Geometric Brownian Motion Model for U.S. Stocks, Bonds and Inflation: Solution, Calibration and Simulation

Transcription

1 Geometrc Brownan Moton Model for U.S. Stocks, and Inflaton: Soluton, Calbraton and Smulaton Frederck Novomestky Comments and suggestons are welcome. Please contact the author for ctaton. Intal Draft: June 7, 001 Abstract Ths paper addresses the problem of desgnng stochastc models for creatng realstc sample paths of U.S. asset class returns and nflaton. These models are used for valung penson plan labltes, expense and portfolo asset values, and, when combned wth a dynamc penson plan model, are also used to construct optmal fundng and asset allocaton strateges. Correlated geometrc Brownan moton processes are used to descrbe the dynamc behavor of the real value of broad fnancal asset class values and nflaton. These real values are lnked to ther nomnal values usng the Fsher effect. A procedure s derved for calbratng the stochastc models and Monte Carlo experments are performed on these models to valdate the correctness of the calbraton procedure. Department of Management, Polytechnc Unversty, Brooklyn, NY, (718) , fnovomes@poly.edu. Ths research was sponsored by the Center for Fnance and Technology of the Polytechnc Unversty.

2 1. Introducton Penson plan labltes, expenses and asset values are complex and dffcult to value because of the nontrval nteracton of nterest rates, nflaton and asset returns. The general framework of the valuaton analyss has three phases whch generalzes the approach taken by (Zenos 1997) for the valuaton and rsk management of portfolos of mortgage-backed securtes. Phase I. Generate realstc asset class return, nflaton and nterest rate scenaros or sample paths consstent wth the prevalng captal market mcrostructure, ncludng the term structure of nterest rates. Phase II. Generate cash flows for each scenaro. Ths requres models that project normal and unantcpated penson expense, beneft payment and lablty measures under a host of economc condtons. Phase III. Use the cash flows and nterest rates along each path to compute expected net present values of the cash flows. Ths phase can be easly extended to calculate holdngperod returns. Optmal fundng strateges and the correspondng asset allocaton strateges requred to meet or satsfy the range of penson plan objectves are best accomplshed n ths rskbased, scenaro approach. The earlest work n ths area was done by (Hll 1978), (ICF 1979), (Keely 1969), (Kngsland 198), (Lenarcc 1977), (Mulvey 1988), (Tepper 1977), (Tepper and Affleck 1974), and (Wnklevoss 198). Recent research has expanded on scenaro based approaches for asset and lablty modelng (Berger and Ruszczynsk 1995), (Carroll and Nehaus 1998), (D'Arcy, Dulebohm et al. 1999), (Mulvey and Vladmrou 199), (Mulvey 1994), (Mulvey and Zemba 1995), and (Sherrs 1994). A number of papers have been publshed that have appled optons-based analyss to the penson plannng problem (Nader 1991) and (Sherrs 1995). 1

3 Ths paper addresses the problem of desgnng dynamc models for creatng realstc sample paths of asset class returns and nflaton. The paper has the followng structure. Secton descrbes the asset class and nflaton model. Ths s a contnuous tme stochastc model based on real asset values and nflaton from whch nomnal values are derved. Secton 3 develops the framework for specfyng the nternal parameters of the stochastc models based on ex ante parameters such that when a large number of sample paths are constructed usng Monte Carlo smulaton methods the ex post realzed statstcs closely agree wth the ex ante values. Secton 4 descrbes the emprcal work done wth the model that valdates the correctness of the calbraton process. Secton 5 provdes concludng comments.. Asset Class and Inflaton Model Rsk-based asset class models descrbe the uncertan behavor of asset values. Penson plan asset-lablty models requre asset class return models that characterze ths behavor over tme. Contnuous models, n partcular, characterze the value of these assets treatng tme as a contnuous varable. The unt of tme measure s a year. Suppose that we have n dstnct asset classes. The nomnal value of the th asset class at tme t s denoted by U ( t ). Let P ( t) represent the value of a prce ndex at tme t whose relatve change over tme best represents the nflaton rate. Let V ( t ) be the real value of the th asset class at tme t. Ths value s related to the nomnal value and the nflaton ndex n the followng manner. U ( t) =P( tv ) ( t) (1)

4 From Fsher's Theory of Interest 1, or the so-called Fsher Effect, appled to the returns of fnancal assets, the nomnal return of the th N asset from tme t to T, denoted by R ( tt, ), R P s related to the real return, R ( tt, ), and the nflaton rate, R ( tt, ), n the followng manner. 1 R N ( tt, ) 1 R R ( tt, ) 1 R P + = é ù é ( tt, ) ù ë + + û ë û () These returns are related to ther underlyng values as follows. N VT ( ) 1 + R ( tt, ) = V ( t) R UT ( ) 1 + R ( tt, ) = U ( t) P P( T) 1 + R ( tt, ) = (5) P ( t) (3) (4) Ths partcular structure provdes an explct relatonshp between nomnal asset values and nflaton. In the context of a penson plan, the nflaton component of the model can be used to ntroduce unantcpated changes n salary over tme. Inflaton drven changes n asset value and salares, n general, wll result n actuaral gans and losses that have an mpact on total penson plan expense and the funded status of the plan. 1 Brealey, R. A. and S. C. Myers (000). Prncples of Fnance. New York, NY, Irwn McGraw- Hll., p , p Levch, R. M. (1998). Internatonal Fnancal Markets: Prces and Polces. New York, NY, Irwn McGraw-Hll., p

