Post on 08-Feb-2023
Dynamically Enhanced Total Return Fund: A Global Bond
Equity Portfolio
Adrien Chenin, Abishek Kumar, Nina Kuklisova,
Stefan Racovita, Amritpal Singh Sidhu *
ABSTRACT
We test portfolio asset allocation strategies. Our portfolio gives 90% weight to assigned to
Strategic Asset Allocation and remaining 10% of capital goes to Tactical Asset Allocation. The
strategy is targeted to clients looking to invest in global equity and bond assets. Portfolio not only
captures global bonds and equities but also tries to enhance the returns with strategic tilt as well
as tactical allocation.
Given that our portfolio is targeted to clients looking to invest in 60%/40% global equi-
ty/bond mix, we have shown that our approach of using strategic tilt and tactical allocation in
standard 60%/40% equity/bond outperform the benchmark, and has higher sharpe ratio compared
to the benchmark portfolio.
*The Class Project for the Multi-Asset Portfolio Management Course was implemented under the guidance ofProfessor Irina Bogacheva, professor Colm OβCinneide, and professor Inna Okounkova, Mathematics Department,Columbia University
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1 Dynamically Enhanced Total return Fund: A Global Bond
Equity Portfolio
1.1 Goal and universe selection
1.1.1 Main goals
We design our total return fund by using traditional assets only, that is Equities and Bonds,
from 2000 to 2013. In fact, since we are not using a forward-looking analysis, we want to be exposed
to only Equities and Bonds, since it was a popular investment approach at the beginning of the
2000s. The goal of this project was to achieve an average return of 6% annually on an absolute
basis, 10% annual volatility, and we want to beat market-capitalization bond-equity benchmarks.
We set these values as our goal was to slightly outperform the market.
1.1.2 Preliminary data analysis
Our primary data sources were Bloomberg, Datastream and Barclays Live. The following
table lists the indices that we used altogether with their corresponding ETFs:
Figure 1. Universe Selection
1.1.3 Technical Notes
The data used for each index and ETF were its monthly returns and market capitalization.
For Equites, we extracted unhedged total return gross dividends. Since we are given a capital in
USD and we calculate our portfolio value in USD, investments in currencies different from the USD
are exposed to unhedged foreign currency risk. For the reason that weβre using a total return fund,
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we are also considering dividends reinvestments. We decided not to consider taxation issues and
gross dividends. For Bonds, we use similarly unhedged total return indices with coupons reinvested
in the index but not taxed. Expense ratios were taken from iShares BlackRock, ETF Database,
Yahoo and SPDRs websites. This data was from the previous year. In testing our final portfolio,
we assume that these expense ratios were constant for the past 3 years.
1.2 Strategic Asset Allocation using the Black-Litterman Approach
1.2.1 Main framework construction
For a more explanation of the Black-Litterman Approach and methodology, see Appendix
A, or resources listed in bibliography.
In our Black-Litterman implementation, we had π = 0.2 and risk aversion 4. In order to
get the prior weights, we took the previous market capital weights and ran optimization. Our
procedure is explained in Appendix B.
Afterwards, we got the following matrices for P and Q:
Uncertainty in the view is represented using the diagonal matrix of the covariance matrix
of the SAA-investment universe.
1.2.2 Further parameter specification
We decided that none of our individual assets should have more than 25 % weight of our
portfolio, total weight of equities should not exceed 45 %, and the total weight of bonds should not
exceed 55 %. Our strategic fund is only based on long positions, so the sum of weights of all assets
is 100 %.
After this procedure, we proceeded to the Tactical Asset Allocation.
1.3 Tatical Asset Allocation
Our work resulted in trading strategies improvement.
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1.3.1 A more sophisticated G10 Carry Trade
We use the G10 currencies correspond to the 10 most traded currencies in the world, as
shown in the table below.
Figure 2. Market shares of the most frequently traded world currencies
The basic G10 carry trade consists of ranking the G10 currencies and taking advantage of
the highest spread in terms of deposit rate. Based on the deposit rates below (1 month Libor for
USD and EUR, 1 month domestic deposit rate (DRA) for the other currencies), the basic strategy
consists of borrowing in the lowest deposit rate, that is in JPY at 0.15%, and investing in the
highest deposit rate, that is in AUD at 4.65%.
Figure 3. Indices
Therefore, we earn the following daily return (in case of JPY and AUD):
Returnday 0βday 1 =πΉπday1
π½ππ / π΄ππ·
πΉπday0π½ππ / π΄ππ·
Γ1 + deposit rate
(π΄ππ·day 0
365
)1 + deposit rate
(π½ππday 0
365
) β 1
In addition to earning the deposit rate spread, we are taking risk in terms of FX. For
example, a depreciation of AUD relatively to JPY could offset this return. In spite of this FX risk,
this strategy delivers consistent positive returns. In fact, its annualized alpha is 0.80%, annualized
risk of 1.30% and active Information ratio of 0.61 over the past 27 years. In addition to this FX
risk, it has further disadvantages. First, the dispersion of opinions among arbitrageurs causes a
synchronization problem. Second, with this strategy, crashes in the FX rates are more likely and
larger. As a result, this FX risk causes heavy left tails on the returns distribution. Moreover, there
is risk of liquidity.
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To overcome this problem, we devise a more sophisticated G10 carry trade. In order to
compensate the FX risk, we rank our currencies in terms of their 2-year swap rate momentum (the
difference between the 2-year swap rate and the 10 last days moving average). This way, we take
into account not only the deposit rate of the currency (2-year swap rate is the fixed rate against the
deposit rate of the domestic currency) at the same time as its trend. Then, the highest momentum
corresponds to the highest trend in terms of 2-year swap rate, which correspond to the highest trend
in FX and in deposit rate (buying the past winners and selling the past losers). Therefore, if we use
these results, we can expect taking advantage of the deposit rate spread, while also earning an FX
return. In order to remove the best ones, we take into account the crash risk in FX rates. Then,
we average our strategy by considering the best 3 momentums (top 3) and the worst 3 momentums
(bottom 3) out of our 10 currencies. For instance, in the following example (Figure 3), the strategy
is to be long in CAD, EUR and JPY and short in SEK, GBP and NZD.
Figure 4. 2-year indices
As a result, decided to incorporate 0.25 bps as daily transaction costs. 10 bps monthly cost
corresponds to 10/22 bps per trading day. However, in this case, we are readjusting our portfolio
less than half of the time. Therefore (10/22)/2 =0.25 bps is a conservative transaction cost.
Afetrwards, we run this strategy from January 2001 to November 2013 and observe the
portfolio value and drawdown, as shown in Figure 5.
Figure 5. Portfolio values and drawdowns
Also, we compute a series of useful statistics for the results of this strategy.
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Figure 6. Statistics of this More sophisticated Carry trade strategy
These results show that our strategy delivers more consistent returns than the basic carry
trade. The annualized return is quite high (5.38%) compared to the annualized volatility (3.76%)
which gives us a very good sharpe ratio (1.43). Significantly more than half of the time, it delivers
positive returns (55.09% percentage of days up). Crashes are rare (skewness of 0.04, kurtosis of 11,
min daily return of -2.20% and max drawdown of -5.45%). Therefore, sortino ratio is even higher
(2.44).
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1.3.2 Yield Curve Trade
The basic idea behind the strategy is to capture the yield spread between bonds with
different durations this is a long/short strategy. Here, we are comparing:
Long US 10 Years Treasury Notes against Short 2 Years Notes, and
Long 10 Year German Bonds (10 Y Bunds) against Short 2 Years German Bonds (2 Y Schatz).
This strategy collects the risk-premia for duration risk. Any sort of movement in the yield (level,
slope and curvature) affects the performance. Rebalancing every time the weights move 2% from
the target weight. Transaction costs are taken into consideration, with 10 bp of the turnover.
Figure 7. Bonds performance
Figure 8. Table of our portfolio performance
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2 Performance
2.1 Strategic Asset Allocation Performance
As we can see in figure 9, our SAA outperformed the benchmark by 1.5% per annum on
average. Consistent use of our strategy clearly leads to better results, since the outperformance
of our portfolio is rising over time. We can see that the overall evolution of the portfolio value is
similar to that of the whole market. Yet, our portfolio did not lose as much as other assets in years
2008-2009. This is also visible in the histogram in figure 10.
