Technical Appendix to Risk, Return, and Dividends

Size: px

Start display at page:

Download "Technical Appendix to Risk, Return, and Dividends"

Jasper Townsend
9 years ago
Views:

1 Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027, ph: (212) ; fax: (212) ; [email protected]; WWW: hp:// aa610. Rady School of Managemen, Pepper Canyon Hall, Room 320, 9500 Gilman Dr, MC 0093, La Jolla, CA ; ph: (858) ; fax: (858) ; [email protected]; WWW: hp://rady.ucsd.edu/cms/showconen.aspx?conenid=183

aa610@columbia.edu; WWW: hp://www.columbia.edu/ aa610.

2 A Relaion of Proposiion 2.1 o Pricing Kernel Formulaions By definiion, given he dividend process D, he price of he sock is given by: [ ] P = E Λ s D s ds, (A-1) under he pricing kernel process Λ, ogeher wih a ransversaliy assumpion. We assume ha he pricing kernel follows: dλ Λ = r f (x )d ξ x (x )db x ξ d (x )db d, (A-2) where r f ( ) is he risk-free rae process, and ξ x and ξ d are prices of risk corresponding o shocks o he sae variable x and dividend growh, respecively. Using equaion (A-1), we can express he price-dividend raio as: [ P ( = E exp D s (r f (ξ2 x + ξd)) 2 du + ξ x dbu x + ξ d dbu d ( s ) ] exp µ d du + σ d dbu d ds, assuming ha σ dx = 0 for simpliciy. This can be equivalenly wrien as: [ P ( s = E Q exp (r f µ d 1 ) ] D 2 (σ d ξ d ) 2 ) du ds, (A-3) where he Radon-Nikodym derivaive defining he risk-neural measure Q is given by: ( dq dp = exp s Noe ha equaion (A-3) is a funcion f( ) of x. ) 1 2 (ξ2 x + (σ d ξ d ) 2 ) du ξ x dbu x (σ d ξ d )dbu d. (A-4) We describe how a paricular choice of a reurn process dr, ogeher wih assumpions on dividends, places resricions on he underlying pricing kernel process dλ hrough he following proposiion: Proposiion A.1 Suppose he sae of he economy is described by x, which follows equaion (1), and a sock is a claim o he dividends D ha are described by equaion (2) wih σ dx = 0. If he sock reurn follows equaion (8) and he pricing kernel process follows equaion (A-2), hen he price-dividend raio P /D = f(x ) saisfies he following relaion: (µ x ξ x σ x )f σ2 xf (r f µ d 1 2 σ2 d + ξ d σ d )f = 1, (A-5) ) 1

3 which deermines he price-dividend raio f. This implies ha he expeced reurn µ r ( ) and volailiy σ rx ( ) of he reurn are given by: µ r = r f + ξ x σ x (ln f) + ξ d σ d, σ rx = σ x (ln f) (A-6) Proof: Equaion (A-5) is he sandard Feynman-Kac pricing equaion. Once he price-dividend raio f is obained from solving equaion (A-5), we can derive equaion (A-6) by equaing erms from he drif erm of dr and he diffusion erm on db x in equaion (8). Proposiion A.1 saes ha, given he dividend sream, he pricing kernel compleely deermines he price-dividend raio f, he expeced reurn of he sock µ r, and he volailiy of he sock σ rx. However, if we specify he price of he sock, he expeced reurn, or he volailiy of he sock (each one being sufficien o deermine he oher wo from Proposiion 2.1), he shor rae r f, he prices of risk ξ x and ξ d, or he pricing kernel Λ are no uniquely deermined. For example, suppose we specify µ r. There are poenially infiniely many pairs of r f and ξ = (ξ x, ξ d ) ha can produce he same µ r. For example, one (rivial) choice of ξ is ξ = (0, 0) corresponding o risk neuraliy, and he sock reurn is he same as he risk-free rae. Whereas Proposiion 2.1 shows ha specifying µ r, σ rx, or f compleely deermines he reurn process, he resul from Proposiion A.1 implies ha a single choice of µ r, σ rx, or f does no necessarily deermine he pricing kernel. B Mulivariae Sae Variables Suppose ha here are K sae variables, so ha x = (x 1,..., x K ) represens a K 1 vecor of diffusion processes. We le x follow he diffusion process in equaion (1), where µ x ( ) is a vecor funcion of x and σ x ( ) is a marix funcion of x. Similarly, dividend growh saisfies equaion (2) where he scalars µ d ( ), σ d ( ) are poenially funcions of x. For exposiional simpliciy, we assume ha σ dx = 0 and denoe he scalar price-dividend raio by P/D = f(x). Suppose ha he reurn R saisfies he following diffusion equaion: dr = µ r (x )d + σ rx (x )db x + σ rd (x )db d K = µ r (x )d + σ rxi (x )db x i + σ rd (x )db d, (B-1) i=1 2