5 We adopt the followng the general notaton for m = n + 1 real value ndces n terms of the nomnal values. U ( t) 1 V ( t) = 1 P ( t) Un ( t) V ( t) = m-1 P ( t) P( t) Vm ( t) = P ( 0) (6) Note that the nflaton component of the model s re-scaled to the value of the prce ndex at tme 0. Ths provdes a consstent defnton across all asset values and nflaton. The real values n equaton (6) are the state varables to be represented by a system of stochastc dfferental equatons. Usng equaton (6), the nomnal asset values are obtaned. We ntroduce the followng vector-matrx notaton that s used throughout ths paper. éu ( t) ù 1 U( t) =, Um ( t) =P( t) U m ( t) êë úû (7) év ( t) ù 1 V( t) = V m ( t) êë úû (8) é R R ( tt, ) ù 1 R R P R ( tt, ) =, Rm ( tt, ) = R ( tt, ) R Rm ( tt, ê ) ë úû (9) 4

6 é N R ( tt, ) ù 1 N N P R ( tt, ) =, Rm ( tt, ) = R ( tt, ) (10) N Rm ( tt, ê ) ë úû We now consder a collecton of equatons that characterze real asset class values and an nflaton ndex. Each equaton descrbes the dynamc behavor of the specfed component as a stochastc dfferental equaton (SDE) for a geometrc Brownan moton (GBM) process. The general form for each SDE s as follows. dv = mvdt + svdz, t ³ 0 = 1,, m (11) The parameters, m and s, are the nstantaneous expected value and volatlty for the growth rate for the th component, respectvely, whose value at tme t s V ( t ) and where n s the number of components. The dfferental dz s a standard Wener process. The tme t = 0 denotes the begnnng of the tme horzon over whch these component values are to be smulated. The soluton to the above SDE can be obtaned drectly or verfed through the use of Ito's lemma from (Neftc 000). éæ 1 ö ù V ( t) = V ( 0exp ) m s t s Z ( t), t ³ êç çè ø ë úû (1) The varable Z ( t ) s a zero mean normal random varable wth varance equal to t. Let r represent the correlaton between the random varables Z ( t ) and Z ( t ). j, j In the next secton, we explore the relatonshp between the parameters of the SDE and the observable values such as nomnal asset values and the nflaton ndex. 5

7 3. Model Calbraton In order smulate sample paths of asset class returns and nflaton rates, we need to estmate or specfy the followng parameters. 1. Instantaneous growth rates m ( = 1,, m). Instantaneous volatltes s ( = 1,, m) 3. Correlatons r, ( = 1,, mj ; = 1,, m j ) These parameters are not drectly observable n asset prces and nflaton, but can estmated from or specfed as a functon of the expected values, varances and covarances of the value relatves Y V ( t) / V( 0) =. For convenence, we drop the explct dependence on tme. Subsequently we wll refer to the m 1 vector of these random varables Y éy ù 1 Y = Y ê m ë úû (13) The expected value of Y s the m 1 vector m. Y m Y émy ù 1 my = E { Y} = m êë Ym úû (14) 6

8 The m m matrx of varances and covarances for Y, denoted by C, s gven by. Y é s s s ù Y YY YY m ì ü s s s YY 1 Y YY m C = E ï í( Y-m )( Y- m ) ï ý= (15) Y Y Y ïî ïþ s s s ê YY m 1 YY m Y ë m úû Prme denotes the transpose of a matrx or vector. We can then re-wrte equaton (1) as follows Y = exp( ) (16) The random varable s normally dstrbuted wth mean µ and standard devaton σ where 1 µ µ σ σ = t = σ t (17) (18) We wll also make reference to the m 1 vector of these normally dstrbuted random varables. é ù 1 = ê m ë úû (19) 7

9 The expected value of s the m 1 vector m. m ém ù 1 m = E { } = m êë m úû (0) The m m matrx of varances and covarances for, denoted by C, s gven by. é s s s ù m ì ü s s s 1 m C = E ï í( -m )( - m ) ï ý= (1) ïî ïþ s s s ê m1 m ë m úû The moment generatng functon (MGF) for the normal dstrbuton can be used to derve expressons for the mean and varance of varable Y. Let m ( ) ω be the MGF for the random. It follows that for the normal dstrbuton, the MGF s as follows (see (Hogg and Tans 1993), p. 5). { } ( ω) = exp( ω ) m E 1 = exp µ ω+ σ ω () Note that Y s a log-normally dstrbuted random varable. From equatons (17), (18) and (), we obtan an expresson for the expected value of Y. 8

