Behavioral finance and the making of an optimal portfolio

This study is to put forward some ideas for an optimal portfolio concerning the asymmetric risk-tolerance of investors, which is supported by the behavioral finance theory. The choice modeling theory is also employed for the sake of various portfolios, thereby investigating the risk-tolerance level of investors.

1 Introduction

Recently, sudden and unpredictable rises and

falls in the VN-Index have caused a lot of worries

for investors This unusual phenomenon

report-edly comes from mentality of investors that

can-not be explained according to efficient market

hypothesis Perhaps it is about time we considered

different views in light of theory of behavioral

fi-nance

As we have known, the optimal portfolio is one

of important contents of modern portfolio theory

with the assumption that investors are acting

ra-tionally (They always choose for themselves a

portfolio that maximizes expected return for a

given amount of portfolio risk, or minimizes risk

for a given level of expected return); and their risk

preferences are symmetrical

Behavioral finance theory has shown that the

risk preferences of investors are asymmetric They

are willing to accept the low rate of return for

high-risk investments in order to avoid losses

Key findings of risk preferences and investors’

ways of making decisions on their choices give

rise to several problems:

- How will asymmetry of investors’ risk

toler-ance affect the shape of their utility curve?

- If any, will the shape of utility curve of

in-vestors in the Vietnamese stock market be similar

to the value function suggested by Kahneman and Tversky in their Prospect Theory?

- Finally, when the true shape of the utility curve is identified, how will the optimal portfolio

be worked out?

The paper aims at analyzing and clarifying the shape of utility curve and applying the value func-tion to the making of the optimal portfolio appro-priate to conditions of the Vietnamese stock market

2 Value function in Prospect Theory of Kahne-man and Tversky

Kahneman and Tversky (1979) carried a long experiment to explore psychology of belief and in-stinctive choice By many experiments, they prove that losses produce a psychological consequence that is more serious than joy brought about by gains although the loss may be equal to gain

Their famous Prospect Theory partly helped Kah-neman win the Nobel Memorial Prize in Econom-ics in 2002 This theory is considered as a step forward of the classical utility theory In nature, Prospect Theory introduces a framework for ex-plaining how a decision is made The theory points out two stage of decision making process, namely, editing and evaluation In those stages,

This study is to put forward some ideas for an optimal portfolio concerning the

asym-metric risk-tolerance of investors, which is supported by the behavioral finance theory.

The choice modeling theory is also employed for the sake of various portfolios, thereby

investigating the risk-tolerance level of investors Besides, the insurance issue is also

taken into account when the optimization of the value of investment portfolio may

max-imize the utility of investors; and then draw a conclusion that in case the insurance

pre-mium seems an impediment to the return rate, the insurance value will enable investors

to cope with a greater risk The insurance, from a comprehensive view, will surely be

use-ful to make up for risks in price depreciation Besides, the value equation is to provide

investors with a better utility level Finally, the study refers to the process of risk

distri-bution so as to manage risks in a portfolio of various assets.

Key words: Behavioral finance, investment portfolio, portfolio insurance

relative value of information is received and

eval-uated subjectively

where pj: target probability of outcome j

xj: absolute sum of money of outcome j (pj): importance of each pj

v(xj) : value of each xj The sum of these values constitutes the value

function and it is known as changes in the utility

curve and its shape is as follows:

Figure 2: Utility curve

This shape shows three principal

characteris-tics of the value function:

- It is identified by losses and gains in relation

to reference point, but not absolute wealth

- The value curse is s-shaped: It is convex when

being lower than the reference point and concave

when being higher, which matches the traditional

theory

- Value function shows a clear asymmetry

be-tween values put in gains and losses (The loss function is steeper than the gain function)

3 Quantitative model for testing the shape of utility curve of Vietnamese investors

To test the shape of utility curve, we use the

“choice model.” We suggest here an experimental model at equilibrium point (where decisions to buy/ sell stocks are carried out) As for changes in utility during the portfolio holding time that may produce imbalance, they are not taken into con-sideration because this requires another experi-mental model This model is used for determining what attributes are considered as most important when selecting their portfolios The three

attrib-utes are: rate of return E(r), loss risk f(s), and holding period f(t)

The choice model tries to model the decision making process through a specifically designed survey sheet Choice Model can predict with great accuracy how individuals would react in a partic-ular situation Unlike a poll or a survey, predic-tions are able to be made over large numbers of scenarios within a context, to the order of many trillions of possible scenarios The Nobel Prize for economics was awarded to a principal exponent of the Choice Modeling theory, Daniel McFadden In this paper, we use JMP - a statistical software de-veloped by SAS - one of the world famous software developers Investor’s utility function under con-straints set for the experiment is as follows: [E(U)|F,W]=pU[aE(r)]+pU [bf(s)]+pU[cf (t)] (2)

