Cha p t e r 4

A t ti t udes T o w ards R i sk

In th e p rev i ou s l ectu r e , w e ex p l o r ed the i m p lica tion s o f e x p ec ted u tilit y m ax im iz atio n. In th is lec t u r e, co n s id er in g t h e lotte r ies o v e r m on e y , I w ill in tro d u c e t h e ba sic n o t ion s reg a r d in g r isk, suc h as risk a v ersio n a n d c ertain t y equ i v a len c e. U n d e r stan d i n g these co ncep ts is essen ti a l to follo w m o s t a r e as in m o d e rn econ om ics.

4. 1 T heory

T a k e the s et of alter n ativ es as X = R w h ic h c orr esp o n d s th e w ealth l ev el o f th e d ecision m a k e r. T h e d e c ision m ak er ha s a n i ncr e asin g v o n N e u m an n - M o r g en ste r n u t i lit y f u n c - tion u : R → R , r ep resen ti n g h i s p refer e n ces o v er the l o tter i es on h i s w ealth l ev el. I w i ll assum e th at u is d i ff eren ti a b le w h enev er n eed ed . S in ce w e h a v e a c on tin u u m o f con se - qu en ces, it is m o re con v en ien t to rep r esen t lotteries b y c um u l ativ e d i s tribu t ion f u n c tion s F : X → [0 , 1] . I w r i t e f f o r t he de ns i t y o f F w h en it exists. T h e exp e cted u t ili t y o f F is giv e n b y

U ( F ) ≡ E F ( u ) ≡ Z u ( x ) dF ( x ) ,

whe r e E F is the e xp ectation op erator u n d e r F . T h e ex p ected w e a l th lev e l u nd er F is

E F ( x ) = Z

xdF ( x ) .

By c o mpa r i n g E F ( x ) to E F ( u ) , o ne can l ea rn a b ou t t h e d ecision m a k e r’s a ttitu d e s to w a rd s r isk.

33

34 CHAPTER 4 . A TTI TUDES TO W ARDS RI SK

A d ecision m a k er i s c a l l e d ri sk aver se i f he a l w a ys pre f e r s s ure w ea l t h l ev el E F ( x ) to th e l ottery F , i . e . ,

E F ( u ) ≤ u ( E F ( x )) ( ∀ F ) .

He i s c a l l e d strictl y risk ave r se if the i n e q u alit y i s a lw a y s strict f o r no nd eg en era te l o t - t e r i e s . H e i s c a l l e d ri sk n e utr a l if he is a l w a y s in di ff eren t:

E F ( u ) = u ( E F ( x )) ( ∀ F ) .

F i na ll y , h e is c a lled ri sk se eki n g (or r isk l o v in g) if h e p r efers l ottery to th e s ure o u t com e , i.e.,

E F ( u ) ≥ u ( E F ( x )) ( ∀ F ) .

C l earl y , b y J e n sen ’s inequa lit y , w h i c h y ou m u st kn o w b y n o w , risk a v ersio n cor r e - sp o n d s to the c o n ca vit y of the u ti lit y f u n ction :

• D M i s r i sk a v erse i f an d o n l y i f u i s con c a v e;

• h e is strictl y ri sk a v erse if an d o n l y i f u is strictly c o n ca v e ;

• he i s ri s k ne ut ra l i f a nd onl y i f u is line a r, a n d

• h e is risk seeking if an d o n l y i f u is con v ex.

A n oth e r w a y to assess the a ttitu des t o w ard s risk i s certain t y e qu i v a l en ce. T h e ce r - ta in ty e q u i va len t of a l ottery F , d en oted b y CE ( F ) , i s a su re w e a l th lev el t h a t y ield s th e s a m e e x p ec ted u tilit y a s F . T h a t i s ,

CE ( F ) = u − 1 ( U ( F )) = u − 1 ( E F ( u )) .

