Real Estate Equity Deivatives Geltne Mille 2 nd Edition Chapte 26 Section 26.3 Real Estate Deivatives (Index Retun Swaps)
Real Estate Equity Deivatives A deivative is an asset whose value depends completely on the value o anothe asset (o a combination o assets). e.g., Stock options. Cuently, pivate equity R.E. deivatives oeed ae essentially utues contacts: No cash changes hands up ont ( notional tade amt. ) The majo poducts ae swaps : e.g., Swap NPI etun o a ixed etun each quate.
Poducts Cuently Oeed NPI Appeciation Swap o Fixed NPI Total Retun Swap o Fixed NPI Popety Type Total Retun Swap Simila poducts on IPD in U.K.
How Deivatives Wok: An Example... Littleton Fieman s Fund is a Pension Fund that wants to invest in diect (pivate) eal estate o potolio divesiication But Littleton is small: They ace high tansaction costs and/o low divesiication within eal estate (ew popeties: noise, basis isk ); and Littleton is woied about lack o pivate R.E. liquidity (0 yea investment?!?...) Southen State Teaches is a lage pension und, aleady invested in eal estate. Southen inds itsel ove-invested in R.E. due to denominato eect (stock mkt decline puts them ove-taget in R.E.). Southen hates to sell any o thei individual popeties because they like these popeties and they hate to incu the high tansactions cost o sale; but They need to educe thei exposue to the R.E. asset class in thei potolio.
How Deivatives Wok: An Example... Both paties can beneit om the NCREIF Appeciation Swap: Littleton takes the long position (swaps ixed etun o NPI appeciation etun). Southen takes the shot position (swaps NPI appeciation etun o the ixed etun). Southen pays Littleton (shot pays long) the NPI appeciation etun ( loating leg ) on the notional tade amount each quate. Littleton pays Southen (long pays shot) the ixed leg (spead) on the notional tade amount each quate. Net cash owed is settled at the end o each quate when the NPI is epoted, o duation o swap contact (typically 2-3 yeas).
Suppose Littleton & Southen agee on a two yea contact to tade a notional amount o $00 million at the end o 2005, with a ixed leg (spead) o 00bps: No cash changes hands at end o 2005Q4. Suppose 2006Q NPI appeciation etun is 2.5%, then: Southen owes Littleton (shot owes long).025*$00 = $2,500,000; Littleton owes Southen (long owes shot).0*$00 = $,000,000; They settle net cash low: Southen pays Littleton $,500,000. Suppose 2006Q2 NPI appeciation etun is then negative.0%: Southen owes Littleton -.0*$00 = -$,000,000 (i.e., Littleton owes Southen $,000,000); Littleton still owes Southen anothe $,000,000 on the ixed spead (as always); They settle net cash low: Littleton pays southen $2,000,000. This pocess continues though 2007Q4. How Deivatives Wok: An Example...
How Deivatives Wok: An Example... Potolio isk impact on Littleton (long position): Gains isk eect o $00 million woth o eal estate investment; Loses isk eect o $00 million woth o iskee bond investment. i.e.: They ve swapped $00 million o iskee bond isk o $00 million o pivate R.E. isk (because vitually all o the quately isk in R.E. investment is in the appeciation etun).
How Deivatives Wok: An Example... Potolio isk impact on Southen (shot position): Gains isk eect o $00 million woth o iskee bonds; Loses isk eect o $00 million woth o pivate R.E. investment (like NPI). i.e.: They ve swapped $00 million o eal estate isk o $00 million o iskee bond isk (i.e., they ve eliminated $00 million woth o R.E. isk exposue, again because vitually all o the quately isk in R.E. investment is in the appeciation etun, and the ixed spead is iskless).
How Deivatives Wok: An Example... Littleton (long position) coves thei exposue to the ixed spead by holding $00 million o iskee bonds in thei potolio. Southen (shot position) coves thei exposue to the loating leg (NPI appeciation etun) by holding $00 million o eal estate (simila to NCREIF popeties) in thei potolio.
Deivatives o Potolio Balance How Littleton might have aived at thei $00 million long puchase: Littleton peviously had total potolio $300 million invested 50%/50% Stocks & Bonds. They want to move to equal shaes Stocks, Bonds & Real Estate (o divesiication): Fist Littleton sells $50 million in Stocks & invests poceeds in bonds, so: Now Littleton has $00 million in Stocks (= taget) and $200 million in bonds (= taget $00 million). Bond investment ove taget is invested in iskee bonds ($00 million to cove ixed spead in R.E. deivative). Next Littleton buys $00 million long position in R.E. appeciation etun swap (equies zeo cash investment). Littleton now eectively has isk exposue like $00 million each in Stocks, Bonds, Real Estate, although actually still owns $200 million in bonds.
