Objective: This assignment is designed to demonstrate the importance of correlation in reducing risk.
Directions:
Part 1 of the assignment
- Complete the tasks listed below:
- In the first worksheet, there are a number of examples computing the expected returns and standard deviations of the different investment.
- You are required to compute the following inWorksheet 1: Handout:
- Expected Return for High Tech
2. Standard deviation for High Tech
3. Expected return for 50% HT and 50% Collections
4. Standard Deviation for a portfolio of 50% HT and 50% Collections
5. Discuss the standard deviation for this last portfolio.
Part 2 of assignment
Go toCorrelation worksheet
6. Discuss what happens when the following correlations are used in cell E5: -1, 1, 0, 0.3
7. Describe why portfolio diversification is important.
- Submit your spreadsheet and responses to this assignment.
handout Exercise for Calculating Expected Return and Risk for an Individual Investment and a Portfolio Assume that you recently graduated and have taken a position working as a financial planner. Your first assignment is to advise a client who wishes to invest $100,000. The information below has been gathered by financial analysts and economists working for your corporation. Your supervisor has restricted you to the following investment alternatives: T-bills, High Tech, Collections, U.S. Rubber, and the Market Portfolio. An analyst working for your investment firm assigned the following probabilities for five possible states of the economy and expected returns for each investment for each of the five states. Using the information below make a recommendation. Estimated Rate of Return on Alternative Investments State ofProbabilityHighU.S.Market2-Stocks Economyof StateT-BillsTechCollectionsRubberPortfolioHT&Coll Recession0.18.0%-22.0%28.0%10.0%-13.0% Below Average0.28.0%-2.0%14.7%-10.0%1.0% Average0.48.0%20.0%0.0%7.0%15.0% Above Average0.28.0%35.0%-10.0%45.0%29.0% Boom0.18.0%50.0%-20.0%30.0%43.0% E(R)8.0%1.7%13.8%15.0% Standard Deviation0.0%13.4%18.8%15.3% I. Calculate the expected return E(R) for High Tech in the space below. where: n = the number of states in the economy (5 in this example) Pi is the probability of state i ocurring (i is recession, below avg, etc.) E(Ri) is the expected return for the investment for state i. T-billsE(RHTbills) = .1(8%) + .2(8%) + .4(8%) + .2(8%) + .1(8%) = 8.0% High Tech CollectionsE(RColl) = .1(28%) + .2(14.7%) + .4(0%) + .2(-10%) + .1(-20%) = 1.7% U.S. RubberE(RUSR) = .1(10%) + .2(-10%) + .4(7%) + .2(45%) + .1(30%) = 13.8% Market PortfolioE(RMkt) = .1(-13%) + .2(1%) + .4(15%) + .2(29%) + .1(43%) = 15.0% II. Calculate the standard deviation for an individual security (High Tech). where: n = the number of states in the economy (5 in this example) Pi is the probability of state i ocurring (i is recession, below avg, etc.) E(Ri) is the expected return for the investment calculated in step I above. III. Calculate the E(R) of a portfolio consisting of 50% in High Tech and 50% in Collections. IV. Calculate the standard deviation for the portfolio of High Tech and Collections. V. Why is the standard deviation of the potfolio of High Tech and Collections so small? Page &P