Elton, Gruber, Brown and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 4

Transcription

1 Elton, Gruber, Brown and Goetzmann Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons to Text roblems: Chapter 4 Chapter 4: roblem 1 A. Expected return s the sum of each outcome tmes ts assocated probablty. Expected return of Asset 1 R1 16% % % 0.5 1% R 6%; R 3 14%; R 4 1% Standard devaton of return s the square root of the sum of the squares of each outcome mnus the mean tmes the assocated probablty. Standard devaton of Asset 1 σ 1 [(16% 1%) (1% 1%) (8% 1%) 0.5] 1/ 8 1/.83% σ 1/ 1.41%; σ / 4.4%; σ / 3.7% B. Covarance of return between Assets 1 and σ 1 (16 1) (4 6) (1 1) (6 6) (8 1) (8 6) The varance/covarance matrx for all pars of assets s: Correlaton of return between Assets 1 and ρ The correlaton matrx for all pars of assets s: Elton, Gruber, Brown, and Goetzmann 4-1 Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons To Text roblems: Chapter 4

2 C. ortfolo Expected Return A 1/ 1% + 1/ 6% 9% B 13% C 1% D 10% E 13% F 1/3 1% + 1/3 6% + 1/3 14% 10.67% G 10.67% H 1.67% I 1/4 1% + 1/4 6% + 1/4 14% + 1/4 1% 11% ortfolo Varance A (1/) 8 + (1/) + 1/ 1/ ( 4) 0.5 B 1.5 C 4.6 D E 7 F (1/3) 8 + (1/3) + (1/3) /3 1/3 ( 4) + 1/3 1/ /3 1/3 ( 6) 3.6 G H 6.7 I (1/4) 8 + (1/4) + (1/4) 18 + (1/4) /4 1/4 ( 4) + 1/4 1/ /4 1/ /4 1/4 ( 6) + 1/4 1/ /4 1/4 0.7 Elton, Gruber, Brown, and Goetzmann 4- Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons To Text roblems: Chapter 4

3 Chapter 4: roblem A. Monthly Returns Month Securty A 3.7% 0.4% -6.5% 1.4% 6.%.1% B 10.5% 0.5% 3.7% 1.0% 3.4% -1.4% C 1.4% 14.9% -1.4% 10.8% 4.9% 16.9% B. Sample Average (Mean) Monthly Returns R A ( 3.7% + 0.4% 6.5% + 1.4% + 6.% +.1% ) 6 1.% R B.95% R C 7.9% C. Sample Standard Devatons of Monthly Returns σ A ( 3.7% 1.% ) + ( 0.4% 1.% ) + ( 6.5% 1.% ) + ( 1.4% 1.% ) + ( 6.% 1.% ) + (.1% 1.% ) % σ B % σ C % Elton, Gruber, Brown, and Goetzmann 4-3 Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons To Text roblems: Chapter 4

4 σ AB D. Sample Covarances and Correlaton Coeffcents of Monthly Returns ( 3.7% 1.% ) ( 10.5%.95% ) + ( 0.4% 1.% ) ( 0.5%.95% ) + ( 6.5% 1.% ) ( 3.7%.95% ) + ( 1.4% 1.% ) ( 1.0%.95% ) + ( 6.% 1.% ) ( 3.4%.95% ) + (.1% 1.% ) ( 1.4%.95% ).17 σ 7.4 ; σ AC ρ AB BC ρ 0.7 ; ρ AC BC 6 E. ortfolo Returns and Standard Devatons ortfolo 1 (X1 1/; X 1/; X3 0): R 1 1/ 1.% + 1/.95% %.09% σ 1 ( 1/ ) ( 1/ ) ( 1/ 1/ / / ) % ortfolo (X1 1/; X 0; X3 1/): R 4.57% σ % ortfolo 3 (X1 0; X 1/; X3 1/): R % σ % ortfolo 4 (X1 1/3; X 1/3; X3 1/3): R % σ % Elton, Gruber, Brown, and Goetzmann 4-4 Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons To Text roblems: Chapter 4

5 Chapter 4: roblem 3 It s shown n the text below Table 4.8 that a formula for the varance of an equally weghted portfolo (where X 1/N for 1,, N securtes) s ( σ σ ) σ σ 1/N + j where σ j s the average varance across all securtes, σ s the average covarance across all pars of securtes, and N s the number of securtes. Usng the above formula wth σ j 50 and σ 10 we have: ortfolo Sze (N) σ Chapter 4: roblem 4 As s shown n the text, as the number of securtes (N) approaches nfnty, an equally weghted portfolo s varance (total rsk) approaches a mnmum equal to the average covarance of the pars of securtes n the portfolo, whch n roblem 3 s gven as 10. Therefore, 10% above the mnmum rsk level would result n the portfolo s varance beng equal to 11. Settng the formula shown n the above answer to roblem 3 equal to 11 and usng σ j 50 and σ 10 we have: σ 1/ N ( 50 10) Solvng the above equaton for N gves N 40 securtes. Elton, Gruber, Brown, and Goetzmann 4-5 Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons To Text roblems: Chapter 4

6 Chapter 4: roblem 5 As shown n the text, f the portfolo contans only one securty, then the portfolo s average varance s equal to the average varance across all securtes, σ j. If nstead an equally weghted portfolo contans a very large number of securtes, then ts varance wll be approxmately equal to the average covarance of the pars of securtes n the portfolo, σ. Therefore, the fracton of rsk that of an ndvdual securty that can be elmnated by holdng a large portfolo s expressed by the followng rato: σ σ σ From Table 4.9, the above rato s equal to 0.6 (60%) for Italan securtes and 0.8 (80%) for Belgan securtes. Settng the above rato equal to those values and solvng for σ gves σ 0.4σ for Italan securtes and σ 0.σ for Belgan securtes. Thus, the rato 0.σ σ 0.σ σ σ σ 4 for Belgan securtes. σ 0.4σ equals σ for Italan securtes and If the average varance of a sngle securty, σ j, n each country equals 50, then σ 0.4σ for Italan securtes and σ 0.σ for Belgan securtes. Usng the formula shown n the precedng answer to roblem 3 wth σ j 50 and ether σ 0 for Italy or σ 10 for Belgum we have: ortfolo Sze (N securtes) Italan σ Belgan σ Elton, Gruber, Brown, and Goetzmann 4-6 Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons To Text roblems: Chapter 4

7 Chapter 4: roblem 6 The formula for an equally weghted portfolo's varance that appears below Table 4.8 n the text s ( σ σ ) σ σ 1/N + j where σ j s the average varance across all securtes, σ s the average covarance across all securtes, and N s the number of securtes. The text below Table 4.8 states that the average varance for the securtes n that table was and that the average covarance was Usng the above equaton wth those two numbers, settng σ equal to 8, and solvng for N gves: 8 1/N ( ) N N Snce the portfolo's varance decreases as N ncreases, holdng 4 securtes wll provde a varance less than 8, so 4 s the mnmum number of securtes that wll provde a portfolo varance less than 8. Elton, Gruber, Brown, and Goetzmann 4-7 Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons To Text roblems: Chapter 4