The Hansen ratio in mean--variance portfolio theory
aa r X i v : . [ q -f i n . P M ] J u l THE HANSEN RATIO IN MEAN–VARIANCE PORTFOLIO THEORY
ALEˇS ˇCERN ´Y
Abstract.
It is shown that the ratio between the mean and the L –norm leads to a par-ticularly parsimonious description of the mean–variance efficient frontier and the dual pricingkernel restrictions known as the Hansen–Jagannathan (HJ) bounds. Because this ratio hasnot appeared in economic theory previously, it seems appropriate to name it the Hansen ratio.The initial treatment of the mean–variance theory via the Hansen ratio is extended in twodirections, to monotone mean–variance preferences and to arbitrary Hilbert space setting. Amultiperiod example with IID returns is also discussed. Introduction
Roy (1952) gave the first formula for the efficient mean-variance frontier in a one-periodmarket spanned by a finite number of assets. Since then a concerted effort has been made tocompute the mean-variance frontier in a dynamic setting, typically in the context of quadratichedging; see Li and Ng (2000), Bertsimas et al. (2001), and Lim (2004, 2005), for example. Theresults in these studies are explicit; yet despite or perhaps because of this, one gets no closer toa good economic understanding of the underlying principles that drive them.The literature also features a parallel stream in an abstract market setting with possibly infin-itely many assets where geometry plays an important role. The geometric approach starts withChamberlain and Rothschild (1983). Its objects take a more explicit form in Hansen and Richard(1987) who identify two important portfolios that fully describe the efficient frontier (the port-folios Y and X below). In this paper, we rediscover and extend the seminal contributions of Hansen and Richard(1987) and Hansen and Jagannathan (1991) in the context of utility maximization. This natu-rally leads to the ratio of mean to L –norm: the Hansen ratio from the title. The ratio plays animportant role in the description of the efficient frontier (Subsection 2.3); it features prominentlyin the Hansen–Jagannathan bound (Subsection 2.5); and last but not least, it is instrumentalin converting the one-period efficient frontier into the dynamically efficient frontier (Section 5and Appendix A). The expected utility approach also yields a clean economic intepretation ofthe Hansen–Jagannathan bound under positivity constraints in terms of the monotone Hansenratio/monotone Sharpe ratio (Section 3). In Section 4, the Hilbert space generalization of the L theory is illustrated on an example from Cochrane (2014). Section 6 concludes. Date : August 3, 2020.
MSC [2020]. (Primary) 91B02, 91B16; 91G10 (Secondary) 60G51.
Keywords.
Hansen ratio, Hansen–Jagannathan inequality, efficient frontier, monotone mean–variance preference.I would like to thank Christoph Czichowsky for helpful discussions. An accessible account of this literature appears in Cochrane (2001, Chapter 5). The L theory Denote by L the collection of all random variables with finite second moment on some fixedprobability space. Let M be a closed linear subspace of L and π a continuous linear functionalon M . We think of elements of M as terminal wealths W of traded positions whose initial priceis given by π ( W ). Denote by M ( c ) all traded wealth distributions available at cost c ∈ R , M ( c ) = { W ∈ M : π ( W ) = c } , and assume M (1) is not empty. The elements of M (1) are commonly known as the fully investedportfolios. With Cochrane (2001), we will refer to the elements of M (0) as zero-cost portfolios. For each W with finite second moment we write µ W = E [ W ] for the mean, ω W = E [ W ]for the second non-central moment, and σ W = E [( W − µ W ) ] for the variance. Observe that ω W = k W k is the L –norm of W . Observe also that µ W , σ W , and ω W are tied together throughthe relationship σ W = ω W − µ W . The Hansen ratio,HR W = µ W ω W , therefore satisfies the inequality HR W ≤ . (2.1)Furthermore, HR W = 1 if and only if W is risk-free.The portfolio theory is concerned with two questions: how to describe all pricing rules on L that are consistent with π and how to identify efficient portfolios in M (1). It turns out there isa special