Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 9
Content:
• Benchmark portfolios
• Ledoit & Wolf shrinkage covariance estimator
• Shrinkage portfolios
• Grouping strategies
Benchmark portfolios
We need to compare portfolios with appropriate
benchmarks to know if the results are satisfying.
A benchmark portfolio surprisingly difficult to beat is the
one created with a naïve 1/N rule that assigns equal
weights to all the assets in all periods. Three strengths:
• immune to estimation errors (as it needs no inputs)
• does not require to perform any optimization procedure
• has a very low turnover, which translates into very low
transaction costs
Benchmark portfolios
Another possible benchmark is a large stock market index,
like the S&P 500. While it is not possible to buy an index, it
is possible to buy an ETF (Exchange-Traded Fund), i.e., a
fund that is traded on the financial markets and which
tries to replicate the index. Advantages:
• only one instrument (the ETF) needs to be bought; no
need to trade all the stocks contained in the index
• over the long run stock markets provide good returns
(in countries with a solid economy)
• no estimation and optimization procedures required
Ledoit & Wolf shrinkage covariance estimator
Basic mean-variance optimization with sample estimates
often struggles to beat common benchmarks, especially
the naïve 1/N rule.
To improve optimization performance we need to reduce
the severity of the estimation errors in the inputs, and/or
reduce the impact of such errors on portfolio formation.
On the estimation side we focus on the covariance matrix:
• while 𝝁 is generally more affected by estimation errors,
Ʃ easier and more effective to improve
• the problem gets worse in Ʃ as 𝑁 grows, due to the
higher number of parameters (𝑁2)
Ledoit & Wolf shrinkage covariance estimator
Shrinkage estimators are arguably the most successful in
dealing with the estimation of the covariance matrix.
Ledoit & Wolf (2004) define the shrinkage estimator
∑ 𝑳𝑾 = 𝛿𝑰𝜇 Ƹ𝑠 + (1 − 𝛿)∑
where ∑ is the usual sample covariance matrix, 𝑰 is the
identity matrix, 𝜇 Ƹ𝑠 is the average sample variance of all the
variables, and 0 ≤ 𝛿 ≤ 1 is the shrinkage parameter.
The product 𝑰𝜇 Ƹ𝑠 gives a diagonal matrix whose elements
on the diagonal are equal to the average sample variance
and all the other elements are equal to zero, and which is
the target matrix toward which the sample covariance
matrix is shrunk.
Ledoit & Wolf shrinkage covariance estimator
An optimal 𝛿 is selected, and this shrunk estimator
improves over the sample covariance, which translate in
better portfolio performance.
Moreover, it gives a nonsingular covariance matrix even
when the number of periods 𝑇 used for estimation is
smaller than the number of assets 𝑁.
The sample covariance matrix, on the contrary, in such
case is singular and therefore not invertible, which makes
it impossible to perform the optimization procedures.
It is now standard to use this estimator instead of the
sample covariance in portfolio optimization.
Shrinkage portfolios
Tu and Zhou (2011) propose a shrinkage portfolio that
shrinks the mean-variance portfolio (optimized, but
affected by estimation errors) toward the naïve 1/N
portfolio (not optimized, but immune from estimation
errors, improving over both.
The weights of such portfolio are given by
𝒘∗ = 𝛿𝒘 𝑵𝑨𝑰𝑽𝑬 + (1 − 𝛿)𝒘
where 𝒘 𝑵𝑨𝑰𝑽𝑬 is the vector of weights of the naïve 1/N
portfolio, 𝒘 is the vector of weights of the optimized
portfolio, and 𝛿 controls the shrinkage intensity.
The value of 𝛿 can be chosen using optimization rules,
heuristics, or cross-validation.
Grouping strategies
Branger et al. (2019) propose a grouping strategy.
The idea is that since the performance of mean-variance
(or minimum variance) optimization suffers more and
more as 𝑁 increases, due to the growing number of
parameters to estimate, one could achieve better
performance by reducing the problem dimension.
Assets are grouped together in a certain number of
groups. The optimization procedure is than performed
between the groups, while within each group the assets
are equally weighted.
Grouping strategies
Possible grouping criteria: how similar they are in terms
of estimated mean, variance or beta.
The number of groups can be chosen using optimization
rules, heuristics, or cross-validation.
The higher the number of groups, the closer we get to
the usual optimization; the smaller the number of
groups, the closer we get to the naïve 1/N rule.
In the extreme case where we only have one group we
obtain the naïve 1/N portfolio; in the extreme case
where the number of groups is equal to N, we get the
usual optimized portfolio.