1 UPPSALA UNIVERSITY Jan 31, 2010 Department of Statistics Uppsala Bachelor’s Thesis Fall Term 2010 Advisor: Lars Forsberg FORECASTING FOREIGN EXCHANGE VOLATILITY FOR VALUE AT RISK CAN REALIZED VOLATILITY OUTPERFORM GARCH PREDICTIONS? David Fallman 1 & Jens Wirf 2 ABSTRACT In this paper we use model-free estimates of daily exchange rate volatilities employing high-frequency intraday data, known as Realized Volatility, which is then forecasted with ARMA-models and used to produce one-day-ahead Value- at-Risk predictions. The forecasting accuracy of the method is contrasted against the more widely used ARCH-models based on daily squared returns. Our results indicate that the ARCH-models tend to underestimate the Value-at- Risk in foreign exchange markets compared to models using Realized Volatility. KEYWORDS: Realized volatility, volatility forecasting, exchange rates, high- frequency data, value-at-risk. 1 [email protected]2 [email protected]
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UPPSALA UNIVERSITY Jan 31, 2010 Department of Statistics Uppsala
Bachelor’s Thesis Fall Term 2010 Advisor: Lars Forsberg
F O R E C A S T I N G F O R E I GN E X CHA N G E VO L A T I L I T Y F O R
V A L U E A T R I S K
CAN REALIZED VOLATILITY OUTPERFORM GARCH PREDICTIONS?
David Fallman1 & Jens Wirf2
ABSTRACT
In this paper we use model-free estimates of daily exchange rate volatilities employing high-frequency intraday data, known as Realized Volatility, which is then forecasted with ARMA-models and used to produce one-day-ahead Value-at-Risk predictions. The forecasting accuracy of the method is contrasted against the more widely used ARCH-models based on daily squared returns. Our results indicate that the ARCH-models tend to underestimate the Value-at-Risk in foreign exchange markets compared to models using Realized Volatility.
3. Data .................................................................................................................................................. 8
3.1 Data Construction ............................................................................................................... 8
3.2 Data Description .................................................................................................................. 9
*Statistic reported: Mean, Standard Devivation of coefficients. **Mean adjusted R-squared. ***Percentage of cases where a null hypothesis of ‘no residual autocorrelation’ is rejected at 5% significance, lag length=5. Bold font indicates highest R-Squared and Italic font is the lowest B-G result.
The second set of models is the ARCH-type models. Note here that these models
use the daily return data and not the higher frequency RV data. We have opted
to fit three different ARCH-model specifications, namely the ARCH(1), the
GARCH(1,1) as well as the EGARCH(1,1). As with the previous set of data we
evaluate to what extent these models are able to capture the information
contained within the in-sample period. Again, the shear amount estimation
outputs associated with the models means we need summarize the results. We
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present selected aspects the models coefficient stability and fit. Results are
found in Table 4.2. The diagnostics indicate, if anything, that the EGARCH model
leaves the least residual autocorrelation. The models otherwise appear very
MAE: Mean Absolute Error, MAPE: Mean Absolute Percentage Error, RMSE: Root Mean Squared Error Bold font indicates lowest error. Note that the logged and square rooted series are anti-logged and squared respectively prior to evaluation.
5 Mean Absolute Error, Mean Absolute Percentage Error and Root Mean Squared Error
17
Next, we have a look at the corresponding results for the ARCH-type models,
Table 4.4. It is clear to see that the GARCH specification most often give the top
forecasts, although the ARCH model is not far behind.
TABLE 4.4 – GARCH FORECAST EVALUATION
ARCH GARCH EGARCH NAIVE
EUR/SEK MAE 3.9E-06 4.5E-06 4.7E-06 1.1E-07
MAPE 0.4844 0.4053 0.4446 0.3506
RMSE 6.2E-06 7.9E-06 8.0E-06 4.9E-06
** *
EUR/JPY
MAE 1.1E-05 1.0E-05 1.2E-05 -3.2E-08
MAPE 0.5383 0.4470 0.5050 0.4210
RMSE 2.5E-05 2.6E-05 2.5E-05 2.0E-05
* **
EUR/GBP
MAE 2.9E-06 2.5E-06 3.3E-06 -3.3E-10
MAPE 0.3648 0.2751 0.3954 0.3788
RMSE 6.2E-06 5.9E-06 6.4E-06 5.7E-06
***
EUR/USD
MAE 4.5E-05 4.5E-05 4.6E-05 -2.2E-08
MAPE 0.7913 0.7954 0.8118 0.3837
RMSE 5.7E-05 5.6E-05 5.7E-05 2.9E-05
* ** MAE: Mean Absolute Error, MAPE: Mean Absolute Percentage Error, RMSE: Root Mean Squared Error. Bold font indicates lowest error.
