Probability distribution of the HPR on the put option and on the stock;
The formula for calculating HPR is
A point of importance that needs a note is that the put option in this case costs $ 12.00. The cost of the index is $ 100 per share.
Explanations (workings)
From the calculations, the negative HPR of the put option shows that there is an advantage inherent in the option and there is presence of priced risks. It can also show that the options have traded for a short time. Options are sensitive and as volatility increases, expected put option returns increases, that is, they become negatives.
What is the probability distribution of the HPR on a portfolio on a portfolio consisting of one share of the index fund and a put option?
In this case, one share of the index fund together with a put option costs $ 112.00. Therefore, the probability distribution of the HPR on the portfolio is as given below;
Explanations
HPR Boom = ($140 – $112 + $4)/$112 = 28.6%
HPR Normal = ($110 – $112 + $4)/$112 = 1.8%
HPR Recession = ($110 + $112 + $4)/$112 = 1.8%
This shows that, during the boom, there is a high Holding Period Return of 28.6% which is a good indicator as compared to 1.8% during poor state of the economy (recession)
Purchase of the put option constituting a purchase of insurance in this case.
When the HPR is at a minimum of 1.8%, it is a complete guarantee no matter what happens to the stock price. At this point, there is protection against decline in prices and therefore, it offers insurance against price decline Barucci (2003).
Analysis
Holding Period Ratio refers to the return received from holding a portfolio of assets (Marcus, Bodie, & Kane, 2010). To obtain HPR, the income at the beginning of any given period which includes capital and dividend income, is divided by the amount that obtained at the beginning of the period. It measures the return one has obtained from a certain investment. Holding period refers to the time an investor owns an asset before selling it while holding period return refers to the return for the period (Marcus, Bodie, & Kane, 2010).
Put option – it is the right; a holder of an underlying asset is given to sell the asset or security at a particular specified time at a certain amount. A put option does not put an obligation to the holder of the security to sell it. On the other hand, a call option allows the holder the right to buy and sell securities (Eugene & Michael 2010). When the price of an underlying asset depreciates, a put becomes more valuable (Subramani, 2011).
A stock refers to a unit of capital held by investors in a company. This is the capital invested in a business by its founders. Stocks are different from other assets held by a business in that the assets may depreciate in value and quantity while stocks do not depreciate Edwin, Martin, Stephen, & William (2009)
The Security Market Line abbreviated as (SML) is a line that benchmarks the expected performance of an investment. Assets that are properly valued plot very well on the SML Eugene & Michael (2010).
a) In this case, what McKay should do is that he should invest in Murray’s existing portfolio. To do this, he should borrow more funds where he will invest in the more risky assets on margin. There will be increased expected return and in addition to this, the portfolio on the CML (Capital Market Line) will increase the risk (variability) because of higher proportion of risky assets in the portfolio Edwin, Martin, Stephen, & William (2009).
Capital Market Line – graphical representation.
According to Separability theorem, all investors, regardless of their attitude towards risk, should hold the same risk assets in their portfolios. CML is usable in capital asset pricing model to show the rate of return for portfolios depending on the standard deviation and the risk free rate of return for a certain portfolio Levich (2001). The line is obtained by drawing a tangent from the intercept on the efficient frontier up to the where the expected return is equal to the risk free rate of return Sercu & Uppal (1995).
Lisa, however, wants to cut the risk exposure of her portfolio. She does not want to do lending or borrowing to achieve her objective. To achieve York’s objective, McKay should do the following;
He should take in higher beta stocks and replace the low beta stocks. Once he does this, he will reduce the total beta of York’s portfolio. When he manages to reduce the overall portfolio beta, he will have succeeded in reducing the systematic risk of the total portfolio and thus, the portfolio volatility relative to the market Bodie, Kane & Marcus (2008). The primary objective of York’s is in the reduction of her exposure to risk. York manages to meet this objective by reducing the beta of the entire portfolio. In other words, York’s objective is not in the maintenance of efficiency. (Subramani, 2011). If McKay sold all her equities in order to use the money received to buy risk free securities, she will not reduce her risk exposure.
This is because; York does not allow lending or borrowing Oster (1994).
