qmim final project

8/3/2019 QMIM Final Project

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QUANTITATIVE METHODS IN INVESTMENT MANAGEMENT

Columbia University

Department of Mathematics

December 2011

Prof. Alex Greyserman

“Trend-following strategies for selected currencies”

Ana Sokoleva as4148Andres Jaime aaj2132Ezequiel Zambaglione eaz2109

Joaquin Tapia jt2670



Index

Introduction

Trading Algorithm

Optimization & Backtesting

Robustness evaluation

Trading Costs

Optimal strategy portfolio

Benchmark

Backtesting vs Benchmark

Risk Management

Conclusions

Bibliography

Appendix Codes

A Trend-following strategy

B Trading strategy – Long/short/neutral

positions

C Markowitz (Benchmark)

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10

13

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2



Introduction

Standard Moving Averages are used to identify trends in financial markets, and coupled with the

establishment of trading rules and the selection of parameters related to trend speed and trade risk

can lead an investor to obtain profits. Our project consisted in developing a Trend-Following tradingstrategy for three currencies, all against the US Dollar:

Euro (EUR),

Swiss Franc (CHF), and

Swedish Krona (SEK)

The strategy’s performance was measured against a Benchmark (BMK) comprising the four currencies

mentioned. The Benchmark was the result of the Classical Mean-Variance optimization process. The

initial allocation reflects the optimal weights of each currency in the optimal portfolio. The RiskManagement tools used were the Relative-VaR (using a 95% level of confidence) and Drawdown. The

Relative-VaR provided a risk measure easy to follow and compute, therefore if risk limits are crossed

we close the position until risk limits are within the tolerable levels.

Each currency in the portfolio includes the overnight rate earned on the investment in each currency

(Merrill Lynch indexes) and the source was Bloomberg.

Trading Algorithm

The trading rule is based on the analytic property of the smoothened time series, in which we used to

parameters:

Slope parameter (M), refers to the time window (measured in days) used to smooth the time series (f).

Then, the daily difference of the smoothed time series between time “t” and “t-1” is taken to compute

the slope of the time series.

(2)

A second smoothing step is required to estimate de curvature of the series.

(3)

This will provide an estimate of the curvature of the time series

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(4)

At time t, the trading rule proposes taking a long/short position on the underlying asset if (2) and (4)

are either both positive or both negative. In other words, we enter a long/short position when not only

the slope but also the curvature indicate a bullish or bearish trend.

Optimization and Backtesting

The parameters that we optimized were the moving averages. We calculate the Sharpe ratio of the

first 60 Moving averages for each currency, choosing the one with the highest to back-test our strategy.

The results for the best 10 Sharpe ratios are shown below:

In the case of the EUR, using out-sample data, we can see that the 18 day moving average performed

remarkably well, with a Sharpe ratio of 1.70. Also, 53.1% of the time the strategy won, while the

average daily annualized return for the strategy was 14.6%.

EUR

2008-2009 2010 to date 6 1 2 3 4 5

Rank MA in sample out sample total trades win % loss % average return

1 18 1.35 1.70 351 186 53.1% 164 46.9% 14.6%

2 15 1.31 1.64 344 186 54.23% 157 45.77% 14.40%

3 19 1.29 1.80 343 185 54.09% 157 45.91% 15.84%

4 5 1.28 -0.82 340 168 49.41% 172 50.59% -8.61%

5 4 1.22 -0.05 362 178 49.17% 184 50.83% -0.50%

6 7 1.18 -0.25 329 163 49.70% 165 50.30% -2.39%

7 13 1.07 1.02 348 183 52.59% 165 47.41% 9.34%

8 17 0.95 1.62 349 189 54.31% 159 45.69% 14.19%

9 3 0.94 -0.84 365 176 48.35% 188 51.65% -8.68%

10 16 0.92 1.24 349 187 53.74% 161 46.26% 11.03%

Regarding CHF, we achieve a Sharpe ratio of 1.21 using the 29 moving average. This strategy won 53.5%

of the time with a 11.5% of average daily annualized return.

