A UNIFIED FRAMEWORK TO ANALYZE CLASSICAL RISK MEASURES IN FINANCE

Transcription

1 A UNIFIED FRAMEWORK TO ANALYZE CLASSICAL RISK MEASURES IN FINANCE FRANCESCO M. ARIS* Unversty of Bresca *Departent of Quanttatve Methods. Contrada S. Chara Bresca. Italy. Tel ; Fax ; e-al: pars@eco.unbs.t Keywords: rsk easures, duraton, systeatc rsk, volatlty, rsk anageent JEL classfcaton: G0, G, G.

2 ABSTRACT Ths paper has the an objectve to show the functonal relatonshps lnkng alternatve rsk easures, frequently eployed n the fnance lterature, to each other. The varance or standard devaton of securtes rates of returns, the systeatc rsk, of a stock and the duraton of fxed ncoe securtes usually refer to dfferent knds of fnancal nstruents to easure alternatve knds of rsks (volatlty, arket rsk, nterest rate rsk. We show that all such ndcators can be ndfferently appled n coputng any knd of rsk ebedded n prtve nstruents and dervatves as well. Ths fact has portant plcatons for a nuber of practcal applcatons such as, for exaple, coputng the Value at Rsk or optzng the balance sheet aturty structure. The proved consstency aong alternatve rsk-easures allows to pleent hedgng strateges wth respect to a global rsk exposure, ndependently of the specfc sources of rsk.

3 3 -Introducton. The proble of defnng coherent rsk easures s certanly a hot ssue n atheatcal fnance. It conssts of deternng the atheatcal propertes akng a rsk-easure a relable rsk ndcator ndependently of the dstrbuton functon descrbng the rsk tself. It s long te, however, snce the fnance lterature, both theoretcal and eprcal, has borrowed fro statstcs ndces lke varance and standard devaton and ade up other coeffcents such as the systeatc rsk,, and the duraton. All these quanttes are eployed as rsk-easures but are appled by developng well separated theoretcal and eprcal odels and tradtonally refer to dfferent asset categores and alternatve rsk typologes. The rgorous separaton characterzng the use of alternatve rsk-easures s consstent wth the dea of dstngushng aong dfferent sources of rsks present n econoc and fnancal systes. Much later scholars and practtoners agreed upon the fact that pleentng an effectve rsk anageent strategy needs the defnton of a unque rsk-easure, referred to as the "global" rsk exposure. Ths s the sae dea underlyng the concept of Value at Rsk, whch s totally ndependent of the nature of the orgnal rsks. The effort of ths paper s to offer a theoretcal contrbuton showng how varance or standard devaton, systeatc rsk and duraton (to be nterpreted n the sequel of the paper as an asset s senstvty easure wth respect to nterest rate varatons can be dealt wth wthn a coon arket odel where any knd of fnancal asset s accounted for. Our attept to analyze alternatve rsk easures wthn a unfyng theoretcal approach has the objectve to pont out how the opportunty to copute the aount of rsk exposure prevals on the knowledge of the source orgnatng any specfc rsk whenever a rsk-anagng strategy has to be pleented. Furtherore, such a knd of analyss could have nterestng eprcal applcatons aed to test the consstency aong alternatve rsk-easures n dfferent arkets. The paper s organzed as follows. In secton the systeatc rsk coeffcent,, s expressed as a functon of duraton of fxed ncoe securtes payng known coupons. Secton 3 extends the sae analyss to other portant knds of prtve and dervatve securtes. Secton 4, after coputng the volatlty of an nvestent yeld as a functon of ts systeatc rsk, derves the Artzner & oth. [], Ebrechts & oth. [7] and Kast & oth. [] are recoended references on the topc.

