1. DEPARTMENT OF ECONOMICS, RENSSELAER POLYTECHNIC INSTITUTE
ARMA-GARCH Model for Financial Assets Return Estimation in
Financial Industry
GE CHEN
661401857

2. Introduction:
At beginning, the behavior of stock return is considered as a random walk following
the brownies motion [Black & Scholes,1973] and this fantastic idea lead to a great
innovation in Financial world: the Black-Scholes model. At the same time, Eugene
Fama published a groundbreaking article: the efficient market theory [Fama, 1970],
which states that all public information is calculated into a stock's current share price,
and only information that is not publicly available can benefit investors seeking to
earn abnormal returns on investments. In other words, beating the market is
impossible. Rational investors will buy undervalued assets and sell overvalued assets
at the same time, letting small arbitrage opportunities surviving in the market.
However, in the year of 1987, the “big jump” in the market firstly questioned those
theories. There is no coincidence. Same accidents occurred successively in 1990,
1997 and 2008, which is well known as the financial crisis, which implies that
investors are not always rational, especially in the downside, investors will runaway
in panic as fast as they can from the unsecured assets even though their risk premium
of those assets are undervalued. Therefore, a financial asset return is not random but
may depend on previous data or information in the market. Also, we could observe
the behavior of market return and find that the volatility cluster is not always constant，
but varies with time. The intention of this paper is to introduce a time-series model
with varying variance that could explain and predict these behaviors.
This paper will choose financial industry as target market, because during the last 10
years commercial banks were suffered a large downside price fluctuations especially
during the financial crisis.

3. 2. Data and Method
2.1 Data Source
The database contains five years weekly stock prices in financial industry from Jan 5th,
2005 to Nov. 20th, 2014. Target companies involved Bank of America (Ticker: BAC),
Capital One (Ticker: COF), Wells Fargo (Ticker: WFC), Citi (Ticker: C) and J.P
Morgan (Ticker: JPM). The Stock prices come from the Yahoo database.
To avoid the stock split effect, which sharply influences the stock return, we use the
adjusted stock close price instead of unadjusted stock close.
2.2 Data Characteristics:
Table-2.2
Variable Name Description
Year Span from 2005 to 2014
Week Span from 1 to 52 (one year)
BAC Stock price of the Bank of America
Citi Stock price of Citi bank
WFC Stock price of Wells Fargo bank
COF Stock price of Capital One
JPM Stock price of J.P Morgan
2.3 Selection:
The stock price used in the paper is weekly data instead of daily return, which has not
a continuing time of return (weekend and holiday). Jeremy Berkowitz and James
O’Brien (2002) recommended to use “large multinational institutions and meet the
Basle ‘ large trader’ criterion – with trading activity equal to at least 10 percent of
total assets or $1 billion.” [Berkowitz & O’Brien, 2002, pp1094].
2.4 Methods:
1. Portfolio
We set up a portfolio contains five stocks of large bank and invest with equal weight.
Therefore the up & down of the portfolio can be regarded as the volatility of the
industry.
2. Autocorrelation Function ACF (h) and Partial Autocorrelation Function
PACF
The correlation between two returns of the time series with a lag of the (h). The return
occurred h weeks ago have an influence on the return of this week. But this influence
will decay with the time lag. The ACF can be shown as follow:
ρ(h) = Cov(Rt ,Rt+h)
Var(Rt ) * Var(Rt+h )
The Partial Autocorrelation Function is simply defined as the coefficient of Rt-h after
regressing the return of the week Rt on Rt-1,Rt-2….,Rt-h. For the rest of the Variables,
the PACF is zero.