The risk return trade of between the return on any security and the systematic risk is what the Security Market Line represents. It is as follows;
Risk
Risk and return – their relationship.
Most of the investors increase the rate of return as the perceived risk increases. SML reflects the risk and return available on the different investments. This involves all the risky assets in the market at any given time. Investors invest in assets that have their preferred levels of risk. This involves investors who invest in low risk investment and investors who invest in high-risk investment Richard & DeFusco (2011). When one begins with an initial SML, three possible changes may occur. To start with, individual investments on the SML may change their position because of changes in the deemed risk of the investment. Secondly, the SML slope may change due to the change in the behavior and attitudes of investors towards risk and finally the SML may have a parallel shift because of change in the expected rate of inflation Richard & DeFusco (2011). With this information, they can be able to make informed decisions.
In consideration of client in problem 19 where A=3.5.
If the investor decides to invest in the passive portfolio, the proportion, y that he would select is as follows;
- The optimal proportion to conduct an investment in the passive portfolio is given by the formula below;
Substituting the following;
E(rM) = 13%; rf = 8%; σM = 25%; A = 3.5:
- The percentage of the investment in the respective fund, subtracted at the year-end, that is, charged to make the client indifferent between the passive strategy and fund affected by the capital allocation decision, that is, his choice of y is not affected by capital allocation. This means that one can charge a fee that is equivalent to the reward to volatility ratio of such a portfolio to your competition Zhang (2009).
Analysis and Definition
Passive portfolio refers to a strategy of investment founded on the principle of diversification. The goal of passive portfolio is to have a market index’s return. Passive managers buy and hold security from the market Solnik (2000).
Most of the holding assets reduce the investment risk since they correlate negatively. The portfolios that have the highest return at any degree of risk are referred to as efficient portfolios. They also provide the lowest degree of risk at any expected return. The investor should make sure that he holds those assets that minimize his risks. A diversifiable risk is a risk that an investor can eliminate if he held an efficient portfolio. Non-diversifiable risks are the risk that still exists in efficient and diversified portfolios. In essence, the investor aims at eliminating the diversifiable risk Reilly & Brown (2007).
Rearranging the table, we obtain the following;
There is a positive correlation between inflation and the returns of assets since they all lean towards zero (Reilly, & Brown, 2007). The degree of the serial correlation is what makes the difference. The cause of the highest positive serial correlation is the large company stocks and inflation. This is followed by government intermediate term of 0.32. The high correlation could be due to government long-term investment. The high correlation could be due to small company investments and treasury bills. It is therefore prudent to note that there is a positive serial correlation between inflation and returns, as all these figures tend to lean towards zero (Wittner, 2003).
Computation of the serial correlation in decade return, the given data is arranged and set up in an excel sheet and then, the function in the excel ”CORREL” is used to do the calculation.
The correlation above is based on 7 observations. This means that, we cannot derive any meaningful statistical conclusions. The above results lead us into making one conclusion that there happens to be persistent serial correlation with the exclusion of the stocks from the large companies. There will be a change in this conclusion when real rates are used as depicted in question 21.
Serial Correlation is a study that is used to tell whether there is a relationship between dependent and independent variables. According to Durbin Watson, the number ranges from zero to four. Number 2 is the pivotal point meaning there is no autocorrelation Edwin, Martin, Stephen, & William (2009). Any figure that is calculated and tends to move towards four shows a negative autocorrelation whereas figures towards 0 show a positive autocorrelation. In the case below, we shall find out if there is an autocorrelation between the various share returns and the inflation rates for the period given Edwin, Martin, Stephen, & William (2009)
When all the other factors are equal and held constant, the active portfolio will have an Information Ratio that doubles. The portfolio (S-square) will have the Sharpe Ratio square being equal to the Sharpe ratio for the (SM-square) market index added to the IR’s square. This translates to the fact that IR will quadruple because it has doubled.
S-square is to SM-square + (4 × IR). In comparison to the previous S-square, then the difference is 3 IR (Marcus, Bodie, & Kane, 2010).
The answer of question 20 Page 277
The table below summarizes the solution to question 20 on page 227.
