Assignment 3 1. Design a multifactor model with at least 2 factors besides the market factor, and answer the following questions. a) What makes your choice of factor a “factor” in multifactor model? b) Does the factor of your choice co-move with the market factor? If yes, should you include it along with the market factor? c) Describe how the stock return would be affected when the factor of your choice changes. d) Describe a scenario where one can benefit from trading on the factor of your choice.

a) There is a variety of factors that can determine the returns on security. The market based factor is the return on the broad market index such as S&P 500. This market return is one of the factors in the model. Other factors include the following: • GDP growth rate. This factors shows overall macroeconomic conditions that tend to affect a stock’s performance. • Risk free rate of return. • 10 year T-bond interest rate that shows the required return on 10-year government bonds. • A company’s ROE as an indicator of its profitability. Therefore, the model will look like this: [pic], where betas show the sensitivity of return to the factor, rm is the market rate of return, rf is the risk-free rate, T-bond is the 10-year T-Bond rate and ROE is the return on equity (ROE). Generally, GDP has a positive effect on a stock return. Higher economic growth leads to more business opportunities and potentially higher profits. This positively affects a stock’s return. Market rate of return has a positive effect assuming a positive beta. Risk-free rate has a negative impact on a stock’s performance. Higher rate leads to lower return on a company’s stock. The same effect comes from 10-year T-Bond rate. Higher rate leads to lower expected return on a stock. A company’s ROE has a positive effect, because higher profitability means higher dividends and a stock price. b). Interest rates move in an opposite direction compared to the market factor. GDP growth rate tends to go inline with the market return though it is not always the case. Sometimes the impact of the market factor diverges from the effect of GDP growth. ROE or any other indicator of a company’s profitability does not always move in the same direction. Therefore, the correlation between this factor and the market return is not absolutely positive. This means that the factor can benefit the model. c). When a growth in GDP goes up, the expected return will also increase, but the growth will depend on its sensitivity to this factor. A drop in GDP growth will signal lower return. Both risk free rate and 10-year T-bond rate will have a negative effect on a company’s stock. The magnitude will also depend on the sensitivity to the factor. The ROE will have a positive effect, as it is already mentioned. Thus, a rise in profitability will drive the return upward while a drop in profitability will hurt the return. The sensitivity to the factor will determine the rise/fall in a stock’s return. d). Investors can benefit from trading on these factors by building the regression model and identifying the exact beta ratios to quantify the model’s effects on a particular stock. The model enables investors to find the sensitivity of a stock return to the overall economic health. If a stock is cyclical, it will respond to a rise in GDP with higher return. Thus, this model will help assess this stock’s performance. Interest rates help identify a stock’s sensitivity to the factor. This also helps in trading decisions. For example, if a stock is too sensitive to the rate, it is possible to estimate the stock’s return if the rates change. The model also ROE helps measure the sensitivity of a stock to a change in profitability. This may also help better estimate a stock return.

2. Select 3 stocks with monthly trading information available for the previous 60 months. Complete the following tasks in Excel and write a simple report with Word. a. Calculate the return and standard deviation of the 3 stocks b. Calculate the covariance and correlation between each pair of stocks c. Construct an optimal portfolio with the 3 stocks under the objective of maximizing the Sharpe Ratio, assuming the risk free rate is 1% annually. d. Write a report that summarizes your work. Submit both Excel and Word files.

There are three stocks chosen – Apple Inc. (AAPL), Exxon Mobil (XOM) and The Boeing Company (BA). These companies are from different sectors and this fact is beneficial for diversification. Adjusted close is chosen because it helps take into account the stock split and effects from dividends. Therefore, this allows me to calculate the company’s expected return. The geometric average is used to calculate monthly expected return of the stocks. It is calculated as (P2015/P2010)^(1/60)-1, where P2015 is the closing adjusted price for October 2015 while P2010 is the closing adjusted piece for October 2010. There are 60 months, so 1/n is 1/60. Stock price data is derived from Yahoo Finance (2015). Portfolio diversification is beneficial for investors, because it allows them to reduce the risk without hurting profitability. The tool to measure portfolio performance is the Sharpe ratio. It is calculated as the expected return divided by the standard deviation. Therefore, it enables investors to measure the risk-return trade off. The optimal portfolio is the one with the highest Sharpe ratio. The Sharpe ratio is reward-to-volatility ratio; ratio of portfolio excess return to standard deviation]. The higher ratio signal better reward-risk trade off. Thus, it enables investors to take into account of both the effect from the expected return and risk measured by standard deviation. Apple operates in technologies sector. The company has been growing in market capitalization in recent 5 years. The adjusted stock price has increased from $40.04 in October 2010 to $118.99 at the end of October 2015. Therefore, Apple has almost tripled since 2010 (a growth of 197.17%). The average expected monthly return for AAPL is 1.83%. However, the risk-free rate is 1% annually, so the monthly excessive return is about 1.75%. To measure the risk associated with the AAPL stock it is required to find its variance and standard deviation. As it is shown by Table 1, the company’s variance is 0.004739 while its standard deviation is 0.068839. Apple’s Sharpe ratio is about 0.254 (25.4%). The company’s stock is the highest among the companies, so is its Sharpe ratio and standard deviation. Exxon Mobil operates on energy market in petroleum and gas industry. The company is among the most capitalized in the world. The company has increased in market value since 2010. By 2014 the company has grown to $96.97 and then it collapsed to less than $74. XOM stock has soared by 42.37% since 2010. The geometric average annual return was 7.32%. The expected monthly return is 0.59%. The excessive expected monthly return is about 0.51%. The stock’s risk is measured by variance and standard deviation. The variance for XOM is 0.002023. Standard deviation for the stock is 0.044981. The stock’s Sharpe ratio is 0.112754 or 11.28%. The company’s stock is the least profitable and also has the lowest Sharpe ratio among the three. Taking into account that Exxon Mobil operates on a different sector, it is exposed to different risks than Apple and Boeing, so it is a good choice for diversification. The Boeing Company (BA) operates in airline manufacturing. The company’s stock price has grown from $62.53 to $147.16 since 2010. Therefore, the stock has jumped by 135.34%. The geometric average annual rate of return has been 18.67%. The expected monthly return is 1.44% while the excessive monthly return is 1.35%. The company’s variance is 0.003217 while the standard deviation is about 0.057. The Boeing’s Sharpe ratio is 0.2386 or 23.86%. The stock is second in terms of profitability, risk and the Sharpe ratio.
| |Return |Variance |
| |AAPL |XOM |BA |
|AAPL |0,004739 |0,000851 |0,00087 |
|XOM |0,000851 |0,002023 |0,000935 |
|BA |0,00087 |0,000935 |0,003217 |