4. 3. ARMA-GARCH model
Autoregressive Processes AR (p)
Regress the return of this week on previous p lag of return, which can be showed as:
pΣ
Rt = c + aiRt−i +σε t
i=1
Rt represents the return of this week and Rt-I stands for the “memory” or the reaction
of investors to the previous return. εt is the noise, which can take it as the “current
information”. [Ruppert, 2011, pp208]
Moving average MA (q)
In the autoregressive process, a very large lag p is needed to fitting the AR model and
also the noise term may correlate to the entire lag of return. Hence, Moving average
model is a remedy for both mentioned above.
qΣ
Rt = ε t + θi
ε t−i
i=1
Combined AR (p) and MA(q) together, ARMA(p,q) is :
qΣ
pΣ
Rt = c + aiRt−i + θ jε t− j
j=1
+σε t
i=1
The ARMA (p,q) model assumes the volatility of the portfolio return is constant, but
actually the volatility varies by time. Hence, if we solely use the ARMA (p,q) to
predict the real portfolio return in the future, the result is not accurate. Therefore, the
time series with varying conditional variance is also needed to forecast the future.
This leading to generalized autoregressive conditional heteroskedasticity (GARCH):
σ t
nΣ
mΣ
2 =α0 + αi (σ t−1ε t−1)2 + β j
j=1
i=1
σ t− j
2
After mixing the ARMA(p,q) and GARCH(m,n) together, we could get the
ARMA(p,q)-GARCH(m,n) model as :
qΣ+σ tε t
pΣ
Rt = c + aiRt−i + θ jε t− j
j=1
i=1
σ t
σ t− j
nΣ
mΣ2 =α0 + αi (σ t−1ε t−1)2 + β j
j=1
i=1
2

5. 2.5 Summary:
Table2.5-1 Stock Price of five banks and Portfolio
Variable Obs Mean Std. Dev. Min Max
BAC 514 22.39806 13.44794 3.06 46.73
WFC 514 29.34182 8.234728 7.69 53.84
JPM 514 38.4937 8.861095 14.21 61.47
Citi 514 170.5578 174.4625 10.26 506.47
COF 514 54.59888 17.2153 7.92 84.32
Portfolio 514 63.07806 38.94875
8.628 138.908
Table 25-2 Stock Return of five banks and Portfolio
Variable Obs Mean Std. Dev. Min Max
BAC 513 -.0014251 .0774166 -.5938175 .6079168
WFC 513 .0015848 .0602823 -.3674881 .4817998
JPM 513 .0013688 .0564013 -.4167235 .3991114
Citi 513 -.0039452 .0941118 -.9263282 .7879235
COF 513 .0001979 .0682063 -.3712152 .5939554
Portfolio
513 -.0014815 .067499 -.5498683 .4876892
Return
Table-2.5-1 and Table-2.5-2 lists the variables used, their definitions and units of
measurement, and summary statistics for each variable. As we can see in the in Table-
2.5-2, the mean of the portfolio return is -.0014815, with a standard deviation of 0.067.
If we assume the stock return follows the Gaussian distribution, with 99% chance, the
minimum loss should be 1.74%. But actually the loss in real world is over 2.5%,
which is much higher than Gaussian distribution expected. Hence if we use Gaussian
distribution to estimate the potential risk of the investment, the loss is underestimated.
In Figure-2.5 we display a density graph regarding to the empirical portfolio return of
five banks. The excess Kurtosis is 24.336, showed the 10 years portfolio return has a
high peak than Gaussian distribution. Also the skewness is -0.496, which means the
real world stock return distribution is left skewed.
Figure-2.5: Empirical Portfolio Return Density

6. 2.6 Empirical Volatility cluster:
Figure-2.6: Empirical Volatility Cluster
In Figure-2.6, the volatility of the stock returns does not keep constant all the time. It
is reasonable to recognize that in calm periods, the market will have a lower volatility,
but when financial crisis falls, the volatility will be very high as shown in the figure.
The Figure-2.6 proves the theory of inconstant variance of financial assets over time
by Engle (1982) and Bollerslev (1986).
3. Result
3.1 ACF plot and Ljung-Box Test
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
1 0.9957 0.9958 512.57 0.0000
2 0.9919 0.0762 1022.3 0.0000
3 0.9881 -0.0118 1529 0.0000
4 0.9844 0.0531 2033 0.0000
5 0.9801 -0.0785 2533.5 0.0000
6 0.9758 -0.0001 3030.7 0.0000
7 0.9714 -0.0595 3524.3 0.0000
8 0.9670 -0.0021 4014.5 0.0000
9 0.9625 -0.0280 4501 0.0000
10 0.9582 0.0195 4984.1 0.0000
11 0.9536 -0.0498 5463.6 0.0000
12 0.9500 0.1156 5940.5 0.0000
13 0.9466 0.0584 6414.8 0.0000
14 0.9425 -0.1068 6886.1 0.0000
15 0.9386 0.0396 7354.3 0.0000
16 0.9336 -0.2066 7818.5 0.0000
17 0.9285 -0.0355 8278.5 0.0000
18 0.9231 -0.0635 8734.2 0.0000
19 0.9183 0.0452 9186 0.0000
20 0.9124 -0.0978 9633 0.0000
.
-1 0 1 -1 0 1
Figure-3.1-1 Portfolio Price ACF

7. LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
1 -0.1424 -0.1424 10.469 0.0012
2 0.0667 0.0473 12.768 0.0017
3 -0.0985 -0.0846 17.789 0.0005
4 0.0815 0.0558 21.236 0.0003
5 0.0532 0.0828 22.706 0.0004
6 0.0438 0.0491 23.708 0.0006
7 -0.0663 -0.0512 26.006 0.0005
8 0.0562 0.0453 27.658 0.0005
9 -0.1108 -0.0996 34.087 0.0001
10 0.1068 0.0580 40.079 0.0000
11 -0.1453 -0.1125 51.186 0.0000
12 -0.1031 -0.1686 56.789 0.0000
13 0.1537 0.1718 69.271 0.0000
14 -0.0327 -0.0071 69.837 0.0000
15 0.2603 0.2706 105.79 0.0000
16 -0.1418 -0.0442 116.47 0.0000
17 0.0259 0.0053 116.83 0.0000
18 -0.0654 -0.0713 119.12 0.0000
19 0.1341 0.0773 128.74 0.0000
20 0.0181 0.0153 128.92 0.0000
-1 0 1 -1 0 1
Figure-3.1-2 Portfolio Return ACF
After ploting the (Autocorrelation Function), we can find in Figure-3.1-1, the asset
price decays to zero very slowly. With the Ljung-Box test, the Q-statistics are very
high during the first 20 lags, which means the null hypothesis is rejected and ρ(1),
ρ(2)…,ρ(20) ≠0. The price of the asset is hard to predict since we hardly fingure out
whether it is stationary or not. However, when testing the Portfolio return (Figure-3.1-
2), the ACF decays quickly to zero and proves that the differenced series is stationary.
[Ruppert, 2011, pp207].
The Augumented Dickey-Fuller (DF) test is also implemented to test the unit root of
the Portfolio Return. The DF with trend test statistic is -26.088 and without trend is -
26.067 which are much lower than 1% critical value of DF test(-3.96% with trend, -
3.43 without trend). The null hypothesis that the time series variable have a unit root
is rejected.
3.2 Akaike information criterion (AIC) and Bayesian information criterion (BIC)
Table-3.2 AIC and BIC for ARMA-GARCH
model with different lags
ARMA(p, q)-GARCH(m,n) AIC BIC
ARMA(1,1)-GARCH(1,1) -1878.3 -1852.9
ARMA(1,1)-GARCH(1,2) -1862.7 -1837.3
ARMA(1,1)-GARCH(2,1) -1872.3 -1846.8
ARMA(1,1)-GARCH(2,2) -1821.5 -1796.1
ARMA(1,2)-GARCH(1,1) -1876.6 -1851.1
ARMA(1,1)-GARCH(1,2) -1859.9 -1834.4
ARMA(1,2)-GARCH(2,1) -1870.8 -1845.3
ARMA(1,2)-GARCH(2,2) -1819.3 -1793.8
ARMA(2,1)-GARCH(1,1) -1876.6 -1851.1
ARMA(2,1)-GARCH(1,2) -1860.1 -1834.6
ARMA(2,1)-GARCH(2,1) -1870.8 -1845.4
ARMA(2,1)-GARCH(2,2) -1819.3 -1793.8
ARMA(2,2)-GARCH(1,1) -1875.9 -1850.4
ARMA(2,2)-GARCH(1,2) -1860.1 -1834.6
ARMA(2,2)-GARCH(2,1) -1868.9 -1843.5
ARMA(2,2)-GARCH(2,2) -1819.2 -1793.8