3

EUR

2008-2009

Rank MA in sample

1 18 1.35

2 15 1.31

3 19 1.29

4 5 1.28

5 4 1.22

6 7 1.18

7 13 1.07

8 17 0.95

9 3 0.94

10 16 0.92

SEK

2008-2009

Rank MA in sample

1 60 1.73

2 45 1.50

3 58 1.37

4 47 1.05

5 59 0.95

6 57 0.90

7 56 0.83

8 55 0.75

9 52 0.58

10 46 0.53

CHF

2008-2009

Rank MA in sample

1 29 0.79

2 38 0.78

3 20 0.53

4 28 0.45

5 31 0.44

6 40 0.41

7 26 0.36

8 16 0.34

9 25 0.33

10 39 0.29



CHF

2008-2009 2010 to date 6 1 2 3 4 5


1 29 0.79 1.21 349 185 53.5% 161 46.5% 11.5%

2 38 0.78 0.41 354 179 51.14% 171 48.86% 4.15%

3 20 0.53 2.41 348 191 55.20% 155 44.80% 24.64%

4 28 0.45 1.35 351 187 53.58% 162 46.42% 12.94%

5 31 0.44 1.77 343 183 53.98% 156 46.02% 16.15%

6 40 0.41 -0.12 343 174 51.48% 164 48.52% -1.48%

7 26 0.36 1.09 337 179 53.43% 156 46.57% 10.38%

8 16 0.34 1.25 341 177 52.52% 160 47.48% 13.75%9 25 0.33 1.27 338 176 52.38% 160 47.62% 13.80%

10 39 0.29 0.66 346 175 51.17% 167 48.83% 6.59%

Finally, the SEK was the worst performer. This was the only currency with a negative sharpe ratio of

-0.72. This strategy lost 50.8% of the times with a -9.7% average daily annualized return.

SEK

2008-2009 2010 to date 6 1 2 3 4 5


1 60 1.73 -0.72 373 183 49.2% 189 50.8% -9.7%

2 45 1.50 -0.04 372 183 49.33% 188 50.67% -0.56%

3 58 1.37 -0.75 374 181 48.53% 192 51.47% -10.19%4 47 1.05 -0.12 371 185 50.00% 185 50.00% -1.52%

5 59 0.95 -0.52 373 180 48.39% 192 51.61% -6.92%

6 57 0.90 -1.01 372 177 47.71% 194 52.29% -13.89%

7 56 0.83 -1.08 374 177 47.45% 196 52.55% -14.47%

8 55 0.75 -0.33 370 180 48.65% 190 51.35% -4.16%

9 52 0.58 -0.49 376 185 49.20% 191 50.80% -6.31%

10 46 0.53 0.13 362 179 49.58% 182 50.42% 1.56%

Robustness evaluation

The performance of the EUR and CHF Sharpe Ratio’s according to the best ranked moving averages for

the in-sample data shows greater stability than for the out-sample period. The following graphs show

this behavior.

Swiss Franc: Sharpe ratio's stability

in-sample versus out-sample data

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

29 38 20 28 31 40 26 16 25 39

a r p e a o s u n s

Best Mo ving Averages (days)

in-sample

out-sample

Euro: Sharpe ratio's stability


-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

18 15 1 9 5 4 7 13 17 3 16

a r p e a o s u n s

Best Moving Averages (days)

in-sample

out-sample

However, for the SEK the Sharpe ratios showed slightly greater stability.

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Swedish Krona: Sharpe ratio's stability


-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

60 45 58 47 59 57 56 55 52 46

a r p e a o s u n s

Best Moving Averages (days)

in-sample

out-sample

Trading Costs

All the trades are assumed to be executed using the price of the WMR fix next business day, thereforethe strategy avoids transaction costs. However, the trade-off of this execution is represented by the

loss of exact timing to execute, i.e. to start-up or stopped the strategy at any given time outside the

London Fixing.

Optimal Strategy Portfolio

With the strategy in place for these currencies, we constructed an optimal portfolio for finding the

appropriate allocation of funds among each currency-strategy. To construct the optimal portfolio for

the in-sample period, we used the annual returns of each currency considered, and then constructed

the return’s covariance matrix and solve for the optimal weights that maximize the Sharpe ratio. We

included non-negativity restrictions for the weights and its sum equal to 100%. The result of the

optimization leaded us to a portfolio of two currencies, EUR and SEK, with weights of 56.62% and

43.38%, respectively.

