LectureUtilityFunctionandPriceofRisk資產(chǎn)組合金融系研究生課程課件

In

this

chapter…….-1How

to

construct

utility

functionDefinition

of

risk

aversionMarkowitz’s

risk

premium=EW-CEWPratt&Arrow’s

RP&RRA(-WU’’/U’)Why

investors

love

right

skewEU

and

mean-variancetargetAll

investors

love

FOSD,Risk

aversioninvestors

prefer

SOSD0.

The

St.

Petersburg

Paradox-2Game

A:

Flip

a

fair

coin

,The

payoff:

Ifthe

head

appears,

you

get

$2.otherwisenothing.You

would

pay

1

$

for

that

game.0.

The

St.

Petersburg

Paradox-3The

game：Flip

a

fair

coin

until

the

first

head

appearsThe

payoff:

If

the

first

head

appears

on

the

k-th

flip,you

get

$2k

How

much

would

you

be

willing

to

pay

for

achanceto

play

this

game?Using

an

expected

value,

you

should

be

willing

to

payat

least

the

expected

value

of

the

payoff

from

playingthe

game.What

is

the

expected

payoff?0.The

St.

Petersburg

Paradox-44

8 ….

2kOutcomes=“Head

appears

in

toss

#”:

1

2

3 ……

kProbability

head

occurs

on

giventoss

?

? 1/8

…

1/kPayoff

=

2Expected

payoff=1+1+1+……+1You

would

pay

infinity

to

attend

thegame1、utility

function:

function

between

consumption

,orwealth

and

investor’s

satisfied

degree

.-52、How

to

construct

utilityfunction(binominal

tree

method)step

1:we

arbitrarily

assign

a

utility

of

–10utiles

to

a

loss

of

$1000,

and

0

to

nothing.Step

2:investor’s

must

decide

the

probabilityα,

which

makes

indifferent

between

0

andgamble

(1000,-1000:

α),

orFor

an

example

:if

α

=0.6 then

U(1000)=6.7-62、How

to

construct

utilityfunction(binominal

tree

method)Step

3:repeat

step2

,and

get

different

utility

ofdifferent

losses

and

gains.such

as:step4:

plot

wealth

~utility

,

and

the

utilitycurve

is

drawn.WealthUtility-73、What

is

Expected

Utility?--utility

of

uncertainJust

like

that

expected

value

is

a

parameter

ofR.V., utility

of

uncertain

outcome

can

bemeasured

by

Expected

Utility.[Von

Neumann-Morgenstern]GE[U(G)]=

αU(100)+(1-α)U(-100)In

general

expected

utility

function

canbedefined

as

followed:-83、What

is

Expected

Utility?-9Example.

For

an

investor,

U(w)

=ln(w),his

initial

wealth

is

500

.

(1)What’s

theutility

of

a

gameG(300,-300,0.5).

(2)What’s

the

utilityof the

game’sexpected

value?(3)Which

is

bigger,EU(G)

or

U[E(G)]4、Attitude

toward

risk:

according

to

utility

functionTypes

of

Attitude

towardrisk(1)Risk

aversion:utility

of

certain

outcome.{Notice:

E( )

is

aconstant, israndom.}–

expected

utility

of

uncertain

outcome.A

certain

outcome

brings

greater

utilityto

the

risk

aversioner

than

uncertainoutcome,

even

though

they

have

equalexpected

value.{E[E(

)]=E(

)}-104、Attitude

toward

risk:

according

to

utility

functionTypes

of

Attitude

towardrisk(1)Risk

aversion:

For

bond,

which

gives

you

$2

interest,its

priceis

100

.{E(

)=2}For

stocks,

which

gives

you

$3

dividend(P=50%),and

$1

dividend

(p=50%,its

price

isbelow

100

.{E(

)=2}-114、Attitude

toward

risk:

according

to

utility

function(2)Risk

neutral:–

A

certain

outcome

brings

same

utility

to

therisk

neutral

with

uncertain

outcome,

iftheyhave

equal

expected

value.{E[E(

)]=E(

)}-124、Attitude

toward

risk:

according

to

utility

function(3)Risk

lover:–

A

certain

outcome

brings

less

utility

to

therisk

lover

with

uncertain

outcome,

even

ifthey

have

equal

expected

value.{E[E()]=E(

)}-134、Attitude

toward

risk:accordingto

utility

function

diagramU’(w)<0,

=0,

or

>0For

almost

every

investor,heis

a

risk

aversioner

.-145、Pricing

of

risk

premium----

Markowitz

Risk

Premium.Certain

Equivalent

Wealth

(CEW

).If U(W*)

is

equal

to W*

iscalledCEW.

