzng Bomedcal Research Through Securtzaton Technques. Nature Botechnology, 2012, onlne. Kannanen, V., and C. Keuschngg. The Optmal Portfolo of Start-Up Frms In Asset Captal Fnance. Journal of Corporate Fnance, Vol. 9, No. 5 (2003), pp. 521-534. Lossen, U. The Performance of Prvate Equty Funds: Does Dversfcaton Matter? Dscusson Papers 192, SFB/TR 15, Unversty of Munch, 2006. To order reprnts of ths artcle, please contact Dewey Palmer at dpalmer@journals.com or 212-224-3675. 48 DIGITAL PORTFOLIOS WINTER 2013