10 µ Y = E = m { exp( ) } ( 1) 1 = exp µ + σ = exp ( µ t) (3) In the same manner we can derve an expresson for the second moment of Y { } = { exp( ) } EY E = m ( ) ( µ σ ) ( µ t σt) = exp + = exp + (4) The varance σ Y can easly be obtaned as follows { } ( ) ( µ t) ( σt) Y exp( t ) 1 σ = EY µ Y Y = exp exp 1 = µ σ (5) Gven estmates of m and Y s by applyng equatons (3) and (5) s, we can derve the correspondng estmates of Y m and m ( my ) 1 = ln t (6) 9

11 é 1 s ù æ ö Y ln s = + 1 t çm ê çè Y ë ø úû (7) In order to compute the matrx of correlaton coeffcents, we need the m m covarance matrx for the jont multvarate normal dstrbuton of. Equatons (3) and (5) gve us the formulas used to compute the dagonal elements. Suppose that we are gven the m m covarance matrx C of Y. From equatons Y (15) and (5), we can compute the m m matrx of second moments and cross moments. E { YY} é EY EYY EYY EYY EY EYY = ê EYY { m 1} EYY { m } EY { m} ë úû { } { } ù 1 1 { 1 m} { 1} { } ú { m } (8) The expresson for these moments s as follows. { } E YY = C + mm (9) Y Y Y where m s the m 1 vector of expected values Y The multvarate probablty densty functon (PDF) for the m 1 vector of normally dstrbuted random varables from (Anderson 1984) s gven by. f é êë -m ( ) ( ) / -1/ - x = p C exp ê- ( x- ) C 1 ( x- ) ú 1 ù m m (30) úû 10

12 Let w be an m 1 vector éw, w,, w ù ë 1 n û Y wth w as exponents are determned as follows. of non-negatve ntegers. The moments of M EY Y Y m Y w ( ) 1 w wm w = { 1 } (31) Substtutng equaton (14) nto equaton (30), we obtan the followng result. w1 w wm { } { exp( ) exp( ) exp( ) 1 m = w w w 1 1 m m } EY Y Y E ì m æ öü = Eïexp w ï í ý çå ïî è = 1 ø ïþ = E { exp( w ) } (3) Let P be the m 1 vector of random varables defned as a zero mean verson of the random varables. P = -m (33) The vector P of random varables has a zero mean and a covarance matrx equal to that of the vector. These random varables are also jontly normally dstrbuted as well. m P = 0 (34) C P = C (35) Substtute equaton (33) nto (3) and factorng the result we obtan the followng equaton. 11

13 M { } ( ) = exp( ) E exp( ) w w m w (36) Y P We can factor the expectaton nto a product of m expectatons of m ndependent random varables through a transformaton of varables. Let A be an m m orthonormal matrx whose columns are the egenvectors of the covarance matrx C. AA = I ( l ) AC A = dag l, l,, m 1 (37) dag ( l, l,, l ) 1 m of the covarance matrx él 0 0 ù 1 0 l 0 = ê 0 0 lm ë úû C. s the dagonal matrx of the egenvalues Let P = AQ. Then, from equaton (37), Q= A P= AP. These m random - 1 varables are ndependent wth zero mean and varances equal to the egenvalues of C. m Q = E { Q} = E = A m = 0 { AP } P (38) 1

14 ì C = E ï íq - Q - ïî ì ü = E ï í ï ( APAP )( ) ý ïî ïþ = E ( m )( m ) Q Q Q { A ( PP ) A} ì ü = A E ï í( P-m )( P-m ) ï ýa (39) P P ïî ïþ = AC A P = AC A ( l ) = dag l, l,, m 1 ü ï ý ïþ b = A w. We now substtute P = AQ nto equaton (36) wth M Y { ë û} { exp bq } E{ exp bq } ( w) exp( w m ) E exp éw ù ( AQ) = ê ú = exp ( w m ) E ( ) m ( w m) ( ) = exp Õ = 1 From equatons () and (39), we note that 1 { exp( )} expç m s Q Q æ ö E bq = b b ç + çè ø æ1 ö = exp ç b l çè ø Usng ths result, we obtan the followng expresson for the w -specfed moment of the random vector Y. 13

