PORTFOLIO THEORY
 When we think about investments, 2 things matter:
1. the expected return
2. the dispersion of possible returns
The Mean - Standard Deviation Rule:
 It is usually assumed that security returns will be described adequately by the mean, and
standard deviation. This is reasonable if security returns are normally distributed.
 The mean standard deviation rule is based upon the following two assumptions:
1) Individuals prefer more return to less
2) Individuals prefer less risk to more
 When choosing a portfolio of assets, investors will seek out the portfolio that provides the
highest return for a given level of risk, or alternatively, the lowest level of risk for a given
level of return. Portfolios with this characteristic are said to be efficient.
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THE RISK OF A PORTFOLIO OF SECURITIES
 A portfolio is a collection of assets such as stocks, bonds, real estate, etc.
 The return to a portfolio of assets is an average return of all of the assets, weighted by the
relative amount invested in each asset. However, the risk of a portfolio of assets is
fundamentally different from the average risk of all assets in the portfolio.
 The process of diversification will reduce the risk of assets when held in portfolios. Think of
the following picture of two asset’s returns:
25
20
15
10
5
0
-5
1
2
3
4
5
6
7
-10
-15
-20
-25
 The riskiness of a portfolio depends not only on the riskiness of each asset in the portfolio,
but also on the relation between the returns from the two assets.
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CORRELATION
 Correlation measures the tendency for the returns of two assets to move together. In
technical terms, it measures the strength and direction of the linear relationship between
two variables.
 For example, if the correlation between assets A and B is equal to 0.30, we can say that the
amount of change in ‘A’ is predictable about 30% of the time for each unit change in ‘B’.
The mathematical formula for portfolio risk
Two Assets: a & b
wa and wb are the fractions of total funds invested in each asset.
a and b are the standard deviations of returns for each asset.
ρab is the correlation between the two assets’ returns. It must lie between –1 and +1.
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The definition of expected return and standard deviation for a 2-asset portfolio:
r p  wa ra  wb rb
 P  wa2 a2  wb2 b2  2 wa wb a b  ab
What about n assets?
n
Portfolio return: r p   wi ri
i 1
n 2 2
n n
Portfolio risk:  p   wi  i    wi w j ij
i 1
i 1 j 1
[ ij  ij i j ]
i j
where wi = the percent of the value of the portfolio in asset i
ri = the expected return for asset i
n = number of assets in the portfolio
i2 = the variance of rates of return for asset i
ij = the covariance between the rates of return for assets i and j
ij =the correlation coefficient between the rates of return for assets i and j
.
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MATRIX THEORY
 A matrix is a rectangular array of numbers. Matrix multiplication allows you to easily
compute the expected return and standard deviation of a multi-asset portfolio.
Ex: Two-asset case:
 r1 
rp  [w1 w 2 ]  
r2 
 w1r1  w 2r2
 The first matrix is a row vector (1 x 2), and the second matrix is a column vector (2 x 1).
 In order to multiply matrices, the number of columns of the first matrix must be equal to
the number of rows of the second. The resulting matrix will have the number of rows
equal to that of the first matrix and the number of columns equal to that of the second
matrix.
Ex: A 2 x 3 matrix multiplied by a 3 x 4 matrix will result in a 2 x 4 matrix.
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Example of Matrix Multiplication for the Two-asset Portfolio Variance:
 11 12   w 1 