4 4 relatonshp exstng between volatlty and duraton, or other equvalent easures, characterzng fnancal assets. Secton 5 derves Value at Rsk accordng to the paraetrc approach as a functon of the alternatve rsk-easures studed n prevous sectons. Secton 6 rephrases the proble of optzng the aturty structure of the balance sheet for a rsk averse entrepreneur by expressng the objectve functon n ters of systeatc rsk nstead of duraton, as done by rsan & Tan [7]. Fnally, secton 7 s devoted to soe concludng rearks. -The systeatc rsk as a functon of the bond-duraton. Assue that the rando rate of return of a fxed ncoe securty payng known coupons, labeled, s descrbed by a lnear arket odel forally defned by the followng equaton: R α + R + ε ( where R s the rate of return on the arket portfolo, representng the unque factor sultaneously nfluencng the perforance of all traded securtes n the arketplace. α s the expected value of securty s "dosyncratc" return, whle ε s ts resdual rate of return such that: E ( ε 0; cov( ε, R 0 and cov( ε, ε cov( R, R var( R systeatc rsk coeffcent whose value s gven by the followng well known forula: j j j. Fnally, s the ( R, R ρ σ ( R R σ ( R cov var (

5 5 where ρ and σ ( are the correlaton coeffcent between the rate of return on securty and the rate of return on the arket portfolo, and the standard devatons of the aforeentoned rates of return, respectvely. An alternatve rsk easure, frequently eployed n the analyss of fxed ncoe securtes, s the duraton, ntroduced by Macaulay [4]. Ths easure has a te denson and s expressed as follows: D H ( ( ( t t t t C y h + h h h th > t : (3 where ( ( t t h H h: th > t C h + y s the prce of securty at current te t, C h s the known aount (ether coupon or prncpal or both pad out by securty at te t h and y s the securty s yeld to aturty. Let us consder now the rando rate of return of a coupon bond wth known payents, t s : R (4 the prce varaton can be convenently expressed (by Taylor s forula as y y + t t. In case of a suffcently sall nterest rate varaton only the frst coponent of the bond prce change s affected; let us denote t as follows: R t t y y. Fro (3 and the bond s prcng forula, the followng Notce that equaton (4 assues an edate renvestent nto the sae securty of any nteredate payent eventually atured.

6 6 equalty s edately verfed: y D ( + y replaced wth the followng one:. Therefore, equaton (4 can be R D ( + y y MD y (5 where MD D ( + y s the odfed duraton. Substtutng (5 nto ( and applyng the propertes of covarances to the yeld varatons, assued to be rando, enables us to express the systeatc rsk coeffcent as a functon of the odfed duraton as expressed by the followng equaton: MD cov ( y, y ( y (6 var where y can be nterpreted as the arket yeld, coputed as a functon of the cash flows pad out by all traded fxed rate bonds. Equaton (6 shows how the arket rsk characterzng bond s crtcally dependent upon the nterest rate rsk, as easured by the bond s odfed duraton. 3-Extendng the systeatc rsk-duraton relatonshp to other knds of securtes. The purpose of ths secton s to explore how the analyss developed n the prevous one wth respect to coupon bonds wth known payents can be adapted to alternatve categores of fnancal assets.

7 7 Coon stocks. One of the ost popular prcng odels of coon stocks s the constant perpetual growth odel. It s based on two assuptons: a the share of net operatng ncoe devoted to nvestent, Η, s constant over te; b the expected yeld on corporate nvestents,, s constant. Accordng to such hypotheses the current prce of a coon stock s: NOI ( Η k Η (7 where NOI s the net operatng ncoe at current te t and k s the rsk-adjusted cost of the equty captal of the fr ssung the stock. Gordon wrtes equaton (7 n an alternatve but equvalent way by explctly defnng two quanttes whch are the current dvdend, dv NOI ( Η, and the constant growth rate, gη. Thus, (7 becoes: dv (8 k g Accordng to Gordon s forula the duraton of the stock s 3 : D + k k + k k g (9

8 8 Expandng (8 accordng to the Taylor s forula lted to the ters of frst order, gets 4 : ( ( ( ( k g k g k dv g g k dv k k g g usng (8 and (9 the last expresson, excludng the resduals, splfes as follows: ( k D g D k + therefore: ( ( k g MD k g k D + (0 Now, substtutng (0 nto ( and assung stochastc varatons, we are able to express the stock s systeatc rsk as a functon of the odfed duraton: ( ( [ ] ( R R k R g MD var, cov, cov ( where R s the rate of return on the arket portfolo. Equaton ( shows how the relatonshp between the undversfable rsk and the odfed duraton of coon stocks depends on the 3 Accordng to the Gordon odel a gven varaton of the cost of equty captal has the sae effect as an equvalent and opposte varaton of the growth rate. Such a fact ples that the stock duraton s not a good easure of the stock senstvty, to the extent that k and g are soehow correlated.