Solution is as follows;
S2P/S2M = b2; (0.2)2/(0.16)2 = 1.56; b=1.25
According to Sharpe ratio, risk and return are directly proportional. For every additional risk, there is an additional return. The usage of Sharpe ratio is to evaluate the adjusted risk of a portfolio. This is then usable in the determination of the additional risk that a portfolio may come with (Porter, 1980). The driving forces of the ratio are rate of return from the portfolio, the risk free rate of return and the portfolios standard deviation. This ratio tells whether high returns are realized because of wise investment or just risk taking. In the above scenario, when alpha doubles, the overall risk of the portfolio increases Edwin, Martin, Stephen, & William (2009)
Alpha is a performance measure ratio that takes into account the volatility of a portfolios return and compares the same with that of a risk free asset or that of a risk-adjusted portfolio. A positive alpha tells or shows an over performance whereas a negative alpha depicts an under performance of an asset. The beta of this asset is expected to increase since it is a measure of risk (Levich, 2001).
Beta, alpha, standard deviation, covariance, and Sharpe’s ratio are some of the parameters used in risk evaluation. All of them are driven by the return on investment and the amount of risk involved in the investment. It is therefore important to note that if alpha doubles or varies, the other measures of risk that are dependent on alpha will also vary appropriately. In the above case, when alpha for S&P 500, HP, DELL, WMT, TARGET, BP and SHELL double, the beta will also vary Porter (1990). This therefore tells us that alpha, covariance, standard deviation, beta and Sharpe’s ratio are related and any variance or change in one of them causes the change of the other variable relatively. From the above computation, HP proves to be more risky with a beta of 2.0348, followed by TARGET at 1.2672, then DELL, HP, SHELL then finally BP at 0.467. This tells us that BP is the least risky if beta is to be used to evaluate its performance. Risk premium still shows that HP is the most risk and the less risky is BP with a risk premium of 0.04 (Marcus, Bodie, & Kane, 2010). This therefore tells any potential investor that if risk is his or her basis of decision making then the most appropriate or convenient investment is BP, which poses the least risk. If return on investment is anything to go by as an investment motive then HP promises the highest return than any other asset in this portfolio. Any investor who considers himself or herself a risk taker is advised to invest in HP as opposed to any other asset since HP promises the best reward in terms of returns (Solnik, 2000). Correlation advises that an investor can only invest in two assets at the same time if there is a correlation between these assets. In this case HP DELL and SHELL have a correlation of 0.08, 0.5 and 0.7 respectively (Sercu, & Uppal 1995).
References
Barucci, E. (2003). Financial Markets Theory: Equilibrium, Efficiency, and Information. New York: Springer.
Bodie, Z, Kane, A., & Marcus, A. (2008). Investments. New York: McGraw-Hill Irwin.
Edwin J., Martin J., Stephen J. & William N. (2009). Modern Portfolio Theory and Investment Analysis. Boston: John Wiley & Sons.
Eugene F. & Michael C. (2010). Financial Management Theory and Practice. London: Cengage Learning.
Levich, M. (2001). International Financial Market, (2nd ed.). New York: McGraw-Hill.
Marcus, A. J., Bodie, Z., & Kane, A. (2010). Investment and Portfolio Management. Florida: McGraw Hill.
Oster, S. (1994). Modern Competitive Analysis. Nueva: Oxford University Press.
Porter, M. (1980). Competitive Strategy: Techniques for Analyzing Industries and Competitors. London: The Free Press.
Porter, M. (1990). The Competitive Advantage of Nations. London: MacMillan Press.
Reilly, K., & Brown, C. (2007). Investment Analysis and Portfolio Management. New York: Southwestern Thomson.
Richard A. & DeFusco, D. W. (2011). Quantitative Investment Analysis. Boston: John Wiley & Sons.
Sercu, P., & Uppal, R., (1995). International Financial Markets and the Firm. New York: SouthWestern.
Solnik, B. (2000). International Investments (4th ed.). New York: Longman.
Subramani, R. V. (2011). Accounting for Investments, Equities, Futures and Options. Boston: John Wiley & Sons.
Zhang, H. (2009). Investmet. Web.