UNDERSANDING RISK ESIMAING HE CONRIBUION O RISK OF INDIVIDUAL BES BY KEMAL ASAD-SYED (INVESMEN OFFICER HE WORLD BANK INVESMEN DEPARMEN) 1818H Street NW, Washngton DC 20433 kasadsyed@worldbank.org (202-473-0798) ARUN S. MURALIDHAR (VICE PRESIDEN & HEAD OF CURRENCY RESEARCH JPMorgan Investment Management) 320 East 46 th Street, Apt. 14-H, New York, NY 10017 arunmura@hotmal.com (212-837-2733) PAOLO PASQUARIELLO (Ph-D CANDIDAE NEW YORK UNIVERSIY SERN SCHOOL OF BUSINESS; contrbuted to ths paper durng an nternshp at J.P. Morgan Investment Management) 40W 4 th Street, 9 th Floor, Room 9-180, New York, NY 10012 ppasquar@stern.nyu.edu (212-998-0369) he vews expressed n the paper are prmarly theoretcal and academc n nature and do not represent the vews of J.P. Morgan or the World Bank, or any of ther afflates. All opnons, theores and errors, as such, are purely those of the authors. 1
UNDERSANDING RISK ESIMAING HE CONRIBUION O RISK OF INDIVIDUAL BES Abstract Portfolo managers may take many bets to outperform a benchmark. hs note provdes two smple methodologes to calculate the contrbuton to total rsk of a specfc bet, whether the rsk measure s an absolute or a relatve one, and demonstrates how nvestors can develop smple n-house models to measure such rsk. In addton, we demonstrate how other measures, that are not derved from fnance theory, but are used as frst approxmatons are ncorrect. hese smple tools can allow nvestors to measure, montor and hence manage the rsks n ther portfolos more effectvely. We would lke to thank Sudhr Krshnamurth, Ramasastry Ambarsh, P.S. Srnvas, Lester Segel and Ronald van der Wouden for helpful comments. 2
1. INRODUCION he ssue of rsk management s becomng more mportant for nsttutonal nvestors, especally penson funds, and a number of workng groups have been formed to evaluate what rsk standards should be adopted by oversght commttees for the management of such plans. However, one of the present shortcomngs n the ndustry s that there s no unform model that has been adopted to measure rsks, whch would then allow management to manage them. In addton, many software provders have focused only on the absolute rsk of a portfolo n measurng the value-at-rsk of a portfolo. Most rsk systems have often not captured the largest rsk that most penson plans are exposed to, namely asset-lablty rsk, and ths was hghlghted n Kemal Asad-Syed and Muraldhar (1998). 1 Further, when the performance of an nsttutonal nvestor s measured relatve to a passve benchmark, t s mperatve to measure not only the absolute rsk of the benchmark and the actual portfolo, but also the rsk relatve to a benchmark. 2 Even those software packages that have focused on relatve rsk have not adequately captured the contrbuton to total rsk of any specfc bet taken by portfolo managers. Ltterman (1996) hghlghts the usefulness of ths measure and gves an ndcaton of how ths measure may be computed; however, the paper does not provde the methodology for the calculaton. hs note provdes two smple methodologes to calculate the contrbuton to total rsk of a specfc bet, whether the rsk measure s an absolute or a relatve one, and demonstrates how plan sponsors can develop smple n-house models to 1 Pensonmetrcs s one of the few software products that s targeted specfcally to penson funds and evaluates the rsk of assets vs-à-vs labltes. 3