Deivatives o Potolio Balance Littleton could accomplish this same esult by buying $00 million woth o popeties o pivate R.E. investment unds. Howeve: Tansaction costs & management ees might be highe than deivative ees. Eectively ewe numbe o popeties (even in a und) would add noise and/o basis isk compaed to deivative that tacks NPI benchmak. Thee might be less liquidity o a longe hoizon ixed commitment (less investment lexibility), cetainly with diect popety investment, possibly with und investment (depending on type o und).
Anothe consideation: Deivative Risk vs Popety Risk om a Potolio Pespective Fo popety etuns to be as liquid as deivative etuns, popety etuns must be based on tansaction pices. Deivative etuns ae based on an appaisal-based index (at least in the case o NPI & IPD), which might have moe avoable isk chaacteistics (due to smoothing ). (This consideation is bette o the long position than the shot.)
How Southen might have aived at thei $00 million shot puchase: Pevious Southen Potolio: Stocks: $3333 M (55%) Bonds: $88 M (30%) R.E.: $ 909 M (5%) Total: $6060 M (00%) Ate 20% Stock Cash, Potolio: Stocks: $2667 M (49%): Δ - $666 M Bonds: $88 M (34%): Δ 0 R.E.: $ 909 M (7%): Δ 0 Total: $5394 M (00%): Δ - $666 M Southen is now below-taget in Stocks, above in bonds & R.E. (aka denominato eect ) Can coect (ebalance) by: Puchase (Sale): Stocks: $300 M (% o pevious) Bonds: ($200 M) (% o pevious) R.E.: ($00 M) (% o pevious) Total Net:$ 0 M (0% o pevious) Southen can accomplish this by... Deivatives o Potolio Balance New Potolio: Stocks: $2667 300 = 2967 M (55%) Bonds: $88-200 = 68 M (30%) R.E.: $ 909-00 = 809 M (5%) Total: $5394 0 = 5394 M (00%)
Puchase (Sale): Stocks: $300 M (% o pevious) Bonds: ($200 M) (% o pevious) R.E.: ($00 M) (% o pevious) Total Net:$ 0 M (0% o pevious) Southen can accomplish the above by: Shot $00 M R.E. Deivative (0 cash low); Deivatives o Potolio Balance New Potolio: Stocks: $2667 300 = 2967 M (55%) Bonds: $88-200 = 68 M (30%) R.E.: $ 909-00 = 809 M (5%) Total: $5394 0 = 5394 M (00%) Cove R.E. shot by eamaking $00 M woth o R.E. (like NCREIF) to cove loating leg (R.E. appeciation), theeby educing R.E. isk exposue to $809 M; Shot $00 iskless bonds (T-Bond utues mkt) ($00 M cash low), coveed by R.E. Deivative ixed spead, so no impact on potolio isk exposue; Sell $200 M bonds ($200 M cash low), educing bond exposue to $68 M; Use esulting $300 M cash low to puchase stocks, binging exposue to $2967 M. Without having to actually sell any popeties.
Deivatives o Potolio Balance Southen could take a moe taditional appoach o simply boowing an incemental $00 million against thei R.E. potolio. Howeve: Covenants o estictions may pevent such boowing; Boowing tansaction costs and ees may exceed those o deivative; Inteest ates may make boowing NPV < 0 tansaction o tax-exempt investo (maginal boowe in debt mkt is taxed); And anyway this will not actually poduce the taget isk & etun allocation
Deivatives o Potolio Balance The esult o simply boowing $00 million against thei R.E. potolio may educe the eal estate equity on Southen s books to $809 million, but it inceases the leveage o thei eal estate, theeby etaining the isk and etun impact o the ull $909 million eal estate asset holding in the potolio. The boowing is like shoting bonds, theeby negating anothe $00 million o bond investment in the potolio isk/etun poile, esulting in the ollowing eective potolio allocation: New Potolio: Stocks: Use poceeds to buy stock: $2667 300 = 2967 M (55%) Bonds: Sell $200 M woth: $88-300 = 58 M (28%) R.E.: Boow $00 M like shoting bonds: $ 909-0 = 909 M (7%) Total: $5394 0 = 5394 M (00%) The only way to poduce the taget esult without the use o the deivative is by actually selling $00 million woth o Southen s R.E. popeties.