It is i m m e diate f r om t he de fi ni t i o ns t ha t

• D M i s r i sk a v erse i f an d o n l y i f CE ( F ) ≤ E F ( x ) for a ll F ;

• he i s ri s k ne ut ra l i f a nd onl y i f CE ( F ) = E F ( x ) fo r a ll F , a n d

• h e is risk seeking if an d o n l y i f CE ( F ) ≥ E F ( x ) for a ll F .

It is so m e ti m e s u seful to qu an tify th e d egr e e o f r isk a v e rsio n. T h ere a re t w o i m p orta n t m e asu r es o f risk a v ersio n . T h e fi rst o ne is absol u t e risk aversi on :

r A ( x ) = − u 00 ( x ) /u 0 ( x ) ,

w h ic h i s a lso c alled A rro w - P r at t c o e ffi ci en t o f a bsol ute r i s k a v e rsi o n. Note that u 00 m e asu r es th e c on ca vit y o f th e u ti lit y f un cti o n , w h ile u 0 n o r m al izes the c on ca vit y as th e u t il it y r ep resen t ati o n i s u n i q u e u p t o a ffi n e tra n sfor m a tion s.

A c on v e nien t a ssu m p tion in econ om ic an alysis is c o nstant absol u t e ri sk aversi on

( C A R A) . A CA RA ut i l i t y f unc t i o n t a k e s t he s i m p l e f o rm of

u ( x ) = − e − αx ,

whe r e α is the c o e ffi cien t o f a b s o l u t e r isk a v e rsio n. T h is u t ilit y f un ctio n b e c om es es p e - cia l ly con v en i e n t w h en th e l otteries a r e d istribu t ed n o rm all y . I n t ha t c a se, th e c erta i n t y equ i v a len t b eco m e s

1

CE ( F ) = μ − ασ 2

2

whe r e μ an d σ 2 are t h e m e a n an d t h e v a rian ce of th e d istrib utio n, re sp ec tiv e l y . W h ile C A R A is a c on v e n i en t a ssu m p tion , s o m e m a y fi nd i t m o re pl aus i bl e t hat a bs ol ut e r i s k a v ersion is decrea sing w i th w e a l th lev e l ( D A R A ), so th a t ric h er p e op le tak e high er risks. In deed , s om e m a y w a n t to n o rm a l ize t h e am ou n t o f r i sk a v ersion w i th resp ect to th e lev e l o f w e a lth . T h is lead s t o t h e c o n cep t o f r e la tiv e r isk a v e r s io n . T h e coe ffi cient o f

r e la tiv e r is k a ver s i o n is

r R ( x ) = − xu 00 ( x ) /u 0 ( x ) .

The c o n s ta n t r e la tiv e r is k a ve rs io n (C R R A ) u t ilit y f u n c t ion t ak es th e f orm o f

u ( x ) = − x 1 − ρ / (1 − ρ ) ,

whe r e ρ is t h e c o e ffi ci en t o f c onstan t r el ati v e r i s k a v e rsi o n. W h en ρ = 1 , i t i s t h e l o g ut i l i t y f unc t i o n : u ( x ) = l o g ( x ) .

U s i n g t he ab o v e c once pt s , one c an al s o c o m p are t he at t i t u des o f t w o dec i s i on m a k e rs to w a rd s r isk. T o this en d , ta k e an y t w o d ecision m a k e rs D M 1 a n d D M 2 w ith u 1 and u 2

an d w rite CE i ( F ) ≡ u − 1 ( E F ( u i )) and r A, i = − u 00 /u 0

f o r t he certain t y e quiv al en t a nd

i i i

co e ffi c i en t o f a bs o l ut e r i s k a v e rs i o n u nde r u i fo r i ∈ { 1 , 2 } .

De fi ni ti on 11 D M 1 i s m o r e r is k a ve r s e t h a n D M 2 if eith e r o f th e e qu iva l en t c o n d i tio n s in th e n ex t p r o p o s itio n h o l d s .

P r op os i t i on 3 T h e f o l l o w i n g a r e e q u i v a l e n t .

