Portfolio MGT Exam 1

(CH1, P3) At the beginning of last year, you invested $4,000 in 80 shares of the Chang Corporation. During the year, Chang paid dividends of $5 per share. At the end of the year, you sold the 80 shares for $59 a share. Compute your total HPY on these shares and indicate how much was due to the price change and how much was due to the dividend income.

$4,000 buys 80 shares (4,000/80)= $50 per share HPR = (Ending Value *including cash flows*)/(Beginning Value) HPR= [(80*$59)+(80*$5)]/$4,000= 5,120/4,000= 1.28 HPY= HPR-1= 1.28-1 = 28% HPR (Price Increase) = (59*80)/4000 = 1.18 HPY (Price) = 1.18 - 1 = 18% HPY Total= HPY(Price) + HPY Dividends .28 = .18 + HPYd HPYd= .10

(CH 1, P6) Find the expected return of the stock Probability. Possible Return .30 -10% .10 0 .30 10% .30 25%

E(R)= Sum[(Return x Probability)] E(R)= (0.3 x -10%) + (.1 x 0%) + (0.3 x 10%) + (0.3 x 25%) = 7.5%

(CH 2, P9) You are given the following long-run annual rates of return for alternative investment instruments: U.S. Government T-bills 3.50% Large-cap common stock 11.75 Long-term corporate bonds 5.50 Long-term government bonds 4.90 Small-capitalization common stock 13.10 The annual rate of inflation during this period was 3 percent. Compute real rate of return on these investments.

Real Rate of Return= [(1+Return)/(1+Inflation)] - 1 T-Bills = (1.035)/(1.03) - 1 = 0.485% or 0.00485 Large Cap Stock = (1.1175)/(1.03) - 1 = 8.495% Corp Bond = (1.055)/(1.03) - 1 = 2.42% Govt Bond = (1.049)/(1.03) - 1 = 1.845% Small Cap Stock = (1.1310)/(1.03) - 1 = 9.81%

(Appendix 1A, P1) Possible Rate of Return Probability −0.10. 0.25 0.00 0.15 0.10 0.35 0.25 0.25 a. Compute the expected return [E(Ri )] on this investment, the variance of this return (σ 2 ), and its standard deviation (σ). b. Under what conditions can the standard deviation be used to measure the relative risk of two investments? c. Under what conditions must the coefficient of variation be used to measure the relative risk of two investments?

a) E(R) = Sum(Probability x Possible Return) E(R) = (-0.1 x 0.25) + (0 x 0.15) + (0.1 x 0.35) + (0.25 x 0.25) E(R) = -0.25 + 0 + 0.035 + 0.0625 E(R) = 0.0725 Variance = Sum: (Probability)(Possible Return - Expected Return) ^2 V= (0.25)(-0.1 - 0.0725)^2 + (0.15)(0-0.0725)^2 + (0.35)(0.1-0.0725)^2 + (0.25)(0.25-0.0725)^2 V= (0.25)(0.02976) + (0.15)(0.0053) + (0.35)(0.0008) + (0.25)(0.0315) V= 0.0164 Std Dev= Sqr Rt(0.0164) = 0.128 Std Dev is useful in comparing risk of assets with similar expected returns

(Ch6, P3) Month Stock A Stock B 1 −0.04 0.07 2 0.06 −0.02 3 −0.07 −0.10 4 0.12 0.15 5 −0.02 −0.06 6 0.05 0.02 Compute the following. a. Average monthly rate of return Ri for each stock b. Standard deviation of returns for each stock c. Covariance between the rates of return d. The correlation coefficient between the rates of return What level of correlation did you expect? How did your expectations compare with the computed correlation? Would these two stocks be good choices for diversification?