Othe Deivative Poducts The Popety Secto Swap Swap total etun o one NPI sub-index o total etun o anothe plus/minus a ixed leg (that might equal zeo). Useul o potolio ebalancing. To bette match benchmak (educe a type o systematic basis isk ). To speculate (make a bet) on one secto whee you eel you have supeio knowledge.
Deivatives o Hedging & Speculation Pevious example showed use o deivative o potolio balancing o taget allocation puposes. Thee is anothe majo use o deivatives: Speculation & Hedging
Using Deivatives to Make Money in a Down Maket Suppose you think the eal estate maket (& NPI) is headed down. You stand to lose money, even though you ae a good eal estate asset manage. You can use the shot position in the NPI appeciation o total etun utue to continue to make money in the down maket
Using Deivatives to Make Money in a Down Maket Example: You think NPI appeciation will be negative %/qt ove next 2 yeas. You eel you have positive alpha (supe-nomal etun due to supeio asset mgt). O it could simply be you have lowe-isk (less cyclical) popeties. (This would be beta ). So you appeciation will be only negative 0.5%/qt ove the same peiod. How can you use the swap contact?...
Using Deivatives to Make Money in a Down Maket Say you shot $00 million o NPI appeciation utues. I you expectations ae bon out, you will eceive $ million pe qt om the deivative, plus the ixed spead (which howeve will pobably be negative, hence paid by you). You lose only about $500,000 pe qt on you popety evaluations, leaving you net positive by $500,000 beoe counting the ixed spead. I the ixed spead is geate than negative 50 basis-points, you will make money on popety appeciation in a down maket! (You o couse also still have you popeties cash yield.)
Even i you don t have positive alpha, shoting the appeciation deivative hedges the down side o the eal estate maket. Even i you don t ully cove you popety position, you can educe downside volatility by shoting the appeciation swap. This is like: Using Deivatives to Make Money in a Down Maket Popety Maket Insuance I you have it when the maket is headed down, you will likely beat the benchmak, and you will beat the peomance o othe potolio manages who haven t hedged.
Picing the appeciation swap Thee ae two majo appoaches to analyzing the picing (o valuation) o the eal estate index swap: Abitage Analysis Equilibium Analysis The two appoaches give identical esults when the undelying index is always valued at the equilibium (liquid maket) value o its constituent popeties. Thee is also a useul methodology o valuing any swap contact: Cetainty Equivalence DCF Analysis (CEQ) CEQ valuation is based on equilibium analysis, but gives esults identical to both abitage and equilibium analysis when the undelying index is at equilibium value.
Abitage Analysis... Fist conside an abitage analysis... Suppose it wee possible to hold and eiciently tade long and shot positions in the NPI diectly (a so-called spot maket o the NPI), such that: Possible to constuct a iskless hedge between the undelying asset and the swap contact: Then we could deive a picing elationship like the classical Futues-Spot Paity Theoem... (Aside note: This implies that the index must itsel diectly elect equilibium pice and etun expectations.) We ll come back to this point late
Abitage Analysis... Let: V t = Value level o the undelying appeciation index at end o peiod t. E[y] = Expected income etun o the undelying index (assumed constant & iskless).* = Riskee inteest ate (e.g., LIBOR). F = Fixed leg (spead) paid om long to shot position (in pecent o notional value). The pice o the swap contact is given by F, the value o the ageed-upon ixed spead. F can be deived by abitage analysis as ollows...
