Investment Analysis (FIN 670) Fall Homework 5

Transcription

1 Investment Analysis (FIN 670) Fall 009 Homework 5 Instructions: please read careully You should show your work how to get the answer or each calculation question to get ull credit The due date is Tuesday, November 10, 009. Late homework will not be graded. Name(s): tudent ID

2 1. Compute the expected return or a three-stock portolio with the ollowing: a. 13.3% b. 14.6% c. 9.3% d. 3.4% 1. b 10*0. + 1* *0.5 = A portolio is considered to be eicient i. a. there is no other portolio with a higher expected return b. there is no other portolio with a lower risk c. there is no other portolio oers a higher expected return with a higher risk d. there is no other portolio oers a lower risk with the same expected return. d 3. Which o the ollowing is (are) most correct concerning a two-stock portolio? a. The portolio should have no company speciic risk. b. Portolio standard deviation can never be a weighted average o the two stocks' standard deviations. c. Portolio return is a weighted average o the two stocks' returns. d. All o the above are correct. 3. c 4. The maximum beneit o diversiication can be achieved by combining securities in a portolio where the correlation coeicient between the securities is. a. between 0 and -1 b. 0 c. -1 d c 5. A portolio is composed o two stocks, A and. tock A has a standard deviation o return o 0% while stock has a standard deviation o return o 30%. tock A comprises 40% o the portolio while stock comprises 60% o the portolio. What is the standard deviation o return on the portolio i the correlation coeicient between the returns on A and is 0.5? a. 3.1% b. 5% c. 6% d. 4.7% 5. a σ = (.4) (.) + (.6) (.3) σ =.053 σ = (.4)(.6)(.)(.3)(.5)

3 6. A portolio is composed o two stocks, A and. tock A has an expected return o 10% while stock has an expected return o 18%. What is the proportion o stock A in the portolio so that the expected return o the portolio is 16.4%? a. 0. b. 0.8 c. 0.4 d a E(Rp) = (Wa)E(Ra) + (1-Wa)E(Rb) = Wa(0.10) + (1-Wa)(0.18) Wa = Which o the ollowing portolios cannot lie on the eicient rontier? 7. b a. Portolio X b. Portolio Y c. Portolio Z d. All portolios should lie on the eicient rontier. 8. The standard deviation o return on stock A is 0.5 while the standard deviation o return on stock is I the covariance o returns on A and is 0.06, the correlation coeicient between the returns on A and is. a. 0. b. 0.6 c. 0.7 d d Correlatio n =.06/[.5(.30)] =.8 9. Careul selection o dierent stocks rom dierent industries can eliminate the risk o a portolio. a. Nonsystematic b. Market c. Total d. All o the above. 9. a 10. A positive covariance between two stocks' returns indicates that the two stocks' returns. a. move in opposite direction b. move in the same direction c. have the same risk d. have no relationship

4 10. b 11. What happens typically to the portolio's risk when more stocks are added to a 5-stock portolio? a. The portolio's market risk would decrease. b. The portolio's total risk would decline. c. The portolio's unsystematic would decrease. d. oth and C above are correct. 11. d 1. Which o the ollowing statements are correct concerning diversiiable risks? I. Diversiiable risks can be essentially eliminated by investing in several unrelated securities. II. The market rewards investors or diversiiable risk by paying a risk premium. III. Diversiiable risks are generally associated with an individual irm or industry. IV. eta measures diversiiable risk. a. I and III only b. II and IV only c. I and IV only d. II and III only e. I, II, and III only 1. a 13. Which o the ollowing statements concerning nondiversiiable risk are correct? I. Nondiversiiable risk is measured by standard deviation. II. ystematic risk is another name or nondiversiiable risk. III. The risk premium increases as the nondiversiiable risk increases. IV. Nondiversiiable risks are those risks you can not avoid i you are invested in the inancial markets. a. I and III only b. II and IV only c. I, II, and III only d. II, III, and IV only e. I, II, III, and IV 13. d 14. Which one o the ollowing is an example o a nondiversiiable risk? a. a well respected president o a irm suddenly resigns b. a well respected chairman o the Federal Reserve suddenly resigns c. a key employee o a irm suddenly resigns and accepts employment with a key competitor d. a well managed irm reduces its work orce and automates several jobs e. a poorly managed irm suddenly goes out o business due to lack o sales 14. b