9 9 covarances of the arket rate of return wth the cost of equty captal, k, and the fr s growth rate, g, separately consdered. Indexed bonds. One of the splest ways to prce ndexed bonds s to let ther value depend on the dynac behavour of a sngle state varable whch s the nstantaneous rsk-free rate of nterest, r(t. Such a dynac can be descrbed by the followng stochastc dfferental equaton: ( r, t dt + b( r tdz dr a, ( where a and b are functons of te and of the current value of the nstantaneous nterest rate, whle z s a standard Wener process. Let us now consder the bond prce, represented by the functon ( r t, dependng on the nstantaneous rsk-free rate and te. Accordng to the analyss developed n De Felce & Morcon [5], Ito s Lea can be appled to ( n order to derve ts dynacs 5 : d µ dt + σdz the dsperson coeffcent nto the last equaton s gven by σ b Ωb, where the quantty r Ω r log r easures the bass nterest rate rsk assocated to the bond, has a tedenson and s equvalent to the Macaulay duraton wthn a stochastc odel n contnuous te. Therefore, the dynacs of the bond prce can be expressed n ters of the bass rsk: 4 Let us pont out the fact that accordng to the assuptons underlyng Gordon s odel, varatons ether n the cost of captal or n the growth rate are the result of unexpected exogenous shocks, after whch these varables are stll assued to be constant.

10 0 d µ dt + Ω bdz (3 slarly to what done n secton, we now solve for the rate of return of the ndexed bond, whch s: R ( µ dt + Ω bdz and sultaneously denote wth z the standard Wener process characterzng the bond arket ndex 6. Fnally, t s easy to show that the systeatc rsk coeffcent can be expressed as a functon of the bass nterest rate rsk of the bond, sply followng the sae procedure appled to obtan expresson (6. Under the hypothess of dscrete varatons of the bond prce, the result s as follows: Ω b cov ( z, z ( z (4 var Fro (6 and (4 the systeatc-nterest rate rsk relatonshp for plan vanlla nterest rate swaps can be deterned, condtonal on the assupton that the floatng coponent of the contract can be prced accordng to the odel presented n ths secton 7 : 5 For the odel s detals see De Felce & Morcon [5]. 6 The obvous underlyng assupton s that the arket ndex follows a stochastc process structurally equvalent to those assocated to the sngle bonds. 7 Equaton (5 can be easly adapted to the case of a fxed vs. floatng currency swap under the assupton that the spot foregn exchange rate follows an Ito process equvalent to the one characterzng the nstantaneous rsk-free rate. The new expresson s: MD ρ yσ ( y ( z Ω b ρ rσ ( zr ( z Ω b ρ sσ ( z s ( z r r s s where all the varables σ σ σ preserve ther eanng referred to the nstantaneous rsk-free rate, r and the spot foregn exchange rate, s, respectvely.

11 MD ρ yσ ( y ( z Ω bρ zσ ( z ( z (5 σ σ where ρ.. s the correlaton coeffcent characterzng the par of varables consdered. 3 Optons. It s well known that opton contracts are characterzed by convex payoffs and non-lognoral prce dstrbutons. Because of these reasons a systeatc rsk coeffcent based on the tradtonal assuptons underlyng the lnear arket odel and the CAM cannot be defned wth respect to such nstruents. Nevertheless, t s eanngful to express the dependence of the opton prce, whch s a dervatve securty, on the behavour of the arket for the underlyng asset and, as a consequence, on the senstvty of the underlyng tself wth respect to unexpected changes of nterest rates. In order to capture such a dependence an opton- coeffcent as a functon of the coeffcent assocated to the underlyng asset wll be defned n what follows. 3. Call opton. Let us start defnng the prce of a standard call opton (ether European or Aercan as the functon C C( S; t ; ts nstantaneous yeld, accountng for the jont effect of the varables S and t, can be expressed through the followng chan of equaltes: ln C ln C ds r C + t S dt C C C + t S ds dt