8. When comparing model fits, we iterated check diffrerent lags combination and select
the minium of the BIC and AIC as best fit for the model, shown in Table-3.2. Both
AIC and BIC recommended the one lag model, ARMA(1,1) GARCH(1,1) is a best fit
for the portfolio return of five banks.
3.3 “Break” test when using AR (1) and ARMA (1,1)
Figure-3.3-1: QLR-Statistic of AR (1)
The AR(1) model has a “break” in forth week of 2009, as shown in Figure-3.3-1. The
null hypothesis that no “break” existed in the model is rejected by significant level of
10% and 5%, (QLR-statistic 5.00 and 5.86 separately). In the time span 2008 and
2009.
Figure-3.3-2: QLR-statistic ARMA (1,1) GARCH (1,1)

9. When testing ARMA (1,1) GARCH (1,1) with the empirical return as raw data, the
maximum QLR-statistic is 5.08 in the year of 2011, 47th week. The model is rejected
in the 10% critic test and accepted in the level of 5%. Therefore, the ARMA (1,1) –
GARCH (1,1) model is more stable than AR (1).
3.4 ARMA (1,1)-GARCH (1,1) parameter fitting for Stock Return
Table-3.4 Result of AR(1), ARMA(1,1) and ARMA (1,1)-GARCH (1,1)
model on Stock Return of five banks
Stock Return (1)AR(1) (2)ARMA(1,1) (3)ARMA(1,1)-
GARCH(1,1)
Intercept -0.00166 -0.00148 0.00204*
(0.00386) (0.00317) (0.000810)
AR(1) -0.142 -0.774*** 0.922***
(0.137) (0.0669) (0.207)
MA(1) 0.665*** -0.950***
(0.0727) (0.167)
ARCH(1) 0.177***
(0.0392)
GARCH(1) 0.818***
(0.0325)
GARCH Intercept 0.0000319
(0.0000167)
Sigma 0.0665***
(0.000756)
Statistic
Number of Obs. 513 513 513
Chi square F (1.085) 269.44 191.0
AIC -1313.9 -1317.8 -1878.3
BIC -1305.4 -1300.9 -1852.0
Log
pseudolikelihood
662.9191 945.1744
These regressions were estimated using (1) AR(1), (2)ARMA(1,1) (3) ARMA (1,1)-
GARCH(1,1) model on Portfolio Returns involving 5 banks. Standard errors are given
in parentheses under coefficients. The individual coefficient is statistically significant
at * 5%, **1% or ***0.1% significant level using a two-sided test.
Table-3.4 calculates the coefficients using AR (1), ARMA (1,1) and ARMA (1,1)
GARCH (1,1) model by five major banks data in U.S.. The AR(1) model with
portfolio data has a low F-statistic and the coefficient is not significant at level of 5%.
AR (1) is not appropriate for portfolio data. Both coefficients of ARMA (1,1), and
ARMA(1,1)-GARCH(1,1) are significant different from zero, when test with 0.1%
significant level. But comparing with ARMA (1,1) model, ARMA-GARCH model’s
pseudo likelihood value is larger. In other words, the ARMA (1,1)-GARCH (1,1)
model is more close to the population one. Also the non-zero coefficients ARCH and
GARCH present that the raw data does have a volatility cluster, which means the
volatility of the raw data is not constant during the whole period. But the ARMA (1,1)
model always assumes the volatility of data is constant during the whole period.
Hence, ARMA(1,1)-GARCH(1,1) model is the best model fitting the raw data among
all these three autoregressive models.