pricing equaion. Once he price-dividend raio f is obained from solving equaion (A-5), we can derive equaion (A-6) by equaing erms from he drif erm of dr and he diffusion erm on db x in equaion (8).

4 where µ r ( ) is a scalar funcion of x, σ rx ( ) is a marix funcion of x, σ rxi represens he ih row of σ rx, and he vecor of Brownian moions db x is pariioned as db x = (db x 1... db x K ). From he definiion of he reurn dr = df /f + dd /D + 1/f d, he diffusion erm of he reurn is given by: ( ) ln f σ rxdb x + σ d db d. Thus, in order for σ rx o represen he diffusion coefficiens of a reurn, we mus have: σ rx = (B-2) ( ) ln f σ x, (B-3) or, equivalenly: σ rxi = K j=1 ln f j (σ x ) ji, where (σ x ) ji is he elemen of σ x in he jh row and ih column. From equaion (B-3), here mus be a funcion f such ha: ln f The necessary and sufficien condiion for his is: = (σ rxσ 1 x ). (B-4) Assumpion B.1 The diffusion coefficiens σ x and σ rx saisfy he inegrabiliy condiion: j (σ rx σ 1 x ) i = i (σ rx σ 1 x ) j. (B-5) Noe ha unlike he univariae case, we canno arbirarily specify he diffusion coefficiens σ rx of he reurn. If σ rx does no saisfy he inegrabiliy condiion, hen equaion (B-1) canno represen a reurn implied from a pricing funcion. The mulivariae version of equaions (8) and (11) in Proposiion 2.1 are: Proposiion B.1 Suppose ha he reurn R follows he diffusion equaion (B-1) and ha diffusion raio σ rx σ 1 x saisfies he inegrabiliy condiion Assumpion B.1. Then, he price-dividend raio and he expeced reurn are deermined up o inegraion consan. Proof: From he inegrabiliy condiion (B-5), i follows from elemenary calculus ha here exiss a funcion f ha saisfies: d ln f = σ rx σ 1 x dx. (B-6) 3

Thus, in order for σ rx o represen he diffusion coefficiens of a reurn, we mus have: σ rx = (B-2) ( ) ln f σ x, (B-3) or, equivalenly: σ rxi = K j=1 ln f j (σ x ) ji, where (σ x ) ji is he elemen of

5 Equaion (B-6) is he mulivariae version of equaion (11). The funcion f is he dividend yield, and i is unique up o a muliplicaive consan (since ln f is deermined up o an addiive consan). The expeced reurn is hen deermined by: µ r (x) = µ x f f σ x σ x f + 1 f Equaion (B-7) is he mulivariae version of equaion (8). + µ d σ2 d. (B-7) The inegrabiliy condiion in Assumpion B.1 imposes srong resricions on mulivariae sochasic volailiy processes. For example, he diffusion process v1 db 1 + v 1 v2 db 2, where v 1 and v 2 are sochasic processes, canno represen he reurn diffusion process of a valid pricing funcion because i violaes he condiion (B-5). 4

The expeced reurn is hen deermined by: µ r (x) = µ x f + 1 2 f σ x σ x f + 1 f Equaion (B-7) is he mulivariae version of equaion (8). + µ d + 1 2 σ2 d.

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian