**PART 1 (Lambda=2)**

Investors could be categorized according to their risk profiles. Some investors are risk averse; some are risk neutral while others are risk takers. The optimization of risk has been performed by optimizing the following expression:

**µ _{ω} – λ σ_{ω}^{2}**

Here Lambda represents the investors dislike for the risk or variance. In the problem the optimal portfolio has been found through excel solver for an investor who is risk averse with a lambda of 2. All the computations are performed in the excel spreadsheet with the portfolio mean return, standard deviation of portfolio, value at risk for 95% and 99% confidence level, the probability for loss on investment in the portfolio. The optimal mix for the two assets in this portfolio which are stock and bonds is to invest 49% and 51% in both of these assets respectively.

**PART 2 (Lambda=1)**

The same optimization problem with respect to the investor’s dislike to risk has been performed using excel solver. However, this time the lambda has been taken as 1 which means that the investor desires to take more risk. Based upon this the optimal mix for investment in stocks and bonds is 91% and 9% respectively. It could be seen that the mean return has increased to 11.2% from 8.4% when lambda was 2. This is because now the investor is willing to take more risk. However, the standard deviation for the portfolio has also increased as high risk is compensated by high return.

**PART 3 (Probability to Lose Money)**

In this problem the optimization objective is to basically attain the asset weights for stocks and bonds which minimizes the probability of losing money and that attains a minimum probability when log return on the portfolio is less than 0. All the computations are performed in the excel spreadsheet. The optimal weights for investment in stocks and bonds are 23% and 77% respectively.

**PART 4 (Probability Shortfall 5%)**

In this part the optimal portfolio weights have been found for a different level of shortfall. The level of shortfall in this part has been set at 5% and this means that this optimization exercise would generate a portfolio which would minimize the probability of attaining a holding period return of less than 5%. The optimal weights for investment for the optimal portfolio based upon a minimum HPR of 5% are 62% and 38% respectively.

**Comparison Table**

If we compare all the four above optimized models of portfolio optimization then it could be seen that, the highest mean return for the portfolio has been achieved by model 2 as in second part above. Nevertheless, the risk is also high for investing in this portfolio. The value at risk which is the maximum loss possible on investment in a particular portfolio is also higher for this portfolio at a 95% and 99% confidence level. However, the expected value of an investment of $ 10,000 is worth more under this portfolio as compared to the first and the fourth portfolio model. Overall, looking at the comparison table and comparing all the parameters for portfolio optimization, it could be said that the dislike of the investor for risk or volatility is of more importance as compared to attaining a minimum holding period return or accepting a shortfall on investment. Investors thus prefer to invest in such portfolios based upon their specific level of risk.Optimal Portfolio of Stocks and Bonds Case Solution

**PART 5 (Probability Shortfall Table)**

The same analysis as performed in part 4 above has been performed in this problem one by one for different levels of H (Holding period returns) ranging from -10% to +10%. The results for mean, standard deviation, probability of losing money, value at risk and expected values of the investments are shown in the table for different shortfall levels. The most suitable portfolio according to me is by minimizing the probability of attaining a holding period return of less than 5%. It has been highlighted in the excel spreadsheet. It could be seen that the mean return for this portfolio is second highest among all the different shortfall levels and the risk is much lower as compared to majority of other portfolios. Similar is the risk with value at risk for 95% and 995 confidence level. Therefore, it is more comfortable for me to invest in this portfolio with an investment of 62% in stocks and 38% in bonds with a mean portfolio return of 9.25%.............................

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