 p 2  w1 w 2  

 w 

 21 22   2 
 w1212  w 2 2 2 2  2w1w 212
Example:
.30 .02 .50
 p  .50 .50 
 
.02 .40 .50
2
 .185
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EFFICIENT PORTFOLIOS
Calculating Efficient N-Asset Portfolios When There Are Short-Sale Restrictions
Use Excel’s Solver to optimize the portfolio model by identifying the best proportion of the
portfolio to invest in each asset without allowing short positions.
o One objective is to minimize the risk of the portfolio σp2
By changing the decision variables wi of the proportion to invest in each asset i
Subject to the constraints:
Σ wi = 100% and that wi > 0
rp = desired return level
This calculates the Minimum Variance Portfolio (MVP+) which is the lowest risk
portfolio on the efficient frontier
o
Efficient Frontier with Short Sales
Portfolio
Return rp
Efficient Frontier without Short Sales
MVP+
Standard Deviation σp
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o A second objective is to maximize the return of the portfolio rp
By changing the decision variables wi of the proportion to invest in each asset i
Subject to the constraints:
Σ wi = 100% and that wi > 0
σp2 ≤ Target variance
This calculates a constrained maximized return portfolio which is the point on the
efficient frontier when the x-axis value is the square root of the target variance
o
A third objective is to maximize the Sharpe Index (rp - rf)/ σp
By changing the decision variables wi of the proportion to invest in each asset i
Subject to the constraints:
Σ wi = 100% and that wi > 0
This calculates the tangency portfolio which assumes that investors can borrow and
lend at a risk-free rate. This assumption results in a linear efficient frontier that begins
at rf on the y intercept and is tangent to the efficient frontier of risky assets.
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VaR as a Measure of Portfolio Risk
 Value at risk (VaR) measures the worst expected loss under normal market conditions
over a specific time interval at a given confidence level.

It answers the question:
o “How much can I lose with x% probability over a pre-set horizon?”
 It is found by calculating the lowest quantile of the potential losses that can occur within
a given portfolio during a specified time period. The basic time period T and the
confidence level (quantile) q are the two parameters that must be specified in context of
the goal of risk measurement.
 For example, a portfolio manager has a daily VaR equal to $1 million at 1 percent. This
means that the designed portfolio can only have a 1 percent chance that a daily loss
bigger than $1 million occurs under normal market conditions. These types of
requirements are set to control downside risk.
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There are several methods for calculating the VaR of a portfolio involving $I initial investment for
a specified investment horizon T:
1. Computational Method:
 Assume the portfolio return in period T is normally distributed with expected value rp
and standard deviation σp. Use the =NORMINV(q, rp,σp) formula to calculate the cutoff
amount in the left tail of the distribution. This formula calculates the ending portfolio
return % value that will have a q% chance of values being less than or equal to it. If the
VaR is expressed in a dollar amount, the VaR is I multiplied by this cutoff amount.
 Example: Assume a manager has designed a new portfolio where the return is
normally distributed with an annual mean return of 20% and a standard deviation of
30%. The current value of the portfolio is $100 million. The portfolio manager is
required to have a daily VaR equal to $1 million at 1 percent. Assume 240 trading
days per year. Does this portfolio meet the requirement?
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Example Solution:
Daily expected return is 20% / 240 = 0.000833
Daily standard deviation is 30% / sqrt(240) = .019365.
The 1% return cutoff is found by programming:
=norminv(.01, .000833,.019365) = -.04422
VaR = I * worst case portfolio return
= - .04422*$100,000,000 = $4,422,000
There is a 1 percent chance that a daily loss bigger than $4,422,000 can occur with
the designed portfolio under normal market conditions. Since the VaR for this
portfolio is larger than the required VaR, the portfolio does NOT meet the
requirements.
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2. Simulate Returns of Individual Assets over multi-periods up to T (i.e. a random walk) and
calculate the simulated portfolio return at period T. Run a Monte Carlo simulation with the
portfolio return in period T as the key output variable and then use descriptive statistics, the
histogram and Excel’s percentile function to determine the VaR. This approach allows you to
model different assumptions besides the normal distribution for portfolio returns (e.g. the
betapert distribution).
3. Model individual asset returns as functions of other variables such as interest rates or
exchange rates. Simulate values for these other variables over the future while you run a
Monte Carlo simulation with the portfolio return in period T as the key output variable. Use
descriptive statistics, the histogram and Excel’s percentile function to determine the VaR.
This approach allows you to try to forecast expected returns for individual assets as opposed
to using historical results or “expert opinions”.
4. Historical simulation involves collecting a data set of past returns, sorting the returns in
descending order and using Excel’s percentile function to determine the return for the q
percentile.
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