Conside a 2-peiod appeciation swap. Constuct a iskless hedge as ollows: Cash Flow Today(t) Cash Flow End o Qt(t) Cash Flow End o Qt(t2) Shot position in Index V t -g t V t - E[y]V t -g t2 V t - E[y]V t - V t Invest isklessly zeocoupon Long position in appeciation swap -V t / ( ) 2 0 V t 0 g t V t - FV t g t2 V t - FV t Hedge Potolio = Sum (-/( ) 2 )V t -(FE[y])V t -(FE[y])V t iskless iskless
O on the othe side, the iskless hedge o the shot Cash Flow Today(t) Cash Flow End o Qt(t) Cash Flow End o Qt(t2) Long position in NPI -V t g t V t E[y]V t g t2 V t E[y]V t V t Boow isklessly zeocoupon Shot position in appeciation swap V t / ( ) 2 0 -V t 0 -g t V t FV t -g t2 V t FV t Hedge Potolio = Sum -(-/( ) 2 )V t (FE[y])V t (FE[y])V t iskless iskless
Suppose you pay a cash pice C t at time t o the long position in the swap. You can constuct the hedge descibed peviously by shoting the index and investing in a iskless zeo-coupon bond o a net cash low at time t o: (- /( ) 2 )V t, and theeby convet you utue cash low steam into a iskless one o -(F E[y])V 0 each utue peiod. Thus, including the C t cost o the swap, the NPV o you tansaction at time t is: Conside a 2-peiod appeciation swap. Constuct a iskless hedge as ollows: ( ) ( ) ( ) ( ) 2 0 0 0 2 ] [ ] [ ) ( t V y E F V y E F C V long NPV = Since in act the swap is puchased o zeo net cash up-ont at time t, C t = 0, and the NPV o the long position in the appeciation swap is: ( ) ( ) ( ) ( ) 2 0 0 0 2 ] [ ] [ ) ( V y E F V y E F V long NPV =
To avoid abitage, this NPV must equal zeo, so the abitageavoidance pice F o the appeciation swap is ound by setting the pevious equation equal to zeo and solving o F : Conside a 2-peiod appeciation swap. Constuct a iskless hedge as ollows: ( ) ( ) ( ) ( ) ( ) ( ) ] [ ] [ ] [ ] [ ] [ 0 2 2 0 0 0 2 y E F y E F V y E F V y E F V y E F V = = = =
In geneal, the pice F is given by the ollowing equation: Extension to a T-peiod swap... F = E[y] ( ) ( ) ( ) ( ) ( ) ( ). ], [ ] [ ] [ ] [ ] [ 0 0 0 0 any T y E F y E F V y E F V y E F V y E F V T T T = = = = L
Abitage Analysis... The ixed spead (F ), pice o the appeciation swap, equals the iskee inteest ate minus the eal estate income etun. Let: E[RP V ] = E[ V ] = Real Estate (index) Risk Pemium E[g] = Real Estate Expected Appeciation Rate E[y] = E[ V ] E[g] = Real Estate Income Retun (constant) Shot position ist buys the notional amount o eal estate and eceives expected total etun o: E[ V ] = E[RP V ] = E[g] E[y] But then swaps g t o F (eplaces g t with F) duing the contact, theeby eliminating all isk duing that time, hence eliminates any need o E[RP V ], giving a equied expected etun o: FE[y] =, which implies: F = - E[y] Note that this is independent o E *
Abitage Analysis... The ixed spead (F ), pice o the appeciation swap, equals the iskee inteest ate minus the eal estate income etun. Note that: F = E[y] F = (E[ V ] E[g] ) = E[g] (E[ V ] ) F = E[g] E[RP V ] The ixed spead (F ), pice o the appeciation swap, equals the expected index appeciation ate ( E[g] ) minus the index expected etun isk pemium ( E[RP V ] ). I E[RP V ] is constant, this implies that the utues pice ( F ) moves one-o-one with the expected index appeciation, E[g].
Abitage Analysis... F = E[y] F = E[g] E[RP V ] I E[RP V ] is constant, this implies that the utues pice ( F ) moves one-o-one with the expected index appeciation, E[g]. Howeve, it is impotant to note that: As long as the undelying index is piced at its equilibium value (which it would have to be i we could eally use it to constuct abitages), F will not necessaily o pimaily elect E[g], but athe: The equilibium swap pice will pimaily elect inteest ates and eal estate income etuns, accoding to: F = E[y]. In othe wods: I the index is piced at equilibium, changes ove time in E[g] may lagely elect coesponding changes in the maket s equilibium E[RP V ] equiement o eal estate, except as elected in changes in E[y].
Abitage Analysis... The complete abitage valuation omula o the t-peiod appeciation swap: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 ] [, ] [ ] [ ] [ ) ( V y E F T o V y E F V y E F V y E F V long NPV T T T = = = = L The NPV o the shot position is just the negative o the above. Note: F should include impact o ees. E[y] can be equivalently expessed as: E[ V ] E[g] (Caution: This assumes V 0 is an equilibium value: no index lag.)
