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 O’Cinneide, 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 we’re 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 𝜏 = 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 0→day 1 =𝐹𝑋day1

𝐽𝑃𝑌 / 𝐴𝑈𝐷

𝐹𝑋day0𝐽𝑃𝑌 / 𝐴𝑈𝐷

×1 + deposit rate

(𝐴𝑈𝐷day 0

365

)1 + deposit rate

(𝐽𝑃𝑌day 0

365

) − 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;

𝐸[𝑅] = [(𝜏∑

)−1 + 𝑃 ′𝜎−1𝑃 ]−1 (𝜏∑

)−1Π + 𝑃 ′Ω−1𝑄]; (1)

where [𝑅] is the new (posterior) Combined Return Vector (N x 1 column vector);

𝜏 is a scalar;∑is the covariance matrix of excess returns (N x N matrix);

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𝑃 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);

Ω is a diagonal covariance matrix of error terms from the expressed views representing the uncer-

tainty in each view (K x K matrix);

Π is the Implied Prior Return Vector (N x 1 column vector);

𝑄 is the View Vector (K x 1 column vector);

𝐾 is the number of views;

𝑁 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

𝑅𝑡+1 (𝑅𝐿𝑉 ) = 𝑓(𝑅𝑡(𝑅𝐿𝑉 ), 𝑅𝑡(𝑅𝐿𝐺), 𝑅𝑡(𝑅𝐿𝑉 ), 𝑅𝑡(𝑅𝑈𝐽), 𝑅𝑡(𝑅𝑂𝑈), ...). (2)

The functional form that is assumed here is

𝑦(𝑥) = 𝑤𝑇𝜑(𝑥) + 𝑏. (3)

We fitted the underlying function was fitted by the vector regression.

We minimize the error function, which has the form:

𝐶

𝑁∑𝑛=1

𝐸𝜖(𝑦(𝑥𝑛) − 𝑡𝑛) +1

2||𝑤||2. (4)

Our procedure could still have some limitations, because there exist 𝜖-insensitive error functions,

such as

𝐿𝜖(𝑦) =

⎧⎨⎩0 for —f(x) - y— < 𝜖;

|𝑓(𝑥) − 𝑦| − 𝜖 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 𝜖 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 𝜏 = 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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