measure plan rsk. he frst approach develops the mathematcal technque suggested by Ltterman (1996); the second provdes a more ntutve approach that s derved from asset prcng theory. In addton, we demonstrate how other measures, that are not derved from fnance theory, but are used as frst approxmatons are ncorrect. 3 hs approach also provdes valuable nsght nto the correlaton of bets wth the entre portfolo of bets, thereby enhancng the evaluaton of rsk-takng actvtes. hese smple tools can allow sponsors to measure, montor and hence manage the rsks n ther portfolos more effectvely. hs paper wll also look at the feasblty of usng such statstcs for the allocaton of captal. he paper s developed n the context of a penson plan, but the concepts and conclusons apply more generally to any nvestor, whether a portfolo manager or an nsttutonal nvestor wth nvestment advsers. 2. PENSION PLAN RISKS Pror to dscussng how one estmates the contrbuton to rsk of a specfc bet, we detal the dfferent rsks that a plan s exposed to. Rsk s generated n penson plans at dfferent levels. At the hghest level, selectng a benchmark for the asset portfolo creates the possblty for rsks from asset-lablty msmatches (or asset-lablty rsk). Alternatvely, selectng an asset benchmark for purely asset reasons mples targetng an absolute rsk pont or a target varablty of returns. At the next level, once target asset class allocatons and benchmarks have been determned, a plan sponsor may create addtonal rsk by nvestng tactcally n the actual portfolo away from these target levels (or tactcal 2 Ambarsh and Segel (1996). 4
rsk). 4 At the smplest level, tactcal rsk s created by under or over weghtng ndvdual asset classes. In ths note, we wll focus only on () the absolute rsk of the benchmark portfolo; () the absolute rsk of the actual portfolo on any gven day (whch f tactcal bets are permtted could be qute dfferent from that of the benchmark); and () the relatve rsk mpled by the actual portfolo vs-à-vs the benchmark or tactcal rsk. hereafter, t s possble to demonstrate the contrbuton to the total rsk or varablty of returns of each asset class n whch the plan has made ether a target allocaton or a tactcal devaton. 5 he concept of the margnal s very well developed n economcs n determnng optmal consumpton, prcng etc. and n an analogous fashon we wll attempt to demonstrate whether the margnal rsk measure can be used n the optmal utlzaton of a rsk budget. 3. DEFINIION OF ERMS. For convenence, we defne two portfolos, the benchmark and the actual portfolo, and three rsk measures, the absolute rsk of the benchmark, the absolute rsk of the actual portfolo, and the relatve rsk of the actual portfolo. (a) Benchmark Portfolo: hs s the strategc long-term asset allocaton of the plan that s descrbed by lstng the varous asset classes n whch the plan s nvested and the 3 Ltterman (1996) makes a smlar pont n a footnote for one of these methods. 4 Mashayekh Beschloss and Muraldhar (1996). 5 For the purpose of ths artcle we wll demonstrate how asset class allocatons at a target or tactcal level can be used to determne contrbuton to rsk. he extenson of determnng the contrbuton of any devaton from a benchmark (e.g., securty, country or currency selecton) s trval. 5
long-term target allocatons. A hypothetcal benchmark portfolo s provded n able 1. (b) Actual Portfolo: hs s the nvestor s portfolo on any measurement day. As a consequence of portfolo managers overweghtng or underweghtng asset classes, the lve portfolo can and wll dffer from the benchmark. For llustratve purposes, we provde a hypothetcal actual portfolo n able 1, whch s relatve to the benchmark. In the last column n able 1, s the percentage devaton of each asset class from ts benchmark; the sum of these devatons s zero. 6 (c) Absolute Rsk of the Benchmark: In asset space, the rsk of the benchmark portfolo s descrbed by the varance or standard devaton of the expected return of ths portfolo. 