# MINI CASE: John & Marsha on Portfolio Selection

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MINI CASE: John & Marsha on Portfolio Selection

John---manages a \$125 million common- stock portfolio for a large pension fund(closet indexer) John’s portfolio returns always seem to track the S&P 500 market index, the correlation between his returns and market returns is over 90% John因為自己投資組合的報酬率和s&p500指數太過相近而擔心 他覺得自己只是一個indexer 但marsha 安慰他 那是因為他的客戶都要求高價股的多樣化投資組合 他告訴 john 他是一個明星證券分析師 且說過pioneer gypsum是一個很好的投資標的(便宜) 並看漲 global mining

HERE’S THE TRICK Take your benchmark, the S&P 500, as security 1.
(what you would end up with as an indexer) Then consider a few securities you really now something about (Pioneer could be security 2,Global could be security 3) Maximize the Sharpe ratio= [E(Rp)－Rf]/σp E(Rp)：投資組合預期報酬率 Rf：無風險利率 σp：投資組合的標準差

MINI CASE 概述 Pioneer Gypsum Global Mining Expected return 11.0% 12.9%
Standard deviation 32% 24% Beta 0.65 1.22 Stock price \$87.50 \$105.00

Question 1 Calculate the expected return, risk premium, and standard deviation of a portfolio invested partly in the market and partly in Pioneer. Does adding Pioneer to the market benchmark improve the Sharpe ratio? How much should John invest in Pioneer and how much in the market?

Market standard deviation=16% Sharpe ratio=[E(Rp)－Rf]/σp E(Rp)：投資組合預期報酬率 Rf：無風險利率 σp：投資組合的標準差

ANSWER 1 β= σ 𝑝𝑚 σ 𝑚 2 = 𝑐𝑜𝑣.(𝑝𝑚) σ 𝑚 2 = 𝜌 𝑝𝑚 . 𝜎 𝑝 𝜎 𝑚 Pioneer β= 𝑐𝑜𝑣.(𝑝𝑚) σ 𝑚 2 = 𝜌 𝑝𝑚∙ 𝜎 𝑝 ∙ 𝜎 𝑚 σ 𝑚 2 =0.65 = 𝜌 𝑝𝑚∙ 0.32 ∙(0.16) (0.16) 2 𝜌 𝑝𝑚 =0.325 Market Pioneer 0.0256= (0.16) 2 0.1664=(0.16)(0.32)(0.325) 0.1664 0.1024= (0.32) 2

Example If invest 99% Market & 1% Pioneer Excepted return Risk premium
=99%*(0.125)+1%*(0.11)= Risk premium =99%*(7.5%)+1%*(11-5)%= Standard deviation(p.175) = 𝜎 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = (0.99) 2 × × 0.99 × 0.01 × (0.01) 2 ×(0.1024) = ( ) = Sharpe ratio =[E(Rp)－Rf]/σp =( %)/(0.1685) =

Excel Market (%) Pioneer E(x) risk premium standard deviation
sharpe ratio WM WP 99 1 0.99 0.01 98 2 0.1247 0.0747 0.98 0.02 97 3 0.97 0.03 96 4 0.1244 0.0744 0.96 0.04 95 5 0.95 0.05 94 6 0.1241 0.0741 0.94 0.06 93 7 0.93 0.07 92 8 0.1238 0.0738 0.92 0.08 91 9 0.91 0.09 90 10 0.1235 0.0735 0.9 0.1 50 0.1175 0.0675 0.5

Question 1 96% in Market and 4% in Pioneer Yes!
Calculate the expected return, risk premium, and standard deviation of a portfolio invested partly in the market and partly in Pioneer. Does adding Pioneer to the market benchmark improve the Sharpe ratio? How much should John invest in Pioneer and how much in the market? Yes! 96% in Market and 4% in Pioneer

Question 2 Repeat the analysis for Global Mining.
What should John do in this case? Assume that Global accounts for 0.75% of the S&P 500 index. (Assume a market standard deviation of 16%)

ANSWER 2 Market Global 0.0312=(0.16)(0.24)(0.813) 0.0312

Example If invest 99% Market & 1% Global Excepted return Risk premium
=99%*(0.125)+1%*(0.129)= Risk premium =99%*(7.5%)+1%*(12.9-5)%= Standard deviation = 𝜎 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = (0.99) 2 × × 0.99 × 0.01 × (0.01) 2 ×(0.0576) = ( ) = Sharpe ratio=[E(Rp)－Rf]/σp =( %)/(0.1604)=

Excel Market (%) Global E(x) risk premium standard deviation
sharpe ratio WM WG 99 1 0.99 0.01 98 2 0.98 0.02 97 3 0.97 0.03 96 4 0.96 0.04 95 5 0.1252 0.0752 0.95 0.05 94 6 0.94 0.06 93 7 0.93 0.07 92 8 0.92 0.08 91 9 0.91 0.09 90 10 0.1254 0.0754 0.9 0.1 50 0.127 0.077 0.5 100 0.125 0.075 1 100.75 -0.75 1.0075

=beta*expected risk premium on market Expected risk premium on Global ( 𝑟 𝑔𝑙𝑜𝑏𝑎𝑙 - 𝑟 𝑓 )=𝛽( 𝑟 𝑚𝑎𝑟𝑘𝑒𝑡 − 𝑟 𝑓𝑟𝑒𝑒 ) =(1.22)(12.5%-5%)=0.0915 ( 𝑟 𝑔𝑙𝑜𝑏𝑎𝑙 -5%)=0.0915 𝒓 𝒈𝒍𝒐𝒃𝒂𝒍 =0.1415=𝟏𝟒.𝟏𝟓%>12.9%

Question 2 Repeat the analysis for Global Mining. What should John do in this case? c) Assume that Global accounts for 0.75% of the S&P 500 index. (Assume a market standard deviation of 16%) 賣掉所持有的Global股份將其轉換至Market Global---(-0.75%) Market---(100%+0.75%) 因為個股的期望溢酬為14.15%，但Global只有12.9%，所以應當將所持有的Global股票賣掉，並將其所得投入市場。

MPT-modern portfolio theory 現代投資組合理論 投資者如何衡量不同的投資風險及合理 組合自己的資金以取得最大收益