ࡱ> J{[ Rbbjbj2ΐΐJ# 448"6ti'^^^p&r&r&r&r&r&r&)+fr&lZ^llr&#'$$$l8p&$lp&$$$,i Ϥb$\&9'0i'$,t b,$$,X%^r$ ^^^r&r&"^^^i'llll,^^^^^^^^^4 L: CHAPTER 8 RISK-AVERSION, CAPITAL ASSET ALLOCATION, AND MARKOWITZ PORTFOLIO-SELECTION MODEL 1.  EMBED Equation.DSMT4  2. The variance of a portfolio consists of two components: a) A weighted average of the variances of the securities in the portfolio. b) A component which measures the relationship between each pair of securities in the portfolio. This relationship is measured by the covariance. For a two security portfolio:  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  implies risk averse  EMBED Equation.DSMT4  implies risk neutral  EMBED Equation.DSMT4  implies risk seeker (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  4. A fair game means that the expected returns given information set  equal the expected returns without the information set. In other words, the returns already reflect all information. Sometimes we define an investor s risk preferences by looking at his preferences towards a fair gamble. A fair gamble is defined as a gamble where the cost equals the expected value. Risk averse investors always reject a fair gamble. Risk neutral investors are indifferent to a fair gamble. Risk seeking investors will always accept a fair gamble. 5. a)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  6. An efficient portfolio is one which has the highest expected return for a given level of risk. Likewise, an inefficient portfolio is one which does not have the highest expected return for a given level of risk. We can use the dominance principle to show that inefficient portfolios will always be dominated by other portfolios. 7. Short selling entails the selling of a security that you dont own. In portfolio theory, the selling short of one asset is often used to provide the funds to purchase another asset. The efficient frontier consists of all portfolios which satisfy the following two conditions. 1) Have the highest expected return for a given level of risk, and 2) Have the lowest risk for a given expected return. Efficient Frontier Efficient Frontier Without Short Selling With Short Selling  8. Margin requirements determine the amount of money an investor can borrow to purchase shares of stock. The Dyl Model, which incorporates margin requirements into the Markowitz Model, causes the efficient frontier to shift to the northeast. [See Figure 8-10B of text.] 9. a)  b)  c)  10. a) Minimum risk portfolio can be found by  EMBED Equation.DSMT4  and solving for WA.  EMBED Equation.DSMT4  So, sell short 23.52% of asset A and place 123.52% in asset B to minimize the risk of the portfolio.  EMBED Equation.DSMT4  Var(RP) = (.2352)2(3)2 + (1.2352)2 (2)2 + 2(.2352)(1.2352)(3)(2)(.8) = 3.81 b) By selling short the risk free asset, you can obtain the additional funds necessary for investing an additional $5,000 in the risky portfolio.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  11. If we assume that Investors can borrow and lend at the risk free rate Rf and if Investors have the same expectations, there will be only one best portfolio.  