Combining Economic Fundamentals to Predict Exchange Rates
final
1. Factor-Based GTAA Strategy
Report
MATH G4078 Final Project
Working Group:
Fengqiao Chen (fc2530) Peijun Dai (pd2485)
Yicheng Wang (yw2761) Yichao Xue (yx2311)
May 6, 2016
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2. 1 Abstract
In this project, we will create and back-test a Factor-Based Long-Short GTAA strategy
with 10% risk and monthly rebalancing. Our ultimate goal is to present a strategy where
there is no exposure to global equity market and exposure to 10 % risk. Our investment
universe is comprised of 10 developed country equity index futures with 10 years of
monthly data. We constructed three 2-factor-mimicking portfolio for the equity universe,
and we found that the model with Volatility and Value factor performed the best.
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3. 2 Data Preparation
2.1 Investment Universe
The first step in this project is to determine the investment universe. As described in
the abstract above, the chosen investment universe is comprised of Investment universe
is comprised of 10 developed country equity index futures from MSCI. The ten countries
are: Australia, Canada, Germany, Spain, France, UK, Italy, Japan, Netherlands, USA.
2.2 Data Transformation and Explanations
The returns to value and momentum strategies have become quintessential to the market
efficiency debate. 1
Based on the Asness, Tobias and Lasse’s conclusion, it is therefore
natural for us to employ value and momentum to build a two factor-mimicking portfolios
for the equity universe.
Our basic scheme of value premium is associated with investors bias to over-extrapolate
past earnings growth and higher risk of value equities. A momentum factor is important
to our project as it is one of the most investigated and persistent effects. Another factor
volatility is included to make pair combination testing. For the weight of volatility factor,
we decide to long asset with low volatility and short asset with high volatility. In order to
identify the underlying outliers, we transformed the data through this formula below, this
formula make sure that the sum of the new weight will be zero, the rank in the formula
is constructed based on the E/P ratio of the asset: the smaller E/P Ratio, the higher
Ranking. Also we can use the adjusting factor Ct to adjust the long-short strategy :
WSignal
t,i = Ct[Rank(Signalt,i) −
1
10
10
j=1
Rank(Signalt,j)].
For the factors like momentum and volatility, we do not need to identify the outliers, we
transformed as this way:
WSignal
t,i = Ct[(Signalt,i) −
1
10
10
j=1
(Signalt,j)].
1. Asness, Clifford S., Tobias J. Moskowitz, and Lasse Heje Pedersen. ”Value and Momentum Every-
where.” The Journal of Finance 68, no. 3 (2013): 929-85. doi:10.1111/jofi.12021.
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4. For the three factors, we get the new transformed weight as below:
WValue
t,i = +{Rank(Fept,i) −
1
10
10
j=1
Rank(Fept,j)},
WMomentum
t,i = +{(Momt,i) −
1
10
10
j=1
(Momt,j)},
WVolatility
t,j = −{(Volt,i) −
1
10
10
j=1
(Volt,j)}.
After the transformation, for value factor: we tend to short the high ranking asset
(Low E/P Ratio) and long the low raning asset (High E/P Ratio); for the momentum
factor, we tend to short the Low momentum asset and long high momentum asset; for
the volatility factor, we tend to short high volatility asset and long low volatility asset.
Rather than optimize the portfolio to build the best possible complex model of future
volatility, or to time volatility, we decide to scale the portfolios, using a simple ex-ante
measure, to roughly constant volatility (10 %). The return on the portfolio could be
calculated by using the ranks and weights of the signals as we mentioned above. We take
the return on a 50/50 equal combination of every pair of 3 factors into consideration.
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5. 3 Risk Model
3.1 Risk Estimation
We choose the 36-month rolling window of returns to create monthly covariance matrix,
use a rule of thumb and multiply each element of the matrix by 12 to annualize the
covariance. The following formula represents the element on the i, j unit of the whole
covariance matrix Σ
σm
ij =
1
36
m
t=m−35
(Rt,i − ¯Ri)(Rt,j − ¯Rj) ∗ 12 .