1. u 1 = g ◦ u 2 fo r s o m e c o n c a ve fu n c tio n g ,

2. CE 1 ( F ) ≤ CE 2 ( F ) for every F ;

3. r A, 1 ≥ r A, 2 ever ywher e .

Pr o o f . Si nc e b ot h u 1 and u 2 are i n c reasin g, th ere e x i sts an i n c reasin g f u n ction g su c h th at u 1 = g ◦ u 2 . T o s ee th e e qu iv a l en ce b e t w een 1 a nd 2, no te th at CE 1 ( F ) = u − 1 ( g − 1 ( E F ( g ( u 2 )))) . B y J en sen ’s ine q ua lit y , g is co nca v e if a n d on ly if E F ( g ( u 2 )) ≤ g ( E F ( u 2 )) fo r e v e r y F . T h u s , g is co nc a v e i f a n d o n ly for e v ery F ,

CE 1 ( F ) = u − 1 ¡ g − 1 ( E F ( g ( u 2 ))) ¢

≤ u − 1 ¡ g − 1 ( g ( E F ( u 2 ))) ¢ = u − 1 ( E F ( u 2 ))

2 2

= CE 2 ( F ) ,

w h ere t he i n equ a l i t y uses also the f a c t t h at g − 1 is inc r ea sing .

T o s e e t he e q ui v a l e nc e b e t w e en 1 a nd 3 , not e that

u jj

g jj u j

+ g j u jj

u jj g jj g jj

H e n ce,

r A, 1 = − 1

1

= − 2 2

g j u j

= − 2 −

2 g j

= r A, 2 − g j .

g jj = g j · ( r A, 2 − r A, 1 ) .

Th us , r A, 1 ≥ r A, 2 ev eryw h e re if an d o nly i f g jj ≤ 0 e v e r ywhe re, w hi c h i s t r ue i f a n d o nl y if g is c o n c a v e .

Si nce one ca n e n v i s i on a nd i n di vi dual wi t h t w o d i ff eren t i nitia l w e alth s a s t w o d i ff er en t d ecision m a k e r s , t he a b o v e c h a ra cterizati o n a l l o w s o n e to ex plor e h o w on e’s a ttitu d e t o w a rds r i s k c ha ng es a s hi s i ni t i al w e al t h l e v e l c hange s . T o do t h i s , l e t us wri t e w for t h e initia l w e a lth l ev el o f a n ind i v i du al an d w rite lo tter i e s a s c h a n ges i n h is w e a l th. T ha t i s, giv en a n y lo ttery z , t h e fi na l w e a l th o f t he i n di vi dual i s x = w + z . D e fi ne u ( ·| w ) by

u ( z | w ) = u ( z + w ) .

The c o e ffi c i e n t o f a b s olu t e r isk a v e rsion u nd er in itial w e a lth w 0 is

r A ( z | w ) = − u jj ( z + w ) /u j ( z + w ) = r A ( z + w ) .

Co ro l l a r y 1 T h e d e c isio n m aker b e c o m e s l ess r isk a verse a gainst the c ha nges in his we a l t h ( z ) w h e n h is in itia l w e a l th in c r e a s e s i f a n d o n ly if h e h a s d e c r e a s in g a bs o l u t e r is k aversion.

Pr o o f . By Prop os i t i o n 3 , i t s u ffi ces t o s ho w t hat r A ( ·| w ) i s de cre a si ng i n w (i.e. r A ( ·| w j ) ≤ r A ( ·| w ) w h en ev er w j ≥ w ) i f a n d o n l y i f r A is decreasing. B ut thi s is im m e d i ate b eca use r A ( z | w ) = r A ( z − w ) by d e fi nition.

O n e c an f u rth e r c on clud e t h a t if th e d e c is io n m a k e r h a s c o n sta n t a bso l u t e r isk a v e r - sio n , t h e n h is a t titu d e to w a r d th e r isk i n c h a n g es in h i s w e a lth ( z ) i s i n d ep en den t o f his in itia l w e a lth .

