FIN4504 CH.6 HW Exam 2

Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following estimates: Original Portfolio Expected Monthly Returns- 0.67% St.Dev of Monthly Returns- 2.37% ABC Company Expected Monthly Returns- 1.25% St.Dev of Monthly Returns- 2.95% The correlation coefficient of ABC stock returns with the original portfolio returns is .40. a. The inheritance changes Grace's overall portfolio and she is deciding whether to keep the ABC stock. Assuming Grace keeps the ABC stock, calculate the: i. Expected return of her new portfolio which includes the ABC stock.

0.728% E(rp) = (900,000/1,000,000)*E(ro) + (100,000/1,000,000)*E(rabc) E(rp) = .9*0.67 + .1*1.25 = 0.728%

Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following estimates: Original Portfolio Expected Monthly Returns- 0.67% St.Dev of Monthly Returns- 2.37% ABC Company Expected Monthly Returns- 1.25% St.Dev of Monthly Returns- 2.95% The correlation coefficient of ABC stock returns with the original portfolio returns is .40. b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government securities yielding .42% monthly. Assuming Grace sells the ABC stock and replaces it with the government securities, calculate the: iii. Standard deviation of her new portfolio which includes the government securities.

2.133 σc = y*σp = .9*2.37 = 2.133

Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following estimates: Original Portfolio Expected Monthly Returns- 0.67% St.Dev of Monthly Returns- 2.37% ABC Company Expected Monthly Returns- 1.25% St.Dev of Monthly Returns- 2.95% The correlation coefficient of ABC stock returns with the original portfolio returns is .40. a. The inheritance changes Grace's overall portfolio and she is deciding whether to keep the ABC stock. Assuming Grace keeps the ABC stock, calculate the: ii. Covariance of ABC stock returns with the original portfolio returns.

2.7966 Cov = corr*σx*σy = 0.4*2.37*2.95 = 2.7966

You hold and equity portfolio consisting of 2 stocks, X and Y. You put 50% of your money in each stock. The possible scenarios of returns for the coming year for each stock are given below. Stock X Boom (40% chance)- 15% Bear (60% chance)- -10% Stock Y Boom (40% chance)- -5% Bear (60% chance)- 20% a) What is the expected return on your portfolio?

5% E(r) = w1*r1 + w2*r2 E(rx) = .4*15 + .6*(-10) = 0 E(ry) = .4*(-5) + .6*(20) = 10% E(rp) = .5*(0) + .5*(10) = 5%

The correlation coefficients between pairs of stocks are: Corr(A,B)=.85; Corr(A,C)=.60; Corr(A,D)=.45. Each stock has expected return of 8% and standard deviation of 20%. You are holding a portfolio consisting of stock A only. If you can add just one more stock to your portfolio, which would you add, B, C, or D?

Choose the one that has the lowest correlation with A: stock D. This will lead to the lowest portfolio standard deviation for a given amount of expected return.

You hold and equity portfolio consisting of 2 stocks, X and Y. You put 50% of your money in each stock. The possible scenarios of returns for the coming year for each stock are given below. Stock X Boom (40% chance)- 15% Bear (60% chance)- -10% Stock Y Boom (40% chance)- -5% Bear (60% chance)- 20% b) What is the correlation coefficient between X and Y?

Correlation coefficient = -1 Cov(rx,ry) = Sum of (P(s)*(r1(s) - E(r1)) x (r2(s) - E(r2)) Cov(rx,ry) = .4*(15 - 0)*(-5 - 10) + .6*(-10 - 0)*(20 - 10) = -150 σx^2 = Sum of (prob*(rx - E(rx))^2) σx^2 = .4*(15-0)^2 + .6*(-10-0)^2 = 150 σx = 12.25% σy^2 = Sum of (prob*(ry - E(ry))^2) σy^2 = .4*(-5-10)^2 + .6*(20-10)^2 = 150 σy = 12.25% corr = Cov /(σx* σy) = -1

Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following estimates: Original Portfolio Expected Monthly Returns- 0.67% St.Dev of Monthly Returns- 2.37% ABC Company Expected Monthly Returns- 1.25% St.Dev of Monthly Returns- 2.95% The correlation coefficient of ABC stock returns with the original portfolio returns is .40. b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government securities yielding .42% monthly. Assuming Grace sells the ABC stock and replaces it with the government securities, calculate the: i. Expected return of her new portfolio which includes the government securities.

E(rc) = 0.645% E(rc) = .9*0.67+.1*0.42 = 0.645%

The standard deviation of a portfolio is always equal to the weighted average of the standard deviations of the assets in the portfolio. True or false?