portfolio in M (1) that forms an important part of the answer in both directions.2.1. Special fully invested portfolio Y . Under our assumptions, there is a unique portfolioin M (1) orthogonal to M (0). Let us denote this portfolio by Y . As M (1) = Y ⊕ ⊥ M (0) , (2.2)we observe that Y is also the unique element of M (1) with the smallest L norm. This means Y is efficient in the sense of having the smallest variance at the fixed mean µ Y , in view of σ Y = ω Y − µ Y and the minimality of ω Y among all fully invested portfolios. The orthogonality(2.2) yields that the linear functional π Y ( W ) = E [ Y W ] ω Y , W ∈ L , (2.3)correctly prices all positions in the marketed subspace M , inasmuch π Y correctly prices Y at 1 and all elements of M (0) at zero. The special pricing kernel π Y goes back to at leastChamberlain and Rothschild (1983). The portfolio Y appears explicitly in Hansen and Richard(1987, Eq. 3.6). The role of π is significant only to the extent that M (1) is a closed subset of L , that zero position is not anelement of M (1) (as linearity yields π (0) = 0), and that we have M (0) = M (1) − M (1). One could therefore use aclosed affine subspace M (1) not containing zero as the primitive input. HE HANSEN RATIO 3
Special zero-cost portfolio X . Let us now consider the quadratic utility function U ( x ) = x − x / . (2.4)The expected quadratic utility preference reads u ( W ) = E [ U ( W )] = µ W − ω W , W ∈ L . One easily verifies that the maximal utility from an optimally scaled investment W equalsmax α ∈ R u ( αW ) = 12 HR W . (2.5)Denote by X ∈ M (0) the zero-cost portfolio with the highest expected utility. After complet-ing U to a square, U ( x ) = 12 −
12 (1 − x ) , we observe that X is the orthogonal projection of the constant payoff 1 (which here signifies thebliss point of the utility U ) onto the zero-cost subspace M (0). This implies 1 − X is orthogonalto all elements of M (0), in particular to X itself, which yields µ X = ω X = HR X . As X has the smallest value of µ − ω / M (0). In particular, for the given mean µ X , there can be no zero-cost portfolio with the secondmoment smaller than ω X . It immediately follows that • all efficient zero-cost portfolios are a constant multiple of X ; • HR X is the highest Hansen ratio available in M (0).The performance of zero-cost portfolios is often quoted in terms of their Sharpe ratio. Bystraightforward calculations, one obtains the following identities/conversions,1 + SR W = 11 − HR W ; SR W = HR W q − HR W ; HR W = SR W q W , W ∈ L . (2.6)Assuming one cannot obtain risk-free profit with zero initial outlay, 1 / ∈ M (0), one obtains thefollowing range restrictions for the efficient zero-cost portfolio X ,0 ≤ HR X < , ≤ HR X ≤ SR X < ∞ . The Hansen ratio on the efficient frontier.
When two random variables are orthog-onal, E [ V W ] = 0, their second moments are additive, ω V + W = ω V + ω W . Because M (0) is a closed subspace of L , this establishes the existence of X . By the same token, the Hansen ratio of the residual 1 − X is complementary to the Hansen ratio of X , i.e., µ − X = ω − X = HR − X = 1 − HR X . ALEˇS ˇCERN ´Y
Because the means are additive in any case, we find that the expected quadratic utility of aportfolio of orthogonal investments is additive and hence each component can be optimizedseparately. From here and from (2.5) one immediately draws two conclusions. • The squared Hansen ratio of a portfolio of orthogonal investments is subadditive u ( αV + βW ) ≤
12 HR αV + βW ≤ max α,β ∈ R u ( αV + βW ) = 12 (cid:16) HR V + HR W (cid:17) ; • By changing the risk aversion in the utility function (2.4), one observes that the maximalsquared Hansen ratio of a sum of two orthogonal investments is attained even if we fix α = 1 and vary only β , provided that µ V = 0. More generally,sup β ∈ R HR V + βW = HR V + HR W . (2.7)The supremum is attained, except for the case µ V = HR V = 0 where it is attainedasymptotically with | β | going to infinity.Now let us apply these observations to the assets on the efficient frontier. Thanks to (2.2), allfully invested portfolios are of the form Y + W , where W costs zero. As the means and secondmoments are additive (the latter due to the orthogonality