Solutions Chapter 11: Example for Calculating Expected Return and Risk for an Individual Investment and a Portfolio Assume that you recently graduated and have taken a position working as a financial planner. Your first assignment is to advise a client who wishes to invest $100,000. The information below has been gathered by financial analysts and economists working for your corporation. Your supervisor has restricted you to the following investment alternatives: T-bills, High Tech, Collections, U.S. Rubber, and the Market Portfolio. An analyst working for your investment firm assigned the following probabilities for five possible states of the economy and expected returns for each investment for each of the five states. Using the information below make a recommendation. Estimated Rate of Return on Alternative Investments State ofProbabilityHighU.S.Market2-Stocks Economyof StateT-BillsTechCollectionsRubberPortfolioHT&Coll Recession0.18.0%-22.0%28.0%10.0%-13.0%3.0% Below Average0.28.0%-2.0%14.7%-10.0%1.0%6.4% Average0.48.0%20.0%0.0%7.0%15.0%10.0% Above Average0.28.0%35.0%-10.0%45.0%29.0%12.5% Boom0.18.0%50.0%-20.0%30.0%43.0%15.0% E(R)8.0%0.1741.7%13.8%15.0%9.6% Standard Deviation0.0%20.0%13.4%18.8%15.3%3.3% I. Calculate the expected return E(R) for High Tech in the space below. where: n = the number of states in the economy (5 in this example) Pi is the probability of state i ocurring (i is recession, below avg, etc.) E(Ri) is the expected return for the investment for state i. T-billsE(RHTbills) = .1(8%) + .2(8%) + .4(8%) + .2(8%) + .1(8%) = 8.0% High TechE(R) = 17.4% CollectionsE(RColl) = .1(28%) + .2(14.7%) + .4(0%) + .2(-10%) + .1(-20%) = 1.7% U.S. RubberE(RUSR) = .1(10%) + .2(-10%) + .4(7%) + .2(45%) + .1(30%) = 13.8% Market PortfolioE(RMkt) = .1(-13%) + .2(1%) + .4(15%) + .2(29%) + .1(43%) = 15.0% II. Calculate the standard deviation for an individual security (High Tech). where: n = the number of states in the economy (5 in this example) Pi is the probability of state i ocurring (i is recession, below avg, etc.) E(Ri) is the expected return for the investment calculated in step I above. III. Calculate the E(R) of a portfolio consisting of 50% in High Tech and 50% in Collections. where: wi = weight or percentage invested in each security E(Ri) is the expected return for the investment calculated in step I above. Expected Return for a portfolio consisting of 50%of High Tech and 50% of Collections = 9.6% IV. Calculate the standard deviation for the portfolio of High Tech and Collections. Note: the standard deviation is calculated using the expected returns for each state of the economy last collumn in above table. V. Why is the standard deviation of the potfolio of High Tech and Collections so small? The two stocks are almost perfectly negatively correlated. Page &P 5 States Example of Portfolio Risk & Return for Two Stocks Assuming Five Possible States of the Economy Note these two stocks are almost perfectly negatively correlated. The losses of one stock are offset by gains of the other. High TechCollections -22.0%28.0% -2.0%14.7% 20.0%0.0% 35.0%-10.0% 50.0%-20.0% 17.4%1.7% 20.0%13.4% Note: The above table and graph illustrate that the gains for one company