12 leadng to the equaton: r ( Θ + (6 C C Sr S ds where r S s the nstantaneous yeld of the call s underlyng and Θ and are the partals of S dt the call preu wth respect to te and the underlyng asset, respectvely. Let us now defne the of the call opton: C cov var ( rc, r ( r where r s the nstantaneous arket yeld. By substtutng (6 nto the last expresson the followng equaton s easly derved: C Ξ C S where C S Ξ C s the elastcty of the call prce wth respect to the underlyng. S C Copeland & Weston [4] and ars [5] show that such an equalty can be alternatvely wrtten, by applyng the results of the Black & Scholes [] analyss, as follows: C S (7 r ( T t ( K S e [ N( d N( d ]

13 3 where the varables at the denonator of (7 are those of the Black & Scholes opton prcng forula. Notce that K < S when the call s n the oney, r( T t e < and ( d N( N. The last d three nequaltes ply that C S when S s postve and C S < when S s negatve. lese note that the denonator of (7 s always postve. In fact ( T t S N( d N( d 0 r Ke > s r( T t equvalent to SN( d Ke N( d 0 > whch s always verfed before the call s expraton, the left hand sde of the nequalty beng the Black & Scholes forula. By substtutng nto (7 S wth the approprate quantty aong those derved n ths secton 8 the call systeatc rsk can be expressed n ters of the duraton or bass nterest rate rsk of the underlyng asset ut opton. The sae knd of analyss developed wth respect to the call opton can be extended to the European put 0 to express the opton senstvty as a functon of the systeatc rsk assocated to the underlyng asset. The result s: Ξ S where Ξ denotes the put elastcty wth respect to the asset on whch the opton has been wrtten. The Black & Scholes results can be stll evoked to rearrange the last equalty as follows : 8 The choce of the approprate quantty obvously depends on the nature of the opton-underlyng. 9 Notce that our analyss s very general. Therefore, t has to be adapted to partcular stuatons such as the case of a call wrtten on a bond. We know, n fact, that the volatlty of a bond s not proportonal to the current prce of the bond tself, as suggested by the geoetrc Brownan oton here assued to descrbe the call-underlyng s dynac. 0 Notce that the results presented n ths secton cannot be extended to the case of an Aercan put, sply because no closed-for soluton s currently avalable to copute ts value. Reeber that equaton (8 s based on the well known property: N ( d ( N d.

14 4 ( S (8 r T t ( K S e [ N( d N( d ] Furtherore, let us consder that ( T t S N( d N( d 0 r Ke < because t s equvalent to say that the quantty SN( d KN( d, whch s the opposte of the Black & Scholes forula for the put prce, s negatve. Ths s always the case before the put s expraton. Ths fact ples that s negatve when S s postve and vce versa. The last result s perfectly consstent wth the theory of opton prcng evdencng that the put s senstvty s the opposte of the one characterzng the underlyng. The -coeffcent of the put can be expressed as a functon of the duraton characterzng the opton-undelyng by substtutng n (8 the approprate for of S. 4 Futures. The followng equaton can be easly derved wth respect to futures contracts by applyng the sae theoretcal approach already developed to deal wth optons: F Ξ F S The current futures prce s gven, as we know, by the followng quantty: F ( T t r Se, whle the futures elastcty wth respect to the spot prce s Ξ F F S S F. The last two equatons edately ply that F S e r ( T t F S and Ξ F and the an result follows: F S (9