between Y and M (0)), the portfolio Y + W is efficient if and only if W is efficient in M (0). This shows the efficient frontier is of theform Y + λX, λ ∈ R ;a result that goes back to at least Hansen and Richard (1987, Lemma 3.3).Formula (2.7) shows that the squared Hansen ratio on the efficient frontier is dominated byHR X + HR Y ; this upper bound is attained for some λ ∈ R if µ Y = 0 and otherwise it is attainedasymptotically for | λ | going to infinity. All this yields a somewhat counter-intuitive result.Because HR X + HR Y is the least upper bound of the squared Hansen ratio on the efficientfrontier, and because the squared Hansen ratio always satisfies the risk-free bound (2.1), weconclude that HR X + HR Y ≤ . (2.8)The restriction on Y is remarkable; it states the square of the Hansen ratio of Y cannot be anarbitrary number below 1, it must be a number below 1 − HR X . Observe that the investmentperformance of X restricts the possible location of Y in the mean–standard deviation diagram.No such restriction flowing from X applies to the minimum variance portfolio, whose propertieswe examine next.2.4. The minimum variance fully invested portfolio Z . Assume for now that the min-imum variance fully invested portfolio exists and denote this portfolio by Z . The minimumvariance portfolio is evidently efficient because for the given mean µ Z there can be no portfoliowith standard deviation smaller than σ Z . Consequently, there exists ˆ λ ∈ R such that Z = Y + ˆ λX. (2.9) HE HANSEN RATIO 5
Conversely, let us now identify the minimum variance portfolio on the efficient frontier (thusalso establishing existence). As is well known, variance is the residual sum of squares from anorthogonal projection of the random variable in question onto constant 1. As we seek Z of theform (2.9), the minimal variance σ Z is therefore the residual sum of squares from an orthogonalprojection of Y onto the span of X and 1, σ Z = min α,λ ∈ R ω Y + λX − α = min α,λ ∈ R k Y + λX − α k = min α,λ ∈ R k Y + ( λ − α ) X − α (1 − X ) k . (2.10)Because X itself is the projection of 1 onto M (0), we have that 1 − X is orthogonal to X .Furthermore, Y too is orthogonal to X so the least squares optimality in (2.10) yields µ Z = ˆ α = ˆ λ. (2.11)The coefficient ˆ α can now be obtained by regressing Y onto 1 − X which yields µ Z = ˆ α = µ Y − HR X . (2.12)From (2.10) it is also immediate that σ Z = k Y k − ˆ α k − X k = ω Y − µ Y − HR X = ω Y − HR Y − HR X ! . (2.13)Non-negativity of σ Z now once again yields the Hansen ratio restriction (2.8).Observe that the knowledge of any two of the three special portfolios X , Y , and Z impliesthe knowledge of the remaining portfolio. It is thus a matter of convenience which two specialportfolios one chooses to identify. We will return to this point in the concluding Section 6.2.5. The Hansen–Jagannathan bound.
By following the logic of Subsection 2.3, it is notdifficult to see that the ‘efficient frontier’ of pricing kernels is of the form m = Yω Y + ηV, η ∈ R , (2.14)where V is an ‘efficient’ element of M ⊥ . From Subsection 2.2 we know V may be taken as thesolution of min W ∈ M ⊥ k − W k . This yields V as the residual from the orthogonal projectionof 1 onto the span of X and Y (in fact onto M ), V = 1 − X − µ Y ω Y Y, with HR V = 1 − HR X − HR Y . Variable V is the realized distance from the bliss point of the quadratic utility (2.4) for anoptimal investment in M with optimally chosen initial wealth µ Y /ω Y .We now conclude from (2.14) and the arguments in Subsection 2.3 thatHR m ≤ HR Y + HR V = 1 − HR X . (2.15) Equation (2.11) in combination with (2.9) gives the remarkable identity Z = Y + µ Z X. ALEˇS ˇCERN ´Y
This is a previously unpublished version of the Hansen–Jagannathan bound. More commonly,provided µ m = 0, the bound is formulated in terms of variance. By taking reciprocals in (2.15)one obtains HR − m − ≥ (1 − HR X ) − − . This then yields, with the help of conversion (2.6), the more commonly encountered formula σ m µ m ≥ SR X ; (2.16)see Hansen and Jagannathan (1991, Eq. 17).3. The monotone Hansen ratio and non-negativity constraints
The next example shows that the Hansen ratio does not preserve the state-wise stochasticdominance ordering.