Both John Doe and Jane Roe will purchase the same portfolio, portfolio M, because the combination of the risk free asset and portfolio M dominates all other combinations. However, Jane Roe will probably place some of her money in the riskless asset and some in portfolio M, i.e. she will be a lender. John Doe, on the other hand, will most likely borrow at the risk free rate to buy more of portfolio M, i.e. he will be a borrower. 12. To determine the optimal weights for a portfolio using the Markowitz approach, the portfolio manager can either: 1) Set a target rate of return and find the portfolio which has the least risk. or 2) Set a level of risk for the portfolio and maximize the expected return for this given level of risk. These two problems can be solved using calculus or linear programming methods on a computer. 13. Based on Appendix 10C, we can use Microsoft Excel to calculate the optimal weights of Markowitz model. In Markowitz Model in equation (8.13), we know  EMBED Equation.DSMT4 . (8.13) Where E* =10%, the required rate of return of the portfolio. From the matrix calculation of Markowitz Model, we can get the optimal weights from multiplying  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Based on Excel function, the values of average and variance are calculated by AVERAGE and VARP functions and the covariance matrix is calculated by Covariance in Data Analysis. JNJIBMBA EMBED Equation.DSMT4 0.00820.00500.0113Variance0.00250.00700.0082Variance Covariance Matrix0JNJIBMBAJNJ0.00250.00070.0007IBM0.00070.00700.0006BA0.00070.00060.0082 Therefore, we can obtain matrix A and K as follows: Matrix A 0.00500.00140.00151.00000.00820.00140.01410.00131.00000.00500.00150.00130.01641.00000.01131.00001.00001.00000.00000.00000.00820.00500.01130.00000.0000  EMBED Equation.DSMT4  Then, from the matrix calculation of Markowitz Model, we can get the optimal weights from multiplying  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 . Where matrix  EMBED Equation.DSMT4  is 96.4952 -47.8974 -48.5978 0.5160 17.4282 -47.8974 23.7749 24.1225 1.5313 -166.1896 -48.5978 24.1225 24.4752 -1.0473 148.7614 0.5160 1.5313 -1.0473 -0.0488 5.5748 17.4282 -166.1896 148.7614 5.5748 -690.3857  The optimal weights in W matrix as follows: W12.2588 W2-15.0877 W313.8289 10.5087 2-63.4638 If short-selling is allowed, the optimal weights for JNJ, IBM and BA are 2.2588, -15.0877, and 13.8289 respectively. If short-selling is not allowed, we can obtain the adjusted weights from equation (8.19) as follows:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  14. There are four companies in the portfolio, therefore the matrix to calculate the optimal weights are shown as follows:  EMBED Equation.DSMT4  Where E* =12%, the required rate of return of the portfolio. The information about the mean and variance of rate of returns for four companies are as follows: JNJIBMBACAT EMBED Equation.DSMT4 0.00820.00500.01130.0169Variance0.00250.00700.00820.0097Variance Covariance Matrix0JNJIBMBACATJNJ0.00250.00070.00070.0011IBM0.00070.00700.00060.0025BA0.00070.00060.00820.0031CAT0.00110.00250.00310.0097 Therefore, we can obtain matrix A and K as follows: 0.00500.00140.00150.00211.00000.00820.00140.01410.00130.00491.00000.00500.00150.00130.01640.00631.00000.01130.00210.00490.00630.01941.00000.01691.00001.00001.00001.00000.00000.00000.00820.00500.01130.01690.00000.0000  