8/7/2019 portfolio_risk
1/2
Portfolio Risk and ReturnPrepared by Pamela Parrish Peterson,
PhD., CFA
The return on a portfolio of assets is calculated as:N
p ii=1
r = w r i
where ri is the expected return on asset i, andwi is the weight
of asset i in the portfolio.
Portfolio risk is calculated using the risk of the individual
assets (measured by the standarddeviation), the weights of the
assets in the portfolio, and either the correlation between oramong
the assets or the covariance of the assets returns.
For a two-asset portfolio, the risk of the portfolio, p, is:
2 2 2 2p 1 1 2 2 1 1 2 2 1
2 2 2 2p 1 1 2 2 1 2 12
1212
1 2
= w +w +2w w
or
= w +w +2w w cov
covsince =
2
where i is the standard deviation of asset is returns,12 is the
correlation between the returns of asset 1 and 2, andcov12 is the
covariance between the returns of asset 1 and 2.
Problem What is the portfolio standard deviation for a two-asset
portfolio comprisedof the following two assets if the correlation
of their returns is 0.5?
Asset A Asset B
Expected return 10% 20%Standard deviation of expectedreturns
5% 20%
Amount invested $40,000 $60,000
Solution p = 13.115%
Calculation
( ) ( ) ( )
2 2 2 2p
p
p
p
= 0.4 0.05 +0.6 0.2 +2(0.4)(0.05)(0.6)(0.2)(0.5)
(0.16)(0.0025) (0.36)(0.04) (2)(0.0012)
0.0004 0.0144 0.0024
0.0172 0.131149 or 13.1149%
= + +
= + +
= =
Portfolio risk and return 1 of 2
8/7/2019 portfolio_risk
2/2
Portfolio risk and return 2 of 2
Portfolio risk and return practice problems
Problem 1 What is the portfolio return and standard deviation
for a two-assetportfolio comprised of the following two assets if
the correlation of theirreturns is 0.5?
Asset C Asset DExpected return 7% 25%Standard deviation of
expected returns 5% 30%Amount invested $50,000 $50,000Correlation
0.40
Problem 2 What is the portfolio return and standard deviation
for a two-assetportfolio comprised of the following two assets if
the correlation of theirreturns is 0.5?
Asset E Asset F
Expected return 5% 50%Standard deviation of expected returns 5%
40%Amount invested $60,000 $40,000Correlation 0.20
Problem 3 What is the portfolio return and standard deviation
for a two-assetportfolio comprised of the following two assets if
the correlation of theirreturns is 0.5?
Asset G Asset HExpected return 10% 25%Standard deviation of
expected returns 8% 40%
Amount invested $40,000 $60,000Correlation -0.30
Solutions to portfolio risk and return practice problems
Problem Expected portfolio return Portfolio risk
1 16% 16.1632%2 23% 16.8582%3 19% 23.2413%