Figure 9. Plot of our portfolio performance
Figure 10. Histogram of portfolio performance
Figure 11 shows that in addition to higher returns, our portfolio showed lower volatility
than the market, and thus also a higher sharpe ratio.
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Figure 11. Statistics of portfolio performance
In figure 12, we can see that our portfolio because less diversified at later time, while it was
also outperforming the market more significantly.
Figure 12. Portfolio Weights and Statistics from Strategic Asset Allocation
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2.2 Total Portfolio Performance
2.2.1 Merging strategies
Our tactical asset allocation assigns 10% of the wights in our investments. Our prefixed
allocation for the three tactical strategies is the following:
Our risk management is exponentially weighted moving average of volatility to asses the risk
of each strategy on a monthly basis. Based on the risk estimate the portfolio volatility was scaled
to 10%. In tactical allocations, we leveraged the by up to 10% annually. In strategic allocation, if
the volatility became higher than 10%, we reduced the portfolio proportionally and the remaining
was invested in cash.
Figure 13. Plot of our portfolio performance
Figure 14. Statistics of our portfolio performance
2.2.2 Performance Attribution
After backtesting our portfolio performance, we found effects to be the following:
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Figure 15. Performance attribution breakdown
Figure 16. Table of performance attributions
For the attribution of the performance, we break down the active portfolio annual returns
into the active equity and active bond returns:
Active Portfolio returns = Active Equity Returns + Active Bond returns
Figure 17. Overall performance attribution
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In 2010 and 2013, most of the portfolio returns is attributed to its equity part while in 2011
and 2012, it is mostly the bonds portfolio. Then, for each active return (Equity or Bond), we break
down their active annual returns into their equity/bond selection part (sum of the equity/bond ETF
returns) and their part attributed to the benchmark (equity/bond over/under weighted portfolio).
Active Equity Returns = Equity Selection Returns + Equity benchmark over/underweight returns
Figure 18. Equity performance attribution
The equity performance is attributed most of the time to the equity benchmark except in
2013 where the equity selection performs more than the equity portfolio, while the equity benchmark
experiences a negative return.
Active Bond Returns = Bond Selection Returns + Bond benchmark over/underweight returns
Figure 19. Bond performance attribution
For the bond portfolio, it is the opposite. The active bond returns are mostly attributed to
the performance of our bond selection.
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3 Conclusion
Our analysis allows the following conclusions:
First, Carefully Designed Strategic Asset Allocation and Tactical Asset allocation do im-
prove the performance as compared to market capitalized weight portfolio.
Second, our strategy also lead to Volatility Reduction (11% to 6.5%) and to a slightly better
return (10% compared to 9.5%).
Last, our strategic asset allocation could be further improved with adding more uncorrelated
input factors. For coming up with tactical allocation strategies, more research would be necessary.
Most importantly of all, even with these very good results, we understand that this strategy
can only insure us against underperforming. Since this model is known and relatively convenient to
implement, it can be used by many other portfolio managers. Therefore, we are aspiring to continue
developing our understanding of financial markets so that we can come with more advanced models.
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A Black-Litterman approach implementation
The Black-Litterman strategy uses a Bayesian approach to combine the subjective views
of an investor regarding the expected returns os one or more assets with the market equilibrium
vector of expected returns. The ability of investors to explicitly express their views adds practical
flexibility to the model. A view can be expresses as either absolute, relative, or a combination of
the two views with an associated confidence level.
The effect of a view on the asset allocation is to tilt the portfolio towards outperforming
sectors and away from the underperforming ones. Views are dampened by the blend with the
equilibrium returns in order to limit the effect of extremes and to ensure greater consistency across
the estimates.