15 M Y m æ1 ö = exp Õexp b l ç çè 1 ø = é m 1 ù = exp( w m ) exp ( b l ) ê å ë ú = 1 û ( w) ( w m) (40) Observe that dag ( l, l,, l ) 1 m b él 0 0ùéb ù él b ù 1 1 úê l 0 úêb l b ê úê ú ê ú = úê = úê úê 0 0 l úê m b l m ê mbm ê úê ú ë ûë û ë úû Usng ths result, we can re-wrte the summaton n equaton (40) n the followng manner. m å = 1 é ù = é ù ê ú ê b ë û ë ú û ( b l ) dag( l, l,, l 1 m) b dag( l, l,, l 1 m) é ù ê ( l, l,, l ) dag ( l, l,, l ) úb ë û = b dag ê 1 m 1 m ú = = ( A w) dag ( l, l,, l 1 n )( A w) ( A w) ( AC A)( A w ) = w C w Substtutng ths result nto equaton (40) we obtan the followng expresson. M æ1 ö w = exp wm exp ç w C w çè ø (41) ( ) ( ) Y 14

16 The parameters requred to smulate the multvarate GBM process drectly can be represented by the followng vector and matrx ém1 ù m m = m ê m ë úû é s s s ù 1 1, 1, m s s s,1, m C = ê s s s m,1 m, m ë úû (4) (43) These parameters are related to the multvarate normal dstrbuton mean vector and covarance matrx as follows. m C é 1 = m- dag êë = t C ( s, s,, s ) 1 m ù t úû (44) Usng equaton (41), we can derve expressons for all of the moments of Y n terms of the mean, varances and covarances of. Let us consder the followng three smple cases. Case 1: w = e where e s an m 1 unt vector wth a one n the th element and zero elsewhere. Ths exponent vector corresponds to the frst moment, or expected value, of Y. 15

17 M ( ) = EY { } Y w 1 ( m ) ç s æ ö = exp exp ç çè ø æ 1 ö = exp ç m s + çè ø (45) Ths corresponds to equaton (3). Case : w = e. Ths exponent vector corresponds to the second moment of Y M ( ) = EY { } Y w é1 ( m ) ( ) êë ( m s ) ù = exp expê s úû = exp + (46) Ths corresponds to equaton (4). From equaton (5), we obtan the varance of Y. w = e + e Case 3:. Ths exponent vector corresponds to the cross-moment between j Y and Y. j M Y ( w) = EYY { 1 } é1 ù ( m m ) ( ) s s s j j ê j ë úû æ 1 ö æ 1 ö m s m s ç ( s è ) ø çè j j ø j exp( s ) = exp + exp + + = exp + exp + exp = m m Y Y j j (47) 16

18 Usng equatons (45) and (47), we obtan the followng expresson for the covarance between Y and Y. j s { j} = EYY -m m YY Y Y j j ( s ) é ù = m m exp 1 Y Y ê - j ú ë j û (48) From equatons (44), and (48), we obtan the followng expresson for s. j, s j, é 1 s ù æ ö YY ln j = + 1 t m m ê ç è Y Y j ë ø úû (49) Note that the covarance matrces, C and gven by. C, have the same correlaton matrx whch s r j, sj, = ss j (50) 17

19 4. Emprcal Analyss The calbraton, smulaton and evaluaton of GBM models for real asset values and nflaton begn wth the statstcal estmaton of means, varances, and covarances of the real returns and nflaton rates for the U.S. captal markets. Nomnal returns and nflaton rates were obtaned from Ibbotson Assocates (Ibbotson 000) for the 74-year perod of 196 to 1999 and for the followng asset classes. 1. company stocks. company stocks 3. Long-term government bonds 4. Intermedate-term government bonds 5. Long-term corporate bonds Monthly real returns are derved from these data usng equaton (). Annual nomnal and real returns are the cumulatve compounded returns derved from the monthly returns. Table 1 contans the annual nomnal return summary statstcs to nclude both arthmetc and geometrc mean returns as well as the annual standard devaton of returns. Table shows the covarance matrx and correlaton matrx. Tables 3 and 4 present comparable statstcs for the annual real returns. The sample of real return relatves s then used to estmate the means, varances and covarances of the value relatve vector, Y, and these estmates appear n Table 5. As expected, the covarance matrces n Tables 4 and 5 are dentcal. The estmates n Table 5 are then used to compute the means, varances and covarances of the multvarate normal vector,, usng equatons (17) and (18). The results are provded n Table 6. As a fnal step n the estmaton process, the nternal model parameters for the stochastc dfferental equatons are computed usng equatons (6) and (7) and the results are tabulated n Table 7. 18

20 The parameters n Table 6 are the nput parameters for a Mcrosoft Excel workbook mplementaton of a Monte Carlo smulaton model usng add-n from Palsade Corporaton 3. The model smulates the values of the vectors, and Y, at the end of one year gven the followng table of ntal values. Asset Class Real Value, U ( 0) Nomnal Value, ( 0) Long-term Intermedate-term Long-term Inflaton V The number of teratons performed s 3,000. The workbook model calculates the followng output varables. 1. Value relatve vector Y. R Real returns, R ( 0,1) 3. N Nomnal returns, R ( 0,1) The nput varables are random samples of the vector. Tables 8-11 present the frst and second moments of the nput and output varables. A seres of charts follow these tables that compare the estmates derved from hstorcal return records to the smulated results. If the model has been properly calbrated, then the smulated results should be reasonably close to the estmated results. For each output measure and nput varable, the followng charts are provded. 3 Palsade Corporaton: To obtan a copy of the Excel smulaton model workbook used n ths paper, please contact the author. 19