Fo total etun swap, the iskless hedge o the shot Cash Flow Today(t) Cash Flow End o Qt(t) Cash Flow End o Qt(t2) Long position in NPI -V t t V t t2 V t V t Boow isklessly zeocoupon Shot position in total etun swap V t / ( ) 2 0 -V t 0 - t V t FV t - t2 V t FV t Hedge Potolio = Sum -(-/( ) 2 )V t FV t FV t iskless iskless
Conside a 2-peiod total etun swap. Constuct a iskless hedge as ollows: Suppose you pay a cash pice C t at time t o the shot position in the swap. You can constuct the hedge descibed peviously by buying the index and isklessly boowing V t /( ) 2 zeo-coupon o 2 peiods, o a net cash low at time t o: -(-/( ) 2 )V t, and theeby convet you utue cash low steam into a iskless one o FV t each utue peiod. Thus, including the C t cost o the swap, the NPV o you tansaction at time t is: NPV ( shot) = 2 0 t ( ) ( ) 2 V C Since in act the swap is puchased o zeo net cash up-ont at time t, C t = 0, and the NPV o the shot position in the total etun swap is: NPV ( shot) = FV t FV ( ) 2 t ( ) 2 V FV t FV t t
To avoid abitage, this NPV must equal zeo, so the abitageavoidance pice F o the total etun swap is ound by setting the pevious equation equal to zeo and solving o F : Conside a 2-peiod total etun swap. Constuct a iskless hedge as ollows: ( ) ( ) ( ) t t t t F FV FV FV V = = = 2 2 2
Equilibium Analysis... In eal estate we cannot constuct the abitage, because we cannot diectly tade the undelying index. Howeve, the abitage analysis gives a picing esult that equates the expected total etun isk pemium pe unit o isk within and acoss the elevant asset makets: so-called linea picing E[] E[RP] Risk This is a chaacteistic o equilibium picing, which may also be viewed as nomative (i.e., ai ) picing.
Littleton s expected oveall net etun is: Fom the Pespective o the Coveed Long Position NPI Appeciation Retun (E L [g NPI ]), minus the Fixed Spead (F ), plus the iskee ate ( ) that they eceive on thei coveing bond investment. Littleton s oveall net isk exposue is that o the NPI: Because almost all etun isk is in the appeciation etun component. Theeoe, Littleton equies: E L [g NPI ] F E[RP NPI ]
Southen s expected oveall net etun is: Fom the Pespective o the Coveed Shot Position Thei expected Total Retun on thei ( eamaked ) coveing eal estate potolio (E S [ S ]) (which is simila to the NPI in isk), plus the Fixed Spead (F ), minus the expected NPI Appeciation Retun (E S [g NPI ] ). Southen s oveall net isk exposue is nil: Because thei exposue to the NPI appeciation obligation is coveed, and eal estate cash yield ( E[y] ) is nealy constant. Theeoe, Southen equies: F E S [g NPI ] E S [ S ]
Equilibium Analysis... Putting the two pevious picing conditions om each side togethe, we have the ollowing easible picing ange o F : E S [ S ] E S [g NPI ] F E L [g NPI ] E[RP NPI ] I all paties have the same expectations, the above picing condition becomes: E[ NPI ] E[g NPI ] F E[g NPI ] E[RP NPI ] ( E[RP NPI ]) E[g NPI ] F E[g NPI ] E[RP NPI ] E[RP NPI ] E[g NPI ] F E[g NPI ] E[RP NPI ] F = E[g NPI ] E[RP NPI ] = E[y NPI ] which is the same as the abitage-based esult*.
Equilibium Analysis... F = E[g NPI ] E[RP NPI ] = E[y NPI ] In this deivation we have not assumed that the undelying index is piced at its equilibium value. Howeve, i the undelying index is piced at its equilibium value, then the same obsevations as beoe apply to the equilibium pice and valuation o the swap contact. In paticula: Swap pice and value ae: Independent o Index Expected Retun E[ V ]. Independent o Index Expected Appeciation E[g]. Except in both cases as these expectations ae elected in E[y]. (i.e., movements in equilibium E[g] & E[RP V ] may lagely cancel.) Independent o the Volatility o the undelying Index.
Equilibium Analysis... F = E[g NPI ] E[RP NPI ] = E[y NPI ] Thee is an impotant coollay to this obsevation: I eal estate swap pices (speads) ae obseved to move noticeably with: Then: Changes in Index Expected Retuns going owad E[ V ], OR Changes in Index Expected Appeciation going owad E[g], Beyond the movements implied by changes in E[y], The undelying index is not piced at equilibium (e.g., the index value may be lagged behind cuent popety maket equilibium values due to the eect o appaisal valuation and/o othe index constuction issues.)