(U(CEW

)=

)(Some

certain

value

which

has

the

same

utilitywith

an

uncertain

event

is

called

CEW)-15CEWCEW-16-175、What

is

Expected

Utility?Solve

the

St.Petersburg

Paradoxa=4-185、Pricing

of

risk

premium----MarkowitzRisk

Premium.(2)

Markowitz

risk

premium

:the

difference

between

EW

and

CEW

；U[E(W)]>EU(W)RPE(W)CEW5、Pricing

of

risk

premium----

Markowitz

Risk

Premium.-19(2)

Markowitz

risk

premium

:

the

differencebetween

E(W)

and

CEW

.Example:

investor’s

utility

function is

U(W)

=ln(W);current

wealth

$10,000,

he

is

faced

potentiallosses

(fire,

earthquake

),

the

loss

is

(-5000,0,0.01),

how

much

would

he

pay

for his

risk.

Solution

:

Expected

Value

of

future

wealth(5000,10,000,0.01)=0.01*5000+0.99*10,000=9950ln(CEW

)=0.01*ln

5000+0.99*ln

10,000=9.2CEW=Exp(9.2)=9887.6,RiskPremium=E(W)-CEW=62.4${total

RP

CEW+50$}Is

insurance

company

risk

lover?-20A

little

more

different

about

riskPure

risk,

loss

100%Speculative

risk

(Martin

Halek&

Joseph

Eisenhauer)–

Speculative

risk

is

binomial

distributed

for

individual

but

normalfor

insurancecompany對(duì)單個(gè)投資者而言，1/100可能的可能遭受火災(zāi)，因此火災(zāi)是不確定事件、風(fēng)險(xiǎn)事件。設(shè)其期望收益與風(fēng)險(xiǎn)為(u,sigma)對(duì)保險(xiǎn)公司而言，1萬(wàn)個(gè)投保人，基本上就是

100次火災(zāi)，不確定性很少，發(fā)生火災(zāi)基本上就是必然的。其單位保單上的風(fēng)險(xiǎn)為sigma/100。1

million

investors,

std

is

almost

0.Binomial

converges

normal,

and Sigma

converges

to

0.Is

insurance

company

risk

lover?-21你的家產(chǎn)為100萬(wàn)，去年火災(zāi)發(fā)生概率萬(wàn)分之一。你的期望收益為-100元。如果保費(fèi)為90元，你都不愿支付，你是風(fēng)險(xiǎn)愛(ài)好者(投保后期望收益-90元，風(fēng)險(xiǎn)為0元，不投保期望收益-100元，風(fēng)險(xiǎn)為1000元，因此投保的期望收益高但風(fēng)險(xiǎn)?。?。絕大部分客戶(hù)愿意支付100元以上的費(fèi)用，人都是

Risk

aversion風(fēng)險(xiǎn)回避者。（或者說(shuō)投資者確定性地花110元去避免一個(gè)預(yù)期虧100元的東西）.保險(xiǎn)公司也是風(fēng)險(xiǎn)回避者，它們?cè)O(shè)計(jì)的保單110元，得毛利潤(rùn)10元。Behavior

Finance:

Sometimespeople

are

risk

loverWhy

does

investor

sell

stock

whosepricegoes

up,and

hold

stock

whose

pricegoesdown?Odean(1998))。Suppose

cost

is50，current

is**

，it

may

go

upor

down

5

withprob=0.5If

current

is

55Investor

sells

it,

utility

is

U

(5)If

he

holds

it,

utility

is

U(0)

or

U(10)If

current

is

45Investor

sells

it,

utility

is

U

(-5)If

he

holds

it,

utility

is

U(0)

or

U(-10)When

facing

loss,

he

prefer

uncertainty-226、Pricing

of

risk

premium----

Pratt

&

Arrow

Risk

PremiumDeducing

of

Pratt

&

Arrow

Risk

Premium

Suppose

current

wealth

is

w

,

denotefuture

uncertain

.usuallyprobability of

bad

condition

is

verysmall.or

.( is

called

risk

neutral

gamble

)Pratt

&

Arrow

define risk

premium(RP)

asfollowed:-236、Pricing

of

risk

premium----

Pratt

&

Arrow

Risk

PremiumCont.(1)

Deducing

ofPratt-Arrow

RiskPremiumright

–hand

side(is

expanded

at

W)

=U(W)-RP*U’(W)+terms

of

higher-order(A)left-hand

side==(B)-246、Pricing

of

risk

premium----

Pratt

&

Arrow

Risk

Premiumby

(A)

and

(B)

,for

risk-aversion

investors

,U’’(w)<0,RP(Risk

Premium

)>0-256、Pricing

of

risk

premium----

Pratt

&

Arrow

Risk

Premium（2）Further

measurement

of

risk

aversion

.（2.1）

Absolute

riskaversion: risk

attitudewhenwealth

changes

(ARA

),wedefine

ARA=for

increasing

functionARAGiven

risk

,of

W,

,it

means

the

investor

will

pay

more

risk

premium

when

hiswealth

is

increased.