Strategy optimal portfolio

ReturnsStandard

Deviation

Optimal

Weights

EUR 16.79% 11.59% 56.62%SEK 21.73% 15.21% 43.38%

CHF 6.83% 12.36% 0.00%

USD 1.27% 0.00% 0.00%

Portfolio 18.93% 9.32% 100.00%

As can be seen in the table above, the risk-return profile of the Swiss Franc was dominated by the

Euro. Also, it is important to mention that the risk-free rate was in historical lows during this period.

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Benchmark

In order to compare the performance of our strategy’s portfolio we constructed a benchmark based on

the aforementioned currencies. The methodology used was the Markowitz Portfolio Selection Model,solving for the lowest portfolio variance for a given portfolio return.

2 , = ′Σ x

s.t

= ′μ

The risk-free rate used was the average rate of the US Treasury 1 month bill during the period i.e. 1.85

percent. We chose US Treasuries over Libor because during the recent financial turmoil (which covers

our analysis period) there were some doubts and skepticism about the quotations of some banks for the

determination of Libor rates. Moreover, since banks participating in Libor setting were downgraded and

thus had lower credit ratings than the US Government, we considered the later a better measure of a

risk-free rate.

The optimal portfolio (highest Sharpe Ratio) suggested showed an average return of 5.20% and a

standard deviation of 6.95% in 5.5 years of sample, with an allocation of 35% in US dollar and Swiss

Franc, 21.37% in Sweden Krona, and 8.63% in Euro.

Optimal Portfolio of selected currencies*

Figures in Percent except for the Sharpe Ratio

Sharpe

ratio**

Target

Return

Effective

Return

Standard

Deviation

of

Returns

EUR CHF SEK USD Total

0.3480 4.2% 4.4% 7.5% 35% 5% 25% 35% 100%

0.3480 4.4% 4.4% 7.5% 35% 5% 25% 35% 100%

0.3879 4.6% 4.6% 7.1% 35% 14% 16% 35% 100%

0.4393 4.8% 4.8% 6.7% 35% 25% 5% 35% 100%

0.4767 5.0% 5.0% 6.6% 28% 32% 5% 35% 100%

0.4817 5.20% 5.20% 6.95% 8.63% 35.00% 21.37% 35.00% 100.00%0.4579 5.4% 5.4% 7.8% 6% 35% 30% 29% 100%

0.4379 5.6% 5.6% 8.6% 11% 35% 33% 21% 100%

0.4212 5.8% 5.8% 9.4% 17% 35% 35% 13% 100%

0.4081 6.0% 6.0% 10.1% 25% 35% 35% 5% 100%

*Sample period: January 2006 to October 2011.

**The risk-free rate used was the 3-month T-bills rate for the corresponding period, i.e. 1.85%.

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The corresponding chart is shown below:

4.0%

4.5%

5.0%

5.5%

6.0%

6.5%

6.0% 7.0% 8.0% 9.0% 10.0% 11.0%

P o r t f o l i o ' s A n n u a l R e t u r n s

Portfolio's Standard Deviation of Annual Returns

Efficient Frontier

( 6.95% , 5.20% )

Portfolio performance versus Benchmark

In order to compare the performance of our strategy portfolio against the benchmark for the out-of-

sample period we constructed an index for each of the portfolio’s returns. The optimal strategy

portfolio in the out-of-sample period outperformed the benchmark, with a total return of 14% and 500

basis point over the Benchmark. Furthermore, the annual return of the strategy was 6.64% with a

Sharpe ratio of 0.63, against the 0.54 of the Benchmark.

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1-2010 04-2010 07-2010 10-2010 01-2011 04-2011 07-2011 10-2011

BMK Index

Strategy

Optimal portfolio: Trend-Following strategy performanc

Including the risk measures that will be detailed in the following sections, the next chart display the

performance of the strategy portfolio including stop-out actions when violation of risks limits occurred

(green line).