Appeciation vs Total Retun Swaps o Fixed Speads F = E[g NPI ] E[RP NPI ] = E[y NPI ] Fo the appeciation swap this picing ule will oten give a negative value o F (that is, shot paty pays the ixed spead to the long paty). This will almost always be tue i the undelying index is piced at its equilibium value. This is because the appeciation etun is only action o the total etun that is equied to compensate investos o beaing eal estate investment isk, And the swap long position that beas that isk only eceives the appeciation etun, not the total etun. I the swap contact is witten on the total etun (E[ NPI ]) instead o just the appeciation, we obtain by the same easoning as above (just substitute E[ NPI ] o E[g NPI ] in the above): F = E[ NPI ] E[RP NPI ] = ( is o couse nomally positive.*)
Typical Numbes F = E[g NPI ] E[RP NPI ] = E[y NPI ] The long-tem histoical aveage quately etun components o the NPI ae as ollows (978-2005): E[g NPI ] = 0.46% E[RP NPI ] = 0.90% =.5% E[y NPI ] =.94% Which implies a long-tem aveage appeciation pice, F o: F = 0.46% 0.90% =.5%.94% = 0.43%
Equilibium Analysis... Anothe coollay: I the appeciation swap is piced with a positive spead (F > 0), It is a stong signal that the undelying index is undepiced elative to the actual cuent maket value o the eal estate it epesents: Index value below its equilibium level implies expected nea-tem utue etuns to the index above thei equilibium ates. This could be due to index lag ate a ecent upsuge in popety maket value. (e.g., 2006.)
Suppose ove the peiod o the contact: Eect o Heteogeneous Expectations The potential long paty expects g NPI will ovepeom 25 bps/qt above the geneal maket expectation: E L [g NPI ] = E[g NPI ] 25bps. The potential shot paty is elatively beaish: E S [g NPI ] = E[g NPI ] 25bps. Suppose uthe that the potential shot paty eels that thei own (coveing) eal estate potolio will beat the NPI total etun by an aveage o 25bps/qt (even though it contains the same isk as the NPI, i.e., the excess is alpha ): E S [ S ] = E S [ NPI ] 25bps
Eect o Heteogeneous Expectations Plugging these expectations into ou picing omula o the appeciation swap, we obtain: E S [ S ] E S [g NPI ] F E L [g NPI ] E[RP NPI ] E S [RP S ] E S [g NPI ] F E L [g NPI ] E[RP NPI ] (E[RP NPI ] 25bps) (E[g NPI ] 25bps) F (E L [g NPI ] 25bps) E[RP NPI ] E[RP NPI ] E[g NPI ] 50bps F E[g NPI ] E[RP NPI ] 25bps Such complementay heteogeneous expectations add 75 bps to the easible pice ange o F. (e.g., instead o F = -0.43bp, we could have: -0.93% F -0.8%, with the exact pice ageed in negotiation.) Similaly, incompatible heteogeneous expectations (beaish long, bullish shot ) will eliminate any possible satisactoy F value, making tading ineasible.
Type o Heteogeneity Useul o Tading Implication o the act that the zeo-npv pice condition can be expessed in eithe o two ways: E[g] E[RP] OR E[y] Apat om alpha consideations, in ode to obtain complementay (ovelapping) pice equiements, we equie: Heteogeneity in g NPI expectations that ae not canceled out by heteogeneity in RP NPI expectations. This equies: Osetting E[g NPI ] & E[y NPI ] expectations. e.g.: I the long paty believes that NPI appeciation will be %/yea highe than aveage it must also believe that NPI income yields will be %/yea lowe than aveage.
Eect o Tansaction Fees Tansaction ees naow the easible tading ange. I the ees ae basis-points o notional value pe peiod (chaged to each paty), then the easible picing ange becomes: E S [ S ] E S [g NPI ] F E L [g NPI ] E[RP NPI ] Fees chaged to both paties togethe must be less than: E L [g NPI ] E S [g NPI ] E S [ S ] E[RP NPI ] This would seem to equie complementay heteogeneous expectations in ode to allow any easible tading ange. Howeve,...
Eect o Tansaction Fees The above analysis ignoes the savings o tansactions costs, investment management ees, and othe advantages o using deivatives vesus taditional methods o accomplishing the potolio balancing o hedging/speculation objectives undelying the shot and long positions. Fo example, suppose the long paty would avoid 20 bp/qt in investment management and und ees, and suppose the shot paty eectively saves 0 bp/qt in costs o boowing against thei potolio (altenative taditional methods to accomplish potolio balancing objectives). Then even with homogeneous expectations, thee exists 30 bp/qt o savings that can be split among the two paties and the bokes and investment banking ees o the deivative pocess.
What about pedictability in the NPI? Appaisal smoothing and stale appaisals (along with tue undelying popety maket sluggishness) give the NPI much moe inetia than typical secuities indexes, making the NPI elatively smooth and pedictable. Wouldn t this type o pedictability esult in an absence o heteogeneous expectations, and theeby an absence o countepaties o tading the deivatives, making a unctioning utues maket impossible?