Example 3.1.
In a three-state model with W d = − W m = 1%, W u = 2% and withprobabilities p d = , p m = , and p d = one obtainsHR W = 1 √ , while for f W = W + 0 . W > . (cid:9) W one hasHR e W = 4 √ < HR W . (cid:3) With Filipovi´c and Kupper (2007), define the monotone Hansen ratio, HR, as the monotonehull of HR, that is, HR : L − L → ( −∞ , ∞ ] withHR W = sup e W ∈ L HR W − e W . (3.1)Intuitively, the monotonization allows to set aside some non-negative cash amount if this leadsto an increase in the Hansen ratio of the remaining wealth. It is shown in ˇCern´y (2020) thatfor W ∈ L − L with positive (possibly infinite) mean and non-zero downside, the supremumin (3.1) is attained. Furthermore, one hasHR( W ) = HR(( ˆ αW ) ∧
1) = HR( W ∧ ˆ α − ) = max k> HR( f W ∧ k ) , (3.2)where ˆ α > E [ W ˆ αW ≤ ] = ˆ α E [ W ˆ αW ≤ ] . (3.3)This shows that one puts away all returns above some fixed threshold k chosen in such a waythat the optimal investment with the truncated investment opportunity only just touches butdoes not reach over the bliss point of the quadratic utility. Example 3.2.
In the setting of Example 3.1, the optimal value of k in max k> HR( f W ∧ k ) is2%, hence HR ( f W ) = HR ( f W ∧ .
02) = 12 . (cid:3) HE HANSEN RATIO 7
In effect, the monotone Hansen ratio is obtained from the maximization of monotonizedquadratic utility U ( x ) = x ∧ − ( x ∧ / , so that for W with non-negative mean one obtains, in analogy to (2.5),max α ∈ R u ( αW ) = max α ∈ R E [ U ( αW )] = 12 HR W . By the Fenchel inequality, one obtains for all non-negative pricing kernels m ∈ L and all W ∈ M (0) E [ U ( W )] ≤ min λ ∈ R E (cid:20)
12 (1 − λm ) (cid:21) . From here we have the lower bound sup W ∈ M (0) HR W ≤ − HR m for all non-negative pricing kernels in L . Under minimal assumptions this lower bound is tight;see Biagini and ˇCern´y (2020, Theorem 4.3).Observe that the monotone Sharpe ratio, SR, is related to HR in the same way the standardSharpe ratio is to HR; see (2.6). The analogon of the classical Hansen–Jagannathan inequality(2.16) for non-negative pricing kernels thus reads σ m µ m ≥ sup W ∈ M (0) SR W . Example 3.3.
We remain in the setting of Example 3.1. A market spanned by the zero-costinvestment f W has the standard squared Sharpe ratio ofSR e W = 11 − HR e W − . , while by Example 3.2, the square of its monotone Sharpe ratio isSR e W = 11 − HR e W − , which leads to a tightening of the variance bound for non-negative pricing kernels. (cid:3) The Hilbert space generalization
Following Cochrane (2014), let us now consider a Hilbert space with an inner product h V, W i .In the classical mean–variance theory, the Hilbert space is L with h V, W i = E [ V W ]. Inparticular, µ V = h V, i . In the generalized theory the role of 1 will be played by a specialelement I that also assumes the role of a ‘risk-free’ payoff. We will set µ V = h V, I i . To make the analogy complete, we will also require
I / ∈ M (0); h I, I i = 1 . ALEˇS ˇCERN ´Y
For ease of notation we continue to write ω V = h V, V i = k V k ; σ V = k V − µ V I k = ω V − µ V , referring to µ V as the mean , ω V as the second moment , and σ V as the variance , even whenthese objects no longer have such classical interpretation. Observe that σ I = 0 so the payoff I is indeed ‘risk-free’.The results in Subsections 2.1–2.5 now translate directly to the Hilbert space setting. Oneonly needs to replace 1 − X (resp., 1 − W ) with I − X (resp., I − W ) and E [ Y W ] (resp. E [ V W ])with h Y, W i (resp., h V, W i ). Example 4.1.
