Portfolio management UNIT FIVE BBS 4th year by Dilli Baral
Paul Cox CQF Final Project with MatLab code
1. PORTFOLIO CONSTRUCTION USING
THE BLACK-LITTERMAN MODEL:
AN ANALYSIS OF THE IVYBYTES
RECOMMENDED PORTFOLIO
Paul Cox
CQF FINAL PROJECT
2. Abstract
This paper demonstrates the Black-Litterman approach to creating
stable, mean-variance efficient portfolios that incorporate analyst
views into portfolio optimization. The diverse, multi asset class
portfolio recommended in the IvyBytes guide to investing will form
the basis of a prior and expected posterior returns will be estimated
based on sensible views.
Introduction
Black-Litterman’s asset allocation model overcomes many of the shortcomings of
the classic Markowitz model where optimal asset allocations are not well behaved. This
paper demonstrates the Black-Litterman framework for incorporating analyst views into
portfolio construction by looking at a very specific portfolio. The portfolio examined is
the diverse group of assets recommended to beginning investors in the IvyBytes Guide
to investing. Mean-variance analysis based on sample estimates alone is shown to
have little value, leading to corner solutions in optimization where portfolios are highly
concentrated and change wildly based on small changes to inputs. Following the
methods outlined by BL a suitable prior is developed using robust inputs - exponential
smoothing of returns and reverse optimization using global market index weights to
estimate market equilibrium returns. Optimizations with respect to indices of satisfaction
will also be considered. Plausible views regarding investment opportunities are
constructed and incorporated into reasonable looking portfolios.
IvyBytes Guide
IvyBytes is a start-up company that allows authors to self-publish concise and
accessible guides on topics in which they have expertise. Founded in 2011 by Alex Frey
3. of Harvard, IvyBytes seeks to provide unbiased information about important topics
which may be surrounded by contradictory advice. In “A Beginner’s Guide to
Investing: How to Grow Your Money the Smart and Easy Way – An IvyBytes Guide” Frey
attempts to filter though the noise of much of the investment advice that is already out
there. The book goes further than most investment books in that a specific portfolio -
complete with ticker symbols and portfolio weights - is recommended.
The IvyBytes guide to investing is a good book for beginners. Basic investing for
retirement concepts are covered including: the power of compound interest,
explanation of stocks and bonds in terms of ownership vs. lending and the concepts of
alpha returns vs. beta returns. Diversification of assets beyond simply owning a
diversified portfolio of U.S. stocks is encouraged and the diverse global mix of asset
classes used by managers of Ivy League endowments is examined. ETF’s are discussed
and the following diversified one-stop portfolio is ultimately recommended.
Asset Class Weight Ticker
U.S. Stocks 25% VTI
International stocks
(developed markets) 14% VEA
Emerging Markets
stocks 14% VWO
U.S. Real Estate 9% VNQ
Foreign Real Estate 7% VNQI
TIPS 12% TIP
Commodities 6% DBC
U.S. Total Bond
Market 7% BND
Gold 6% GLD
Table 1. IvyBytes recommended portfolio. Source: A Beginner’s Guide to Investing by
Alex Frey, Table 6.
This IvyBytes recommended portfolio will be our focus.
4. The Data and Mean-Variance
Data for each of the nine asset classes are shown in Figure 1. Specifically, daily
adjusted close prices from May 30, 2011 through May 29, 2015 are plotted. The MatLab
code for these time series plots is in the ‘Paul_Cox_CQF_Project_fts_plots.m’ file.
Figure 1. Daily adjusted close prices for each of the nine asset classes in the IvyBytes
recommended portfolio – 5/30/11-5/29/15. Source: Yahoo Finance.
Putting aside the IvyBytes recommended weights for a moment, let us consider
the portfolio selection problem among these nine risky assets using the mean-variance
optimization criterion first proposed by Markowitz. In this case the portfolio optimization
is defined as the following minimization problem:
min
1
2
Σ
A. C.
D = F
5. 퉐ҫ
And,
= 1
Where µ is the vector of asset expected returns, w is the vector of weights and Σ
is the covariance matrix obtained by pre and post multiplying the correlation matrix by
the diagonal standard deviation matrix. This minimizes the variance of the portfolio
(scaled by ½) subject to a return objective, m and the budget equation where the sum
of all weights must equal 1. At this point there is no risk-free asset and all our wealth
must be fully invested in a combination of the 9 assets. This is a minimization problem
with equality constraints and we can solve it by using the method of Lagrange:
, , =
1
2
Σ + F − D + 1 −
Solving for the first-order condition and noting that the second-order condition is
the covariance matrix which is positive definite we get w*, the optimal weight vector:
, , = Σ − D − = 0
∗
= Σ D +
In the problem at hand, Σ-1 ≈
314.8569 -129.786 -36.4128 -78.4726 4.334775 49.18255 -20.4729 114.1341 18.31203
-129.786 243.4848 -29.4895 8.587049 -110.034 6.197318 -7.26272 48.25263 -9.69668
-36.4128 -29.4895 120.0844 -1.91062 -68.592 5.304265 -10.4257 -33.8259 -5.4227
-78.4726 8.587049 -1.91062 90.95625 -24.7135 -0.37697 5.985156 -102.038 0.216958
4.334775 -110.034 -68.592 -24.7135 228.0567 -37.9958 0.455646 16.81511 2.356732
49.18255 6.197318 5.304265 -0.37697 -37.9958 940.5886 -33.0412 -1266.79 -14.6392
-20.4729 -7.26272 -10.4257 5.985156 0.455646 -33.0412 82.41821 69.34746 -22.6943
114.1341 48.25263 -33.8259 -102.038 16.81511 -1266.79 69.34746 2993.343 -41.648
18.31203 -9.69668 -5.4227 0.216958 2.356732 -14.6392 -22.6943 -41.648 41.26716
6. ψ
And µ ≈
0.131373
0.05051
-0.01117
0.096225
0.063823
0.024071
-0.13384
0.029942
-0.06779
where the vector of returns is in the same order as Table 1.
Now we need to find λ and γ. We can express the constraints as:
D = F
1 = 1
Substituting w* into these constraints:
D′ Σ D + = D
Σ D + D Σ = F
′ Σ D + =
Σ D + Σ = 1
Defining the scalars A, B, and C for convenience:
= ′ Σ
! = D′ Σ
" = D′ Σ D
And are equal to:
7. ϋ
The constraint equations become:
" + ! = F
! + = 1
Then, solving for λ and γ:
=
F − !
" − !#
=
" − !F
" − !#
Substituting these values back in to the equation for w* we can define the risk
minimizing portfolio for a given level of return. The MatLab code for this base case
mean-variance optimization in the ‘Paul_Cox_CQF_Project_Part_1.m’ file. Figure 2
graphs the efficient frontier for the nine risky asset classes in the IvyBytes portfolio (curve
in red).
8. 堊ω
Figure 2. Markowitz analysis of IvyBytes recommended portfolio.
Figure 2 also pinpoints the global minimum variance portfolio. The global
minimum variance portfolio is obtained by solving the unconstrained problem:
min
$
%&
#
F
where %&
#
expressed as a function of m is:
%&
#
=
F2
− 2!F + "
" − !2
Solving for the first-order condition we get:
'%(
2 F
'F
=
2 F − 2!
" − !#
= 0
Efficient
Frontier
for 9 risky
assets
Global Minimum Variance Portfolio
Capital
Market
Line
Tangency Portfolio
9. And solving for m we get:
F) =
!
where mg is a minimum given that the second-order condition is positive (2A/(AC-B2)>0)
specifically, in our IvyBytes problem mg ≈ 0.039475. Knowing mg we can solve for the
vector of weights in the global minimum variance portfolio as:
) =
Σ
And for the IvyBytes analysis, wg ≈
0.148923
0.012797
-0.03835
-0.06431
0.006751
-0.22216
0.040637
1.135892
-0.02019
For the variance,
%)
#
=
1
Giving a standard deviation of σg ≈ 0.02514 for the global minimum variance portfolio.
This point is shown by the blue asterisk in Figure 1.
Taking into account borrowing and lending at the risk-free rate the optimization
problem becomes:
min
1
2
Σ
Subject to:
* + D − * = F
10. ϊ
Any wealth that is not invested in the risky assets will be invested in the risk-free
asset. The Lagrangian for this problem is:
, , =
1
2
Σ + F − * − D − *
Solving for the first-order condition by taking the derivative with respect to the
vector w we can solve for the optimal weight vector w*:
= Σ − D − * = 0
∗
= λΣ D − *
Plugging this equation into the constraint we get:
=
F − *
μ − * ′ Σ μ − *
Finally we get the equation for w*:
∗
=
F − * Σ μ − *
μ − * ′ Σ μ − *
This equation along with the risk-free rate traces out the Capital Market Line
(shown in blue in Figure 1) for various levels of returns. The CML is drawn from the risk-
free rate on the y-axis through the Tangency Portfolio on the efficient frontier. In the
Markowitz paradigm all investors hold a combination of the risk-free asset and the
tangency portfolio. In this analysis the risk-free rate is assumed to be the average of the
five year Treasury note rate over the same period (5/30/11 – 5/29/15). In this IvyBytes
analysis the risk-free rate r ≈ 0.0120224. The vector of weights in the tangency portfolio
wt is given by:
- =
Σ−1
μ − *
! − *
Where wt for the IvyBytes problem is, wt ≈
11. σ
0.718318 VTI
-0.17583 VEA
-0.24327 VWO
-0.12282 VNQ
0.165604 VNQI
-0.03208 TIP
-0.26055 DBC
0.923641 BND
0.026987 GLD
The mean and standard deviation of this portfolio are given by:
D- = D =
" − !*
! − *
≈ 0.1469
And,
%- = 2 ′-Σ - ≈ 0.055715
Taking a closer look at these weights we see that the optimal portfolio is
unintuitive and somewhat of a corner solution. In this example where short selling is
allowed the optimization is heavily long U.S. stocks and U.S. Bonds and (with the
exception of some minor holdings in foreign real estate and gold) this optimum portfolio
sells short every other asset class to do it. If we impose a restriction on short-selling the
optimal frontier reduces to holding only some combination of U.S. stocks and U.S. bonds
and has zero weights in the remaining 7 asset classes as shown in Figure 3 below.