Figure 20. Black-Litterman Approach
The scheme of the Bayesian approach is the following:
With the Black-Litterman Approach, we construct a combined Return Formula;
πΈ[π ] = [(πβ
)β1 + π β²πβ1π ]β1 (πβ
)β1Ξ + π β²Ξ©β1π]; (1)
where [π ] is the new (posterior) Combined Return Vector (N x 1 column vector);
π is a scalar;βis the covariance matrix of excess returns (N x N matrix);
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π is a matrix that identifies the assets involved in the views (K x N matrix or 1 x N row vector in
the special case of 1 view);
Ξ© is a diagonal covariance matrix of error terms from the expressed views representing the uncer-
tainty in each view (K x K matrix);
Ξ is the Implied Prior Return Vector (N x 1 column vector);
π is the View Vector (K x 1 column vector);
πΎ is the number of views;
π is the number of assets in the portfolio.
When implementing this scheme, the original static view that we considered was the State
Street forecast for each asset category. Then, we adjust it for the Asset Recent Performance, and
got the weighted average of Static view and expected annualized return, based on the last 1 year.
The view based on predictive regression assumes the future return of each asset to be a
function of the SAA-Investment universe could be mathematically stated as
π π‘+1 (π πΏπ ) = π(π π‘(π πΏπ ), π π‘(π πΏπΊ), π π‘(π πΏπ ), π π‘(π ππ½), π π‘(π ππ), ...). (2)
The functional form that is assumed here is
π¦(π₯) = π€ππ(π₯) + π. (3)
We fitted the underlying function was fitted by the vector regression.
We minimize the error function, which has the form:
πΆ
πβπ=1
πΈπ(π¦(π₯π) β π‘π) +1
2||π€||2. (4)
Our procedure could still have some limitations, because there exist π-insensitive error functions,
such as
πΏπ(π¦) =
β§β¨β©0 for βf(x) - yβ < π;
|π(π₯) β π¦| β π otherwise.(5)
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Figure 21. LIBSVM library implementation
Afterwards, we got the following matrices for P and Q:
Uncertainty in the view is represented using the diagonal matrix of the covariance matrix
of the SAA-investment universe.
In our LIBSVM library implementation, we used the Radial Basis function as Kernel, and
chose the π value 0.001.
B Implementation of Black-Litterman
B.1 Introduction
Typically, we would like to do a lot of fundamental research supported with quantitative
analysis to come up with views and uncertainty around the views. However, for the purpose of this
project, we decided to use a purely systematic approach to construct our view. Our framework
is based on predictive regression. We decided to use Support Vector Regression as our prediction
tool. The main reasons for using it was that we are comfortable with using it, and this tool is also
very robust around noise (due to its epsilon-insensitive loss function).
Essentially, our prediction framework is based on the fact that our investment universe
(stock and bond indices) broadly reflects the market state. With this assumption, we can state a
hypothesis that the next period return of each asset can be predicted using the current state of the
market.
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B.2 Procedure
Our construction methodology was the following:
1. We used 59 months of monthly data to train our predictive regression model.
2. We built 11 different predictive models to treat each regression independently.
3. For each of these, we ran regression with an input vector of the standardized returns of each
asset class in our investment universe. E.g. if we are fitting our model for may 2001 return
of Russel 2000, we take the April 2001 returns for every asset we have in our investment
universe. Then, we use the in-sample mean and standard deviation of each asset to find the
standardized the input vector. We did not perform any standardization for output vector.
4. We chose a RBF (radial basis function) kernel to define the mapping between our input and
output.
5. Due to a lack of sufficient history, we did not do a cross-validation at every stage. We
performed an initial cross-validation to decide on our hyper parameters. (Generalization
parameter (10) and Epsilon (0.001).
6. At the end of our training period we obtain a regression model, which we used afterwards to
generate the forecast of next period return for each asset separately.
7. After we started our predictive regression framework , we kept sliding the historical window
by 1 month and forecasted the next period return.
Note: Support vector regression also provides us a nice and intuitive way to associate uncertainty
for each prediction that we make. However, we have not used the uncertainty associated with our
prediction at this stage. We simply used the asset covariance matrix with a π = 0.5 to generate
our uncertainty in the view.
References
The Intuition Behind Black-Litterman Model Portfolios; Goldman Sachs Investment Management
Division; December 1999
Making Risk Additive: The Alpha and Beta of Risk Attribution and Risk Delta; MSCI; March
14, 2012.
Making Risk Additive: Marginal Contribution to Risk & Correlation Risk Attribution; MSCI;
February 16, 2012.
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