Figure 1: Editing and evaluation of information

where:

F is information about available choices and

their attributes

W is a set of characteristics (personality, age,

and education) of groups of investors As for

indi-vidual investors, W is not a condition

pU denotes partial utility for E( r ), f(s) and

f(t)

a, b, c are weights

4 Data collecting method

Data are collected from a survey in the form of

questionnaires sent directly to individual

in-vestors The questionnaire includes basic factors

affecting the investment choice and its questions

are specifically designed to suit the choice model

and allow the making of a diagram Samples are

100 individual investors who are asked to select

five preset portfolios with standardized risks and

rates of return Among these samples, there are

many undergraduates and postgraduates along

with individual investors in major trading floors

in HCMC run by Bảo Việt, Đông Á, and FPT

groups

In designing the questionnaire, the authors

re-ferred to operations of Australian Stock Exchange

Five classes of portfolios are offered to investors:

high growth, diversified, balanced, defensive and

capital guarded ones They are all standardized

and popular portfolios in the Australian Stock

Ex-change The author, however, has adjusted

compo-nents of the five portfolios with a view to making

them appropriate to conditions in Vietnamese

stock market Their selection will provide us with

clues to their thoughts when making decision on

investment in relation to their utility

Conjoint analysis based on choices is used to

run the choice model thereby finding partial

util-ity of Vietnamese investors ; and utilutil-ity of a choice

will be the sum of partial pU[.]

5 Modeling results Utility of components of portfolio (produced by JPM) allows us to calculate degrees of investors’

satisfaction of their portfolio by adding up all com-ponent utilities

JMP-produced results are as follows:

Figure 3: JMP-produced results

These data allow us to draw the following 3-D investors’ utility line:

pU of risk 1.273 0.356 -0.272 -0.393 -0.964

pU of return rate 0 13% – 14.5% 20% 25% 33%

-1.021 10.3% –10.5% 0.25 -0.67 -1.29 -1.41 -1.99

-0.045 12.5 %–12.8% 1.23 0.31 -0.32 -0.44 -1.01

0.056 13% –13.7% 1.33 0.41 -0.22 -0.34 -0.91

0.159 13.7% – 14.% 1.43 0.52 -0.11 -0.23 -0.81

0.851 14.5% – 15.5% 2.12 1.21 0.58 0.46 -0.11

Table 1: Utility of components

Figure 4: 3-D diagram of investors’ utility

We can reduce the utility functions to a 2-D

level and turn it into a diagram of risk and return

rate because of investors’ indifference to future

in-vestment period Resulting indifference shown in

Figure 5 is almost analogous to the theoretical

in-vestor indifference curve and perhaps more

simi-lar to the value curve presented by Kahneman and

Tversky Particularly, these indifference curves

seem to be in a quadratic form reflecting

differ-ences in values of losses and gains Effects of

re-sults on their decisions are values they perceive

as linked with their expectation of portfolio

per-formance

Figure 5: 2-D diagram of investor’s utility

6 Working out the optimal portfolio under

Viet-namese conditions using value function

Effort to find an optimal portfolio based on

modern portfolio theory reveals the following

ad-equacies:

- Failure to point out explicitly changes in the

return rate during the holding period

- Standard deviation of the return rate only

represents a single period, which requires

re-esti-mation of expected rate of return after a period

That is why I want to present here a new model that could be used for working out the opti-mal portfolio based on the above empirical tests

It is point of intersection between value function and efficient frontier that can help work out the optimal portfolio Thus, the optimal portfolio could

be determined by “value,” or its ability to provide investors with the highest satisfaction

Presentation of the new model is based on the following arguments:

- Efficient frontier: Fama asserts that there is

no other model that can predict market behavior better than the efficient frontier His explanation

of this phenomenon is based irrational an

unsys-Figure 6: Point of intersection between efficient

frontier and indifference curve

tematic behavior, that is, investors can make any

unreasonable decision at will – and there will be

another decision based on an opposite view This

phenomenon raises questions of who will make

ad-justments and how they can be sure of their

ab-solute rationality

- It is the value function that describes

individ-ual investor’s decision through “value” that leads

them to more rational decisions and prevents

them from a situation in which they find their

de-cision wrong and have to sell off their assets Each

investor sets his/her own value to optimal

portfo-lios according to their subjective evaluation, and

they will react asymmetrically to the gain they

re-ceive and the loss they suffer That is why

distri-bution of outcome with a standard deviation will

affect the “value” differently despite the fact that

all optimal portfolios are on the efficient frontier

Thus, different optimal portfolios on the efficient

frontier will have different “values” in the eyes of

each investor and they will choose the portfolio

that may bring them in the biggest value As a

re-sult, the set of values of optimal portfolio is

writ-ten as follows:

The above formula is the product of values of

“gain” and “loss” in each period of investment time

in which Vg is value of gain and Vl, of loss

In practice, we can identify portfolios on the

ef-ficient frontier at any degree of risk aversion we

choose This can be certainly determined with

help from common Excel software After finding

efficient portfolios, we can run “choice model” on

professional software to fund the highest value of

efficient portfolio And of course, it is the optimal

portfolio we will choose In running the model, the

software will automatically design questions about

specific choices, and after making choices, the

model will produce outcomes In this research, I

use the JMP to find values of efficient portfolios

As mentioned above, however, values produced by

this process are based on data from 100 surveyed

investors and the software used is not the latest

version that allows us to identify the value for

each choice Thus, application of this software to

the search for optimal portfolios is a great help to

individual investors because it is convenient,

cheap and able to produce exact outcomes

7 Some new contributions to the effort to work out portfolios in Vietnamese stock market

a Portfolio insurance: By Prospect Theory

of Kahneman and Tversky, we learn that losses produce a psychological consequence that is more serious than joy brought about by gains Empirical demonstrations of this theory point out that in-vestors tend to estimate losses twice as high as gains By applying this result to the development

of portfolios, we see that portfolio insurance can help reduce distribution of losses This means that

it can help investors feel better when predicting the future of portfolios they hold, thereby increas-ing the “value” the portfolio brincreas-ings about And this increase may be greater than what they expect

These discoveries provide Vietnamese with useful knowledge that help them discuss more ef-fective with investors and improve ways of design-ing portfolios for their clients Recent realities show that investors have developed their own method of preventing risks when building their portfolios This proves that the optimal portfolio helps increase the potential “value” when portfolio insurance can help reduce losses Analysis of ben-efits from portfolio insurance shows that develop-ing a market for portfolio insurance is inevitable

This development requires efforts from the gov-ernment to perfect its macroeconomic and micro-economic management

b Risk tolerance: In the past, mean-variance

optimization method was used for identifying the investor’s risk tolerance in an effort to maximize the excess return rate (EERp) of assets and mini-mize portfolio variance of the mean excess return

And we have the following formula: EU = EERp

-Vp / rt It could be deduced that rt = (EERp - Vp

)/EU After determining the covariance matrix and

excess return rate of all assets in the portfolio,

finding the risk tolerance becomes an easy task

However, as we know, the investors’ risk tolerance

is asymmetric, therefore we should make some

ad-justment to the return rate In the past when we

worked out the risk tolerance through the excess

return, the risk tolerance corresponds with

posi-tive rate of return At present, we will find the

risk tolerance that corresponds to known negative

rates of return This means that the risk tolerance

at present will correspond to negative rate of

re-turn This approach reflects more exactly the

process of identifying the optimal portfolio

c Optimal investment time: By above tests,

we find that investor’s utility in relation to

port-folio holding period makes almost no difference in

a period of time varying from one to ten years In

reality, investors will make investment when they

have some capital surplus and withdraw it when

they need cash Thus, they can withdraw their

capital if they care about short-term value, and in

any other cases, which produces the holding

pe-riod

Two complexities occur when designing

portfo-lios for a pool of investors:

- Investors have not similar holding periods

- Investors have different ways of selecting

in-vestment timing and different payables

The investment in the pooled portfolio usually

implies different investment timing, therefore,

any design of portfolio is always a compromise,

and to ensure the optimal holding period for

pooled investors is impossible

Techniques to make decisions on the

appropri-ate time for a pooled portfolio can be borrowed

from the bond theory, and calculation of Macaulay

duration could be used for working out the holding

period

Using calculation of Macaulay duration,

deduct-ing the cash flow with risk-free interest rate, the

expected portfolio period may be modeled as

fol-lows:

where is the average expected holding

pe-riod for net free cash flow in the portfolio, E(FIt) and E(FOt) are time of capital inflow and outflow

at the time t, during the whole period N, and Rf

is risk-free rate of return

It will be a practical application for managers

of investment companies to deal with the question

of how to satisfy all shareholders when conditions and needs of investors are quite different

8 Conclusion This research introduces new applications of the behavior finance to the making of optimal portfolio Such new idea is considered as a pro-gressive development of portfolio theory Identify-ing the optimal portfolio by “value” can help work out more exact and reliable portfolios and help in-vestors make more reasonable decisions, thereby encouraging the market to perform better its “ef-ficient” functionn

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