12. Figure 3. Composition of optimal IvyBytes recommended portfolio with no short-selling.
Clearly the Markowitz asset allocations are not well behaved.
Black-Litterman
“How can we make the standard portfolio optimizer better
behaved?” - Litterman1
1 Modern Investment Management – An Equilibrium Approach by Bob Litterman and the
Quantitative Resources Group, Goldman Sachs Asset Management, p. 76.
13. This is the financial engineering question that Fisher Black and Bob Litterman
asked themselves as they began to develop their model. Their solution was a market
equilibrium based approach that still allowed portfolio managers to incorporate analyst
views about investment opportunities. In the BL model, optimal portfolios are a
weighted combination of the market capitalization equilibrium portfolio and an
“investor views” portfolio. Rather than starting with a vector of asset returns, BL starts
with the equilibrium expected returns that would lead to the optimal portfolio having
the observed global market capitalization weights – a reverse optimization. In the
following paragraphs we will develop robust inputs for the IvyBytes portfolio using
exponentially weighted moving averages (EWMA) of returns. We will establish a prior by
solving the reverse optimization problem. We will examine alternatives to mean-
variance. We will consider 2 views on asset classes that differ from the views of the
market. Finally, we will demonstrate a posterior optimal portfolio from revised excess
returns by using the BL formula.
Robust Inputs
BL starts with the market model of N asset classes whose returns are normally
distributed:
5 ~ 7 8, 9
Using econometric methods we can make covariance matrix input more robust.
One way is by using exponential smoothing of past returns with a EWMA model. The
MatLab code for robust inputs and the implementation of BL is in the
‘Paul_Cox_CQF_Project_Part_2.m’ file.
We will focus on weekly excess returns (returns over the risk-free rate) of the
IvyBytes portfolio. In this case we have 208 observations for each asset class. EWMA for
the variances is calculated as shown in the following example for VTI.
14. 宐τ
This EWMA is a conditional variance for VTI. The calculation in the above
example reduces to an equation in which each week’s variance is a function of the
previous week’s variance times the parameter λ (which we take as 0.94 as used in
RiskMetrics) plus (1 – λ) times the previous week’s squared return. Coding this example
in MatLab we can calculate robust standard deviations of all 9 asset classes, as seen
below.
15. EWMA can be used to calculate a correlation matrix. In the following example
the excess returns of VTI compared to VEA has a correlation coefficient of
approximately 0.8094. The correlation coefficient is the sum of VTI’s excess returns times
VEA’s excess returns times the weights all divided by the product of their standard
deviations as shown below.
17. Reverse Optimization and the Prior
Now that we have the covariance matrix, Σ, we need to get the prior expected
excess returns. The BL model uses market equilibrium returns as a neutral starting point.
BL states that µ is normally distributed with mean π and standard deviation τΣ.
8 ~ 7 :, ;9
Where π is the implied vector of excess returns that clear the market and τΣ is the
estimation error on µ. To get π we solve the following maximization problem:
argmax
@
{@ : − @ 9@}
Setting the derivative of this with respect to w and doing the reverse optimization by
solving for π using market weights wMKT we get:
ΠDEF = 2 G DEF
Now we just need a source for global market capitalization rates of the asset classes in
the IvyBytes portfolio. One such source is a report titled “Global Invested Capital
Market” released by Hewitt EnnisKnupp, An Aon Company. Their June, 2014 study lists
asset classes along with their proportions of the global invested capital market. Note
that the Hewitt study considers more asset classes than the IvyBytes portfolio so we will
have to adjust the weights accordingly.2 Table 2 below shows the market weights used
for our estimation of the prior.