15 5 Equaton (9 allows to apply to futures contracts the sae forulas already derved wth respect to the varous knds of prtve assets consdered n ths work. 4-Duraton and volatlty. The relatonshp between systeatc rsk and duraton wdely dscussed n the prevous paragraphs enables us to nterpret the duraton as a quantty affectng alternatve volatlty easures (.e. varance and standard devaton referred to the yeld of fnancal assets. The well known propertes of the lnear arket odel, thoroughly dscussed by Elton & Gruber [6], lead to the followng expresson for the varance of securty : σ σ + σ ε (0 By substtutng the coeffcent nto equaton (0 wth the approprate expresson aong those derved n sectons and 3 the volatlty as a functon of duraton can be copletely specfed. Nevertheless, the task of expressng the yeld-volatlty as a functon of duraton can be accoplshed ore drectly by defnng the duraton tself as the "seelastcty" of the bond prce wth respect to the dscount rate. The specfc structure of the fnancal asset and the stochastc nature of the rsk-factor can be adequately consdered by applyng such a lne of reasonng. The nstantaneous yeld of a bond prced applyng Ito s Lea to the dynacs of the nstantaneous rsk-free rate s gven by the followng, well known stochastc dfferental equaton: The followng equalty: σ σ, edately derved fro ( s an alternatve to equaton (0. It can be ρ sad that equatons ( and (0 have the sae eanng by recognzng that the hydosncratc volatlty of a yeld decreases for ncreasng values of the correlaton coeffcent.

16 6 d ( r, t dt σ ( r, tdz µ + ( where σ ( r, t, b s the dsperson of the nstantaneous bond-yeld due to an unexpected r change of the rsk-free rate, r. The quantty Ω s the seelastcty of the bond-prce r wth respect to the nstantaneous rsk-free rate and s coonly referred to as the bass rsk of the fxed ncoe securty. Therefore, the followng equalty holds: ( r, t Ω b σ ( accordng to ( the dsperson coeffcent characterzng the dynacs of the nstantaneous bond yeld depends on the bass rsk assocated to the securty, together wth the unexpected varatons of the stochastc factor (the local rate of nterest coon to all the bonds traded n the arket. Let us consder a bond payng known coupons at regular te ntervals; applyng contnuous d copoundng the followng equalty s verfed: Ddδ, where δ denotes the contnuously copounded yeld to aturty of the whole bond, assued to be constant. It follows that s: D. Consequently, equaton ( ebodes the bond s duraton n the followng way: δ ( r, t D b σ (3 Equaton ( s based on a rsk easure whch can be adapted to ndexed bonds and other knds of securtes. Such a quantty has the advantage to preserve the sae theoretcal eanng of duraton. As an exaple equaton ( can be extended to the case of a dervatve securty payng a stochastc payoff at expraton, T. In ths case the quantty Ω can be nterpreted as the dervatveprce ntensty of varaton wth respect to unexpected changes of the rsk-factor. The new equaton s 3 : 3 Equaton (4 assues the prce of the underlyng asset to follow an Ito process of the knd ds µ s dt + σ s dz. t t

17 7 σ d ~ d, ~ (4 s d s ( s t, T sσ s Ω ssσ s s where s and σ s are the prce and the nstantaneous volatlty of the underlyng asset observed at current te t and d s the current dervatve prce (.e. futures, opton. As far as coon stocks are concerned, an explct relatonshp between ther volatlty and odfed duraton can be nterestngly derved fro the constant growth prcng odel ntroduced n secton 3. The followng equaton s verfed sply assung a dscrete and unexpected varaton of the stock prce to be consstent wth the dynacs descrbed n footnote 3: S S MD ( g k St + σsz µ (5 leadng to: z MD( g k µ t varatons 4 : σ and, akng explct the dvdend and dscount rate {[ ag α( k ] t + ( χg x z} t σ z MD t µ (6 assung the ters n t equal to 0, as pled by the Gordon odel, the fnal result s: ( g x σ MD χ (7 Equaton (7 shows that the nstantaneous volatlty paraeter of the stock prce s proportonal to the odfed duraton of the stock. Such a result s copletely consstent wth equaton (0. 4 Equaton (6 s based on the assupton that the stock-dvdend and the dscount rate varatons are subject to the sae source of rsk. The sae result holds under the ore realstc assupton that the rsks drvng the two dynacs are perfectly correlated.