False. The portfolio standard deviation equals the weighted average of the component-asset standard deviations only in the special case that all assets are perfectly positively correlated. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations.

Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rf. The characteristics of two of the stocks are as follows: Stock A E(r)- 8% St.Dev- 40% Stock B E(r)- 13% St.Dev- 60% Could the equilibrium rf be greater than 10%? (Hint: Can a particular stock portfolio be substituted for the risk-free asset?)

Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio, in equilibrium, will be the risk-free rate. To find the proportions of this portfolio [with the proportion wA invested in Stock A and wB = (1 - wA ) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation is: sigmaP = |wA x sigmaA - wB sigmaB| 0 = 40wA - (60 x (1 - wA )) wA = 0.6 The expected rate of return for this risk-free portfolio is: E(r) = (0.6 x 8) + (0.4 x 13) = 10% Therefore, the risk-free rate has to be 10%.

Consider the following stocks: Stock W Expected Return %- 15 Standard Deviation %- 36 Stock X Expected Return %- 12 Standard Deviation %- 15 Stock Z Expected Return %- 5 Standard Deviation %- 7 Stock Y Expected Return %- 9 Standard Deviation %- 21 a) Which stock would you definitely not invest in?

Stock Y. Stock X has both higher return and less risk compared to Y. So stock X dominates Y.

Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following estimates: Original Portfolio Expected Monthly Returns- 0.67% St.Dev of Monthly Returns- 2.37% ABC Company Expected Monthly Returns- 1.25% St.Dev of Monthly Returns- 2.95% The correlation coefficient of ABC stock returns with the original portfolio returns is .40. d. Based on conversations with her husband, Grace is considering selling the $100,000 of ABC stock and acquiring $100,000 of XYZ Company common stock instead. XYZ stock has the same expected return and standard deviation as ABC stock. Her husband comments, "It doesn't matter whether you keep all of the ABC stock or replace it with $100,000 of XYZ stock." State whether her husband's comment is correct or incorrect. Justify your response.

The expected return on the portfolio will stay the same, 0.728% However, the portfolio risk might go up or down depending on whether stock XYZ has higher or lower correlation with the original portfolio than stock ABC does.

Consider the following stocks: Stock W Expected Return %- 15 Standard Deviation %- 36 Stock X Expected Return %- 12 Standard Deviation %- 15 Stock Z Expected Return %- 5 Standard Deviation %- 7 Stock Y Expected Return %- 9 Standard Deviation %- 21 b) If you can also invest in a risk free T-bill with return of 4.5%, which stock would you prefer?

The one that has the highest Sharpe ratio. Sw= (15-4.5) / 36 = 0.2917 Sx= (12-4.5) / 15 = 0.5 Sz = (5-4.5) / 7 = 0.0714 So, the best is stock X.

Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following estimates: Original Portfolio Expected Monthly Returns- 0.67% St.Dev of Monthly Returns- 2.37% ABC Company Expected Monthly Returns- 1.25% St.Dev of Monthly Returns- 2.95% The correlation coefficient of ABC stock returns with the original portfolio returns is .40. b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government securities yielding .42% monthly. Assuming Grace sells the ABC stock and replaces it with the government securities, calculate the: ii. Covariance of the government security returns with the original portfolio returns.

WHY zero? 0

You hold and equity portfolio consisting of 2 stocks, X and Y. You put 50% of your money in each stock. The possible scenarios of returns for the coming year for each stock are given below. Stock X Boom (40% chance)- 15% Bear (60% chance)- -10% Stock Y Boom (40% chance)- -5% Bear (60% chance)- 20% c) What is the standard deviation of your portfolio return?

σp = 0 σp^2 = (wx^2*σrx^2)) + (wy^2*σry^2)) + 2*wx*wy*Cov(rx,ry) σp^2 = .5^2*150 + .5^2*150 + 2*.5*.5*(-150) = 0

A three-asset portfolio has the following characteristics: Asset X E(r)- 15% St.Dev.- 22% Weight- 0.5 Asset Y E(r)- 10% St.Dev.- 8% Weight- 0.4 Asset Z E(r)- 6% St.Dev.- 3% Weight- 0.1 What is the expected return on this three-asset portfolio?

12.1% E(r) = 0.5*15% + 0.4*10% + 0.1*6% = 12.1%

A fund manager is considering 3 mutual funds. The first is a stock fund, the second is a bond fund, and the third is a money market mutual fund that yields a sure 5.5%. The probability distributions of the risky funds are: Stock Fund (S) Expected Return- 15% Standard Deviation- 32% Bond Fund (B) Expected Return- 9% Standard Deviation- 23% The correlation between the fund returns is 0.15. a) Consider the investment opportunity set for the two risky funds. What is the expected return and standard deviation of the minimum variance portfolio?

