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.
RESEARCHES & DISCUSSIONS 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 Besides, the insurance issue is also taken into account when the optimization of the value of investment portfolio may maximize the utility of investors; and then draw a conclusion that in case the insurance premium 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 useful 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 distribution so as to manage risks in a portfolio of various assets Key words: Behavioral finance, investment portfolio, portfolio insurance Introduction Recently, sudden and unpredictable rises and falls in the VN-Index have caused a lot of worries for investors This unusual phenomenon reportedly comes from mentality of investors that cannot be explained according to efficient market hypothesis Perhaps it is about time we considered different views in light of theory of behavioral finance As we have known, the optimal portfolio is one of important contents of modern portfolio theory with the assumption that investors are acting rationally (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 tolerance affect the shape of their utility curve? - If any, will the shape of utility curve of in- * University of Economics - HCMC 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 function to the making of the optimal portfolio appropriate to conditions of the Vietnamese stock market Value function in Prospect Theory of Kahneman and Tversky Kahneman and Tversky (1979) carried a long experiment to explore psychology of belief and instinctive 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 Kahneman win the Nobel Memorial Prize in Economics in 2002 This theory is considered as a step forward of the classical utility theory In nature, Prospect Theory introduces a framework for explaining how a decision is made The theory points out two stage of decision making process, namely, editing and evaluation In those stages, Economic Development Review - April 2011 35 RESEARCHES & DISCUSSIONS relative value of information is received and evaluated subjectively tween values put in gains and losses (The loss function is steeper than the gain function) Figure 1: Editing and evaluation of information 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 characteristics 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- 36 Economic Development Review - April 2011 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 consideration because this requires another experimental model This model is used for determining what attributes are considered as most important when selecting their portfolios The three attributes 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 particular situation Unlike a poll or a survey, predictions 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 developed by SAS - one of the world famous software developers Investor’s utility function under constraints set for the experiment is as follows: [E(U)|F,W]=pU[aE(r)]+pU [bf(s)]+pU[cf (t)] (2) RESEARCHES & DISCUSSIONS 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 individual investors, W is not a condition will be the sum of partial pU[.] 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 component utilities Table 1: Utility of components pU of risk pU of return rate 1.273 0.356 -0.272 -0.393 -0.964 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 pU denotes partial utility for E( r ), f(s) and f(t) a, b, c are weights JMP-produced results are as follows: Figure 3: JMP-produced results Data collecting method Data are collected from a survey in the form of questionnaires sent directly to individual investors 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 referred 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 Exchange The author, however, has adjusted components 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 utility of Vietnamese investors ; and utility of a choice These data allow us to draw the following 3-D investors’ utility line: Economic Development Review - April 2011 37 RESEARCHES & DISCUSSIONS Figure 4: 3-D diagram of investors’ utility That is why I want to present here a new model that could be used for working out the optimal 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: 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 investment period Resulting indifference shown in Figure is almost analogous to the theoretical investor indifference curve and perhaps more similar to the value curve presented by Kahneman and Tversky Particularly, these indifference curves seem to be in a quadratic form reflecting differences in values of losses and gains Effects of results on their decisions are values they perceive as linked with their expectation of portfolio performance Figure 5: 2-D diagram of investor’s utility Working out the optimal portfolio under Vietnamese conditions using value function Effort to find an optimal portfolio based on modern portfolio theory reveals the following adequacies: - 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-estimation of expected rate of return after a period 38 Economic Development Review - April 2011 Figure 6: Point of intersection between efficient frontier and indifference curve - 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- RESEARCHES & DISCUSSIONS 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 adjustments and how they can be sure of their absolute rationality - It is the value function that describes individual investor’s decision through “value” that leads them to more rational decisions and prevents them from a situation in which they find their decision wrong and have to sell off their assets Each investor sets his/her own value to optimal portfolios according to their subjective evaluation, and they will react asymmetrically to the gain they receive and the loss they suffer That is why distribution 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 result, the set of values of optimal portfolio is written 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 efficient 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 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 investors 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 increasing the “value” the portfolio brings about And this increase may be greater than what they expect These discoveries provide Vietnamese with useful knowledge that help them discuss more effective with investors and improve ways of designing 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 benefits from portfolio insurance shows that developing a market for portfolio insurance is inevitable This development requires efforts from the government to perfect its macroeconomic and microeconomic 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 minimize portfolio variance of the mean excess return Economic Development Review - April 2011 39 RESEARCHES & DISCUSSIONS 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 adjustment to the return rate In the past when we worked out the risk tolerance through the excess return, the risk tolerance corresponds with positive 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 return 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 portfolio 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 period Two complexities occur when designing portfolios for a pool of investors: - Investors have not similar holding periods - Investors have different ways of selecting investment 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 appropriate 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, deducting the cash flow with risk-free interest rate, the expected portfolio period may be modeled as follows: N ! E (HP) = MEAN (/ 71/^1 + R f ht * ^ E]FItg - E]FOtgh * t A2 t=1 where E _! HP i is the average expected holding pe- 40 Economic Development Review - April 2011 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 Conclusion This research introduces new applications of the behavior finance to the making of optimal portfolio Such new idea is considered as a progressive development of portfolio theory Identifying the optimal portfolio by “value” can help work out more exact and reliable portfolios and help investors make more reasonable decisions, thereby encouraging the market to perform better its “efficient” functionn References Fama, E (1997), “Market Efficiency, Long-Term Returns and Behavioural Finance”, Journal of Financial Economics, 49, pp 283-306 Kahneman, D., Tversky, A (1979), “Prospect Theory: An Analysis of Decision under Risk”, Econometrica, Volume 47, Issue 2, pp.263-292 Livanas, J (2006a), “How Can the Market be Efficient if Investors Are not Rational?”, Journal of the Securities Institute of Australia, No 2, pp 20-24 Livanas, J (2006b), “Are Investors Rational and Does It Matter?”, Paper presented to the 14th Colloquium of Superannuation Researchers “Choice in Retirement Funding”, Sydney, ( 20–21st July 2006) Livanas, J (2006c), “It’s a Utility But not as We Know It!”, Paper presented at the 19th Banking and Finance Conference, Sydney,( December 2006) Livanas, J (2008), “Empirical Analysis of Investor Utilities in Investment Choice”, Paper presented to the Institute of Actuaries of Australia, Sydney (2008) Markowitz, H (1952), “Portfolio Selection”, The Journal of Finance, Vol VIII, No 1, March 1952, pp 7791 McCulloch, B and D Leonova (2005), “The Market Equity Risk Premium’, New Zealand Treasury, May 2005 Sharpe, W F., (2006), ‘Expected Utility Asset Allocation’, unpublished paper www.wsharpe.com 10 Simon, H.A (1955), “A Behavioral Model of Rational Choice”, Cowles Foundation Paper 98, The Quarterly Journal of Economics, Vol LXIX, pp 99-118 ... diversified, balanced, defensive and capital guarded ones They are all standardized and popular portfolios in the Australian Stock Exchange The author, however, has adjusted components of the five portfolios... applications of the behavior finance to the making of optimal portfolio Such new idea is considered as a progressive development of portfolio theory Identifying the optimal portfolio by “value” can help... choose the portfolio that may bring them in the biggest value As a result, the set of values of optimal portfolio is written as follows: The above formula is the product of values of “gain” and