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1. Does her selection today indicate that Julie’s choices are consistent with economic rationality? Why or why not? PS – consider these in terms of the ‘Allais Paradox’ and ‘Framing’

2. Explain ‘Prospect Theory’ and how it differs from normative theories such as ‘Utility Theory’.

## Expected Return of Stocks A and B

Expected Return of Stock A:

 State of economy Probability Stock A Expected return Boom 0.35 0.13 0.0455 Normal 0.35 0.09 0.0315 Recession 0.3 -0.05 (0.0150) Expected return 0.0620

Expected Return of Stock B:

 State of economy Probability Stock B Expected return Boom 0.35 16% 0.0560 Normal 0.35 2% 0.0070 Recession 0.3 -1% (0.0030) Expected return 0.0600

Variance of Stock A:

 Return on stock A Average expected X-Y (X-Y)2 Probability Probability *(X-Y)2 0.1300 0.0620 0.06800 0.0046 0.35 0.0016 0.0900 0.0620 0.02800 0.0008 0.35 0.0003 (0.0500) 0.0620 (0.11200) 0.0125 0.3 0.0038 Variance 0.0057

Variance of Stock B:

 Return on stock B Average expected X-Y (X-Y)2 Probability Probability *(X-Y)2 0.1600 0.0600 0.10000 0.0100 0.35 0.0035 0.0200 0.0600 (0.04000) 0.0016 0.35 0.0006 (0.0100) 0.0600 (0.07000) 0.0049 0.3 0.0015 Variance 0.0055

Standard Deviation of Stock A:

 Return on stock A Average expected X-Y (X-Y)2 Probability Probability *(X-Y)2 0.1300 0.0620 0.06800 0.0046 0.35 0.0016 0.0900 0.0620 0.02800 0.0008 0.35 0.0003 (0.0500) 0.0620 (0.11200) 0.0125 0.3 0.0038 Standard deviation 0.0752

Standard Deviation of Stock B:

 Return on stock B Average expected X-Y (X-Y)2 Probability Probability *(X-Y)2 0.1600 0.0600 0.10000 0.0100 0.35 0.0035 0.0200 0.0600 (0.04000) 0.0016 0.35 0.0006 (0.0100) 0.0600 (0.07000) 0.0049 0.3 0.0015 Standard deviation 0.0744

Coefficient of Variation of Stock A:

 Coefficient of Variation of Stock A Value Expected return 0.0620 Standard deviation 0.0752 Coefficient variation 1.2130

Coefficient of Variation of Stock B:

 Coefficient of Variation of Stock B Value Expected return 0.0600 Standard deviation 0.0744 Coefficient variation 1.2394

Covariance of Stocks A and B:

 Return on stock A Average expected Z= X-Y Return on stock B Average expected Z= X-Y Z*Z Probability Probability *(Z*Z) 0.13 0.062 0.06800 0.16 0.06 0.1 0.0068 0.35 0.00238 0.09 0.062 0.028 0.02 0.06 -0.04 -0.00112 0.35 -0.000392 -0.05 0.062 -0.112 -0.01 0.06 -0.07 0.00784 0.3 0.002352 Covariance 0.00434

Correlation Coefficient of Stocks A and B:

 Correlation Coefficient of Stocks A and B Values Covariance A and B 0.0043 Standard deviation Stock A 0.0752 Standard deviation Stock B 0.0744 Correlation Coefficient of Stocks A and B 0.7760

Portfolio Return:

 Portfolio Return Stock A Stock B Portfolio invested 0.60 0.40 Expected return 0.0620 0.06 Returns 0.0372 0.0240 Portfolio returns 0.0612

Portfolio Standard Deviation and Variance:

 l) Portfolio Standard Deviation and Variance Values Stock A Weight 60% Stock B Weight 40% Standard deviation Stock A 0.07521 Standard deviation Stock B 0.07436 Correlation Coefficient of Stocks A and B 0.77602 Portfolio Variance 0.00500 Portfolio Standard deviation 0.07074

Weights of the Minimum Variance Portfolio:

 Weights of the Minimum Variance Portfolio Value Standard deviation Stock A 0.075206383 Standard deviation Stock B 0.07436397 Variance of Stock A 0.005656 Variance of Stock B 0.00553 Covariance between Stock A and Stock B 0.00434 Minimum Weight of Stock A 47.49% Minimum Weight of Stock B 52.51%

Proof that these weights lead to the Minimum Variance Portfolio:

 Proof that these weights lead to the Minimum Variance Portfolio Value Stock A Weight 47% Stock B Weight 53% Standard deviation Stock A 0.07521 Standard deviation Stock B 0.07436 Correlation Coefficient of Stocks A and B 0.77602 Portfolio Variance 0.00496 Portfolio Standard deviation 0.07046

The change in portfolio weight has mainly decreased the overall standard deviation of the company, which could eventually help in reducing risk of investment.

