Portfolio Risk and Return - it-educ.jmu.edu
Portfolio Risk and Return - it-educ.jmu.edu
Portfolio Risk and Return - it-educ.jmu.edu
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<strong>Portfolio</strong> <strong>Risk</strong> <strong>and</strong> <strong>Return</strong><br />
Prepared by Pamela Parrish Peterson, PhD., CFA<br />
The return on a portfolio of assets is calculated as:<br />
N<br />
r= p ∑ w i ri<br />
i=1<br />
where ri is the expected return on asset i, <strong>and</strong><br />
wi is the weight of asset i in the portfolio.<br />
<strong>Portfolio</strong> risk is calculated using the risk of the individual assets (measured by the st<strong>and</strong>ard<br />
deviation), the weights of the assets in the portfolio, <strong>and</strong> e<strong>it</strong>her the correlation between or<br />
among the assets or the covariance of the assets’ returns.<br />
For a two-asset portfolio, the risk of the portfolio, σp, is:<br />
2 2 2 2<br />
σ p= w1σ 1+w2σ 2+2w1σ1w2σρ 2 12<br />
or<br />
2 2 2 2<br />
σ p= w1σ 1+w2σ 2+2w1w2cov12 cov<br />
since ρ 12<br />
12=<br />
σσ 1 2<br />
where σi is the st<strong>and</strong>ard deviation of asset i’s returns,<br />
ρ12 is the correlation between the returns of asset 1 <strong>and</strong> 2, <strong>and</strong><br />
cov12 is the covariance between the returns of asset 1 <strong>and</strong> 2.<br />
Problem What is the portfolio st<strong>and</strong>ard deviation for a two-asset portfolio comprised<br />
of the following two assets if the correlation of their returns is 0.5?<br />
Solution σp = 13.115%<br />
Calculation<br />
Asset A Asset B<br />
Expected return 10% 20%<br />
St<strong>and</strong>ard deviation of expected<br />
returns<br />
5% 20%<br />
Amount invested $40,000 $60,000<br />
2 2 2 2<br />
σ p=<br />
0.4 0.05 +0.6 0.2 +2(0.4)(0.05)(0.6)(0.2)(0.5)<br />
( ) ( ) ( )<br />
σ p = (0.16)(0.0025) + (0.36)(0.04) + (2)(0.0012)<br />
σp = 0.0004 + 0.0144 + 0.0024<br />
σp = 0.0172 = 0.131149 or 13.1149%<br />
<strong>Portfolio</strong> risk <strong>and</strong> return 1 of 2
<strong>Portfolio</strong> risk <strong>and</strong> return practice problems<br />
Problem 1 What is the portfolio return <strong>and</strong> st<strong>and</strong>ard deviation for a two-asset<br />
portfolio comprised of the following two assets if the correlation of their<br />
returns is 0.5?<br />
Asset C Asset D<br />
Expected return 7% 25%<br />
St<strong>and</strong>ard deviation of expected returns 5% 30%<br />
Amount invested $50,000 $50,000<br />
Correlation 0.40<br />
Problem 2 What is the portfolio return <strong>and</strong> st<strong>and</strong>ard deviation for a two-asset<br />
portfolio comprised of the following two assets if the correlation of their<br />
returns is 0.5?<br />
Asset E Asset F<br />
Expected return 5% 50%<br />
St<strong>and</strong>ard deviation of expected returns 5% 40%<br />
Amount invested $60,000 $40,000<br />
Correlation 0.20<br />
Problem 3 What is the portfolio return <strong>and</strong> st<strong>and</strong>ard deviation for a two-asset<br />
portfolio comprised of the following two assets if the correlation of their<br />
returns is 0.5?<br />
Asset G Asset H<br />
Expected return 10% 25%<br />
St<strong>and</strong>ard deviation of expected returns 8% 40%<br />
Amount invested $40,000 $60,000<br />
Correlation -0.30<br />
Solutions to portfolio risk <strong>and</strong> return practice problems<br />
Problem Expected portfolio return <strong>Portfolio</strong> risk<br />
1 16% 16.1632%<br />
2 23% 16.8582%<br />
3 19% 23.2413%<br />
<strong>Portfolio</strong> risk <strong>and</strong> return 2 of 2
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