21 1. Estmated versus smulated mean values. Estmated versus smulated standard devatons 3. Estmated versus smulated correlaton coeffcents. Close agreement s observed n all of the varables of nterest. The next experment performed on the results of the Monte Carlo smulaton s to compare the expected return and rsk of portfolos derved usng estmated parameters to the correspondng statstcs computed from the smulaton output. Eght (8) portfolos were derved from the estmated mean, varances, and covarances of nomnal returns n Tables 1 and by solvng the followng Markowtz portfolo optmzaton problem: determne a set of nvestment weghts for the fve asset classes lsted above that mnmze the expected portfolo varance subject to the followng two constrants. 1. The portfolo expected return equals a gven target return.. The sum of the nvestment weghts equals one. The algorthm used to solve ths problem s presented n Appendx A. The results of solvng the mnmum varance, targeted-return portfolo optmzaton problem, for eght dfferent target returns, appear n Table 1. The top panel shows the nvestment weghts for each of the portfolos. The second panel shows the portfolo mean, varance and standard devaton for each of these portfolos based on the estmated parameters n Tables 1 and. The thrd panel presents the correspondng results usng the smulaton outputs n Table 10. The bottom panel shows the percentage dfference n each of the statstcs from whch we make the followng observatons. 1. The percent errors for the portfolo mean do not exceed.3%. For the portfolo standard devatons, percent dfferences are generally less than %. 3. Percentage varance dfferences whch compare farly small quanttes are less than 5% Fgure 13 compares the estmated and smulated effcent fronters, further valdatng the accuracy of the model. 0

22 5. Concluson A system of stochastc dfferental equatons for the real value of fnancal assets and nflaton has been presented whch correspond to correlated geometrc Brownan moton processes. The soluton to these dfferental equatons s provded whch s a lognormally dstrbuted vector random process. Gven estmates of the means, varances and covarances of the value relatves of these random varables, a procedure s derved to estmate the nternal model parameters that are used to perform Monte Carlo smulaton experments. These experments result n sample paths of these random processes that have ex post means, varances and covarances that are reasonably close to the ex ante parameters used to calbrate the model. The correspondng smulated nomnal return statstcs are found to be qute close to the statstcs estmates from hstorcal returns. The stochastc model developed n ths paper s one of several types of models that, when smulated, produce realstc and consstent sample paths of asset class returns and nflaton. The nomnal asset class returns are affected by both random shocks and by changes n nflaton. In the context of penson plan asset-lablty modelng, the nflaton component can be used to generate realstc sample paths of plan partcpant salary growth. Appendx A: Portfolo Optmzaton Problem Suppose that we are gven a one-year ahead forecast of the expected returns of n asset classes represented by the followng vector. R = ér ù 1 R R ê n ë úû (A.1) 1

23 In addton, we also gven a one-year ahead forecast of the expected covarance matrx. éc C C ù és s s ù 1,1 1, 1, n 1 1, 1, n C C C,1,, n s s s,1, n C = = C C C ê n,1 n, nn, ë úû ê s s s n,1 n, n ë úû (A.) We wsh to determne the n 1 vector w of nvestment proportons n these n asset classes that mnmzes the expected portfolo varance subject to the followng two constrants. 1. The expected portfolo return, R, equals a gven target return, P. The sum of the nvestment proportons equals one. R. T Ths constraned portfolo optmzaton problem can be expressed as an unconstraned problem usng the method of Lagrange multplers (Bryson and Ho 1969). The correspondng objectve functon s gven by. 1 f ( w, l, l 1 ) = wcw -l1( Rw -R ) ( 1) T -l 1w - (A.3) The n 1 vector 1 s a vector of all ones. l and 1 l are the Lagrange multplers. The necessary condtons for an optmal soluton are obtaned by settng the n + partal dervatves of the objectve functon equal to zero. Cw-Rl - 1l = 0 Rw = 1w = 1 R T 1 (A.4)

24 From the frst equaton, we can express the nvestment proporton vector n terms of the Lagrange multplers. w = C Rl + C 1 l (A.5) Substtutng equaton (A.5) nto the two other equatons n (A.4) gves the followng system of equatons to solve for the Lagrange multplers. -1 ( RC R) l -1 ( RC 1) -1 ( 1C R) l -1 ( 1C 1) l + l = 1 + = 1 1 R T (A.6) We can re-wrte ths system of equatons n vector-matrx format n the followng way. a a él ù éb ù é 1,1 1,ù 1 1 a a = ê ê,1, l b ë ú ûê ú ê ú ë û ë û (A.7) The soluton to ths system of equatons s l l 1 ba = a a -ba 1, 1, -a a 1,1, 1,,1 a b = a a -a b 1,1,1 1 -a a 1,1, 1,,1 (A.8a) (A.8b) Equatons (A.5) and (A.8) are then solved parametrcally, varyng the target return over a range of values. The correspondng optmal nvestment proportons defne the portfolos along the effcent fronter. 3