10. 4. BackTests
Table-4 Result of ARMA (1,1)-GARCH (1,1) model on Stock
Return of five banks in period (2010w47 to 2013w46)
Stock Return
Intercept 0.00446
(0.00249)
AR(1) -0.682
(2.465)
MA(1) 0.668
(2.507)
ARCH(1) 0.131*
(0.0559)
GARCH(1) 0.850***
(0.0595)
GARCH Intercept 0.0000234
(0.0000427)
Statistic
Number of Obs. 157
Chi square 0.22
AIC -605.5
BIC -587.2
Log pseudolikelihood 308.7463
These regressions were estimated using ARMA (1,1)-GARCH (1,1) model on
Portfolio Returns involving 5 banks in period (2010w47 to 2013w46). Standard errors
are given in parentheses under coefficients. The individual coefficient is statistically
significant at * 5%, **1% or ***0.1% significant level using a two-sided test.
The Figure-4-1 and Figure 4-2 below shows a 1000 times simulation by using the
period span 46th week of 2010 and 46th week 2013 to forecast the period span 47th
week 2013 and 46th week 2014. The coefficient is showed in Table-4. Also the
empirical return at the same period is presented in Figure-4-3 and Figure 4-4. In
Figure-4-2, the simulation shows a volatile return with some extremely values. Also
the density figure in Figure-4.3 shows a high peak distribution with a value of excess
25.25 on kurtosis and -0.3158 on skewness. After reducing the amount of estimated
observations without affecting the original distribution, the excess kurtosis reduces to
8.47 with skewness on -1.45. The empirical distribution has much lower kurtosis
Figure-4-1:Forecast Portfolio Return with 1000 times simulations

11. (2.634) and skewness (-2.44) , as we can find in the Figure-4-3 and Figure 4-4.
Figure-4-2: Distribution of Forecast Return with 1000 times
simulations
Figure-4-3: Distribution of Empirical Return
(2013w47 to 2014w46)
Figure-4-4: Empirical Return (2013w47 to
2014w46

12. The mean return of the forecast is 0.0027 with average standard deviation
of 0.0540. The prediction interval is [-0.0613 0.0315], which is very large.
But actually, the volatility of the forecast data is varying for the whole
period. In term of high volatility, the interval will be widened and in the
period of low volatility, the interval will be narrowed. Comparing with
the forecast data, the empirical return during the period of 2013, 47th
week to 2014, 46th week, has a mean of 0.00228, which is little lower
than the forecast data mean. At same time, the average volatility of
empirical t data is 0.0215, which is a little higher than forecast average
volatility 0.0189.
5.Conclusion
This study presented a research using ARMA-GARCH model to forecast the financial
bank return and volatility. ARMA-GARCH is a better model to estimate the future
return and volatility than AR or ARMA model, since the latter assumes the volatility
of the portfolio return is constant. ARMA-GARCH assumes the volatility is also
varying with the time and also the error, which can be called “new information” in the
market. As we can see in both Table-3.4 and Table-4, the coefficient of one lag
GARCH is much large than ARCH part. In real world, this effect can be explained
that commercial industry investor’s behaviors are sensitive to most “recently news”,
rather than memory of the past volatility.
The study also compared the stability of ARMA (1,1)-GARCH (1,1) model and AR
(1) model. The result is ARMA-GARCH model for financial industry is more stable
than AR(1), because in financial crisis(2008-2009), investors pay a high attention on
market’s “new information”, especially for investors on the equity of bank. Hence, the
autoregressive model is not reasonable to use in this kind of special period.
The fact that AR (1) and MA (1) coefficients are not significant from zero may lead to
our estimate bias. The coefficients probably are not estimate precisely because of
small numbers of samples. We also add more samples in our estimation, but the mean
is more deviated from empirical one since the samples of crisis are involving.

13. RENFERENCES
Jansky,I,M, 2012, Value-at-risk Forecasting with ARMA-GARCH family of models
during the recent financial crisis, Charles University in Prague
BerKowitz,J & O’Brien,J, 2002, How accurate are Value-at-Risk Models at
Commercial Bank, The Journal of Finance Vol. LVII, NO.3, pp1039-1111
Rachev,T,Svetlozar & Menn,C & Fabozzi,2005, J, F, Fat-Tailed and Skewed Asset
Return Distribution, Jon Wiley & Sons,Inc, ISBN13 9780471718864, pp121-140
Gang,J, Advanced STATA with Time Series Data 1, Beihang University
Ruppert,2010, Statistics and Data Analysis for Financial Engineering. Springer Texts
inStatistics. Springer, 2010. ISBN 9781441977861, pp201~pp248 & pp477~pp500
Baum,F,C,2013,Time series estimation and forecasting, Boston Colledge & DIW
Berlin & University of Mauritius
Stock,H,J & Watson,W,M, 2013, Introduction to Econometrics 3rd ed, Addison-
Wsley,ISBN-13:9780138009007,pp516~pp583