7 Mathematcally, the absolute rsk s estmated by takng the benchmark or target weghts and multplyng them through a varance-covarance matrx.e., σ 2 (benchmark) = (v Γv),...(1a) where v = matrx of benchmark asset class weghts (v beng the transpose of v) and Γ s the assumed varance-covarance matrx, and v s the target weght of the th asset class. he square root or the standard devaton s also a rsk measure, as t captures the dsperson of the portfolo return around ts mean. Usng the hypothetcal benchmark portfolo n able 1 and the assumed varance-covarance matrx n Appendx 1, the standard devaton (.e., rsk) of ths portfolo s provded n able 1. 8 6 We are assumng unleveraged devatons from the benchmark, but the results would be unaffected f leverage s approprately captured. 7 See for example Markowtz (1952). 8 Snce numercal smulatons are provded to llumnate the key ponts of ths artcle, we provde a varance-covarance matrx of the varous asset classes. Every nsttutonal nvestor can select ther own matrx; these values were based on estmates from hstorcal data. 6
(d) he Absolute Rsk of the Actual Portfolo: he varance of ths portfolo s calculated n a fashon dentcal to that of the benchmark,.e., σ 2 (actual) = (w Γw)...(1b) where w = matrx of actual asset class weghts, and w s the actual weght of the th asset class. he square root or standard devaton s an alternatve expresson of ths rsk measure and s provded n able 1. (e) Relatve Rsk of the Actual Portfolo: hs s the rsk engendered by off-benchmark postons and Ambarsh and Segel (1996) demonstrate why ths measure should be used when a portfolo s measured relatve to a benchmark. he relatve rsk or varance of the actve portfolo s calculated n a fashon dentcal to those above,.e., σ 2 (relatve) = (z Γz)......(1c) where z = matrx of the dfferences between the actual and target asset class weghts, and z s the devaton from benchmark n the th asset class. Any component of the z matrx can be postve or negatve, as the nvestment team could have chosen to underweght or overweght a partcular asset class. he square root of σ 2 (relatve) per unt of tme s referred to as the trackng error of a portfolo. Mathematcally, trackng error = z Γ z = z z Γz Γz...(1d) he trackng error measures the amount by whch the performance of the actual portfolo can devate from the benchmark and s provded n able 1. 7
4. HE MAHEMAICAL SOLUION FOR MARGINAL CONRIBUION he margnal contrbuton to total rsk from an ndvdual bet s nothng but a functon of the frst dervatve of the rsk measure vs-à-vs the bet under consderaton. Ltterman (1996) defnes t loosely as, the margnal rate of change n rsk per unt change n the poston (at the current poston sze) tmes the poston sze tself, can be thought of as the rate of change n rsk wth respect to a small percentage change n the sze of the poston. For smplcty, we wll use the trackng error for ths estmaton. Margnal contrbuton of the bet n the th asset class (z ) to trackng error : = z * (trackng error)/ z (such that z * (trackng error)/ z = total trackng error) 9 = z * z Γz z = [z * ( z z Γ Γz ) ]..(1e) where ( z z Γ Γz ) s a 1xN matrx measurng the margnal rsk per unt of devaton. Notce that the denomnator n the second term s nothng but the trackng error, thereby normalzng the calculaton. 9 See also Ltterman (1996). 8
he same approach can be followed to measure the margnal contrbuton of each ndvdual poston to the total absolute rsk of the portfolo. In ths case, the margnal contrbuton of the poston n the th asset class to the portfolo s rsk s gven by: = w * (stdev)/ w = [w * ( w Γ w Γw ) ]...(1f) where {(w Γ)/ (w Γz)} s a 1xN matrx measurng the margnal rsk per unt of the postons. Fnally, the margnal contrbuton of the poston n the th asset class to the benchmark s rsk s gven by: = v * (stdev)/ v = [v * ( v Γ v Γv ) ]...(1g) where {(v Γ)/ (v Γv)} s a 1xN matrx measurng the margnal rsk per unt of the postons. 9