5 15. Which o the ollowing risks are relevant to a well-diversiied investor? I. systematic risk II. unsystematic risk III. market risk IV. nondiversiiable risk a. I and III only b. II and IV only c. II, III, and IV only d. I, II, and IV only e. I, III, and IV only 15. e 16. Which one o the ollowing is an example o systematic risk? a. the price o lumber declines sharply b. airline pilots go on strike c. the Federal Reserve increases interest rates d. a hurricane hits a tourist destination e. people become diet conscious and avoid ast ood restautants 16.c 17. Which one o the ollowing is an example o unsystematic risk? a. the inlation rate increases unexpectedly b. the ederal government lowers income taxes c. an oil tanker runs aground and spills its cargo d. interest rates decline by one-hal o one percent e. the GDP rises by percent more than anticipated 17. c 18. Which o the ollowing actions help eliminate unsystematic risk in a portolio? I. spreading the retail industry portion o a portolio over ive separate stocks II. combining stocks with bonds in a portolio III. adding some international securities into a portolio o U.. stocks IV. adding some U.. Treasury bills to a risky portolio a. I and III only b. I, II, and IV only c. I, III, and IV only d. II, III, and IV only e. I, II, III, and IV 18. e

6 19. All things equal, diversiication is most eective when a. securities' returns are positively correlated. b. securities' returns are uncorrelated. c. securities' returns are high. d. securities' returns are negatively correlated. e. A and C. 19. d All things equal, diversiication is most eective when securities' returns are negatively correlated. 0. An investor who wishes to orm a portolio that lies to the right o the optimal risky portolio on the Capital Allocation Line must: a. lend some o her money at the risk-ree rate and invest the remainder in the optimal risky portolio. b. borrow some money at the risk-ree rate and invest in the optimal risky portolio. c. such a portolio cannot be ormed. d. invest only in risky securities. e. and D 0. e An investor who wishes to orm a portolio that lies to the right o the optimal risky portolio on the Capital Allocation Line must borrow some money at the risk-ree rate and invest in the optimal risky portolio. This also implies that the investors must invest only in risky securities. 1. Portolio theory as described by Markowitz is most concerned with a. the elimination o systematic risk. b. the identiication o unsystematic risk. c. the eect o diversiication on portolio risk. d. active portolio management to enhance returns. e. none o the above 1. c Portolio theory as described by Markowitz is most concerned with the eect o diversiication on portolio risk.. ecurity C has expected return o 1% and standard deviation o 0%. ecurity D has expected return o 15% and standard deviation o 7%. I the two securities have a correlation coeicient o 0.7, what is their covariance? a b c d e a Cov(r X, r Y ) = (.7)(.0)(.7) =.0378.

7 The ollowing inormation is or question 3-6 Abigail Grace has a $900,000 ully diversiied portolio. he subsequently inherits AC Company common stock worth $100,000. Her inancial advisor provided her with the ollowing estimates: The correlation coeicient o AC stock returns with the original portolio returns is The inheritance changes Grace s overall portolio and she is deciding whether to keep the AC stock. Assuming Grace keeps the AC stock, calculate the: i. Expected return o her new portolio which includes the AC stock. E(r NP ) = w OP E(r OP ) + w AC E(r AC ) = ( ) + ( ) = 0.78% ii. Covariance o AC stock returns with the original portolio returns. Cov = r σ OP σ AC = = iii. tandard deviation o her new portolio which includes the AC stock. σ NP = [w OP σ OP + w AC σ AC + w OP w AC (Cov OP, AC )] 1/ = [( ) + ( ) + ( )] 1/ =.673%.7%