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0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1-2010 04-2010 07-2010 10-2010 01-2011 04-2011 07-2011 10-2011

BMK Index

Strategy

Strategy with stop

Optimal portfolio: Trend-Following strategy performanc

Individually, the performance for the out-sample Trend-Following strategy for each currency looks as

follow:

EuroEuro: Trend-Following strategy performance

Sharpe Ratio

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

Jan-10 Apr-10 Jul-10 Oct-10 Jan-11 Apr-11 Jul-11 Oct-11

BMK Index TF Strategy

Swiss Franc

As can be seen, most of the time the performance of the out-sample strategy outperformed the BMK

Swiss Franc: Trend-Following strategy performance

Sharpe Ratio

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50


BMK Index strategy

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Swedish Krona

Swedish Krona: Trend-Following strategy performance

Sharpe Ratio

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25


BMK index TF strategy

Risk Management

Drawdowns

One of the methods used to measure portfolio risk is the maximum drawdown, which was used in our

strategy to enhance its risk assessment aspect. Maximum drawdown measures the percentage drop in

cumulative portfolio return from a previously reached high. This metric gives a good indication of the

possible losses that the strategy portfolio can experience at any given point in time. The drawdown is

calculated according to the following formula:

W(t) – Max W(P)/Max W(P),

where W(t) is a portfolio value at time t Є [0, T], and Max W(P) is the maximum between W(t) and the

value of the portfolio at time t-1.

Months to recover, on the other hand, give a good indication of how quickly a portfolio can recover

losses. It means that at some point in time, the current maximum is identified, and then, given that

the portfolio is dropping in the drawdown, it could be found out, how long it takes for it to come again

to previously reached maximum.

For instance, one could make an investment in a strategy portfolio in January, 2010. In May, 2010, the

drawdown reaches a level of minus 7%. So, if the investor took his money back in May, 2010 that would

be his approximate loss. In order to recover, he needs to wait until the level of January, 2010 will be

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reached again. So, the drawdown model also enables to approximate the average period to recover. If

the drawdown is sharp, it should take longer to recover. The key is to understand the speed and depth

of a drawdown with the time it takes to recover these losses.

In real trading strategies, risk managers usually use the following rules regarding magnitude andduration of strategy’s drawdowns:

- all positions should be closed if 20% of drawdown is reached;

- there’s a warning for the risk manager if the level of 15% drawdown is reached;

- time to get out of a drawdown shouldn’t be longer than 1 year.

As can be seen in the graph below, the largest absolute values of our strategy’s drawdowns were

reached in May, 2010 and September, 2010, which correlates with the result that we got for the

assessment of relative-VaR. At the same time, the largest absolute drawdown value in out-of-sample

period was approximately 8.7% which, according to commonly used rules, is not a signal for closing

down our positions. The time to recovery was approximately from 6 to 9 months, which is not very

good, but still also doesn’t give us the signal to close positions. So, it could be concluded that, in our

case, relative-VaR metrics is more sensitive to strategy volatility, and drawdown metrics can be used

for identifying the most significant drops of the portfolio and estimating time to recovery.

Our drawdown assessment was carried out comparing to the benchmark’s drawdowns. And it could be

seen that the benchmark is not as risky as the strategy portfolio is and has the maximum absolute valueof drawdown = 6.3%. On, the other hand, both the strategy and the benchmark have quite long, 9 or

10-months, time to recovery (for example, for the benchmark it’s from May, 2010, till February, 2011).

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Relative VaR

Another risk management tool employed was the Relative VaR of our strategy portfolio against the

benchmark. In this way, we can assess how the portfolio may underperform the benchmark in terms of

risk and set the maximum level of risk that we want to incur with our strategy.

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

02/10 04/10 06/10 08/10 10/10 12/10 02/11 04/11 06/11 08/11 10/11

Relative value at risk over $100 invested-1 day and 0.95 confidence-

The maximum relative VaR reached for the strategy during the out-of-sample period was 140 basis

points. This means that the strategy could under-performed the benchmark by 140 basis points in one

over twenty days.

On the other hand, the relative VaR was used to determine “when” to close a position if the strategy

hit a certain relative-VaR level. We established a maximum level of relative VaR of 80 basis points for

the strategy, i.e. every time this level was overpassed, all the positions were closed. As can be seen in

the previous graph, the position was closed nine times during the out-of-sample period.

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Portfolio BenchmarkPortfolio

with stop

Returns 6.64% 4.05% 5.25%

SD 10.40% 7.32% 9.52%

Sharpe Ratio 0.632 0.544 0.545

This tool allowed us to reduce the strategy’s portfolio volatility with the cost of reducing the annual

return as well. Hence, the Sharpe ratio was also reduced, reaching the same level as the benchmark.