25 References Anderson, T. W. (1984). An Introducton to Multvarate Statstcal Analyss. New York, NY, John Wley & Sons. Berger, A. J. and R. Ruszczynsk (1995). A new scenaro method for large-scale stochastc optmzaton. Operatons Research 43: Brealey, R. A. and S. C. Myers (000). Prncples of Fnance. New York, NY, Irwn McGraw-Hll. Bryson, A. E. and Y.-C. Ho (1969). Appled Optmal Control: Optmzaton, Estmaton, and Control. Waltham, MA, Blasdell Publshng. Carroll, T. J. and G. Nehaus (1998). Penson Plan Fundng and Debt Ratng. Journal of Rsk and Insurance 65(3): D'Arcy, S. P., J. H. Dulebohm, et al. (1999). Optmal Fundng of State Employee Penson System. Journal of Rsk and Insurance 66(3): Hll, J. (1978). Penson Fund Management: A Framework for Investment and Fundng Decsons, Syracuse Unversty. Hogg, R. V. and E. A. Tans (1993). Probablty and Statstcal Inference. Englewood Clff, NJ, Prentce Hall. Ibbotson (000). Stocks,, Blls, and Inflaton: Valuaton Edton 000 Yearbook. Chcago, IL, Ibbotson Assocates. ICF (1979). A Prvate Penson Forecastng Model, Labor-Management Servces Admnstraton. 4

26 Keely, R. H. (1969). Penson Plan Decsons and Fnancal Polcy, Stanford Unversty. Kngsland, L. (198). Projectng the fnancal condton of a penson plan usng plan smulaton analyss. Journal of Fnance 37(): Lenarcc, M. (1977). Forecastng Penson Plan Cash Flows n an Inflatonary Envronment, Harvard Unversty. Levch, R. M. (1998). Internatonal Fnancal Markets: Prces and Polces. New York, NY, Irwn McGraw-Hll. Mulvey, J. M. (1988). A surplus optmzaton perspectve. Investment Management Revew 3: Mulvey, J. M. (1994). An Asset-Lablty Investment System. INTERFACES 4(3): -33. Mulvey, J. M. and H. Vladmrou (199). Stochastc network programmng for fnancal plannng problems. Management Scence 38(11): Mulvey, J. M. and W. T. Zemba (1995). Asset and lablty n a global envronment. Fnance. R. Jarrow, V. Maksmovc and W. T. Zemba. Amsterdam, North Holland: Nader, J. S. (1991). Ratonal Decson Rules for Early Retrement Inducements Contaned n Penson Plans. Journal of Rsk and Insurance 58: Neftc, S. N. (000). An Introducton to the Mathematcs of Fnancal Dervatves. New York, NY, Academc Press. 5

27 Sherrs, M. (1994). The Stochastc Valuaton of Superannuaton Benefts. Transactons of The Insttute of Actuares of Australa: Sherrs, M. (1995). The Valuaton of Opton Features n Retrement Benefts. Journal of Rsk and Insurance 6: Tepper, I. (1977). Rsk vs return n penson fund nvestment. Harvard Busness Revew. 55: Tepper, I. and A. R. P. Affleck (1974). Penson plan labltes and corporate fnancal strateges. Journal of Fnance 9: Wnklevoss, H. E. (198). PLASM: penson lablty and asset smulaton model. Journal of Fnance 37(): Zenos, S. A. (1997). Valuaton and Portfolo Rsk Management wth Mortgage-Backed Securtes. Advances n Fxed Income Valuaton Modelng and Rsk Management. F. J. Fabozz. New Hope, PA, Frank J. Fabozz Assocates:

28 Table 1: Annual Nomnal Return Summary Statstcs Nomnal Return Statstcs Intermedate- Term Long- Term Inflaton Rate Arthmetc Mean 13.8% 17.55% 5.50% 5.37% 5.94% 3.17% Geometrc Mean 11.35% 1.61% 5.1% 5.% 5.61% 3.07% Standard Error.34% 3.90% 1.08% 0.67% 1.01% 0.5% Medan 16.66% 0.09% 3.57% 4.06% 4.0%.87% Standard Devaton 0.14% 33.58% 9.30% 5.75% 8.71% 4.45% Sample Varance Kurtoss Skewness Range 97.3% 00.86% 49.53% 34.3% 50.65% 8.48% Mnmum % % -9.18% -5.13% -8.09% % Maxmum 53.97% 14.85% 40.35% 9.10% 4.56% 18.18% 7