Si mi l a r f ac t s ca n b e o bt ai ne d a b o ut the d ec i s i o n m a k e r ’ s at t i t u de s t o w ard t he ri s k i n m u ltip licat io n o f h is w e alth , u sin g relativ e r i s k a v e r sion in stea d. T o d o th at, w rite y fo r th e m u l t i p licat io n o f h is initial w ea lth s o t ha t h is fi nal w eal t h l ev el i s x = yw . D e fi ne u y ( ·| w ) by

u y ( z | w ) = u ( yw ) .

The c o e ffi ci en t o f a bs o l ut e r i s k a v e rs i o n a g a i ns t y unde r i ni ti al w e a l t h w is

r A, y ( z | w ) = − u jj ( y | w ) /u j ( y | w ) = − w 0 u jj ( yw ) /u j ( yw ) = r R ( yw ) .

y y

H e n ce,

Co ro l l a r y 2 D M ’ s ri sk aversi on agai n s t t he m u lt i p li c a t i o n y i n h i s w e a l t h i s d e c r e a s i n g in h i s i n itia l w e a lth w 0 if h e h a s d e c r e a s in g r ela tiv e r is k a ver sio n r R ; D M ’ s r isk a ve r s io n agai n s t t he m u lt i p li c a t i o n y in h i s w e a lth i s i n d ep e n d e n t o f h i s i n itia l w e a l th w 0 if h e h a s c o n s ta n t r e la tiv e r is k a ver s i o n r R .

4 . 2 A p p lic a t io n s

4. 2. 1 I nsurance

C o nsid er a d ecision m a k e r w h o h a s i nitial w e alth o f w an d m a y lose 1 un it o f h i s w ealth w i th pro b ab ilit y p . H e c a n b u y an insu ra nce , w h ic h i s a divisible g o o d . A unit insurance

co sts q an d c o v ers o ne un it o f lo ss in case of a l oss. W e w a n t to un d e rsta nd h i s d em a n d f o r i nsurance. L et λ b e the a moun t o f i ns ura n c e he buys . H i s e xp ec t e d u t i l i t y i s

U ( λ ) = u ( w − qλ ) ( 1 − p )+ u ( w − qλ − (1 − λ )) p.

Fi rs t c ons i der t he cas e of ac tuari a l l y unf ai r p ri c e q > p , w hic h is na tur a l g iv en th at th e insura nce c o m p a n y needs t o c o v er its o p e ratio n a l costs. In th a t ca se, h e b u ys on l y a p a rtia l i n s u r an ce , i.e., λ < 1 . I ndeed,

U j (1) = ( p (1 − q ) − q (1 − p )) u j ( w − q ) < 0 ,

i.e. , U is strictly in crea sing at th e f u l l i n s u r an ce lev el, an d h en ce o p ti m a l λ mu s t b e less tha n 1 . T h erefo r e, he b e a r s s om e o f t he risks n o m atter h o w risk a v erse h e is a n d ho w l o w t h e m ark u p q − p is. T h i s i s b ecau se w h en th e a m o u n t o f r i sk g ets l o w er a n d lo w e r, u b e com es a p p r o x im ately lin ear a nd the d ecisio n m a k e r b eco m es a pp ro xim a tely ri s k ne ut ra l .

N o w c on sider the ca se o f q = p , t h e a c tu a r ia lly fa ir pric e. T h is ca se is im p o rta n t in th e litera t u r e b ecau se it cor r esp o n d s to th e c om p etitiv e p r ice ( assu m i n g insura nce co m p an ies d o n ot h a v e a n y o th er costs). I n t ha t c a se, h e b u ys full insu ra nce ( i. e. λ = 1 ). T o see t h i s, no te tha t un d e r a ctua rially fair p r ice, h i s e xp ected w ea l t h i s E λ [ x ] = w − q fo r e a c h λ . H ence, f or an y λ < 1 ,

CE ( λ ) < E λ [ x ] = w − q = CE (1) ,

whe r e CE ( λ ) is the c erta i n t y equ i v a len t of w e a l th w h en h e b u ys λ units o f i nsurance. Th us , λ = 1 yield s high er certa in t y equiv a l e nce t h a n a n y o t h e r λ .