What about pedictability in the NPI? Answe: Not necessaily. As seen in ou Littleton & Southen example, deivative tades may have easons othe than speculation o tading deivatives. Heteogeneous expectations may not be necessay. Pedictability in the NPI will simply come out in the equilibium deivative pice, F. Recall that F is a unction o the maket paticipants expectations about the utue NPI appeciation etuns: E L [g NPI ] and E S [g NPI ]. e.g., i NPI is headed down, then E[g NPI ] will be negative, making F moe negative than it would othewise be (meaning the shot position must pay the long position moe in the ixed leg). Well-unctioning utues makets have long existed o vaious commodities and inancial poducts whose utue pice diections ae oten athe pedictable in advance (e.g., con, wheat, oil, oeign exchange, among othes).
What about pedictability in the NPI? O couse, pedictability in the NPI means that deivatives tades who don t want to get taken advantage o must use all the elevant inomation as skillully as possible to be educated about what ae easonable expectations o whee the NPI is headed. Fo example (inte alia ), the MIT tansactions-based index (which tends to lead the NPI ) Roughly speaking: g NPI (y t) = (/2)g TBI (y t-) (/2)g TBI (y t).
What about pedictability in the NPI? Eect o Index Lag on Swap Picing and Valuation: Index Lag Index will oten not be valued at its equilibium value. That is, ealistic expected etuns on the index die in the nea-tem om long-un equilibium ates. In such cicumstances, the Abitage Picing and Valuation Fomula o the swap no longe holds. The zeo-npv picing condition will still be well appoximated by: F = E[g] E[RP V ] but not by E[y], and only povided that E[RP V ] in the above omula elects the maket s long-un equilibium isk pemium, not the cuent disequilibium pemium pesented by the index (while E[g] elects the non-equilibium appeciation in the index).
Altenative Picing Pespectives The pevious picing analysis assumed tading by coveed paties on both sides. Altenative picing pespectives ae possible Suppose neithe paty is coveed at all, and both view themselves as equiing a etun as i they wee actually making the notional investment. The esulting picing condition would be: E S [g NPI ] E[RP NPI ] F E L [g NPI ] E[RP NPI ] Which implies a easible picing ange o F o width: E[g L NPI ] E[gS NPI ] 2 This equies that the long position be substantially moe bullish than the shot position. But this pespective is not based on an equilibium amewok...
Altenative Picing Pespectives We can analyze equilibium picing o the uncoveed swap by using cetaintyequivalence valuation (CEQ) Conside a -peiod binomial wold; V 0 = Cuent value o the R.E.index & notional amt o swap tade; p = Pobability o the up move in index next peiod (eal pob) V up, V down = Value o index next peiod in up o down outcomes; g = -peiod appeciation etun on index; F = Appeciation swap pice (ixed leg) in action o notional tade amount. V 0 p -p V up V down V up V down = ( g up )V 0 = ( g down )V 0 E[V ] = ( E[g])VE 0 = [p(g[ up ) (-p)(g = [ pg up (-p)g down E[g] ] = pg up (-p)g down (g down )]V 0 down ]V 0
Altenative Picing Pespectives Cash low amts o appec swap long position: CF up = (g up F)V 0 Expected CF next peiod is: E[CF ] = [pg[ up (-p)g down = ( E[g] F )V 0 down ]V 0 FV 0 Cetainty Equivalent CF next peiod is: 0 p -p CF down = (g down F)V 0 CEQ[CF CF ] = E[CF ] E[RP V ](CF CF up up CF CF down )/( )/(V up /V 0 V down /V 0 ) = E[CF ] ( E[ V ] )(g up g down )V 0 / (g( up g down ) = E[CF ] ( E[ V ] )V 0 down /V Hence, pesent value o uncoveed swap CF next peiod is: PV[CF ] = [ E[CF ] ( E[ V ] )V 0 ] / ( ) In equilibium, this must equal the 0 net CF o the tade today: PV[CF ] = [ E[CF ] ( E[ V ] )V 0 ] / ( ) = 0. E[CF ] = ( E[ V ] )V 0
Altenative Picing Pespectives Thus, we have the equilibium condition: p CF up = (g up F)V 0 E[CF ] = ( E[] )V 0 ( E[g] F )V 0 = ( E[] )V 0 F = E[g] E[] ] = E[g] E[RP] 0 -p CF down = (g down F)V 0 = E[y] This is the same equilibium picing condition that we obtained beoe: F = E[g] E[RP] = E[y]
PV[ C T CEQ0[ C ] ] = T T ( ) = Cetainty Equivalence Valuation E [ C 0 T STD C 0[ ] ] ± STD 0[ V ] o Deivatives In geneal, we can use cetainty-equivalence discounting to evaluate the swap with heteogeneous expectations, and whethe o not the undelying index is valued at its equilibium level The geneal cetainty equivalence valuation (CEQ) omula o valuing a deivative cash low is as ollows: Let: C t = The deivative value (cash low) t peiods hence, E[ V ] = Equilibium expected total etun on undelying asset (pe peiod). Then the Cetainty Equivalence PV omula is as ollows o a single sum T peiods in the utue, with isk accumulation thoughout T ):* T T ( E[ ]) ( ) ) ( ) T V The isk adjustment is subtacted o long positions, added o shots.