Cochrane (2014) considers, among others, the Hilbert space of sequences ofrandom variables V = { v n } n ∈ N with k V k = β − β X n ∈ N β n E [ v n ]for some fixed positive β <
1. The special ‘risk-free’ payoff I is taken to be the constant cashflow 1 at every date. (cid:3) The Hansen ratio in dynamic models
Consider an n –period model with independent, identically distributed (IID) returns. Denoteby e X , e Y the unconditionally optimal portfolios in the dynamic model and by { X t , Y t } nt =1 theirone-period counterparts running from t − t for each t ∈ { , . . . , n } . By the IID assumptionthe collection { X t , Y t } is also IID. By dynamic programming one obtains e Y = n Y t =1 Y t and 1 − e X − µ e Y ω e Y e Y = n X j =1 µ Y ω Y ! n − j − X j − µ Y ω Y Y j ! n Y t = j +1 Y t , where an empty product is defined to take the value of 1.This yields µ e Y = µ nY , ω e Y = ω nY , HR e Y = µ Y ω Y ! n , − HR e X − HR e Y = (cid:16) − HR X − HR Y (cid:17) n − X t =0 HR tY . (5.1)Hence, the knowledge of µ Y , ω Y , and HR X determines the values of µ e Y , ω e Y , and HR e X in amultiperiod model with IID returns.By Subsection 2.3, the unconditional efficient frontier in the ( µ, ω )–space reads ω = ω e Y + HR − e X ( µ − µ e Y ) . HE HANSEN RATIO 9
If desired, one can now calculate the values of µ e Z , σ e Z via (2.11)–(2.13). On converting HR e X toSR e X by means of (2.6), one obtains the unconditional efficient frontier in the ( µ, σ )–space, σ = σ e Z + SR − e X ( µ − µ e Z ) . A fully worked numerical example is presented in Appendix A.6.
Concluding remarks
The Hansen ratio arises naturally in the description of the mean–variance efficient frontierthrough its link to expected utility maximization. The latter takes on special significance in thedynamic setting. We have seen that the efficient frontier is fully described by any two of thethree special portfolios e X , e Y , and e Z . Yet only the computation of e X and e Y is time-consistent ina dynamic setting. This points to a dichotomy in an effective evaluation of the efficient frontier.When a risk-free asset is assumed, e Z is known a-priori and only e Y needs to be calculated.This is the situation in ˇCern´y and Kallsen (2007). The much more difficult case when e Z isto be calculated in a dynamic setting has received very limited attention in the literature; seeLi and Ng (2000) and Lim (2004, 2005). The approach proposed here, namely computing e X and e Y (or instead of e X a suitable mix, such as e X + µ e Y ω e Y e Y ), leads to substantial simplification insuch circumstances as illustrated in Section 5.In this paper, the Hansen ratio emerges as a long lost twin of the Sharpe ratio. That it hasremained hidden for so long is likely due to an early disconnection between the mean–varianceanalysis on the one hand and the expected utility maximization on the other. The missing linkwas finally uncovered in Filipovi´c and Kupper (2007) who characterize the mean–variance utilityas the cash-invariant hull of the expected quadratic utility. This is what ties the Hansen ratioand the Sharpe ratio inextricably together and explains why the Hansen ratio is to orthogonalinvestments what the Sharpe ratio is to uncorrelated investments. Both quantities are clearlyfundamental. It may have been hidden for a long time but the Hansen ratio is here to stay. References
Bertsimas, D., L. Kogan, and A. W. Lo (2001). Hedging derivative securities and incompletemarkets: an ǫ -arbitrage approach. Operations Research 49 (3), 372–397.Biagini, S. and A. ˇCern´y (2020). Convex duality and Orlicz spaces in expected utility maxi-mization.
Mathematical Finance 30 (1), 85–127.ˇCern´y, A. (2020). Semimartingale theory of monotone mean–variance portfolio allocation.
Math-ematical Finance, 30 (3), 1168–1178.ˇCern´y, A. and J. Kallsen (2007). On the structure of general mean–variance hedging strategies.