8 4. I Grace sells the AC stock, she will invest the proceeds in risk-ree government securities yielding 0.4 percent monthly. Assuming Grace sells the AC stock and replaces it with the government securities, calculate the: (remember the standard deviation o government securities = 0) a. Expected return o her new portolio which includes the government securities. E(r NP ) = w OP E(r OP ) + w G E(r G ) = ( ) + ( ) = 0.645% b. Covariance o the government security returns with the original portolio returns. Cov = r σ OP σ G = = 0 c. tandard deviation o her new portolio which includes the government securities. σ NP = [w OP σ OP + w G σ G + w OP w G (Cov OP, G )] 1/ = [( ) + (0.1 0) + ( )] 1/ =.133%.13% 5. Determine whether the systematic risk o her new portolio, which includes the government securities, will be higher or lower than that o her original portolio. Explain briely Adding the risk-ree government securities would result in a lower beta or the new portolio. The new portolio beta will be a weighted average o the individual security betas in the portolio; the presence o the risk-ree securities would lower that weighted average.

9 6. ased on conversations with her husband, Grace is considering selling the $100,000 o AC stock and acquiring $100,000 o XYZ Company common stock instead. XYZ stock has the same expected return and standard deviation as AC stock. Her husband comments, It doesn t matter whether you keep all o the AC stock or replace it with $100,000 o XYZ stock. tate whether her husband s comment is correct or incorrect. Justiy your response. The comment is not correct. Although the respective standard deviations and expected returns or the two securities under consideration are equal, the covariances between each security and the original portolio are unknown, making it impossible to draw the conclusion stated. For instance, i the covariances are dierent, selecting one security over the other may result in a lower standard deviation or the portolio as a whole. In such a case, that security would be the preerred investment, assuming all other actors are equal. 7. George tephenson s current portolio o $.0 million is invested as ollows: (K) tephenson soon expects to receive an additional $.0 million and plans to invest the entire amount in an index und that best complements the current portolio. tephanie Coppa, CFA, is evaluating the our index unds shown in the ollowing table or their ability to produce a portolio that will meet two criteria relative to the current portolio: (1) maintain or enhance expected return and () maintain or reduce volatility. Each und is invested in an asset class that is not substantially represented in the current portolio. (K) tate which und Coppa should recommend to tephenson. Justiy your choice by describing how your chosen und best meets both o tephenson s criteria. No calculations are required. Fund D represents the single best addition to complement tephenson's current portolio, given his selection criteria. First, Fund D s expected return (14.0 percent) has the

10 potential to increase the portolio s return somewhat. econd, Fund D s relatively low correlation with his current portolio (+0.65) indicates that Fund D will provide greater diversiication beneits than any o the other alternatives except Fund. The result o adding Fund D should be a portolio with approximately the same expected return and somewhat lower volatility compared to the original portolio. The other three unds have shortcomings in terms o either expected return enhancement or volatility reduction through diversiication beneits. Fund A oers the potential or increasing the portolio s return, but is too highly correlated to provide substantial volatility reduction beneits through diversiication. Fund provides substantial volatility reduction through diversiication beneits, but is expected to generate a return well below the current portolio s return. Fund C has the greatest potential to increase the portolio s return, but is too highly correlated to provide substantial volatility reduction beneits through diversiication. The ollowing data apply to question 8--3 A pension und manager is considering 3 mutual unds. This irst is a stock und, the second is a long-term government and corporate bond und, and the third is a T-bill money market und that yields a rate o 8%. The probability distribution o the risky unds is as ollows Expected return tandard deviation tock und () 0% 30% ond und () 1% 15% The correlation between the und returns is What are the investment proportions in the minimum variance portolio o the two risky unds, what is the expected value and standard deviation o its rate o return? (The weight o stock in the minimum variance portolio is calculated using the ollowing ormula) σ Cov( r, r ) w Min () = σ + σ Cov( r, r ) The parameters o the opportunity set are: E(r ) = 0%, E(r ) = 1%, σ = 30%, σ = 15%, ρ = 0.10 From the standard deviations and the correlation coeicient we generate the covariance matrix [note that Cov(r, r ) = ρσ σ ]: onds tocks onds 5 45 tocks The minimum-variance portolio is computed as ollows:

11 σ Cov(r,r ) 5 45 w Min () = = = σ + σ Cov(r,r ) ( 45) w Min () = = The minimum variance portolio mean and standard deviation are: E(r Min ) = ( ) + ( ) = 13.39% σ Min = [ w σ + w σ + w w Cov(r,r )] = [( ) + ( ) + ( )] 1/ = 13.9% 1/ 9. Now, assume you decide to include T-bill into your complete portolio. olve numerically or the proportions o each asset and or the expected return and standard deviation o the optimal risky portolio (the optimal risky portolio is the combination o stock and bond und that gives investor the best risk-return trade-o when combining with T-bill) (The proportion o the optimal risky portolio invested in the stock und is given by: w [E(r ) r ] σ [E(r ) r ]Cov(r,r ) = [E(r ) r ] σ + [E(r ) r ] σ [E(r ) r + E(r ) r ]Cov(r,r ) The proportion o the optimal risky portolio invested in the stock und is given by: w [E(r ) r ] σ [E(r ) r ]Cov(r,r ) = [E(r ) r ] σ + [E(r ) r ] σ [E(r ) r + E(r ) r ]Cov(r,r [(0 8) 5] [(1 8) 45] = = [(0 8) 5] + [(1 8) 900] [( ) 45] w = = The mean and standard deviation o the optimal risky portolio are: E(r P ) = ( ) + ( ) = 15.61% σ p = [( ) + ( ) + ( )] 1/ = 16.54% )

12 30. What is the reward-to-variability ratio o the best easible CAL? E(r ) r p σ p = = uppose now that your portolio must yield an expected return o 14% and be eicient, that is, on the best easible CAL. a. What is the standard deviation o your portolio? I you require that your portolio yield an expected return o 14%, then you can ind the corresponding standard deviation rom the optimal CAL. The equation or this CAL is: E(r E(r ) r p C ) = r + σc = σ P etting E(r C ) equal to 14%, we ind that the standard deviation o the optimal portolio is 13.04%. b. What is the proportion invested in the T-bill und and each o the two risky unds? σ C To ind the proportion invested in the T-bill und, remember that the mean o the complete portolio (i.e., 14%) is an average o the T-bill rate and the optimal combination o stocks and bonds (P). Let y be the proportion invested in the portolio P. The mean o any portolio along the optimal CAL is: E(r C ) = (l y)r + ye(r P ) = r + y[e(r P ) r ] = 8 + y( ) etting E(r C ) = 14% we ind: y = and (1 y) = (the proportion invested in the T-bill und). To ind the proportions invested in each o the unds, multiply times the respective proportions o stocks and bonds in the optimal risky portolio: Proportion o stocks in complete portolio = = Proportion o bonds in complete portolio = = 0.434

13 3. I you were to use only the two risky unds and still require an expected return o 14%, what would be the investment proportions o your portolio? Compare its standard deviation to that o the optimal portolio in the previous problem (#31). What do you conclude? Using only the stock and bond unds to achieve a portolio expected return o 14%, we must ind the appropriate proportion in the stock und (w ) and the appropriate proportion in the bond und (w = 1 w ) as ollows: 14 = 0w + 1(1 w ) = 1 + 8w w = 0.5 o the proportions are 5% invested in the stock und and 75% in the bond und. The standard deviation o this portolio will be: σ P = [( ) + (0.75 5) + ( )] 1/ = 14.13% This is considerably greater than the standard deviation o 13.04% achieved using T-bills and the optimal portolio. 33. tocks oer an expected rate o return o 10% with a standard deviation o 0%, and gold oers an expected return o 5% with a standard deviation o 5%. a. In light o the apparent ineriority o gold to stocks with respect to both mean return and volatility, would anyone hold gold? Explain your answer Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as a part o a portolio. I the correlation between gold and other stocks is suiciently low (or negative), gold will be held as a component in a diversiied portolio, b. How would you answer (a) i the correlation coeicient between gold and stocks were 1.0? Could these expected returns, standard deviations, and correlation represent an equilibrium or the security market (i.e., i no one chooses to hold gold, would the situation be there or long time)? I the correlation between gold and stocks equals +1, then no one would hold gold since gold would move exactly in the same direction o stocks and stocks are better than gold in both return and standard deviation. O course, this situation could not persist. I no one desired gold, its price would all and its expected rate o return would increase until it became suiciently attractive to include in a portolio.