2 Comparing the IvyBytes portfolio to the asset classes in the Hewitt study shows that the only
large asset class missing from the IvyBytes portfolio is non U.S. Bonds (Developed) which,
according to the Hewitt study, 22.4% of the global invested capital market.
18. ω
Asset Class
Weight
from
Hewitt
study
Adjusted
Weight for
IvyBytes
Portfolio
U.S. Stocks 18.00% 27.78%
International stocks
(developed markets) 13.70% 21.14%
Emerging Markets
stocks 4.00% 6.17%
U.S. Real Estate 6.10% 9.41%
Foreign Real Estate 5.00% 7.72%
TIPS 2.50% 3.86%
Commodities 0.15% 0.23%
U.S. Total Bond
Market 15.20% 23.46%
Gold 0.15% 0.23%
Total 64.80% 100.00%
Table 2. Market weights for the asset classes in IvyBytes recommended portfolio. Source
for weights: Global Invested Capital Market, June, 2014 published by Hewitt EnnisKnupp, An Aon
Company, Table 2.
Substituting these weights into the reverse optimization and for various levels of
risk aversion, λ, ΠMKT ≈
λ = 0.01 λ = 1.2 λ = 6 λ = 2.71
0.000148 0.017725 0.088626 0.04003
0.000172 0.020589 0.102947 0.046498
0.000241 0.028864 0.144319 0.065184
0.000146 0.01753 0.087648 0.039588
0.000155 0.018638 0.09319 0.042091
1.46E-05 0.001752 0.008762 0.003957
8.36E-05 0.010034 0.050169 0.02266
3.95E-06 0.000474 0.002372 0.001071
2.88E-05 0.003457 0.017285 0.007807
λ = 0.01 corresponds to the near Kelly investor, λ = 1.2 is the risk aversion level
used by BL and λ = 6 is for a very risk-averse investor. The last one, λ = 2.71 is that level of
19. 며ϊ
risk aversion that sets the risk premium of U.S. stocks to approximately 4% and this is the
one we will use. This is the method of estimating λ is used by Litterman in his book
Modern Investment Management when he discusses the risk aversion coefficient as a
scaling factor of excess returns. With these inputs the composition of the Prior for our
IvyBytes mean-variance optimization problem is shown below.
We can see that this set of optimal portfolios is much more reasonable. It is not a
corner solution as all nine assets are represented at some point along the volatility
vector.
In addition to mean-variance we can optimize by maximizing the Sharpe ratio as
shown below:
20. Views and Calculation of Posterior
BL assumes that the views are normally distributed by:
H8~ 7 I, J
where P is a “pick” matrix with 1 row for each view and 1 column for each asset we will
have 2 views so P will be 2 X 9. And,
J = 'KLM H ;9 H
Is the uncertainty on the views. In the IvyBytes problem, let’s assume we have 2 views.
The first view is that, perhaps due to the Fed raising rates, TIPS outperform the U.S. total
bond market by 1%. The second view is that, perhaps due to a stronger dollar,
21. ϊ
International developed stocks do 4% better than U.S. stocks. We will assume full
confidence in our views although BL allows us to set various confidence levels.
Therefore,
H = N
0 0 0 0 0 −1 0 1 0
1 −1 0 0 0 0 0 0 0
O
I = N
−0.01
−0.04
O
BL takes advantage of Bayes’ formula for updating probabilities based on new
information:
P Q|S =
P{S|Q}
P S
× P Q
for event E and information I. This says that the Posterior probability P{Q|S} (probability of
event E given information I) is equal to the updating weight
UVSWQX
U Y
x P(E), the prior
probability. Through much tedious working Meucci (2010) shows that the posterior
market model is:
5|I; J ∼ 7 D], Σ]
Where,
D] = ( + ;ΣP ;PΣP + Σ ^ − P(
Σ] = 1 + ; Σ − ;2
ΣP′
;PΣP + Ω PΣ
This is the model that incorporates the views. Figure 4 shows the comparison of
the Prior compared to the Posterior.
22. 䨀
Note how the optimal weights have incorporated the information of the views.
The area for VTI is reduced as VEA is viewed as more of an opportunity. Also, the area
for TIP is larger at the expense of BND. Also note that the BL efficient frontiers are
reasonable and lack corner solutions.
In summary, the largest drawback to the Markowitz model is that a straight
mean-variance optimization leads to unintuitive portfolios concentrated in just a few
assets. Mathematically the solutions are correct but they are highly sensitive estimations
of returns which are also the hardest input to estimate. BL doesn’t use sample data of
returns. By solving the reverse optimization problem and backing into the equilibrium
returns that give observed market weights, BL allows portfolio managers to construct
23. reasonable prior without corner solutions. Informed views that differ from the market
can then be incorporated into well balanced, reasonable looking posterior portfolios.