The risk of each portfolio is calculated by its weight and the covariance matrix:
Ex-ante risk =
√
wT Σw,
then scale to the target risk:
wnew
t = wt ∗
Target Risk
Ex-ante risk
The following graphics show the holdings for each portfolio in terms of different factors.
Figure 1: Value Factor
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Value Factor Holdings
AUD CAD DEM ESP FRF GBP ITL JPY NLG USD
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7. 3.2 Discussion of the client zero-equity-beta constraint
To accommodate the request of zero equity beta from the client, we orthogonalized factor
portfolios with respect to some liquid proxy for the global equity market. In our case, we
use the equal-weighted portfolio of 10 country MSCI futures index as the proxy. Since
there are 10 countries’ futures index, we set the equal weight as 0.1. The orthogonalization
is shown below.
Orthogonalized Factori,t = Original Factori,t − Proxy Weight × ˆβt
Where i represents the i − th asset and t represent the t month, and
ˆβt =
Cov(Original, Proxy)
V ar(Proxy)
=
(wnew
t )T
Σ(wProxy
t )
(wProxy)T Σ(wProxy
t )
The following graphics show the β for each portfolio:
Figure 4: Betas
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Equity Betas
Value Momentum Vol
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8. 4 Conclusion
The following two tables are the required statistics we need. Value Factor, Momentum
Factor and Volatility Factor are correspondent to 1,2,3 indicators (i.e. Combined 1&
2 represents the combination of Value Factor and Momentum Factor). The Realized
correlation in Table represents the correlation between each portfolio and equal weighted
global equity index (MSCI). According to the Table Results, we can see that Combined
Factors Portfolio (Value and Volatility Factor) performed best among all portfolios. Its
annualized return is 7.24% with 13.01% drawdown. The Return to Risk Ratio is 0.68.
Table 1
Value factor Momentum factor
Annulized Return 0.58% 0.21%
Annulized realized risk 1.05% 1.10%
Average drawdown 0.44% 1.04%
Return to risk ratio 0.55 0.19
Realized correlation 0.19 -0.20
Table 2
Combined 1 & 2 Combined 1& 3 Combined 2 & 3
Annulized Return 3.63% 7.24% 4.88%
Annulized realized risk 10.93% 10.64% 10.74%
Average drawdown 24.29% 13.01% 10.19%
Return to risk ratio 0.33 0.68 0.45
Realized correlation -0.03 0.15 0.03
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13. ,file="correaltion.xlsx")
# write.xlsx(covar,file="covar.xlsx")
combined.weight <- matrix(NA,314,10)
for (i in 1:314){
a <- 0.5*(value.new[i,]-0.1*value.beta[i]+momentum.new[i,]-0.1*momentum.beta[i])
b <- sqrt(a%*%(cov(returns[(i+1):(i+36),])*35/36*12)%*%a)
m <- 0.1/b
combined.weight[i,] <- as.numeric(m)*a
}
write.xlsx(combined.weight,file="combined weight.xlsx")
combined.weight1 <- matrix(NA,314,10)
for (i in 1:314){
a <- 0.5*(value.new[i,]-0.1*value.beta[i]+vol.new[i,]-0.1*vol.beta[i])
b <- sqrt(a%*%(cov(returns[(i+1):(i+36),])*35/36*12)%*%a)
m <- 0.1/b
combined.weight1[i,] <- as.numeric(m)*a
}
write.xlsx(combined.weight1,file="combined weight value vol.xlsx")
combined.weight2 <- matrix(NA,314,10)
for (i in 1:314){
a <- 0.5*(momentum.new[i,]-0.1*momentum.beta[i]+vol.new[i,]-0.1*vol.beta[i])
b <- sqrt(a%*%(cov(returns[(i+1):(i+36),])*35/36*12)%*%a)
m <- 0.1/b
combined.weight2[i,] <- as.numeric(m)*a
}
write.xlsx(combined.weight2,file="combined weight momentom vol.xlsx")
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