5. HE INUIIVE APPROACH here s another approach to estmatng the contrbuton of an allocaton to total rsk that s derved from asset prcng theory. For smplcty, we call ths the ntutve approach. Defne the contrbuton of a stock I to the total rsk of a portfolo of N stocks (P) as r. Defne the contrbuton of an asset class I to the total rsk of a portfolo of N asset classes (P) as c. From the bascs n fnance, we know that the contrbuton to total rsk of a stock I to a total portfolo of N stocks (P) or r s equal to r = s * covarance(i,p)...(2a) Mathematcally, ths s equvalent to r = s *σ(i,p) = s *ρ I,P= σ(i)*=σ(p),...(2b) where s s the weght of stock n portfolo P, ρ I,P s the correlaton between I and P, σ(i,p) s the covarance between I and P and σ(p) and σ(i) represent the standard devatons of P and I respectvely. Usually, the correlaton among stocks s known and stable, whle that of an ndvdual stock to a specfc portfolo s uncertan. Where the correlaton factor s unknown ex-ante, the contrbuton to rsk can be calculated by the followng: r = s * s *=σ(,),...(2c) 10
where s the summaton operator for = 1 through N stocks and σ(,) s the covarance of stocks and. he sum of all r n the portfolo must equal σ 2 (P) and hence n percentage terms, the contrbuton of stock I to the varance of portfolo P would be r /σ 2 (P). In an analogous fashon to 2(a), 2(b) and 2(c), the contrbuton to total rsk of an asset class for ether absolute or relatve rsk can be defned as above. However, n the case of asset class structurng, the correlaton between that of a specfc asset class and the total portfolo (or those of asset class bets wth the portfolo of bets) s dffcult to determne ex-ante and probably changng as the portfolo composton changes. he correlaton between two asset classes s easer to estmate. Hence an adaptaton of equaton (2c) s appled to estmate the contrbuton of an asset class to portfolo rsk. hus, we have: c (actual) = w * w *=σ(,),...(3a) n the case of the actual rsk of the portfolo and where σ(,) s the covarance between the th and the th asset class and s the summaton operator for = 1 through N asset classes. For the absolute rsk of the benchmark portfolo, we defne: c (benchmark) = v * v *=σ(,),...(3b) In the case of the relatve rsk calculatons, we wll be concerned wth the correlaton of a tactcal bet n an asset class wth the portfolo of tactcal bets. 11
c (tactcal devaton) = z * z *=σ(,),...(3c) It s clear that the c are calculated usng varance as a measure of rsk. o normalze for the standard devaton beng the measure of rsk and usng (1d) we defne: c (actual) = c (absolute)/ (w Γw) = w * Σ w * σ (, )....(3d) w Γw c (benchmark) = c (benchmark)/ (v Γv) = v * Σ v * σ (, )...(3e) v Γv c (tactcal) = c (tactcal)/ (z Γz) = z * Σ z * σ (, ) (3f) z Γz Note that the last equaton descrbes the margnal rsk of a sngle bet to total trackng error. For the portfolos n able 1 (benchmark, actual and devaton), we provde n ables 2 (a) and 2 (b) the margnal contrbuton to total rsk (n percentage ponts) and the percentage contrbuton of each asset class or asset class bet to total rsk. 6. CORRELAIONS OF ASSE ALLOCAIONS O PORFOLIO ALLOCAIONS An nterestng statstc that can be derved from the above s the correlaton of an asset class allocaton to the overall allocaton (as dfferentated from the asset class correlatons n Appendx I) or a specfc asset class bet to a portfolo of bets. In ths secton we develop the analytcal solutons for estmatng these correlatons. hs statstc s useful 12