Weights of the Optimal Risky Portfolio with a risk-free asset:

 Weights of the Optimal Risky Portfolio with a risk-free asset Values Stock A return 0.06200 Stock B return 0.06000 Risk free rate 0.04000 Standard deviation Stock A 0.07521 Standard deviation Stock B 0.07436 Variance of Stock A 0.00566 Variance of Stock B 0.00553 Covariance between Stock A and Stock B 0.00434 Optimal portfolio weight of Stock A 66.40% Optimal portfolio weight of Stock B 33.60%

p) Proof that these weights lead to the Optimal Risky Portfolio:

 Proof that these weights lead to the Minimum Variance Portfolio Value Stock A Weight 66% Stock B Weight 34% Standard deviation Stock A 0.07521 Standard deviation Stock B 0.07436 Correlation Coefficient of Stocks A and B 0.77602 Portfolio Variance 0.00505 Portfolio Standard deviation 0.07110
 Portfolio Return Stock A Stock B Portfolio invested 66% 34% Expected return 0.0620 0.0600 Returns 0.0412 0.0202 Portfolio returns 0.0613

The derivation of optimal portfolio weights mainly increased the overall portfolio return and maximised the overall standard deviation of the portfolio.

Discussion on what you would do with this portfolio:

The use of minimum variance portfolio could eventually help in maintain a constant return and reduce the overall risk from investment.

Calculating the CAPM of different stocks:

 Market return Asset A Asset B Asset C Asset D Asset E Asset F Expected returns 12.0000% 7.0000% 5.1000% 12.0500% 10.5500% 14.5000% 7.3000% Variance 0.00141 0.00043 0.001299 0.00190475 0.00199475 0.002395 0.001131 Standard deviation 3.7550% 2.0736% 3.6042% 4.3643% 4.4663% 4.8939% 3.3630% Covariance 0.00073 0.00126 0.001625 0.001645 0.00179 0.00126 Beta 0.52 0.89 1.15 1.17 1.27 0.89 Rm 12.0000% Rf 0.05 CAPM 8.62% 11.26% 13.07% 13.17% 13.89% 11.26%

The above table mainly depicts the relevant expected return and CAPM of different stocks, which could help in gauging the exact return that needs to be delivered from investment.  Thus, the CAPM return is mainly higher than the expected return of different stocks, as it helps in improving the overall profitability from investment.

Summarising the risky stock in the basis of Coefficient of Variation:

 Market return Asset A Asset B Asset C Asset D Asset E Asset F Expected returns 12.0000% 7.0000% 5.1000% 12.0500% 10.5500% 14.5000% 7.3000% Standard deviation 3.7550% 2.0736% 3.6042% 4.3643% 4.4663% 4.8939% 3.3630% coefficient of variation 0.31 0.30 0.71 0.36 0.42 0.34 0.46 Rank 1 6 3 4 2 5

From the evaluation of coefficient of variation stock A is mainly identified as the least risky stock, which could provide a constant return from investment. The ranking is mainly based on the minimum coefficient, which is achieved by different stocks.

Explaining the Prospect Theory and depicting the difference from normative theories such as Utility Theory:

The overall prospect theory mainly depicts the relevant behavioural economic theory, which mainly depicts the probabilistic alternatives that involve risk. Moreover, the theory mainly focuses on real life choices rather than optimal decisions, which might be taken or considered as normative models. Some researchers mentioned that the theory mainly allows researchers to depict the relevant real life exposures, which might help in generating realistic data from population. On the other hand, other researchers criticises that prospect theory mainly engulfs as a descriptive model, where the no psychological explanation is provided in the theory.

Moreover, the use of Prospective theory is mainly a realistic approach and is conducted in diffident applications for addressing the valid points. Conduct behaviours such as risk aversion or risk seeking caudal effectively from the help of prospect or theory. This depiction of human psychology mainly allows researcher to gauge into the behavioural economic theory. Furthermore, the prospective theory mainly depicts the relevant theory with probability, which could eventually help in attaining realistic data. On the contrary, researchers argued that Prospective theory does not include factors that relevant important in the decision making process, such as emotions.

## Variance of Stocks A and B

The Utility theory mainly depicts the relevant economic concepts, which could measure the utilities derived from goods or services. In addition, the theory mainly focuses on both ranking the possibilities that is preferred or considered on certain objectives. The theory focuses on rational decision, which is considered while selection of relative goods and services. Some researchers mentioned that the rational approach of the theory mainly helps in understanding the overall behaviour of consumers while making any kind decisions.

The major difference between prospective theory and utility theory are mainly depicted as follows.

• The overall assumption of utility theory mainly depicts the relevant maximize, which is needed for depicting expected behaviour of individuals. In addition, the theory mainly states that individuals mainly choose the options, which helps in maximising their expected utility. However, the prospective theory mainly depicts that individuals rely on risk-averse, where dislike risk in some situations while liking risk in others.
• The expected utility value mainly depicts the risk preferences, which are captured by the shape utility functions decision maker are risk averse and risk seeking. However, the prospect theory mainly depicts that investors are risk averse for moist gains and risk seeking for most losses.
• The expected theory mainly depicts the invariance assumptions, which could depict the relevant preference between prospects. However, the prospective theory mainly indicates that different frame needs to be provided with adequate choices, as it depicted that choices does not follow the invariance assumptions.
• The expected theory mainly depicts and assumes choices, which reflect outcomes. The theory mainly indicates the chances that it is assumed to reflect adequate outcomes. However, the prospect theory mainly indicates that aversion, which is mainly used for increasing overall viability of the losses. The theory mainly states that individuals mainly use aversion techniques and make adequate decisions, which reduces risk.
• The overall expected theory mainly depicts that adding common consequence two prospects could when not change the alternative decisions, which depicts that principles are interdependence axiom. However, the prospect theory mainly depicts those common consequences changes preferences, which are in contrary to utility theory.