This risk constrains makes the strategy more versatile in the sense that it can be used for several

investors with different risk aversion profiles. Moreover, an improvement of the strategy can be done

with an optimal risk level in order to maximize the Sharpe ratio in the sample.

Conclusions

The EUR and CHF showed an outstanding performance in the out-of-sample period using the trend

following strategy. This could be because both currencies (specially the EUR) are used as “anti-dollar”

currencies, and USD has clear trends. In the case of the SEK, the underperformance could be due to the

high correlation with commodities (Oil), which present high volatility.

The portfolio approach let us compare the performance of the portfolio strategy which outperformed

the benchmark by 500 basis points, with a Sharpe ratio of 0.64 against 0.54 of the later. Furthermore,

including the stop-out rule the strategy also outperformed the benchmark.

Regarding risk management, the relative VaR was used to determine the stop-out signal, and the

drawdowns helped us to compare riskiness of the portfolios, i.e. as already mentioned the strategy

portfolio was riskier than the benchmark, and for both the time to recovery was quite long (9 or 10-

months). An alternative risk management tool that could be considered in other exercises is to set

maximal values for Drawdowns.

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Bibliography

Kauffman, Perry. The New Commodity Trading Systems and Methods. John Wiley and Sons. 1987.

Covel, Michael. Trend Following. Pearson Education. 2009.

Pardo, Robert. Design, Testing, and Optimization of Trading Systems. Wiley. 1992.

FX timing indicators. Societe Generale Cross Asset Research. April 1, 2010.

Trend Strategies in EM FX. Diversifying the carry trade. Barclays Capital Systematic Strategies. June

3,2011.

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Appendix

a. Matlab code: Trend-following strategy

%the asset has to be ordered from newest to oldest

numMA=60;

MMA=zeros(length(x)-numMA,numMA);

for i=1:numMA %Matrix with the moving averages of the selected asset

for j=1:length(x)-i+1

average=0;

for k=j:j+i-1average=average+x(k);

end

MMA(j,i)=average/i;

end

end

fd=zeros(length(x)-1,numMA); %matrix with the first difference (MA(t)-MA(t-1))

for j=1:numMA

for i=1:length(x)-j

fd(i,j)=MMA(i,j)-MMA(i+1,j);

end

end

sizefd=size(fd);

MMA_fd=zeros(sizefd(1,1),numMA); %Matrix with the moving averages of the first difference

for i=1:numMA

for j=1:sizefd(1,1)-2*i+2

average=0;

for k=j:j+i-1

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average=average+fd(k,i);

end

MMA_fd(j,i)=average/i;

end

end

sizeMMA_fd=size(MMA_fd);

sd=zeros(sizeMMA_fd(1,1),numMA); %matrix with the second difference (MMA_fd(t)-MAA_fd(t-1))

for j=1:numMA

for i=1:sizeMMA_fd(1,1)-2*j+1

sd(i,j)=MMA_fd(i,j)-MMA_fd(i+1,j);

end

end

b. Trading strategy - Signals long/short/neutral position

days = min(731,sizeMMA_fd(1,1)-2*numMA+1); %days that we are going to use the backtest

signal=zeros(min(days,sizeMMA_fd(1,1)-2*j+1),numMA);

for j=1:numMA

back=days;

for i=1:min(days,sizeMMA_fd(1,1)-2*numMA+1)

if fd(back,j)>0 & sd(back,j)>0

signal(back,j)=1;

elseif fd(back,j)<0 & sd(back,j)<0

signal(back,j)=-1;

else

signal(back,j)=0;

end

back=days-i;

end

end

%backtest P/L; creates an index with the Yield to date

back=days;

PL=zeros(min(days,sizeMMA_fd(1,1)-2*numMA+1),numMA);

initial_investment=1;

PL(back,:)=initial_investment;

for j=1:numMA

back=days;

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:=False, Transpose:=False

Sheets("Modelo MV (usd)").Select

Range("G4").Select

Selection.Copy


Cells(5 + escenarios, 31).Select

Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks _

:=False, Transpose:=False


Range("C4:C13").Select

Application.CutCopyMode = False

Selection.Copy


Range(Cells(5 + escenarios, 32), Cells(5 + escenarios, 55)).Select

Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks _

:=False, Transpose:=True

Next escenarios

Application.ScreenUpdating = True

End Sub

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