It

is

called

Increasing

ARA{to

keep

RPconstant,

the

second

term

of

first

equation

will

belower.}{Even

Wealth

goes

up,

risky

asset

goes

down

}Decreasing

ARA，Constant

ARA.-266、Pricing

of

risk

premium----Pratt

&

Arrow

Risk

Premium（2.2）Relative

risk

aversionWe

define

RRA(Relative

risk

aversion)=Decreasing

RRAwhen

the

ratio

of

risk

towealth is

fixed

,positive means

risk

premium

increaseswhenwealth

increases.(Notice

the

condition).This

typeofpeople

is

called

Increasing

RRA.{錢(qián)增加后，股票持倉(cāng)比例下降to

keep

RP

constant,the

second

term

offirst

equation

will

be

lower.}Decreasing

RRAConstant

RRA.-276、Pricing

of

risk

premium----

Pratt

&

Arrow

Risk

Premium-28

Example

(The

background

is

thesameas

before).

investor’s

utility

function

isU(W)

=ln(W);

current

wealth

$10,000,he

is

faced

potential

losses

(fire,earthquake

),

the

loss

is

(-5000,0,0.01),how

much

would

he

pay

for his

riskunder

Pratt

&

Arrow

?6、Pricing

of

risk

premium----

Pratt

&

Arrow

Risk

Premiumsolution

:He

would

pay62.28$-297,Property

of

Risk

AversionIncreasing

ARA

will

be

IncreasingRRAProof

：（

a

）

Suppose

a

investor

with

utilityfunction

，and

he

is

a

increasing

ARA，thenSo（by

RP=we

know

that）-307,Property

of

Risk

AversionDecreasing

RRA

belongs

to

DecreasingARAProof:

Suppose

a

investor

with

utility

function，and

he

is

a

decreasing

RRAthenSinceSo

,or,So

the

investor

must

decreasing

ARA-318.Hyperbolic

Absolute

Risk

Aversion(HARA)

family-328.(HARA)

family-339.Utility

function

and

high

moment-34-359.Utility

function

and

high

momentRight

skew

meansr>0,ln(W)investors

love

it9.Utility

function

and

high

moment-3610.First-order

stochastic

dominance-3710.First-order

stochastic

dominance-3810.First-order

stochastic

dominance-3911.Second-order

stochasticdominanceF(.)

second

order

stochastically

dominates

G(.)

if

and

only

ifIf

F

second-order

stochastically

dominates

G

,

and

if

F

and

Ghave

the

same

mean,

then

F

has

a

smaller

variance

than

G

.All

risk-averseexpected-utility

maximizers

prefer

a

second-order

stochastically

dominant

lottery

to

a

dominated

lottery.-40-4111.Second-order

stochasticdominanceFOSD

&

SOSD-42N(0,1)

dominances

N(0,2)?02sosd.xls測(cè)測(cè)你的風(fēng)險(xiǎn)容忍度（選擇完后計(jì)算總分）-43??１你剛剛在電視游戲中獲得一個(gè)大獎(jiǎng)！你會(huì)選擇哪一個(gè)？A１０００元現(xiàn)金B(yǎng)５０％的機(jī)會(huì)獲得４０００元C，２０％的機(jī)會(huì)獲得１００００元D，５％的機(jī)會(huì)獲得１０萬(wàn)元。２有一個(gè)很好的投資機(jī)會(huì)，但是你得借錢(qián)。你會(huì)接受貸款嗎？A絕對(duì)不會(huì)B也許C是的３你所在的公司是一家小型計(jì)算機(jī)公司。公司為融資要把股票賣(mài)給職工，公司管理層計(jì)劃在三年后使公司上市，在上市之前，你不能出售手中的股票，也沒(méi)有分紅，但公司上市時(shí)，你的股票可能會(huì)翻１０倍，你會(huì)投資買(mǎi)多少錢(qián)買(mǎi)股票？–

A一點(diǎn)兒也不買(mǎi)B半年工資 C一年的工資D兩年工資。

４你住的單位公寓房要變成產(chǎn)權(quán)房。你可以花８萬(wàn)元買(mǎi)下，或者單位給你２萬(wàn)元了結(jié)。房子買(mǎi)后在市場(chǎng)上可賣(mài)１２萬(wàn)，但必須在６個(gè)月以后，這其間你每個(gè)月得多開(kāi)支１２００元。–

A拿２萬(wàn)元了事。B買(mǎi)后再賣(mài)。5.

5

，你繼承了一套１０萬(wàn)元的房子。房子很破舊了，但房子在市中心很-44可能升值。如果出租的話(huà)每個(gè)月凈賺１０００元，如果裝修后再租的話(huà)每個(gè)月可賺１５００元，但這得貸款。你會(huì)：–

A賣(mài)了。B出租。C裝修后再租。

6

你的朋友是地址勘探家，發(fā)現(xiàn)了一個(gè)油田，準(zhǔn)備集資開(kāi)發(fā)。成功的話(huà)利潤(rùn)可能是５０倍。成功的概率是２０％。但如果是一個(gè)枯井的話(huà)大家

將一無(wú)所獲。–

A一點(diǎn)也不買(mǎi)。B半年工資C一年的工資D兩年工資。7你在一場(chǎng)賭博中輸了５００元。A不再玩了。B再拿１００元玩玩。C再拿２００元玩玩。D再拿５００元玩玩。8,

你外出時(shí)：A