Cetainty Equivalence Valuation o Deivatives Fo the index swap, the CEQ valuation omula o a given utue cash low o the swap t peiods in the utue (C t ) with cuent value o the undelying index (= notional amount o tade) V 0 is: PV [ C 0 t ] = CEQ0[ Ct ] = = t ( ) 0 E [ C t ] ± E0[ Ct ] ± = E [ C 0 ( E[ RP ]) t ( ) ( E[ RP ]) t ( ) V V t ] ± ( E[ RP ]). t ( ) STD[ V ] V STD[ V ] V 0 V STD[ C STD[ V whee E[RP V ] is the mkt equilibium isk pemium. 0 t ] ] Whee the isk adjustment in the numeato is subtacted o long positions and added o shot positions (negative coel betw swap & index). Thee is only peiod o isk accumulation (just pio to the cash low), because the cash low is based solely on the index etun in peiod t times a notional amount that is ixed up ont at time 0. Hence, the isk adjustment in the numeato is o just one peiod.
Conside a -peiod example with F = -60bp and the ollowing expectations: Long Pespective: = 0.75% /qt Shot Pespective: E L [ NPI ] = 2.00% /qt E S [ NPI ] = 2.00% /qt E L [g NPI ] = 0.75% /qt Cetainty Equivalence Valuation o Deivatives E S [g NPI ] = 0.55% /qt E L [RP NPI ] =.25% /qt E S [RP NPI ] =.25% /qt E L [C ] = (0.0075 (-0.0060))$00 E S [C ] = (-0.0055 (-0.0060))$00 = $0.75 $0.60 = $.35. = -$0.55 $0.60 = -$.5. Applying the cetainty equivalence DCF valuation omula: PV PV S L ( E[ RP ]) V $.35 (.025) L E [ Ct ] V $00 $.35 $.25 [ Ct ] = = = R.0075.0075 0 = ( E[ RP ]) V $.5 (.025) S E [ Ct ] V $00 $.5 $.25 [ Ct ] = = = R.0075.0075 0 = $0.0993 $0.0993 Because o heteogeneous expectations, The tade allows both sides to ace a positive NPV ex ante.
PV PV S L Cetainty Equivalence Valuation o Deivatives ( E[ RP ]) V $.35 (.025) L E [ Ct ] V $00 $.35 $.25 [ Ct ] = = = R.0075.0075 0 = ( E[ RP ]) V $.5 (.025) S E [ Ct ] V $00 $.5 $.25 [ Ct ] = = = R.0075.0075 0 = $0.0993 $0.0993 PV[C] = CEQ[C] / ( ) = $0.0 /.0075 = $0.0993. Even though the expected cash low is $.35 o the long position, - $.5 o the shot position, the cetainty equivalent cash low is only $0.0 in both cases. The cetainty-equivalence opeation eveses the sign o the shot position cash low expectation, because the isk in the shot position is negative the isk in the undelying index, because the two ae peectly negatively coelated. Note that you should always employ the maket equilibium isk pemium o the undelying index.
NPV L = $0.0993. Same thing using the abitage ( F Ey) ( ) NPV long = NPI T ( ).0060.025 = $00 (.0075).0075 = 0.00744*0.3333*$00 = $0.0993 0 omula Given the same conditions as beoe with F = -0.60%: = 0.75% /qt E L [ NPI ] = 2.00% /qt E S [ NPI ] = 2.00% /qt E L [g NPI ] = 0.75% /qt E S [g NPI ] = 0.55% /qt Deine: Ey L = E L [ NPI ] E L [g NPI ] = 2%.75% =.25% Ey S = E S [ NPI ] E S [g NPI ] = 2%.55% =.45% Applying the abitage valuation omula to the $00 notional tade: ( F Ey) NPV ( long) = NPI T ( ).0060.045 = $00 (.0075).0075 = 0.00744*0.3333*$00 = $0.0993 NPV S = $0.0993. Same answe as beoe. (But this omula only woks i the undelying index is in equilibium.) See downloadable Excel ile example on book CD 0