as t allows the portfolo managers to determne whether bets are postvely, negatvely or uncorrelated wth other bets somethng that s not obvous at the tme of constructng portfolos. Measures of correlaton of a sngle poston wth the total portfolo and of a sngle bet wth the total portfolo of bets can be explctly obtaned from the followng defntons. Frst, for the absolute portfolo: cov ρ P N N N 2 ( ω y, y ) = cov ω y, ω y = ω var( y ) + ω ω cov( y, y ) = ω ω cov( y, y ) cov = σ P ( ω y, y ) ω y σ P P = N = 1 = 1 ω ω cov ω σ * ( y, y ) ω Γω c = ω σ * ( actual) = 1 = 1...(4a) ω Γω where y s the return from the th asset class. hen, for the correlaton of an ndvdual bet wth the portfolo of bets: cov N N N 2 ( y, y ) = cov z y, z y = z var( y ) + z z cov( y, y ) = z z cov( y, y ) Z Zp = 1 ( y, y ) z z cov ρ cov( yz, yzp ) = 1 c ZZP = = σ Zσ = ZP z σ * z Γz N ' = 1 ( tactcal) z σ = 1...(4b) where y z s the spread expected return from the th asset class. Alternatvely, usng the ntutve approach, snce the correlaton between an asset class and the portfolo s unknown ex-ante, from 3(a), 3(b) and 2(b), the correlaton coeffcent 13
of each asset class to the benchmark portfolo (or ρ,b=)=can also be mpled by the followng: = ρ,b== c ( benchmark) σ ( P) * σ ( I ) * v.....(4c) he correlaton coeffcent of each asset class to the actual portfolo (or ρ,p=)= = == c ( actual) σ ( Actual) * σ ( I) * w.....(4d) or the correlaton of the bet n asset class I to the portfolo of bets (ρ Z,ZP) == c ( tactcal) σ ( E) * σ ( I ) * z.....(4e) I n able 3, we provde the mpled correlaton coeffcent of each asset class bet to portfolo of bets based on ther respectve allocatons. 10 hs table shows that the bets n four asset classes are negatvely correlated wth the portfolo of bets, at the current poston. 10 As the allocaton weghts change, the total rsk of a portfolo and hence the mpled correlaton wll also change. 14
7. RESULS here are a number of useful nsghts from these dagnostcs. Frst, notce that whle the portfolo of able 1 s overweght US equtes, overweghtng ths asset class reduces the trackng error, as ths bet s negatvely correlated wth other bets n the portfolo thereby lowerng total relatve rsk (ables 2 (a) and (b)). Second, the absolute or relatve sze of a bet may mask the true contrbuton to total rsk. For example, whle the 2% overweght n US Equtes actually lowers trackng error, the same absolute bet n Hgh Yeld (+2%) contrbutes postvely to relatve rsk (3.3%). 11 In addton, the 2% underweght n Non- US Fxed Income has a neglgble mpact on trackng error, whle the same absolute and relatve devaton n Prvate Equtes contrbutes 20% of total trackng error. Whle Prvate Equtes are more volatle, there s a more complex relatonshp at work whch ncludes the relatonshp wth other bets n the portfolo. 12 hrd, n evaluatng the correlaton of bets wth the overall portfolo of bets, t s revealng to notce that the bets n U.S. Equtes, Non-U.S. Equtes, Non-U.S. Fxed Income and Prvate Equtes are all negatvely correlated wth the portfolo of bets. One could ask f all these are therefore rsk reducng by offsettng other bets n the portfolo. However, where the portfolo s long wth respect to the benchmark and s negatvely correlated, the contrbuton to trackng error s negatve (as n U.S. Equtes). On the other hand, where the portfolo s short wth respect to the benchmark (Non-U.S. 11 hs pont has been made elsewhere, more specfcally wth respect to managng the rsks of currency overlays. See Mashayekh Beschloss and Muraldhar (1996). 15