E(r) = 10.89%, StDev = 19.94% 1. Find the weights using the equation we derived in class. W(s) = (St.DevB^2 - Cov(rs,rb))/(StdevS^2 + StDevB^2 - 2Cov(rs,rb)) W(s) = (529 - 110.4)/(1024 + 529 - (2 x 110.4)) = 0.3142 W(b) = 1 - .3142 = .6858 2. Find the E(r) of MVP E(rmin) = (ws x E(rs)) + (wb x E(rb)) E(r) MVP = .3142 x 15 + .6858 x 9 = 10.89% 3. Find StDev of MVP Stdev(min) = ((w^2s x sigma^2s + w^2b x sigma^2b) + (2ws x wb x Cov(rs,rb)))^1/2 Stand dev MVP = 19.94% (from portfolio variance equation)

A fund manager is considering 3 mutual funds. The first is a stock fund, the second is a bond fund, and the third is a money market mutual fund that yields a sure 5.5%. The probability distributions of the risky funds are: Stock Fund (S) Expected Return- 15% Standard Deviation- 32% Bond Fund (B) Expected Return- 9% Standard Deviation- 23% The correlation between the fund returns is 0.15. b) Now consider the various combinations of the mutual funds. What is the maximum possible Sharpe ratio that can be attained?

1. Find the weights of each fund W(s) = ((E(r1) - rf)*sigma2^2 - (E(r2) - rf)*Covar(r1,r2))/((E(r1) - rf)*sigma2^2 + (E(r2) - rf)*sigma1^2 - (E(r1) - rf + E(r2) - rf)*Cov(r1,r2)) W(s) = 64.66%; w(b) = 35.34% 2. Find the expected return of the portfolio E(r) = w1*r1 + w2*r2 E(r) of the tangency portfolio is 12.88% 3. Find the St.Dev of the portfolio Stand dev = 23.34% Stdev= ((w^2s x sigma^2s + w^2b x sigma^2b) + (2ws x wb x Cov(rs,rb)))^1/2 4. Find the Sharpe Ratio S = (E(rp) - Rf)/sigma_p Max Sharpe = (12.88 - 5.5) / 23.34 = 0.3162

A portfolio manager is considering two mutual funds. A stock fund has expected return of 20% and standard deviation of 30%. A bond fund has expected return of 12% and standard deviation of 15%. The correlation between the fund returns is 0.10. What are the investment proportions in the minimum variance portfolio of the two risky funds, and what are the expected value and standard deviation of its return?

Wmin(S) = 0.1739 Wmin(B) = 0.8261 E(rmin) = 13.39% Stdev(min) = 13.92% Explanation: 1. Find the weights of each mutual fund of the minimum variance portfolio. Wmin(S) = (St.DevB^2 - Cov(rs,rb))/(StdevS^2 + StDevB^2 - 2Cov(rs,rb)) Wmin(S) = (225 - 45)/(900 + 225 - (2 x 45)) = 0.1739 Wmin(B) = 1 - Wmin(S) Wmin(B) = 1 - 0.1739 = 0.8261 2. Find the minimum variance portfolio mean E(rmin) = (ws x E(rs)) + (wb x E(rb)) E(rmin) = (0.1739 x 20) + (0.8261 x 12) = 13.39% 3. Find the portfolio standard deviation Stdev(min) = ((w^2s x sigma^2s + w^2b x sigma^2b) + (2ws x wb x Cov(rs,rb)))^1/2 = ((0.1739^2 x 900) + (0.8261^2 x 225) + (2 x 0.1739 x 0.8261 x 45))^1/2 = 13.92%

Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following estimates: Original Portfolio Expected Monthly Returns- 0.67% St.Dev of Monthly Returns- 2.37% ABC Company Expected Monthly Returns- 1.25% St.Dev of Monthly Returns- 2.95% The correlation coefficient of ABC stock returns with the original portfolio returns is .40. a. The inheritance changes Grace's overall portfolio and she is deciding whether to keep the ABC stock. Assuming Grace keeps the ABC stock, calculate the: iii. Standard deviation of her new portfolio which includes the ABC stock.

σp = 2.267 σp^2 = .9^2*2.37^2 + .1^2*2.95^2 + 2*.9*.1*(2.7966) = 4.55 + 0.087 + 0.5034 = 5.14 σp = 2.267