18 8 5-Frst applcaton: the Value at Rsk. Value at Rsk (VaR gves a sple and very general ndcaton of the rsk exposure characterzng an nvestent over an assgned te-nterval wth respect to a prespecfed confdence level. Its aount denotes the axu expected loss fro the nvestent, wth respect to a gven te-horzon for an arbtrary confdence percentle. Joron [] dscusses n depth the paraetrc approach to the VaR calculaton and presents an explct VaR forula related to the specfc nature of the fnancal asset consdered (.e. ether prtve or dervatve, wth a lnear or a convex payoff. Such an expresson s edately applcable to assets wth norally dstrbuted yelds; t can be extended, however, under suffcently general assuptons, to other knds of dstrbutons. Accordng to the analyss developed by Joron the VaR of a prtve nstruent s: VaR W 0 ασ t (8 where W0 s the wealth ntally allocated to the nvestent, α s a paraeter expressng the devaton fro the ean, whose value depends on the for of the dstrbuton at hand, σ s the volatlty of the nvestent yeld whle t s the assgned te-nterval. Equaton (8 changes f we consder a dervatve contract wth a lnear payoff, such as a futures contract, for exaple: VaR sασ t (9 n (9 s the frst dervatve of the futures prce wth respect to the spot prce of the underlyng nstruent. Let us fnally consder an opton contract as an exaple of a dervatve asset wth a nonlnear payoff (the payoff s convex n ths case. The aount of VaR can be now approxated by the followng quantty:

19 9 VaR ( Γ s α s σ + σ (30 where and Γ are, respectvely, the frst and second dervatves of the opton preu wth respect to the value of the underlyng asset. These forulas of VaR can be easly odfed accordng to the analyss developed n ths paper. Equaton (8 can be adapted to the case of a fxed-coupon bond by expressng the volatlty paraeter as a functon of the systeatc rsk and the systeatc rsk, n turn, as a functon of duraton. The new expresson s: ( y MD cov( y, y t W0α ρ σ ( y ρ σ VaR W0 α t (3 Alternatvely, applyng equaton ( leads to the followng result: VaR W 0 αωb t (3 As an exaple of a dervatve nstruent wth a lnear payoff the nterest rate swap can be consdered. In ths case equaton (9, based on equaton (5, transfors as follows: [ MD ρ σ ( y + Ω bρ σ ( z ] y VaR W0 α t (33 ρ z Another case of lnear dervatve contract s the futures, whose VaR, accordng to equaton (4 too, s: VaR sα Ω b t σ (34 s d d

20 0 Fnally, as to the opton contracts, equaton (30 can be transfored nto one of the followng expressons, dependng on the theoretcal background consdered between those presented n sectons 3 and 4, respectvely: ( y, y ( y ρ ( y y ( y ρ MD cov cov, MD VaR α s + Γs σ σ (35 VaR ( Γs Ω α s Ω sb + sb (36 Equatons fro (3 to (36 clearly show how the VaR easure can be expressed ndfferently as a functon of each of the rsk ndcators currently used n fnance, volatlty, and duraton. The reason s sply the fact that VaR s a synthetc rsk easure, ndependent of the specfc nature of the rsk consdered. The sae knd of approach can be extended to the case of portfolos of assets belongng to alternatve categores. The specfc results wll depend on the partcular ethodology adopted to copute the paraetrc VaR of coplex postons. 6-Second applcaton: optzng the aturty structure of the balance-sheet. Several studes on duraton and unzaton 5 nvestgated how the assets and labltes aturtes of a corporate balance sheet are related to unexpected changes of nterest rates and how they affect, n turn, the fr value. Other authors, see Grove [9] and rsan & Tan [7] aong others, derved the optal aturty structure of the balance sheet for a rsk-averse entrepreneur, defnng the condtons under whch the balance sheet unzaton represents the optal soluton. Grove [9] defnes the functon to be axzed to attan the optal aturty structure of the balance sheet, based on the assupton of rando parallel shfts of a flat ter structure. The control varable s the Macaulay duraton and the objectve functon s: 5 See, aong others, Fsher & Wel [8], Hcks [0], Redngton [8] and Sauelson [9].