Equtes, Non-U.S. Fxed Income and Prvate Equtes), the negatve correlaton, n conuncton wth the short poston contrbutes postvely to trackng error. 8. USEFULNESS OF HIS MEASURE Any ablty to drll down nto a total rsk measure and attrbute the value to ts components s useful for portfolo managers. As hghlghted n the results, when the margnal contrbuton s negatve, all else equal, a margnal unt ncrease n the drecton of the current bet lowers total trackng error. 13 Only the US Equty bet changes the rsk posture by effectvely beng rsk reducng. herefore, ths breakdown can be used to sze bets more effectvely and capture maxmum alpha for a gven rsk tolerance. 14 In addton, the portfolo manager determnes whether the bets are all correlated and s able to dsaggregate how dversfed ther bets may be. For example, n able 1 there are 8 asset class bets; however, the three bets n Non-US Equty, Emergng Markets and Prvate Equty contrbute 99% of the rsk exposure. If the margnal contrbuton s concentrated n a few bets even though a large number of bets may have been mplemented suggests that rsks are not dversfed. Smlarly, a negatve correlaton s nsuffcent nformaton to know whether bets are rsk ncreasng or rsk reducng as demonstrated above. 12 Ltterman (1996) makes a smlar observaton and terms the pont where rsk contrbuton s zero as a canddate for a best hedge. 13 Up to a pont. If the bet sze ncreases, ths becomes the domnant bet n the portfolo and wll contrbute postvely to trackng error. 14 One cautonary note any rsk analyss depends on the correlatons and varances remanng stable over tme and a volaton of ths assumpton would put any rsk analyss nto doubt. Also, once the postons are changed, the statstcs wll need to be recalculated for the new portfolo. 16
he margnal contrbuton s dependent on current allocatons, hence a slght change to a poston mples very dfferent results. For example, by shftng 2% more to U.S. Equtes from Non-U.S. Fxed Income (.e., extendng the prevous bet), the contrbuton from U.S. Equtes to trackng error turns postve and the correlaton of U.S. Equtes and Non-U.S. Fxed Income to other bets n the portfolo are now postve. However, now the contrbuton to trackng error from Non-U.S. Fxed Income turns mldly negatve as the correlaton has shfted sgn. herefore, the senstvty of the margnal rsk analyss to portfolo changes would make t very dffcult to allocate rsk captal on ths bass, for t would requre a constant fne tunng and each asset class manager wll need to be cognzant not only of the vew that they may have on ther specfc market, but also ts mpact on other vews. 9. COMPARISON WIH OHER MEHODOLOGIES. In ths secton, we demonstrate other methodologes that are appled n standard rsk management software and explan the defcences of each. In ables 4 a and b, we compare the contrbuton to total rsk usng these three methods to provde an estmate of the magntude of the error of not capturng the dversfcaton benefts of an asset class. (a) Contrbuton Assumng an Identty Correlaton Matrx: Under ths method, t s assumed that dagonal elements are unty and off-dagonal elements n the correlaton matrx are zero. hs s done to make the calculaton smple. herefore, assumng ndependence between assets would provde a varance estmate whereby addng the 17
weghted varance of each asset class equals the portfolo varance. he problem wth assumng that off-dagonal elements s zero s that the true benefts of dversfcaton are never captured n these analyses. In addton, as able 4a demonstrates the total rsk of the benchmark portfolo s ms-estmated and hence ths approach s ncorrect (7.73% versus 11.69%). (b) Margnal Contrbuton : Under ths methodology, embedded n the most commonly avalable software, the user calculates the varance usng all assets and then extracts one asset class at a tme from the portfolo and recomputes the varance or standard devaton. hs new standard devaton (excludng a partcular asset class) s compared to the full portfolo rsk to gve an estmate of the contrbuton of that partcular asset class. 