a) Expected Return (A) = (-0.04 + 0.06 + -0.07 + 0.12 + -0.02 + 0.05) / 6 Return (A) = 0.1 / 6 = 0.0167 Expected Return (B) = (0.07 + -0.02 + -0.1 + 0.15 + -0.06 + 0.02) / 6 E(Rb) = 0.06 / 6 = 0.01 b) V(A) = (-0.04-0.0167)^2 + (0.06 - 0.0167)^ 2 + (-0.07 - 0.0167)^2 + (0.12 - 0.0167)^2 + (-0.02 - 0.0167)^2 + (0.05 - 0.0167)^2 V(A) = (-0.0567) ^2 + (0.0433)^2 + (-0.0867) ^2 + (.1033)^2 + (-0.0367)^2 + (0.0333)^2 V(A) = 0.00322 + 0.001875 + 0.00752 + 0.010671 + 0.001347 + 0.001109 = 0.025742 / 6 = 0.00429 Std DevA = SqrRt(0.00429) = 0.0655 V(B) = (0.07-0.01)^2 + (-0.02-0.01)^2 + (-0.1 - 0.01)^2 + (0.15-0.01)^2 + (-0.06-0.01)^2 + (0.02 - 0.01)^2 V(B) = 0.0412/6 = 0.006867 Std DevB= SqrRt (0.006867) = 0.08287 c) COV = [Sum:(ReturnA - E(Ra))(ReturnB - E(Rb)]/N =(-0.0567 x 0.06) + (0.0433 x -0.03) + (-0.0867 x -0.11) + (.1033 x 0.14) + (-0.0367 x -0.07) + (0.0333 x 0.01) = 0.0222 / 6 = 0.00367 d) Correlation = COV / StdDevA x StdDevB r = 0.00367/ (0.0655 x 0.08287) = 0.6761

(Appendix 2, P1) Year Stock A Rate of Return. Stock B Rate of Return 2007 5 5 2008 12 15 2009 −11 5 2010 10 7 2011 12 −10 A. Your manager suggests that because these companies produce similar products, you should continue your analysis by computing their covariance. B. Calculate coefficient of correlation Show all calculations.

a) Stock A E(R) = (5+12+(-11)+10+12)/5 = 28/5 E(Ra)= 5.6 Return - E(R): 2007. 5-5.6 = -0.6 2008 12-5.6 = 6.4 2009 -11-5.6 = -16.6 2010 10-5.6 = 4.4 2011 12-5.6 = 6.4 Stock B E(R) = (5+15+5+7+(-10))/5 = 22/5 E(Rb)= 4.4 Return - E(R): 2007 5-4.4= 0.6 2008 15-4.4 = 10.6 2009 5-4.4 = 0.6 2010 7-4.4 = 2.6 2011 -10-4.4 = -14.4 COV = [Sum: (ReturnA-E(Ra))x(ReturnB-E(Rb))] / N COV = [(-0.6)(0.6) + (6.4)(10.6) + (-16.6)(0.6) + (4.4)(2.6) + (6.4)(-14.4)]/5 = -23.2/5 Cov = -4.64 b) Variance(A) = (-0.6)^2 + (6.4)^2 + (-16.6)^2 + (4.4)^2 + (6.4)^2 V(A)= 0.36 + 40.96 + 275.56 + 19.36 + 40.96 = 377.2 / 5 V(A) = 75.44 Std Dev = SqrRt(75.44) = 8.69 V(B) = (0.6)^2 + (10.6)^2 + (0.6)^2 + (2.6)^2 + (-14.4)^2 V(B) = 327.2 / 5 = 65.44 Std Dev = SqrRt(65.44) = 8.09 Correlation = COV / (StdDevA x StdDevB) r = -4.64 / (8.69 x 8.09) r= -0.066