29 Table : Annual Nomnal Return Varances, Covarances and Correlatons Intermedate- Term Long- Term Covarance Matrx Inflaton Rate Intermedate- Term Inflaton Rate Intermedate- Term Long- Term Correlaton Matrx Intermedate- Term Inflaton Rate

30 Table 3: Annual Real Return Summary Statstcs Real Return Statstc Intermedate- Term Long- Term Arthmetc Mean 10.00% 14.05%.50%.3%.9% Geometrc Mean 8.03% 9.7% 1.98%.08%.46% Standard Error.36% 3.83% 1.3% 0.81% 1.16% Medan 11.4% 16.05% 1.50% 1.65%.91% Standard Devaton 0.30% 3.94% 10.56% 7.01% 9.97% Sample Varance Kurtoss Skewness Range 90.91% 00.89% 50.58% 38.8% 5.71% Mnmum % -59.7% % -14.5% % Maxmum 53.5% 141.6% 35.13% 4.30% 37.6% 9

31 Table 4: Annual Real Return Varances, Covarances and Correlatons Intermedate- Term Long- Term Covarance Matrx Inflaton Rate Intermedate- Term Inflaton Rate Intermedate- Term Long- Term Correlaton Matrx Intermedate- Term Inflaton Rate

32 Table 5: Means, Varances, and Covarances of the Value Relatve Vector Y Asset Class, m Y s Y Intermedate-Term Inflaton Rate Covarance Intermedate -Term Inflaton Rate Intermedate -Term Inflaton Rate

33 Table 6: Means, Varances, Covarances and Correlatons of Vector Asset Class, m s 7.86% 18.30% 9.15% 8.30% 1.94% 10.7% Intermedate-Term.06% 6.84%.4% 9.66% Inflaton Rate 3.03% 4.3% Covarance, C Intermedate -Term Inflaton Rate Intermedate -Term Inflaton Rate Correlaton Intermedate -Term Inflaton Rate Intermedate -Term

34 Table 7: Estmated Internal Model Parameters Asset Class, µ σ 09.53% 18.30% 13.15% 8.30%.47% 10.7% Intermedate-Term.9% 6.84%.88% 9.66% Inflaton Rate 3.1% 4.3% Intermedate- Term Covarance, C Inflaton Rate Intermedate- Term Inflaton Rate Intermedate- Term Correlaton Intermedate- Term Inflaton Rate

35 Table 8: One-Year Horzon, Contnuous Smulaton - the Vector Y Asset Class, m Y s Y Intermedate-Term Inflaton Rate Covarance Matrx Intermedate -Term Inflaton Rate Intermedate -Term Inflaton Rate Correlaton Intermedate -Term Inflaton Rate Intermedate -Term

36 Table 9: One-Year Horzon, Contnuous Smulaton - Real Returns Asset Class Mean Sgma 10.03% 0.50% 14.14% 33.55%.51% 10.61% Intermedate-Term.3% 7.05%.93% 10.0% Inflaton Rate 3.17% 4.47% Covarance Matrx Intermedate -Term Inflaton Rate Intermedate -Term Inflaton Rate Correlaton Intermedate -Term Inflaton Rate Intermedate -Term

37 Table 10: One-Year Horzon, Contnuous Smulaton - Nomnal Returns Asset Class Mean Sgma 13.3% 0.51% 17.64% 34.47% 5.50% 9.5% Intermedate-Term 5.38% 5.71% 5.94% 8.56% Inflaton Rate 3.17% 4.47% Covarance Matrx Intermedate -Term Inflaton Rate Intermedate -Term Inflaton Rate Correlaton Intermedate -Term Inflaton Rate Intermedate -Term

38 Table 11: One-Year Horzon, Contnuous Smulaton - Vector Asset Class Mean Sgma 7.87% 18.36% 9.17% 8.40% 1.95% 10.30% Intermedate-Term.06% 6.87%.4% 9.70% Inflaton Rate 3.03% 4.33% Covarance Matrx Intermedate -Term Inflaton Rate Intermedate -Term Inflaton Rate Correlaton Intermedate -Term Inflaton Rate Intermedate -Term