15 he most mportant problem s that the contrbuton to rsk to a total portfolo s to be computed when portfolos are complete (.e., wth all asset classes, ncludng the one whose contrbuton we are tryng to estmate) and not usng subsets of portfolos. herefore, even f correct, the sum of all margnal estmates should equal the true varance of the portfolo (.e., n the case of the benchmark portfolo = 11.69%). As s evdent from able 4a, ths s not the case and the margnal method overestmates the total rsk of the portfolo (12.9%). he rebalancng approach s clearly ncorrect as we obtan a negatve varance whch s an nfeasble result for an asset portfolo. 15 here are two ways to perform ths calculaton; namely, to not rebalance the remanng asset class weghts (.e., so that the sum of the assets need not total 100%) and to rebalance the remanng assets. I would lke to thank Mr. P.S. Srnvas for pontng ths out. he rebalancng method s clearly ncorrect as t excludes the possblty of the asset class ever beng n the portfolo. 18
(c) rackng Error of each bet s solaton: Under ths method one assumes that the bet beng evaluated s the only bet n the portfolo and looks at ts rsk n solaton. hs assumes that the bets are ndependent and s dentcal to (a). Clearly, the margnal method assumng rebalancng and the method assumng ndependence of assets s ncorrect n estmatng ether the total varance or the percentage contrbuton of an asset class or asset class bet. Smlarly, we show numercally that these alternatve methods are nadequate to estmate the percentage contrbuton to relatve rsk. For completeness, we show the results of the Margnal-No Rebalancng calculaton vs-à-vs the proposed method for the trackng error calculaton n able 4 (b). Not only are the resultng totals wrong, but also the magntude and often the sgns are ncorrect for the portfolo bets hence provdng the user wth ncorrect statstcs about the contrbuton of bets to the rsk of the overall portfolo. 11. CAVEAS In the case of the two absolute measures of benchmark rsk and actual rsk, the mpled correlatons are meanngful. However, the mpled correlatons of the asset class bet to the portfolo of bets are based on the assumpton that the varance-covarance matrx for asset classes appled also for asset class bets (whch need not always be true), but ths s an acceptable frst approxmaton and assumes no bas n the bets away from the respectve benchmarks. 19
12. CONCLUSIONS hs artcle set out to demonstrate two smple methods by whch the contrbuton of an asset class allocaton or asset class bet to the total absolute or relatve rsk of the portfolo could be determned. In addton, ths methodology s superor to other methodologes that may be mplemented by rsk management software companes that are avalable to nsttutonal nvestors. More mportant, ths analyss has shown that the sze of bet need not be a good ndcator of contrbuton to total rsk. It s possble to be overweght an asset class and have that bet contrbute negatvely to total rsk. hs follows as the correlaton of an asset class bet to the total portfolo of bets could be negatve even where the correlaton of that asset class to others s postve. Wth respect to correlatons of postons, rather than assume statc correlatons between asset classes and portfolos, we have demonstrated how these can be mpled and examned ex-post. Fnally, we have only demonstrated how contrbutons of asset class allocatons to total rsk are determned; the extensons to estmatng the contrbuton of a selecton of a benchmark of a manager or an ndvdual manager s securty selecton (n ether equtes, bonds or currences) to an entre portfolo s a smple extenson of ths methodology. 20