F i n a ll y , c o n sid e r a m o re risk a v er se decision m a k e r w ith c erta i n t y equ i v a len c e o p e r - ator CE j . I f t he fo rm er d ecisio n m a k e r b uys f u l l i nsu r an ce, s o w ill th e n ew on e. Ind e ed, fo r a n y λ < 1 ,

CE j ( λ ) ≤ CE ( λ ) < C E (1 ) = CE j (1) ,

whe r e t he fi rst i n e qu alit y i s t h e fact tha t th e n ew d ecision m a k e r i s m ore r isk a v e rse, th e secon d i n e qu alit y i s b y t h e fact tha t fu ll insu ra nce w a s o p tim a l f o r th e o rigin a l d ecision mak e r a nd t h e e qua l i t y i s b y t he f a ct t h at t h e r e i s no r i s k u nde r f ul l i ns ura n c e .

4. 2.2 O p t im al P o rtfol i o C h oi ce

C o nsid er a d ecision m a k e r w ith i n i ti a l w e alth w . T h e re i s also a r isky asset th at yield s z for e a c h d o llar i n v e s ted . W r ite F for t h e cd f o f z . W e w a n t t o u nde rs t a nd ho w m uc h th e d ecisio n m a k e r w ou l d in v e st in th e r i s ky asset. W rite α f o r t h e l e ve l o f i nve s t m e nt an d α ∗ for t he op tim a l α . T h e ex p ect ed u t ilit y i s

U ( α ) = Z

u ( w + α ( z − 1)) dF ,

whi c h i s a co nc a v e f unc t i o n . T he opt i m a l i n v e s t m e n t i s det e r m i ned b y t he fi rst-order co nd ition

U j ( α ∗ ) = Z u j ( w + α ∗ ( z − 1)) ( z − 1) dF = 0 .

F i r s t o b s e r v e t h a t h e w i l l n o t t a k e a n y r i s k i f t h e e x p e c t e d r e t u r n E [ z ] − 1 is n o t p o s it iv e .

Ind e ed, i f E [ z ] − 1 ≤ 0 ,

U j (0) = Z

u j ( w ) ( z − 1) dF = u j ( w ) ( E [ z ] − 1) ≤ 0 .

O n the o th er h a n d , h e w il l i n v est a p o s i t i v e a m o u n t a s l on g a s a n y p o sitiv e exp e cted return ( E [ z ] − 1 > 0 ):

U j (0) = u j ( w ) ( E [ z ] − 1) > 0 .

T h is is , a ga in, b e c au se h e is a p p r o x im a t ely r isk n eu tral a g a i n s t s m a ll risk s.

A m a i n fi n d in g i n t h i s e xam p l e is tha t m o re r i sk a v erse ag en ts in v est less in th e risky a sset. I w i l l sho w this i n tuitiv e f act f o r m a ll y n ext. C o nsi d er t w o d ecision m ak ers D M 1 a nd D M 2 w ith u tilit y f u n c tion s u 1 and u 2 , r esp ectiv el y , su c h th at D M 1 i s m o r e risk a v erse th an D M 2. H e nce, u 1 = g ◦ u 2 f o r s ome c onca v e i n cre a s i ng f unc t i o n g wi t h g j ( w ) = 1 . D e n ot e t he v a ri abl e s f or de ci s i on mak e r i b y sub scrip t i , e .g., b y w r itin g

∗ an d α ∗

fo r t he o p ti m a l i n v estm en ts of D M 1 a n d D M 2, resp ecti v e l y . N o w , f or a n y

