fm - modern portfolio theory
TRANSCRIPT
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o ern ort o o eory- ar ow z c en ron er
nanc a o e ng
Prof. Doug Blackburn
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Some Assumptions
There are many types of portfolios that meet alarge number of investor needs.
The optimal portfolio is directly related to
.
The portfolios we will be discussing are based
on the following assumptions: Investors want return
Investors hate risk
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What is risk?
This seems like an easy question yet it isdifficult to answer.
Individuals dislike uncertainty
This implies that individuals like variances to besmall so that actual returns are relatively close to
expected returns.
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Measuring return and variance
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Using Linear Algebra
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Using Linear Algebra
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Setting up the problem
We need to find the optimal portfolio weightsbut we first need to identify the objective.
We have two choices:
.
Minimize variance for a given level of return.
Mathematically, it is more convenient to
minimize the uadratic variance function.
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Changing to matrix notation
It is much more convenient to write theproblem using matrix notation.
matrixcovariance-variancetheisVV11Min 2 wwT
.ts
weightsportfolioofvectortheis11
returnsexpectedofvectortheis
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p
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Objective and first order conditions
sMultiplierLagrangeofmethodUse
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Solving the problem
01V(1) .bymultiplyleftthenandVbymultiplyleft(1),Using 21
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Solving the problem
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Solving the problem
This gives us two equations with two unknowns the two Lagrange Multipliers
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TTT rw
)5( 21 rAB p Simplify
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A,B, and C.
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Solving the problem
r
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The Optimal Portfolio Weights
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Minimum variance portfolios for given returns
Return Two portfolios with the
same return.
Efficient Frontier
E r1
o a n ar ance or o o
Standard deviation
1
2
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Global Minimum Portfolio
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Global Minimum Portfolio
1r
1
mv
rVTReturn of the MV portfolio
mv ar ance o e por o o
11
11
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VwTmv
Weights of the MV portfolio
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Extending the model
Assume that the investor has access to a riskfree asset.
n f rm h m r l m n in h hrisk-free asset does not affect the objective
function.
Risk-free asset has zero variance and zero
correlation with all other assets.
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Extending the model
However, including the risk free asset doesaffect the two linear constraints
The expected portfolio return
The sum of the portfolio weights
11
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Combining the two conditions yieldsN
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1
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Extending the model
The new problem has the Lagrangian:
fpfTT rrrrwwwL
1V
2
This can be solved in a similar fashion as the
prev ous pro em. I will let you work out the details or see
er on .
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Maximizing the Sharpe Ratio
Finding a closed-form solution is always nice,but sometimes it is not possible.
n fin n im l r f liwith the additional constraints on the portfolio
wei hts.
No short selling (weights are non-negative)
asset (weights must be less than x%).
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Maximizing the Sharpe Ratio
We must use numerical methods for obtainingthe portfolio weights.
In hi n n i r n ifunction that looks like
MaximizeP
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(The Sharpe Ratio)
s.constra ntsu ect to
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The Sharpe Ratio and the CAPM
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An alternative / flexible approach
To solve this problem, we can simply useExcels solver functionality to find the
optimal portfolios weights.
Notice that this is a well-behaved objective
function as a function of wei hts
Portfolio return is linear
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Inputs to the model
To use the model, all we need is historicalreturn data forNstocks.
Expected Returns for each stock can be estimated from
the historic mean.
Variance/Covariance matrix can be estimated by taking
the historic variance from the returns of each stock and
the historic covariance from all pairs of stocks.
Risk-free rate can be estimated using the yield on a 3-
month T-bill.
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Where to get data?
Prices / Returns
Yahoo!, Bloomberg, or other data source.
- St. Louis Federal Reserve FRED database.
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Expected returns
The classic approach is to use the historic
mean as a proxy for the expected return.
n l n ili ri m m l f returns (e.g. CAPM) as a proxy.
We will investigate this approach next time.
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Some things to think about
We have derived the optimal portfolio of a
mean-variance optimizer.
There are many other objective functions.
or o os can e orme o:
Consider skewness
Maximize dividend yield Preserve capital
And so on
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Portfolios and Higher Moments
Modern portfolio theory considers only mean
returns and variances/covariances of returns.
Quadratic utility function
Normal distributions
Returns, however, are not normal
ega ve s ewness ncreases e pro a y oextreme bad events.
extreme events (good or bad)
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Which security is preferred?
For a cost of $1 you can by either of the two
following gambles:
1. $40 with =0.1 or-$1 with pL=0.9
2. $1,000,000 with pW=0.1
-$1 with =0.9
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Which security is preferred?
Suppose we maximize Sharpe Ratio:
Shar e Ratio for choice 1 = 0.013
Sharpe Ratio for choice 2 = 0.000
The increase in volatility is in the positive
direction The probability of the same loss is equal across
the two choices
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Skewness Constraints
It is simple to minimize the portfolio variance
subject to both a constraint on the mean and
on skewness.
We can then compare the Sharpe ratios across
the various ortfolios
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Skewness Constraints
10
Skew= -0.2
6
8
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eRatio
Skew= 0.1
Skew= 0.3
2
4Sha
Skew= 0.5
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
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Which Portfolio is Best?
Portfolio 1 Portfolio 2 Portfolio 3
Mean 1.50% 1.50% 1.50%
Skewness -0.49 0.00 0.00
Kurtosis1.15 0.69 0.00
Sharpe Ratio 8.48 5.88 4.72
Variance 0.18% 0.25% 0.32%
Min Ret -14.97% -12.87% -12.77%Max Ret 11.34% 17.98% 18.17%
uar e 4.64% 4.68% 5.53%
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Minimum Variance Portfolios
All three cases, variance was minimized.
Portfolio 1:
= .
Portfolio 2:
ean = . , ew = Portfolio 3:
Mean = 1.5%, Skew = 0, Kurtosis = 0
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Minimum Variance Portfolio
All Variance is not equal.
Large upward movements in prices is good.
bad particularly during down markets.
Perhaps maximum Sharpe Ratio is not the
es r s re urn s a s c.
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Semi-Variance
Some have considered an alternative measure
for risk that captures downside volatility
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A New Measure
Perhaps it is time for a new measure for
portfolio risk.
A HALLEN ECan you develop a theory such that investors
, ,
skewness, and some reasonable amount of