BIBLIOGRAPHY Ambarsh, R., and Segel, L. (1996). me s the Essence. Rsk, August 1996, 9 (8). Asad-Syed, K. and Muraldhar, A.(1998). An Asset-Lablty Approach to Value-at-Rsk. he World Bank, IMD Workng Paper Seres, 98-023. Ltterman, R. (1996). Hot Spots M and Hedges. Goldman Sachs Rsk Management Seres, October 1996. Markowtz, H (1952). Portfolo Selecton. Journal of Fnance, March 1952. Mashayekh Beschloss, A., and Muraldhar, A.(1996). Managng the Implementaton Rsks of a Currency Overlay. Journal of Penson Plan Investng, Wnter 1996, 79-93. Frst draft: 16/12/96 7 pm Second Draft: 06/07/1999 3 pm hrd Draft: 13/07/1999 1 pm 21
Appendx 1 Data on Asset Classes Asset Standard Correlatons Classes Devaton USEQ NUSEQ EMEQ USFI NUSFI HY PE Cash US Equtes 15.0% 1.0 0.5 0.3 0.4 0.4 0.5 0.4-0.08 Non-US Equtes 19.5% 0.5 1.0 0.3 0.2 0.3 0.2 0.1-0.13 Emergng Equtes 23.3% 0.3 0.3 1.0 0.3 0.3 0.2 0.1-0.10 US Fxed Income 5.2% 0.4 0.2 0.3 1.0 0.4 0.3 0.0-0.02 Non-US Fxed Inc. 4.5% 0.4 0.3 0.3 0.4 1.0 0.3 0.0-0.05 Hgh Yeld 9.8% 0.5 0.2 0.2 0.3 0.3 1.0 0.0-0.07 Prvate Equtes 27.0% 0.4 0.1 0.1 0.0 0.0 0.0 1.0 0.00 Cash 1.0% -0.08-0.13-0.10-0.02-0.05-0.07 0.00 1.00 22
ABLES ABLE 1 ABSOLUE AND RELAIVE PORFOLIOS Asset Benchmark Actual Devaton Classes Portfolo Portfolo Portfolo (v) (w) (z) US Equtes 30.0% 32.0% 2.0% Non-US Equtes 35.0% 29.0% -6.0% Emergng Equtes 5.0% 8.0% 3.0% US Fxed Income 7.0% 9.0% 2.0% Non-US Fxed Inc. 10.0% 8.0% -2.0% Hgh Yeld 2.0% 4.0% 2.0% Prvate Equtes 10.0% 8.0% -2.0% Cash 1.0% 2.0% 1.0% otal 100.0% 100.0% 0.0% Standard Devaton 11.69% 11.37% 1.24% 23
ABLE 2 (a) - CONRIBUION O PORFOLIO SANDARD DEVIAION Asset Absolute Rsk Relatve Classes Benchmark Actual Portfolo (%) (%) (%) US Equtes 3.8% 4.1% -0.045% Non-US Equtes 5.8% 4.9% 0.783% Emergng Equtes 0.5% 0.8% 0.241% US Fxed Income 0.1% 0.2% 0.019% Non-US Fxed Inc. 0.2% 0.2% 0.001% Hgh Yeld 0.1% 0.1% 0.040% Prvate Equtes 1.2% 1.0% 0.203% Cash 0.0% 0.0% 0.000% otal Standard Devaton 11.69% 11.37% 1.24% ABLE 2 (b) - PERCENAGE CONRIBUION O PORFOLIO SANDARD DEVIAION Asset Absolute Rsk Relatve Classes Benchmark Actual Portfolo (% of otal) (% of otal) (% of otal) US Equtes 32.2% 36.2% -3.7% Non-US Equtes 49.7% 43.4% 63.1% Emergng Equtes 4.4% 7.4% 19.4% US Fxed Income 1.1% 1.5% 1.5% Non-US Fxed Inc. 1.6% 1.3% 0.1% Hgh Yeld 0.6% 1.3% 3.3% Prvate Equtes 10.5% 8.8% 16.4% Cash 0.0% 0.0% 0.0% otal 100.0% 100.0% 100.0% 24
ABLE 3 IMPLIED CORRELAION OF ASSE CLASS BE O PORFOLIO OF BES Asset Classes Relatve Portfolo US Equtes (0.152) Non-US Equtes (0.670) Emergng Equtes 0.343 US Fxed Income 0.179 Non-US Fxed Inc. (0.009) Hgh Yeld 0.206 Prvate Equtes (0.376) Cash 0.000 25
ABLE 4 (a) - COMPARING HE MEHODS BENCHMARK RISK Asset Assumng Margnal Margnal Proposed Classes Independence No-Rebalancng Rebalanced Method (% Contrbuton) (% Contrbuton) (% Contrbuton) (% Contrbuton) US Equtes 71.5% 40.67% -61.2% 32.2% Non-US Equtes 21.5% 37.7% -70.5% 49.7% Emergng Equtes 1.6% 5.9% 4.2% 4.4% US Fxed Income 1.8% 2.7% 92.7% 1.1% Non-US Fxed Inc. 0.1% 2.1% 70.8% 1.6% Hgh Yeld 0.5% 2.4% 27.8% 0.6% Prvate Equtes 3.0% 8.5% 8.8% 10.5% Cash 0.0% 0.0% 27.4% 0.0% otal 100.0% 100.0% 100.0% 100.0% otal Varance of Portfolo 0.60% 1.66% -0.26% 1.37% Standard Devaton 7.73% 12.90% N/A 11.69% ABLE 4 (b) - COMPARING HE MEHODS RELAIVE RISK Asset Margnal Proposed Classes No-Rebalancng Method (% Contrbuton) (% Contrbuton) US Equtes 45.3% -3.7% Non-US Equtes -34.6% 63.1% Emergng Equtes 35.3% 19.4% US Fxed Income -3.3% 1.5% Non-US Fxed Inc. -1.0% 0.1% Hgh Yeld 6.4% 3.3% Prvate Equtes 51.9% 16.4% Cash 0.0% 0.0% otal 100.0% 100.0% otal rackng Error 1.06% 1.24% 26