Table 2: The Covariance Matrix. Portfolio diversification implies the spreading risk among different investment option, in this case, stocks. There are three stocks but there is an opportunity to create a portfolio with lower risk at a given rate of return or a portfolio with higher rate of return at a given level of risk. Therefore, this portfolio lies on the left and northern sides of the efficient portfolio frontier. An optimal portfolio is the one with the highest Sharpe ratio and return per risk. Therefore, it is the most efficient and beneficial portfolio. To find the optimal portfolio it is important to use portfolio optimization using the linear algebra. This allows investors to find the optimal weights for their stock portfolios. First step is to find the inverse matrix, which is shown by Table 3 below. This is the inverse covariance matrix used for optimization.
|Inverse Matrix | |
|232,6042 |-79,4361 |-39,8264 |
|-79,4361 |598,1043 |-152,394 |
|-39,8264 |-152,394 |365,8804 |

Table 3: The Inverse Matrix. Next step is to find the so-called Z-ratios which are shown by Table 4. The ratios will be used to find the optimal weights and the most efficient portfolio. As it is seen from the table, two ratios are positive while one is negative. This implies that the optimal portfolio involves short selling.
|Z ratio |
|3,125235 |
|-0,41789 |
|3,482344 |
|Sum of Z |
|6,189687 |

Table 4: Z-ratios. Final section is about finding the optimal weights from Z-ratios. The optimal weight for a stock is its Z-ratio divided by the sum of the ratios. Table 5 shows the optimal portfolio. As it is seen from the table, the optimal portfolio implies that 56.26% of capital is to be invested into Boeing’s stock, 50.49% will be invested in Apple’s stock and 6.75% of capital has to be borrowed by selling Exxon Mobil’s stock.
|w1 |0,50491 |
|w2 |-0,06751 |
|w3 |0,562604 |
|Sum |1 |

Table 5: Optimal weights. Therefore, the optimal portfolio will benefit the investors with the higher Sharpe ratio. Table 6 shows performance of the optimal portfolio. As it is seen, the portfolio’s expected return is 1.61% - it is slightly lower than for Apple and higher than Boeing’s return. The portfolio’s risk has lower than Boeing’s level. As suggested by the Sharpe ratio, the optimal portfolio is the highest than that of any stocks. Therefore, portfolio diversification really benefits investors. This is suggested by better risk-reward trade off. Therefore, although investing in Exxon Mobil is the least risky option, it is the least efficient due to extremely low rate of return. On the other hand, the optimal portfolio is not the least risky though its Sharpe ratio indicates the best possible reward-risk trade off.
|Portfolio | |
|Return |0,0161 |
|Variance |0,002601 |
|SD |0,051001 |
|Sharpe |0,31568 |

Table 6: The Optimal Portfolio Performance. 3. Consider a bond selling at par with modified duration of 12 years and convexity of 265. A 1% decrease in yield would cause the price to increase by 12%, according to the duration rule. What would be the percentage price change according to the duration-with- convexity rule?

ΔP/P = -D * Δy + ½ * Convexity * (Δy)^2 = -12 * -.1 + ½ * 265 * (0.1) ^2 = 0.1332 or 13.32%

4. Discuss the differences in writing covered and naked calls. Are risks involved in the two strategies similar or different? Explain.

Writing a covered call is selling a call on stock that the investor owns, is very conservative; the investor receives the premium income from writing the call. If the call is exercised, the stock is called away from the investor; thus the investor has limited his or her upside potential. Writing a naked call is a very risky strategy. The investor sells a call on a stock the investor does not own. If the price of the stock increases, the option will be exercised and the investor must go into the open market and buy the stock at the prevailing market price. Theoretically, the price to which the stock can increase is unlimited; thus, the investor's potential loss in unlimited.