Choosing a Portfolio
of Risky Assets
By: Jassim P. Isik
Markowitz efficient portfolios
• A diversification in the manner suggested
previously leads to the construction of portfolios
with the highest expected return at a given level of
risk.
Theory assumptions:
1. It assumes that only two parameters affect an investor’s
decision: the expected return and the risk.
2. It assumes that an investor is risk averse.
3. It assumes that an investor seeks to achieve the highest
expected return at a give level of risk.
Calculating the Portfolio Risk using
Historical Data
• The variance of a portfolio’s return which we shall simply refer to as
the portfolio variance is calculated from historical data, generally
monthly.
• Variance of two-asset portfolio formula:
𝒗𝒂𝒓 𝑹𝑷 = 𝑾𝟐𝒊 𝒗𝒂𝒓 𝑹𝒊 + 𝒘𝟐𝒊 𝒗𝒂𝒓 𝑹𝒋 + 𝟐𝒘𝒊 𝒘𝒋 𝒔𝒕𝒅 𝑹𝒊 𝒔𝒕𝒅 𝑹𝒋 𝒄𝒐𝒓(𝑹𝒊 , 𝑹𝒋 )
Variance of two-asset portfolio
• var(𝑅𝑝 ) = portfolio variance
• 𝑊𝑖 = percentage of the portfolio′s funds invested in asset i
• 𝑤𝑗 = percentage of the portfolio′s funds invested in asset j
• 𝑣𝑎𝑟(𝑅𝑖 ) = variance of asset i
• 𝑣𝑎𝑟 𝑅𝑗 = variance of asset j
• 𝑠𝑡𝑑 𝑅𝑖 = standard deviation of asset 𝑖
• 𝑠𝑡𝑑 𝑅𝐽 = standard deviation of asset j
• 𝑐𝑜𝑟 𝑅𝑖 𝑅𝑗 = correlation between the return for assets i and j
The extension to three assets formula:
• var Rp = wi2 var R i + wj2 var R j + wk2 var R k +
2wi wj std R i std R j cor R i R J + 2wi wk std R i std R k cor R i , R k +
2wj wk std R j std(R k )cor R j , R k
• Where 𝑤𝑘 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑓𝑢𝑛𝑑𝑠 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑖𝑛 𝑎𝑠𝑠𝑒𝑡 𝑘
𝑠𝑡𝑑 𝑅𝑘 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡 𝑘
𝑐𝑜𝑟(𝑅𝑖 , 𝑅𝑗 ) = 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓𝑜𝑟 𝑎𝑠𝑠𝑒𝑡𝑠 𝑖 𝑎𝑛𝑑 𝑘
𝑐𝑜𝑟 𝑅𝑗 , 𝑅𝑘 = 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓𝑜𝑟 𝑎𝑠𝑠𝑒𝑡𝑠 𝑗 𝑎𝑛𝑑 𝑘
Constructing the Markowitz Efficient
Portfolios
• An investor who is constructing a portfolio will calculate the portfolio risk
(as measured by the portfolio variance) and expected return.
• The procedure for determining the maximum expected return for a given
level of portfolio risk can be found by using a management science
technique called quadratic programming.
Constructing the Markowitz Efficient
Portfolios
•
•
•
•
Any portfolio that can be created is called feasible portfolio.
The collection of all feasible portfolios is called the feasible set of portfolios.
Markowitz efficient portfolio is also said to be a mean-variance efficient portfolio.
The collection of all efficient portfolios is called the Markowitz efficient set of
portfolios.
• The Markowitz efficient set of portfolios is sometimes called the Markowitz efficient
frontier because graphically all the Markowitz efficient portfolios lie on the boundary
of the set of feasible portfolios with the maximum return for a give level of risk.
Choosing a Portfolio in the Markowitz
Efficient Set
After constructing the Markowitz efficient set of portfolios, the next step is to
determine the optimal portfolio.
• The best portfolio to hold of all those on the Markowitz efficient frontier is called
the optimal portfolio.
• The optimal portfolio should depend on the investor’s preference or utility as to
the trade-off between risk and expected return.
• All possible portfolios that can be created from the available assets. Any
portfolio that can be created is called feasible portfolio.
• The collection of all feasible portfolios is called the feasible set of
portfolios.
• Markowitz efficient portfolio is also said to be a mean-variance
efficient portfolio.
• The collection of all efficient portfolios is called the Markowitz
efficient set of portfolios.
• The Markowitz efficient set of portfolios is sometimes called the
Markowitz efficient frontier because graphically all the Markowitz
efficient portfolios lie on the boundary of the set of feasible portfolios
with the maximum return for a give level of risk.
Thank you 