5-Utility and Capital Allocation

8/19/2019 5-Utility and Capital Allocation

1/24

Slide 1 W. Suo

Charactering Investors

And Capital Allocation


2/24

Slide 2 W. Suo

Objective

How to describe an investor?Risk aversion and utility function

Capital AllocationRisk free asset and risky portfolio


3/24

Slide 3 W. Suo

Investor’s Life Cycle

Accumulation !ase"arly to middle yearsLow net wort!Lon# time !ori$on

%earin# risk for !i#!er returns is reasonableConsolidation !ase

&id'point and later "arnin#s e(ceed needs&oderate risk to protect investments

)pendin# !ase"arnin#s !ave mostly ceased*ime !ori$on is s!orter

rotection of assets importantInflation an immediate concern


4/24

Slide 4 W. Suo

How to C!aracteri$e an Investor?

"conomic t!eory assumes every person seeks toma(imi$e t!eir !appiness or utility

*!e more wealt! a person !as+ t!e more utility t!ey !ave*!e !i#!er an investment’s return+ t!e #reater a person’swealt!+ t!erefore+ t!e #reater t!eir utility

However+ uncertainty about e(pected returns makest!e decision more comple(

,e must ma(imi$e our e(pected utility


5/24

Slide 5 W. Suo

*ypes of Investors

Risk'averse investors*!ey re-uire !i#!er returns to encoura#e t!em to accept #reater risks

At !i#!er levels of risk+ investors are less willin# to accept risk unless t!eyreceive increasin#ly lar#er risk premiums

Risk'lovin# investorsrefer to enter risky #ambles even #iven t!e !i#! likeli!ood of diminis!in#

wealt!A t!eoretical possibility but not a reality wit! rational investors

Risk'indifferent investorsrefer !i#!er returns+ but #reater risk doesn’t affect t!eir !appiness In investments, we always assume people are risk averse


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Slide 6 W. Suo

How to describe an investor?

.arious factors determine an investor’s risk'return preferences

A#e/t!e elderly are usually more risk averse"conomy/people are usually more risk averse durin# a recession

essimists are usually more risk averse t!an optimists*raumatic evens tend to make people more risk averseetc

Risk *olerance 0 ,illin#ness 1 Ability


7/24Slide 7 W. Suo

2ominance rinciple 3or Risk averse Investors

1

2 34

"(pected Return

.ariance or )tandard 2eviation4 dominates 56 !as a !i#!er return4 dominates 76 !as a lower risk 8 dominates 76 !as a !i#!er return,!at about 5 and 7+ or 4 and 8+ or 5 and 8?


8/24Slide 8 W. Suo

9tility 3unction

represents an investment opportunity*!e constant measures t!e de#ree of risk aversion

Risk Aversion : 9tility


9/24Slide 9 W. Suo

rofit 0 ;<

rofit 0 '4< 5


10/24Slide 10 W. Suo

Risk Aversion and .alue

*!e )ample Investment

0 44= ' >; B78= 4Risk Aversion 9tility

Hi#! ; 'D> <

7 8>DD Low 5 5D>44

*'bill 0 ;=



9tility and Indifference Curves

Represent an investor’s willin#ness to trade'off return and risk "(ample Bfor an investor wit! A08

"(p Return B= )t 2eviation B=

5< 4<5; 4;>;4< 7<

4; 77>

44

4

4

44

4

4



Indifference Curves

"(pected Return

)tandard 2eviation

Increasin# 9tility



Indifference Curves in σ '"Br )pace

Indifference curves represent points at w!ic! a person is e-ually!appy Bindifferent

In all cases+ Investoris !i#!ly

riskaverse>

Investor ismoderatelyrisk averse>

Investoris a risklover> Investor is

indifferentto risk>



Allocatin# capital between risky assets and risk free assetsossible to split investment funds between safe and risky

assetsRisk free asset pro(y6 *'bills

erfectly price'inde(ed bond E t!e only risk free asset in real terms6*'bills are commonly viewed as Ft!eG risk'free asset6&oney market funds ' t!e most accessible risk'free asset for most investors>

Risky asset stock Bor a portfolio

Can represent t!e total investments in risky assets"(amine risk return tradeoff

2emonstrate !ow different de#rees of risk aversion will affectallocations between risky and risk free assets

Capital Allocation



ortfolios of One Risky Asset

and One Risk'3ree AssetAssume a risky portfolio P defined by

E Br p 0 5;= and

*!e available risk'free asset !asr f 0 @= and

*!e proportions invested y= in P and B5' y = in r f

"(pected returns : standard deviation for combinations

0 complete or combined portfolio



or If +

or If

or

If

Combinations ,it!out

Levera#e



ossible Combinations

5D>;=



CAL (Capital Allocation Line)


19/24

Slide 19 W. Suo

%orrow at t!e Risk'3ree Rate and invest in stock 9sin# ;


20/24

Slide 20 W. Suo

Optimal Allocation

iven an investor’s risk aversion+ !ow to determinet!e optimal allocation?


21/24

Slide 21 W. Suo

CAL wit! Risk references

"Br

@= Lender

%orrower


22/24

Slide 22 W. Suo

reater levels of risk aversion lead to lar#er

proportions of t!e risk free rateLower levels of risk aversion lead to lar#er proportions of t!e portfolio of risky assets,illin#ness to accept !i#! levels of risk for !i#!levels of returns would result in levera#edcombinations

Risk Aversion and Allocation


23/24

Slide 23 W. Suo

&at!ematically J

*!e optimal investment in is


24/24

l d W Suo

CAL with Higher

Borrowing Rate

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5-Utility and Capital Allocation