INVESTMENTS REVIEW: WEEK 1: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS:

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1 INVSTMNTS RVIW: WK 1: RISK AVRSION AND CAITAL ALLOCATION TO RISKY ASSTS: xpected Retun: - Thee is a consideable uncetaint about the utue pice o a stock and the dividend income that it pas - To quanti ou belies about utue state o eth econom and stock maket, we usuall assign pobabilities to each scenaio that we might have o the econom and stock maket - Then, the expected etun is given b: p s s s - Whee: s is a scenaio, ps is the pobabilit o each scenaio, and s is the expected etun geneated b that scenaio - Also, etun toda can be shown to be: R t t t1 t1 - See example in lectue notes Standad Deviation: - To ind the standad deviation o the expected etun when pobabilities ae pesent, we use the ollowing omula: s p s s otolio Constuction: - Thee ae two main steps in the pocess o constuction o a potolio: i Selection o the isk assets, such as shaes and bonds ii Decision o how much to invest in the isk potolio against the isk-ee asset - We need to know the expected etun o the potolio and the degee o isk to decide how much to allocate between the isk-ee asset and the isk potolio - The decision o how much to invest ultimatel depends on the individual investo s pesonal peeences about isk and expected etun Concept o isk avesion: - A isk avese individual is one who pees less isk o the same expected etun i.e. investos must be compensated to bae isk - Most investos ae isk avese - Risk-avese investos eject investment oppotunities with a isk pemium o zeo o less

2 Utilit: - We have to put some stuctue on the concept o isk avesion. In this wa we will bette undestand the dnamics in the investment pocess - Although investos ae pesumed isk avese, each investo will ace dieent tade-o decisions between dieent isk and expected etuns - What will inluence the tade o decisions? i Such actos as dieent degees o unwillingness to bea isk which will be elected in how much additional expected etun the investo equies o taking on an additional unit o isk ii Anothe acto will be how much the investment could aect the investo s total wealth A potential loss o $1000 would pobabl wo a millionae less than someone with eanings o $100 pe week Wh is the concept o utilit impotant? - Knowledge o the investo s utilit unction enables him/he to choose between secuities - In a single measue we have the investo s attitudes to isk and etun at each level o wealth - It is diicult without knowledge o the utilit unction to make decisions between dieent secuities with dieent expected etuns and dieent isks - i.e. Utilit allows us to seach o the optimal potolio Utilit unction: - Thee ae countless utilit unctions. An example is: U 0.5A - Whee i U = the utilit value ii A = the coeicient o isk avesion highe coeicient o isk avesion, moe isk avese iii = vaiance - Utilit is inceased b highe expected etuns and loweed b highe isk - Utilit o a isk-ee potolio is equal to its ate o etun - Moe isk-avese investos i.e. those moe unwilling to take isk, o those who equie highe expected etun to take on one unit o isk, as noted above, will have lage values o A - Investos will assign highe utilit values to potolios with moe attactive isk-etun poiles as shown via this example o a utilit unction above - Fo the given utilit unction i I: A=0, the investo is isk neutal onl inteested in expected etun A>0, then the investo is isk avese standad assumption A<0, then the investo is isk love ve unlikel - See example in lectue notes Tadeo between isk and etun:

3 Nothwest unambiuguousl peeed diection I II III IV - Unambiguousl pee to move nothwest o potolio i.e. quadant I i I move nothwest then ae gaining geate expected etun o less o the same isk, theeoe beneicial! - Unambiguousl pee not to move southeast o potolio i.e. quadant IV i I move southeast, then ae gaining geate isk o the same o less expected etun, theeoe detimental What about quadents II and III? - otolios in these two quadants will depend on the natue o the investo s isk and etun peeences o isk avesion - ossible that thee ae othe potolios in these quadents that ae equall attactive to the investo - This means othe potolios should have the same utilit level and the investo is indieent among such potolios same utilit level, then investo is indieent! Indieence cuves: - The indieence cuves epesent a set o isk and expected etun combinations that povide the investo with the same amount o utilit. The indicate an investo s peeence o isk and etun - Dawn on two-dimensional gaph whee the hoizontal gives isk and the vetical shows expected etun nom - An indieence cuve connects all potolios with the same utilit level will give an indication o which potolios ae the best o an given investo and thei given isk peeences - Some o these potolios will pobabl be high isk, high etun potolios and othes will be low isk, low etun potolios xample o indieence cuve:

4 Indieence cuve Q - An investo with the above indieence cuve will be in indieence apathetic between an o the potolios that lie on it; i.e. the investo would be indieent between potolio and potolio Q - Note that: Given investos peeence to move NW, investos will alwas want to be above the cuve Capital allocation: - Investos constuct potolios b consideing secuities om man dieent asset classes - The will also choose how much to invest in each asset class - See example in lectue notes - Given these weights 70% in isk assets and 30% in the isk-ee assets, the pupose o the investo is to educe isk b changing the isk/isk-ee asset mix - To ind the optimum potolio o the investo, we need to conside two things: i The isk-etun combination available to the investo ii esonal isk-etun peeences o the investo i The isk-etun combinations available to the investo: otolio o a isk and a isk-ee asset: - It s possible to split investment unds between sae and isk assets Risk ee asset: ox T-Bills Note: iskee = 0, cov iskee, x = 0 Risk asset: Stock o potolio - See example in lectue notes The investment oppotunit set:

5 Leveaging o shot selling CAL: Capital Allocation Line.. S Shape atio p. = expected isk pemium Capital Allocation line CAL: - CAL illustates all the isk-etun combinations available to the investo - Inceasing the action o the oveall potolio invested in the isk asset inceases expected etun, but also inceases potolio isk - The exta etun pe exta isk is then 8/ = 0.36 see example This measue is called the Shape Ratio Capital Allocation Line with Leveage: - What about points on the CAL to the ight o potolio? I investos boow at the isk ee ate, we can extend the CAL without changing its slope Howeve, no one can boow at the isk-ee ate except the govenment, thus we will get a kink in the CAL at point will be latte to show less expected etun pe unit isk, eectivel showing the deault isk posed b the boowe S>1 = 0.7 CAL S 1= Note: Flatte is wose; as taking on less pe unit o exta total isk

6 esonal eeences and Asset Allocation: - Let s intoduce pesonal isk-etun peeences - The investo, now, needs to choose among man possible potolios on the CAL - And the choice involves a tade-o between isk and etun - Dieent investos will choose dieent potolios dieence combinations o the isk and isk-ee assets - Moe isk avese investos will choose to hold moe o the isk-ee asset and less o the isk asset - While choosing the optimal potolio, the investo ties to maximize utilit U 0.5A - Note that utilit can eithe incease o decease o a given potolio Utilit as a unction o allocation to the isk asset: Utilit Allocation to isk asset - Hence, om the gaph above, it is quite eas to see that o a given isk peeence given value o A, an investo will have an optimal allocation to the isk and theeoe isk-ee asset maximum value o this gaph - This can be done mathematicall see next page - Mathematicall, we can maximize utilit:

7 Whee is the amount invested in the isk asset: Given: = x + 1- = x x, = x - Utilit maximization omula: - Gaphicall, we use indieence cuves with the CAL - See example in lectue notes Indieence cuves o dieent coeicient o isk avesions dieent values o A: A=4 A=4 A= A= U=0.09 U=0.05 * : 0.5 : potolio potolio A A MaxU A Let A U A U * A