39 Table 14: Estmated versus Smulated Effcent Portfolos Target Return Optmal Portfolo Weghts Intermedate- Term 6% 7.88%.13% % % % 7% 16.58% 4.01% % % % 8% 5.8% 5.90% % % -0.13% 9% 33.97% 7.79% % 10.8% 16.61% 10% 4.67% 9.67% % 104.9% 33.35% 11% 51.37% 11.56% % 87.76% 50.09% 1% 60.06% 13.45% % 71.4% 66.83% 13% 68.76% 15.33% -1.37% 54.71% 83.57% Portfolo Statstcs based on Ex Ante Means, Varances and Covarances Target Return Estmated 6% 7% 8% 9% 10% 11% 1% 13% Portfolo 4.51% 5.48% 7.% 9.31% 11.56% 13.89% 16.7% 18.68% Standard Devaton Portfolo 6.00% 7.00% 8.00% 9.00% 10.00% 11.00% 1.00% 13.00% Return Portfolo Varance Portfolo Statstcs based on Ex Post Means, Varances and Covarances Target Return Smulated 6% 7% 8% 9% 10% 11% 1% 13% Portfolo 4.46% 5.48% 7.9% 9.45% 11.76% 14.15% 16.59% 19.06% Standard Devaton Portfolo 6.01% 7.0% 8.0% 9.0% 10.03% 11.03% 1.03% 13.04% Return Portfolo Varance Percent Dfference Portfolo Standard Devaton Portfolo Return Portfolo Varance Percent Dfference between Ex Post and Ex Ante Portfolo Statstcs Target Return 6% 7% 8% 9% 10% 11% 1% 13% -1.03% 0.06% 0.99% 1.48% 1.74% 1.89% 1.97%.03% 0.% 0.3% 0.4% 0.5% 0.6% 0.7% 0.7% 0.7% -.05% 0.11% 1.98%.98% 3.51% 3.81% 3.98% 4.10% 38

40 Fgure 1: Estmated versus Smulated Average End of Year Real Value Inflaton Rate Intermedate-Term Estmated Mean Annual Value Relatve Smulated Fgure : Estmated versus Smulated Standard Devaton of End of Year Real Value Inflaton Rate Intermedate-Term Estmated Smulated 39

41 Fgure 3: Estmated versus Smulated Correlatons of End of Year Real Values IT vs Inflaton Rate LT vs Inflaton Rate LT vs Inflaton Rate vs Inflaton Rate vs Inflaton Rate vs IT vs LT vs LT vs IT vs LT vs LT vs LT vs IT IT vs LT LT vs LT Estmated Smulated 40

42 Fgure 4: Estmated versus Smulated Average Real Return Inflaton Rate 3.17% 3.17%.93%.9% Intermedate-Term.3%.3%.51%.50% 14.14% 14.05% 10.03% 10.00% Average Annual Real Return Estmated Smulated Fgure 5: Estmated versus Smulated Standard Devaton of Real Returns Inflaton Rate 4.47% 4.45% 10.0% 9.97% Intermedate-Term 7.05% 7.01% 10.61% 10.56% 33.55% 3.94% 0.50% 0.30% Annual Standard Devaton Estmated Smulated 41

43 Fgure 6: Estmated versus Smulated Correlatons of Real Returns IT vs Inflaton Rate LT vs Inflaton Rate LT vs Inflaton Rate vs Inflaton Rate vs Inflaton Rate vs IT vs LT vs LT vs IT vs LT vs LT vs LT vs IT IT vs LT LT vs LT Estmated Smulated 4

44 Fgure 7: Estmated versus Smulated Average Nomnal Returns Inflaton Rate 3.17% 3.17% 5.94% 5.94% Intermedate-Term 5.38% 5.37% 5.50% 5.50% 17.64% 17.55% 13.3% 13.8% Estmated Smulated Fgure 8: Estmated versus Smulated Standard Devaton of Nomnal Returns Inflaton Rate 4.47% 4.45% 8.56% 8.71% Intermedate-Term 5.71% 5.75% 9.5% 9.30% 34.47% 33.58% 0.51% 0.14% Estmated Smulated 43

45 Fgure 9: Estmated versus Smulated Correlatons of Nomnal Returns IT vs Inflaton Rate LT vs Inflaton Rate LT vs Inflaton Rate vs Inflaton Rate vs Inflaton Rate vs IT vs LT vs LT vs IT vs LT vs LT vs LT vs IT IT vs LT LT vs LT Estmated Smulated 44

46 Fgure 10: Estmated versus Smulated Mean Vector Inflaton Rate 3.03% 3.03%.4%.4% Intermedate-Term.06%.06% 1.95% 1.94% 9.17% 9.15% 7.87% 7.86% Estmated Smulated Fgure 11: Estmated versus Smulated Standard Devaton of the Vector Inflaton Rate 4.33% 4.3% 9.70% 9.66% Intermedate-Term 6.87% 6.84% 10.30% 10.7% 8.40% 8.30% 18.36% 18.30% Estmated Smulated 45

47 Fgure 1: Estmated versus Smulated Correlatons of the Vector IT vs Inflaton Rate LT vs Inflaton Rate LT vs Inflaton Rate vs Inflaton Rate vs Inflaton Rate vs IT vs LT vs LT vs IT vs LT vs LT vs LT vs IT IT vs LT LT vs LT Estmated Smulated 46

48 Fgure 13: Estmated versus Smulated Effcent Fronter 15% 14% 13% 1% Portfolo Return 11% 10% 9% 8% 7% 6% 5% 4% 6% 8% 10% 1% 14% 16% 18% 0% Portfolo Standard Devaton Estmated Smulated 47