21 F b 0 ( DAt u[ W + h( LDL ADA ] f ( h a 0 dh (37 where u:u >0 and u"<0, s the entrepreneur s utlty functon, W0 s the ntal dfference between assets and labltes, A and L are the assgned values of assets and labltes, respectvely, D 0 L s the value of the lablty-duraton, whch s gven for a specfc ter structure. DA s the control varable, h s the sze of the rando addtve shock ncurred by the ter structure, whose densty functon s f(h. Fnally, a and b are the lower and upper bounds of the sze of the shock tself. Accordng to the results derved n secton, equaton (37 can be rewrtten as a functon of a ore generc rsk ndcator such as the systeatc rsk coeffcent. The assets undversfable rsk s now the new control varable and the new objectve functon s: ( + y σ ( h ~ ~ [ A σ ( R L σ ( R ] f ( hdh b h F( A u W 0 + A t L0 0 a ρ (38 h derved by substtutng n (37 the assets duraton wth ts value as expressed by rearrangng equaton (6. Equaton (38 s perfectly consstent wth Grove s odel and has the advantage to depend on arket rsk whch certanly expresses a wder concept of rsk copared to the nterest rate rsk easured by duraton. Dervng Grove s results applyng the new rsk easure would be an nterestng exercse. Grove s odel s extended by rsan & Tan [7]. They optze the balance sheet aturty structure referrng to a generc ter structure, not necessarly flat, subject to rando shocks whch can even be non-parallel. Under such hypotheses the ter structure of nterest rates can be approxated by a polynoal of degree J (naely the Taylor forula appled to the bond prce and stopped at the ter of degree J, whle the objectve functon to be axzed nvolves J dfferent senstvty easures, usually referred to as J th order duratons. rsan & Tan defne the followng optzaton proble:

22 ax E u ω T s. t. ω ω ω 0 T T [ L ( ω + L ( ω Σ D( L α] { } (39 where L0 s the te 0 value of the unque lablty reported nto the balance sheet, ω 0 0 A L and A0 are the ntal total assets. ( T A ω ω, ω,, ω n where ω 0 x L s the share of the ntal 0 lablty nvested nto the th asset, whch s a bond. Σ s the n J atrx whose row has the J duraton easures related to the th bond as coponents, D ( L s the colun-vector wth J coponents representng the J duraton easures characterzng the lablty. α [ ] the j th α j 0, where α j s coeffcent of the polynoal structure used as an approxaton of the ter structure varatons, s an n-densonal colun-vector wth unt coponents and D H j r ( t th th Che j h s the j th coponent of the th row of Σ. In other words t s the j-order duraton of the th asset wrtten nto the balance sheet. roble (39 s based on the assupton that the shock of nterest rates nstantaneously occurs after t0 and s expressed by the functon h( t α [ h( t, α t] J, α j t j j. It follows that the functon exp ebodes the effect of the unexpected shock of nterest rates nto the fr s assets and lablty valuaton forula. The McLaurn expanson s appled to such a functon wth respect to the α s, yeldng: exp j [ h( t, α t] + α t J j j Based on such an analyss, the sae optzaton proble can be defned as a functon of the arket nstead of the nterest rate rsk. Let us observe that equaton (40 s satsfed for a suffcently sall h: d J j α j MD r (40 j

23 3 By substtuton of (40 nto (5 the rando yeld of the th bond s defned as a functon of the j th duraton easure. The correspondng systeatc rsk coeffcent, such a quantty: j, can be easly derved fro αmd j ρ rσ ( r ( r j (4 σ equaton (4 edately leads to the followng result 6 : D j s (4 j where ( r ( + r σ ( r σ s αρ r. We are able now to rephrase proble (39 wrtng the objectve functon n ters of the arket rsk as follows: σ ( r ( r where s + T T [ L ( ω + L s ( ω Β ( L ] { u } ax E (43 ω, whle Β s the n J atrx whose ρ σ th row has the J systeatc rsk r ( r coeffcents assocated to the th asset as coponents and ( L s the J-densonal colun-vector of systeatc rsk coeffcents related to the unque lablty. The constrants are obvously unchanged. Once agan, the rsan & Tan results too, could be verfed on the bass of our odfcatons to ther orgnal odel. 7-Conclusons. 6 Wth respect to the unque lablty equaton (4 transfors as follows: D j L s j L