(CH 1, P5) During the past five years, you owned two stocks that had the following annual rates of return: Year. Stock T Stock B 1 0.19 0.08 2 0.08 0.03 3. −0.12. −0.09 4. −0.03 0.02 5 0.15 0.04 a. Compute the arithmetic mean annual rate of return for each stock. Which stock is most desirable by this measure? b. Compute the standard deviation of the annual rate of return for each stock. (Use Chapter 1 Appendix if necessary.) By this measure, which is the preferable stock? c. Compute the coefficient of variation for each stock. (Use the Chapter 1 Appendix if necessary.) By this relative measure of risk, which stock is preferable? d. Compute the geometric mean rate of return for each stock. Discuss the difference between the arithmetic mean return and the geometric mean return for each stock. Discuss the differences in the mean returns relative to the standard deviation of the

a) Arithmetic Mean = Sum Returns/ N AM(T)= [(.19)+(0.08)+(-.12)+(-0.03)+(.15)]/5 AM(T)= .27/5= 0.054 AM(B)= [(.08)+(.03)+(-.09)+(.02)+(.04)]/5 AM(B) = .08/5 =0.016 Stock T has a higher mean return than B b) St Dev= Sq Rt(Variance) Variance = (Sum([Actual Return-Expected Return]^2))/N V(t) = [(0.19-0.054)^2+ (0.08-0.054)^2+ (-0.12-0.054)^2+ (-0.03-0.054)^2+ (0.15-0.054)^2] V(t) = [(0.136)^2 + (0.026)^2 + (-0.174)^2 + (-0.084)^2 +(0.96)^2] V(t)= [0.0185+ 0.00068+0.03028+0.00706+0.00922]/5 V(t)= 0.6574/5 = 0.01315 St Dev = SqrRt(0.01315) =0.11467 V(b) = [(0.08-0.016)^2 + (0.03-0.016)^2 + (-0.09-0.016)^2 + (0.02-0.016)^2 + (0.04-0.016)^2] V(b) = [0.0041+0.002+0.1124+0.0002+0.00058]/5 V(b) = 0.1614/5 = 0.00323 Std Dev = SqrRt(0.00323) =0.05681 Stock B has a smaller Std Dev, and therefore, risk. c) Coefficient of Variation = Std Dev/Expected Return CV(t)= 0.11467/ 0.054 =2.123 CV(b) = 0.05681/0.016 =3.551 CV is a relative measure of risk. Stock t has less risk by this measure d) Geometric Mean= [Product of all HPRs]^(1/N) - 1 GM(t)= [(0.19+1)x(0.08+1)x(-0.12+1)x(-0.03+1)x(0.15+1)].... GM(t) = [(1.19)x(1.08)x(0.88)x(0.97)x(1.15)]^(1/5) - 1 GM(t) = [1.2616]^1/5 = 1.04757 - 1 = 0.04757 GM(b) = [(1.08)x(1.03)x(.91)x(1.02)x(1.04)]^1/5 - 1 GM(b)= [1.07383]^1/5 = 1.01435 - 1 =0.01435

(Ch2, P8) Investment Category. Ar.Mean Geo.Mean. StdDev Common stocks 10.28% 8.81% 16.9% Treasury bills 3.54 3.49 3.2 Long-term gvt bonds 5.10 4.91 6.4 Long-term corp. bonds 5.95 5.65 9.6 Real estate 9.49 9.44 4.5 a. Explain why the geometric and arithmetic mean returns are not equal and whether one or the other may be more useful for investment decision making. b. For the time period indicated, rank these investments on a relative basis using the coefficient of variation from most to least desirable. Explain your rationale. c. Assume the arithmetic mean returns in these series are normally distributed. Calculate the range of returns that an investor would have expected to achieve 95 percent of the time from holding common stock

a) Arithmetic Mean Assumes simple interest. Geomoetric mean assumes compounding interest. Internal Rate of Return is an important measure for geometric mean b) CV = Std Dev/ Arithmetic Mean 1. Real Estate = 4.5/9.49 = .4742 2. T-Bills = 3.2/3.54 = 0.904 3. Gvt Bonds = 6.4/5.1 = 1.255 4. Corp Bonds= 9.6/5.95 = 1.613 5. Common Stock = 16.9/10.28= 1.65 c) Expected Return +/- two Std Deviations 10.28% +/- (2 x 16.9%) = 10.28% +/- 33.8 = -23.52% through 44.08%