off-set the losses for the other company in every state of the economy. In this example, High Tech and Collections are perfectly negatively correlated. How stocks are correlated with one another determines how diversified the portfolio will be. &A 5 States 1 Standard Deviation of Portfolio E(R) E(R) for Portfolio given different weights invested in two stocks, High Tech and Collections 1 Correlation High Tech Collections State of Economy E(R) Portfolio of 50% HT & 50% Coll. OH Example of Portfolio Risk & Return for High Tech & Collection Graph Illustrates the impact of Correlation for Two Stocks Click on cell c13 and d13 to see the formulas for calculating the expected return and risk for two risky assets. Change the correlation coefficient between -1 and 1 to see the impact on the Return & Risk graph for the portfolio. Correlation for High Tech & Collections:-1.0 E(R)Std Dev. High Tech17.4%20.0% Collections1.7%13.4% Portfolio % in% inPortfolioExpected High TechCollectionsRiskReturn 0%100%13.4%1.7% 10%90%10.1%3.3% 20%80%6.7%4.8% 30%70%3.4%6.4% 40%60%0.0%8.0% 50%50%3.3%9.6% 60%40%6.6%11.1% 70%30%10.0%12.7% 80%20%13.3%14.3% 90%10%16.7%15.8% 100%0%20.0%17.4% &A OH Standard Deviation of Portfolio E(R) E(R) for Portfolio given different weights invested in two stocks, High Tech and Collections 11 Correlation!#REF! Correlation!#REF! State of Economy E(R) Portfolio of 50% HT & 50% Coll. Estimated Rate of Return on Investments State ofProb.HighU.S.Market2-Stocks Economyof StateT-BillsTechColl.RubberPortfolioHT&Coll Recession0.18.0%-22.0%28.0%10.0%-13.0% Below Average0.28.0%-2.0%14.7%-10.0%1.0% Average0.48.0%20.0%0.0%7.0%15.0% Above Average0.28.0%35.0%-10.0%45.0%29.0% Boom0.18.0%50.0%-20.0%30.0%43.0% E(R)8.0%1.7%13.8%15.0% Standard Deviation0.0%13.4%18.8%15.3% ( ) å = = n i i i i R E P R E 1 ) ( ) ( n i iii REPRE 1 )()( ( ) å = - = n i i i i i R E R P 1 2 ) ( s n i iiii RERP 1 2 )( ( ) ( ) ( ) ( ) ( ) % 0 % 8 % 8 1 . % 8 % 8 2 . % 8 % 8 4 . % 8 % 8 2 . % 8 % 8 1 . 2 2 2 2 2 = - + - + - + - + - = Tbill s %0%8%81.%8%82.%8%84.%8%82.%8%81. 22222 Tbill ( ) ( ) ( ) ( ) ( ) % 0 % 8 % 8 1 . % 8 % 8 2 . % 8 % 8 4 . % 8 % 8 2 . % 8 % 8 1 . 2 2 2 2 2 = - + - + - + - + - = Tbill s %0%8%81.%8%82.%8%84.%8%82.%8%81. 22222 Tbill % 8 . 18 %) 8 . 13 30 ( 1 . %) 8 . 13 % 45 ( 2 . %) 8 . 13 % 7 ( 4 . %) 8 . 13 % 10 ( 2 . %) 8 . 13 % 10 ( 1 . 2 2 2 2 2 = - + - + - + - - + - = USR s %8.18%)8.1330(1.%)8.13%45(2.%)8.13%7(4.%)8.13%10(2.%)8.13%10(1. 22222 USR å = = n i i i P R E w R E 1 ) ( ) ( n i iiP REwRE 1 )()( = HighTech s HighTech % 3 . 15 %) 15 % 43 ( 1 . %) 15 % 29 ( 2 . %) 15 % 15 ( 4 . %) 15 % 1 ( 2 . %) 15 % 13 ( 1 . 2 2 2 2 2 = - + - + - + - + - - = Mkt s %3.15%)15%43(1.%)15%29(2.%)15%15(4.%)15%1(2.%)15%13(1. 22222 Mkt % 4 . 13 %) 7 . 1 % 20 ( 1 . %) 7 . 1 % 10 ( 2 . %) 7 . 1 % 0 ( 4 . %) 7 . 1 % 7 . 14 ( 2 . %) 7 . 1 % 28 ( 1 . 2 2 2 2 2 = - - + - - + - + - + - = Coll s %4.13%)7.1%20(1.%)7.1%10(2.%)7.1%0(4.%)7.1%7.14(2.%)7.1%28(1. 22222 Coll % 02 . 20 = HighTech s %02.20 HighTech % 3 . 3 %) 6 . 9 % 15 ( 1 . %) 6 . 9 % 5 . 12 ( 2 . %) 6 . 9 % 10 ( 4 . %) 6 . 9 % 4 . 6 ( 2 . %) 6 . 9 % 3 ( 1 . 2 2 2 2 2 & = - + - + - + - + - = Coll HT s %3.3%)6.9%15(1.%)6.9%5.12(2.%)6.9%10(4.%)6.9%4.6(2