α , s i n c e u j ( w + α ( z − 1) ) = g j ( w + α ( z − 1)) u j ( w + α ( z − 1)) , u j ( w + α ( z − 1)) ≥

1 2 1

u j ( w + α ( z − 1)) if an d o nly i f z ≤ 1 . H en ce, [ u j ( w + α ( z − 1)) − u j ( w + α ( z − 1))] ( z − 1) ≤

0 ev erywhere. T h u s, f o r e v e ry α ,

U j ( α ) − U j ( α ) = Z

[ u j ( w + α ( z − 1)) − u j ( w + α ( z − 1))] ( z − 1) dF ≤ 0 .

T h erefor e, α ∗ ≤ α ∗ . ( O n e w a y to see t his i s t o o b serv e th a t U j ( α ∗ ) ≤ U j ( α ∗ ) = 0 .

1 2 1 2 2 2

H e n ce, U 1 is decrea sing at α ∗ a n d m ust h a v e b een m axim i z ed at a l o w er v a lue.)

T o g e th er w i th C o rollary 1, th e a b o v e fi ndi ng yi elds the f ollo w i ng m o notone com p ar - ati v e s ta tics o n th e o pti m al in v estm en t l ev el as a f u n ction o f i n i tial w e alth :

• if the a g e n t h a s d e c re asing a bso l ute r isk a v e rsion , t h en α ∗ is in crea sing w i th th e in itia l w e a lth l e v e l w ;

• i f t h e a ge n t has c ons t an t a bs ol ute r i s k a v e rs i o n, t h e n α ∗ i s i n de p e nden t o f t he in itia l w e a lth l e v e l w .

The o pt i m al l e v el o f i n v es t m en t a s a prop ort i on of t h e i ni t i al w e al t h i s re l a t e d t o t he rela tiv e risk a v ersion . T o see this, w rite β = α/w , a n d ob serv e th a t th e fi n a l w ealth l ev el is

x = w + βw ( z − 1) = w · (1 + β ( z − 1)) .

H e nc e , t h e r i s k i s a b o ut t h e m ul ti pl i c a t i on 1+ β ( z − 1) of hi s i ni t i al w e al t h . F rom t he ab o v e fi n d in g a nd C o rollary 2 , w e can c o n c l u d e fo llo w i n g :

• If D M h a s d e c re asin g r e l ativ e r isk a v e rsio n, th en th e o p t im a l in v e s t m e n t lev el β ∗

a s a p rop o rtio n o f t h e in itial w e a lth w is inc r eas i n g in w ;

• If D M ha s c o n sta n t r e l a t iv e r isk a v e r s io n , th en th e o ptim al in v e stm e n t lev e l β ∗ as a p ro p o rtion o f t he in itial w e a lth w i s i nde p e nde n t of w ; i.e. α ∗ = bw fo r s o m e constan t b .

4. 2.3 O p t im al R i sk Sh ari n g

C o nsid er a s et of a g en ts N = { 1 ,..., n } . E a c h i ha s a co nc a v e, di ff er en tiab le, a nd b o unde d u ti l i t y f unct i o n u i . T here is an unkno w n state s ∈ S . E a c h a g e n t i ha s a ri s k y

asset x ¯ i : S → R , w h o se ou tco m e dep e n d s o n the sta te. A f ea sible a llo cati o n i s a list

( x 1 ,..., x n ) o f c o ns umpt i o n p l ans x i : S → R su c h th at

x 1 ( s )+ ··· + x n ( s ) ≤ x ¯ 1 ( s )+ ··· + x ¯ n ( s ) (4 . 1 )

fo r e a c h s . W e w an t t o e xp lore th e P areto-opti m al a l l o c ations. T o t his e nd , w rite A fo r th e set of a l l f ea sible a ll o c atio ns. N o te t h a t A is a c on v ex s e t. W rite a l so

V = { ( E ( u 1 ( x 1 )) ,...,E ( u n ( x n ))) | ( x 1 ,..., x n ) ∈ A }