The Annals of Probability 35 (4), 1479–1531.Chamberlain, G. and M. Rothschild (1983). Arbitrage, factor structure, and mean-varianceanalysis on large asset markets.
Econometrica 51 (5), 1281–1304.Cochrane, J. H. (2001).
Asset Pricing . Princeton University Press.
Cochrane, J. H. (2014). A mean–variance benchmark for intertemporal portfolio theory.
Journalof Finance 69 (1), 1–49.Filipovi´c, D. and M. Kupper (2007). Monotone and cash-invariant convex functions and hulls.
Insurance: Mathematics & Economics 41 (1), 1–16.Hansen, L. P. and R. Jagannathan (1991). Implications of security market data for models ofdynamic economies.
Journal of Political Economy 99 (2), 225–262.Hansen, L. P. and S. F. Richard (1987). The role of conditioning information in deducingtestable restrictions implied by dynamic asset pricing models.
Econometrica 55 (3), 587–613.Li, D. and W.-L. Ng (2000). Optimal dynamic portfolio selection: Dynamic multi-period for-mulation.
Mathematical Finance 10 (3), 387–406.Lim, A. E. B. (2004). Quadratic hedging and mean-variance portfolio selection with randomparameters in an incomplete market.
Mathematics of Operations Research 29 (1), 132–161.Lim, A. E. B. (2005). Mean–variance hedging when there are jumps.
SIAM Journal on Controland Optimization 44 (5), 1893–1922.Roy, A. D. (1952). Safety first and the holding of assets.
Econometrica 20 (3), 431–449.
Appendix A. Efficient frontier in a multiperiod model
The numerical values in this example are based on Li and Ng (2000, Example 1). There are 3risky assets with IID one-period total returns whose mean and variance, respectively, are givenby µ R = . . . , Σ R = . . . . . . . . . . This yields the second (co)moment matrixΩ R = Σ R + µ R µ ⊤ R = × − . We have HR X + HR Y = µ ⊤ R Ω − R µ R = 28 147 713 78128 448 540 506 ≈ . . Furthermore, µ Y ω Y = ⊤ Ω − R µ R = 12 123 548 00014 224 270 253 ≈ . ω Y = 1 ⊤ Ω − R = 14 224 270 25316 329 740 000 ≈ . , yielding HR Y = µ Y ω Y = 7 349 020 805 415 20011 613 931 746 061 211 ≈ . , together with HR X = µ ⊤ R Ω − R µ R − ( ⊤ Ω − R µ R ) ⊤ Ω − R = 582 3991 632 974 ≈ . , HE HANSEN RATIO 11 µ Y = ⊤ Ω − R µ R ⊤ Ω − R = 3 030 8874 082 435 ≈ . . In a dynamic setting with n = 4 periods one obtains from (5.1)HR e X = 1 − HR nY − HR Y HR X ≈ .
815 50 ,µ e Y = µ nY = (cid:18) (cid:19) ≈ .
303 81 ,ω e Y = ω nY = (cid:18)
14 224 270 25316 329 740 000 (cid:19) ≈ .
575 71 . The efficient frontier in ( µ, ω )–space then reads ω = ω e Y + HR − e X (cid:16) µ − µ e Y (cid:17) ≈ .
575 71 + 1 .
226 25( e µ − .
303 81) . One may alternatively choose to evaluate the parameters of the minimum variance portfolioas shown in Subsection 2.4, µ e Z = µ e Y − HR e X ≈ .
646 63 ,σ e Z = ω e Y − HR e X (cid:16) − HR e Y − HR e X (cid:17) ≈ .
075 446 , SR − e X = HR − e X − ≈ .
226 25 , which yields the efficient frontier in the ( µ, σ )–space, e σ = σ e Z + SR − e X (cid:16) µ − µ e Z (cid:17) ≈ .
075 446 + 0 .
226 25( e µ − .
646 63) . All numerical values shown here are precise to the last digit, subject to rounding. Note,however, that the value od µ e Z in Li and Ng (2000, p. 403) has a small rounding error. Aleˇs ˇCern´y, Business School, City, University of London
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