24 4 Ths paper s a frst attept to deal wth rsk easures tradtonally eployed n fnance (e.g. varance or standard devaton, systeatc rsk coeffcent and duraton, to be nterpreted as the senstvty of the asset prce to shfts of the nterest rate curve wthn a unfed theoretcal fraework. The an objectve s to show that rsk ndcators usually referrng to dfferent assets and rsk categores are, n fact, related to one another n a way allowng the to be analyzed wthn a hoogeneous arket odel. Such a result s not trval at all and has portant plcatons. Frst of all t allows for ore flexblty n applyng atheatcal odels to rsk-anageent, as shown n sectons 5 and 6. The sae proble, n fact, can be properly defned usng ndfferently alternatve varables, dentfable wth the dfferent rsk-easures. Secondly, t ponts out how the aount of rsk exposure plays a pronent role n pleentng a rsk anageent strategy wth respect to the qualty (.e. the source of rsk. Market, nterest rate and credt rsks can be coputed applyng the sae rskeasure whose structure ust be clearly understood. A wde range of theoretcal lterature extensvely explaned the lts of our analyss f referred to rsks related to non-norally dstrbuted rando varables 7. Even though we are aware of the fact that such a lne of research wll lead to rearkable and useful results, we argue that the classcal approach developed n our study wll be stll followed n practcal applcatons for a whle. The reasons are that t s easy to pleent and to extend to the cases of "non-noral" rsks, even f the results derved under the assupton of non-noralty of the probablty dstrbuton ust be prudently consdered as an approxaton of the true values. All we can say s that ths paper contrbutes to show the consstency of alternatve "classcal" rsk-easures wthn the fraework of a coon arket odel. 7 See Ebrechts & oth. [7], aong others, for a thorough dscusson of ths proble.

25 5 REFERENCES [] ARTZNER., DELBAEN F., EBER JM., HEATH D. Coherent easures of rsk, preprnt, 998. [] BLACK F., SCHOLES M. The prcng of optons and corporate labltes, Journal of oltcal Econoy, n.8, 973, [3] BOQUIST J.A., RACETTE G.A., SCHLARBAUM G.G. Duraton and rsk assessent for bonds and coon stocks, The Journal of Fnance, n.30, 975, [4] COELAND T.E., WESTON J.F Fnancal theory and corporate polcy, 3 rd edton, Addson Wesley, Readng (Ma, 988. [5] DE FELICE M., MORICONI F. La teora dell unzzazone fnanzara, Il Mulno, Bologna, 99. [6] ELTON E.J., GRUBER M.J. Modern portfolo theory and nvestent analyss, 5 th edton, John Wley & Sons, New York (NY, 995. [7] EMBRECHTS., MCNEIL A., STRAUMANN D. Correlaton and dependency n rsk anageent: propertes and ptfalls, preprnt, 999. [8] FISHER L., WEIL R.L. Copng wth the rsk of nterest rate fluctuatons: returns to bondholders fro nave to optal strateges, Journal of Busness, n.44, 974, [9] GROVE M.A. On duraton and the optal aturty structure of the balance sheet, Bell Journal of Econocs and Manageent Scence, n.5, 974, [0] HICKS J.R. Value and captal, nd edton, The Clarendon ress, Oxford, 946. [] KAST R., LUCIANO E., ECCATI L. VaR and optzaton, preprnt, 999. [] JORION. Value at rsk: the new benchark for controllng arket rsk, McGraw-Hll, New York, 997. [3] LANSTEIN R., SHARE W.F. Duraton and securty rsk, Journal of Fnancal and Quanttatve Analyss, roceedngs ssue, 978, [4] MACAULAY F.R Soe theoretcal probles suggested by the oveents of nterest rates, bond yelds and stock prces n the U.S snce 856, Natonal Bureau of Econoc Research, New York (NY, 938.

26 6 [5] ARIS F.M. Gl Effett della quotazone sul costo del captale d rscho, Workng aper, Departent of Quanttatve Methods, the Unversty of Bresca, n.6, 990. [6] ARIS F.M, ZUANON M. Eleent d fnanza ateatca, CEDAM, adova, 999. [7] RISMAN E.Z., TIAN Y. Duraton easures, unzaton and utlty axzaton, Journal of Bankng and Fnance, n.7, 993, [8] REDINGTON F.M. Revew of the prncples of lfe offce valuatons, Journal of the Insttute of Actuares, n.78, 95, [9] SAMUELSON.A. The effect of nterest rate ncreases on the bankng syste, Aercan Econoc Revew, n.35, 945, 6-7.