Discussing about stocks for the long run where stocks' long-term advantage to work out over your particular investment horizon is evaluated and second is about whether an investment as risky as stocks belongs in a retirement portfolio in the first place:

The discussion mainly focuses on the overall validation of stocks as a more reliable source of income in the end. There are many critics, which state that stock does not provide a relative return, which could effectively help in generating higher revenue from investment. In addition, there are relative critics where the investors are not relying on the stock returns to generate long-term profits from investment. The stock return mainly provided a higher return form investment, as it helps in generating the required level of income that it higher than investment in bonds or fixed securities. Some researchers mentioned that bond investment mainly provides a constant income, which does not comprehend with the rising return provided by stock investment. On the other hand, other researchers criticises that bond investment does not have any kind of risk, which mainly helps in maintaining a constant level of income with no risk associated with investment.

The relevant growth in income and return, which is provided by investment from stocks. The return from investment mainly indicates the overall profits, which is mainly generated from investment. The figure mainly depicts that during the bear market the overall return of the stock was less than the US-20 year bonds. Therefore, after the bear market the return from investment exponentially increased. Some researchers stated that bond mainly provides a constant return from investment, while the use of stock market allows investors to gauge into the rising bull market. However, other researchers criticises that bond returns only provide constant income, which us reduces by the rising inflation attend by the economy. Thus, it could be understood that bond investment in long-term mainly reduces the overall return from investment.

## Standard Deviation of Stocks A and B

The first debate mainly centres on, whether stocks in long-term investment is an advantage to work out over your particular investment horizon. The statement mainly depicts that whether stock investment could be relied on and might provide long-term income from investment. The overall investment in stocks mainly provide a higher return from investment, as it helps in generating maximum number of return from investment. Some researchers mentioned that stock market mainly provides a higher income from investment, as it taps into the growing economy, which results in higher profits from investment. On the other hand, other researchers criticises that investment in stock market mainly increases the risk from investment, which might hamper investment capital. The debate that investment in long-term situation mainly increases the overall return from investment, as depicted in figure 1. The 20-year bond mainly depicts a reduced retune on investment, whereas the overall return from investment in stock market increased exponentially. The stock market return mainly increased relatively higher than the returns that are generated from US bonds. However, the bond mainly provides a constant level of income with reduced risk. However, the particular investment horizon mainly increased the return from stock market, while the bonds return gradually increased. However, from 2000 to 2009 the return from market mainly increased and tested a higher level of retune. However, the return from 2000 to 2009 was mainly same, whereas the return from bonds gradually increased.

The return from investment from stock market is mainly depicted in the above figure, which states the return that might be generated from investment. It is mainly depicted that in short term period return of 10 years was negative, whereas the bond provide higher return from investment. Thus, it could be understood that long term, investment in stock market could provide a higher return from investment. Some researchers argued that bonds only provide a constant income with no risk from investment. Therefore, it could be understood that stock market investment is not that reliable, as compared to the bond returns.

The second critics are mainly about whether an investment as risky as stocks belongs in a retirement portfolio in the first place. The evaluation of risk and return of stock market could be depicted from the evaluation of figure 1 and 2. Therefore, the return from investment from stock market could be analysed. The S&P evaluation could mainly help in depicting the relevant income generated from stock market, which might help portfolio to increase their return. The decline in stock market in the pension portfolio could be highly risky, as pension funds are mainly focused in providing a constant income from investment. The overall investment of pension funds in the stock market could increase the risk and raise the chances of declined retune from investment. This lingering risk attribute of the stock market could mainly hamper the return from investment that is actually needed for the pension fund. The use of bond and property investment mainly increases the chance of continuous income that is generated from bond investment. Some researchers mentioned that pension funds during the 2008 crises mainly declined in return due to the high increments conducted in property market. Thus, it could be understood that for pension funds investment in bonds is the safest bet, as it allows them to increase the overall return from investment.

Thus, it could be understood that investors having a long-term investment perspective and can contribute to the overall risk might invest in stock market. However, if the investment portfolio is related to pension funds then it mainly needs to have a constant income from investment. Thus, it could be evaluated that use of stock market return could eventually help the investor in generating higher return from investment. Some researchers mentioned that stock market return during 2008 to 2010 mainly declined exponentially, as market was inflated by unethical means. However, the bonds value is not inflated, as it provides a constant income from investment. This non-inflationary attribute of bond mainly allows the pension fund portfolio to retain and achieve the required level of income to support its objectives.

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