Question

Passive Stock Market Investor (PSMI) holds one stock market index ETF in his/her portfolio, but has learned that investing in a bond market index ETF as well may reduce his/her portfolio risk due to diversification. The annual returns for both ETFs for the last ten years are listed below. Measured by the standard deviation of returns, by how much would PSMI’s portfolio’s historical risk been reduced if PSMI invested 70% in the stock market index ETF and 30% in the bond market index ETF rather than his current allocation of 100% in the stock market index ETF? Enter your answer rounded to two decimal places. Do not enter % in the answer box. For example, if your answer is 0.12345 or 12.345% then enter as 12.35 in the answer box. Year Stock ETF Bond ETF 2009 39.00% 5.00% 2010 18.00% 6.00% 2011 1.00% 8.00% 2012 16.00% 4.00% 2013 34.00% -3.00% 2014 13.00% 5.00% 2015 0.00% 0.00% 2016 13.00% 2.00% 2017 21.00% 4.00% 2018 -5.00% 0.00%

Answer 1

Standard Deviation =$\tiny \sqrt{\sum }$[(r_i-r_avg)²/n-1]

Stock ETF Average = [39+18+1+16+34+13+0+13+21-5]/10

= [150]/10

=15

Bond ETF Average = [5+6+8+4-3+5+0+2+4+0]/10

= 31/10

= 3.1

Standard Deviation of Stock ETF

= $\small ^{\sqrt{}}$ [(39-15)²+(18-15)²+(1-15)²+(16-15)²+(34-15)²+(13-15)²+(0-15)²+(13-15)²+(21-15)²+(-5-20)²] / 10-1

= $\small ^{\sqrt{}}$ 0/9

=0

Standard Deviation of Bond ETF

= $\small ^{\sqrt{}}$ [(5-3.1)²+(6-3.1)²+(8-3.1)²+(4-3.1)²+(-3-3.1)²+(5-3.1)²+(0-3.1)²+(2-3.1)²+(4-3.1)²+(0-3.1)²]/10-1

= $\small ^{\sqrt{}}$ 0/9

=0