EMBED Equation.DSMT4  By excel function, the inverse of matrix A can be calculated as: 99.9015 -55.8418 -36.9166 -7.1432 0.8552 -24.9153 -55.8418 42.3581 -3.2389 16.7226 0.7365 -66.8810 -36.9166 -3.2389 64.7865 -24.6310 0.1239 2.3676 -7.1432 16.7226 -24.6310 15.0516 -0.7156 89.4287 0.8552 0.7365 0.1239 -0.7156 -0.0148 1.3243 -24.9153 -66.8810 2.3676 89.4287 1.3243 -159.0383  Then, from the matrix calculation of Markowitz Model, we can get the optimal weights from multiplying  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  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If short-selling is not allowed, we can obtain the adjusted weights from equation (8.19) as follows:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  15. Use the same way and same information in question13 to calculate the optimal weights when E* =0.5% and inverse matrix of A is the same as in question13.  EMBED Equation.DSMT4  Then we can obtain optimal weight matrix W as follows: W10.6031 W20.7003 W3-0.3035 1-0.0209 22.1228 If short-selling is allowed, the optimal weights for JNJ, IBM and BA are 0.6031, 0.7003, and -0.3035 respectively. 16. Use the same way and same information in question14 to calculate the optimal weights when E* =1% and inverse ma 0{m $$1$Ifa$gd= $$1$Ifa$gdUvkdR$$If,0-i <  f<   aa tx 644 Ba pyt=028J{m $$1$Ifa$gd= $$1$Ifa$gdUvkd$$If,0-i <  f<   aa tx 644 Ba pyt=JLRb{{ $$1$Ifa$gd=vkd$$If,0-i <  f<   aa tx 644 Ba pyt=bdj~ $$1$Ifa$gd=gkd$$If,0-i <  f<   tx 644 Ba pyt=~(D`aQT\{r $$Ifa$ $$1$Ifa$gdUgdqgd=gd8gkdd$$If;0-i <   f<    tx 644 Ba pyt= \]^_`deQST\Ϳvk[H>h>OJQJ^J$h5$h>B*OJQJ^Jhphh>B*OJQJ^Jhphh=OJQJ^JhjhlHhqEHU jvM hqCJUVaJo( hqo(jhqUh>OJQJ^Jhh>hqhqOJQJ^JhhqhqOJQJ^JhhJOJQJ^Jhh=jh=Ujjh=h=EHUj1T h=CJUVo(\]`h{r $$Ifa$ $$1$Ifa$gdUvkd$$If;0-i <   f<    aa tx 644 Ba pytU\]_`hiklu 02B"PRTĺĺĺĺ矛wugWjh>OJQJU^Jhh>hJOJQJ^JhUhJOJQJ^Jhh=OJQJ^Jhh=h=OJQJ^Jhh>h>OJQJ^Jhh>B*OJQJ^Jhphh>OJQJ^J$h5$h>B*OJQJ^Jhphh>B*OJQJ^Jhph0h5$h>B*CJOJPJQJ^JaJhph hilu{r $$Ifa$ $$1$Ifa$gdUvkd$$If,0-i <  f<   aa tx 644 Ba pytUuv{r $$Ifa$ $$1$Ifa$gdUvkd$$If,0-i <  f<   aa tx 644 Ba pytU 0 $$Ifa$ $$1$Ifa$gdUgkd`$$If,0-i <  f<   tx 644 Ba pytU02PRw $$Ifa$ $$1$Ifa$gdUgd>gd8gkd$$If;0-i <   f<    tx 644 Ba pytU trix of A is the same as in question14.  EMBED Equation.DSMT4  Then we can obtain optimal weight matrix W as follows: W10.6061 W20.0677 W30.1476 W40.1787 1-0.0015 2-0.2661 Whether short-selling is allowed or not, the optimal weig(*.0@BFHXZ\^`rtvxȺ~t[~t[~t[K~t[K~t[Kh>B*OJQJ^Jhph0h=h>B*CJOJPJQJ^JaJhphh>OJQJ^J$h5$h>B*OJQJ^Jhphh>B*OJQJ^Jhphh>OJQJ^Jhh8h8OJQJ^Jhh8hJOJQJ^Jhjh>OJQJU^Jh(jh=h>EH|OJQJU^Jh$j*1T h>OJQJUV^Jh({r $$Ifa$ $$1$Ifa$gdUvkd$$If;0-i <   f<    aa tx 644 Ba pytU(*0@{r $$Ifa$ $$1$Ifa$gdUvkd$$If,0-i <  f<   aa tx 644 Ba pytU@BHX{r $$Ifa$ $$1$Ifa$gdUvkd$$If,0-i <  f<   aa tx 644 Ba pytUXZ`r{r $$Ifa$ $$1$Ifa$gdUvkdb$$If,0-i <  f<   aa tx 644 Ba pytUrtz $$Ifa$ $$1$Ifa$gdUgkd.$$If,0-i <  f<   tx 644 Ba pytUxzQRSUVXY[\_`abʿhUhqhAFbjhAFbUh>h8OJQJ^JhUh>h>OJQJ^Jh0h=h>B*CJOJPJQJ^JaJhphh>OJQJ^J$h=h>B*OJQJ^JhphRTUWXZ[]^_`abgd8gkd$$If;0-i <   f<    tx 644 Ba pytUhts for JNJ, IBM, BA, and CAT are 0.6061, 0.0677, 0.1476 and 0.1787 respectively.     01h2P. 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