We now turn our attention to comparing the alternative forecasting
frameworks. Recall that the GARCH(1,1) is widely used as the standard for
many financial practitioners and it is hence the benchmark against which the
competing models should measure up. In Table 4.5 we have documented the
improvement in forecasting errors for a selection of the top models against the
GARCH(1,1).
TABLE 4.5 – MODEL COMPARSION AGAINST GARCH
ARCH EGARCH AR-RV LOG-ARMA SQ-ARMA
EUR/SEK 13% -4% -55% 42% 49%
EUR/JPY -10% -20% 13% 18% 19%
EUR/GBP -16% -32% 0% 8% 4%
EUR/USD 0% -2% 60% 62% 60%
Statistics reported: Percentage improvement in RMSE (i.e. lower forecast error) over the benchmark GARCH(1,1) result. Bold font indicates highest improvement for respective series.
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4.3 VALUE-AT-RISK: PRACTICAL APPLICATION
We now turn to the practical application of the results attained in the paper.
Take, for instance, a portfolio with 100’000 Euro invested in each of the four
currencies. The following information is necessary to calculate the desirable
VaR:
Portfolio size, which in this case is 100’000 Euro for each currency.
Volatility forecast for given day (e.g. √ from a Log-ARMA(1,1)
which we recognized as the ideal model (see Table 4.3) for one day
ahead forecasts).
Confidence level which in this case is 99% since we will be calculating
the standard VaR (1%).
Time period of trading (we will be using one day ahead).
The distribution for each exchange rate series (see Table 3.1).
As seen in Table 3.1 all currencies pass as normally distributed except the
Japanese Yen series. Since the Yen series was not normal we proceeded with a
robustness check in the form of a Monte Carlo simulation6. However, it is
noteworthy that only the left tail of the distribution matters in VaR calculations.
Therefore, it is only necessary for the Monte Carlo values on that particular side
that need to correspond to the Gaussian distribution if you want use it. The
conclusion from the simulation is that the critical value taken from the Monte
Carlo simulation is only marginally different to the Gaussian’s 1 percent
quantile (2.33).When comparing VaR(1%) calculated with the Gaussian
distribution, the student-t distribution and with the results of a Monte Carlo
simulation we get what is seen in Graph 4.1.
6 Monte Carlo simulation is a method which calculates distributions with information from the sample. As this did not change our results we will not dwell on it, but the interested reader is referred to Jackel, P. (2002), Monte Carlo – Methods in finance, John Wiley & Sons, 1st edition, Somerset, NJ, USA.
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GRAPH 4.1 – VAR DISTRIBUTION SIMUALTION COMPARISON
We can now calculate one day ahead VAR (1%) for each series with a rolling in-
sample with one trading years observation with volatility forecasts taken from
the log-ARMA(1,1) model.
GRAPH 4.2 – 1% VAR WITH 100’000 EURO PORTFOLIO
400
800
1,200
1,600
2,000
2,400
2,800
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Monte Carlo
Gaussian distribution
T distribution with 10 d.f
EUR / JPY
1% VAR with 100 00 Euro portfolio with different distribution assumtions:
1,000
1,500
2,000
2,500
3,000
3,500
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
EUR / USD
400
600
800
1,000
1,200
1,400
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
EUR / SEK
400
800
1,200
1,600
2,000
2,400
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
EUR / JPY
500
600
700
800
900
1,000
1,100
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
EUR / BGP
1% VAR with 100 000 EUR portfolio
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This is only the probability for each series in an absolute value for each day,
remember that it is only 1% for this to actually happen a given day. As seen in
the graph (note the axis scale) above is it much safer to invest in British Pound
compared to U.S Dollar as when the actual 1-percent-day come so will the losses
for USD probably exceed GBP.
TABLE 4.6 – LOG-ARMA/GARCH VAR COMPARISON
The Log-ARMA model produces much higher and precise volatility estimates
which as seen in Table 4.6 leads to much higher VaR-values. The GARCH model
is heavily underestimates the risk which is particularly worrying since forecast
models have tendencies to predict lower values that the actual observation. So
the GARCH model will have an even higher differential against the actual VaR-
value compared to when it is matched against the Log-ARMA.
5. CONCLUSION
The objective of this paper was to document whether forecasting models based
on realized volatility could outperform those of the more widely used ARCH-
family, in the case of exchange rate series. Furthermore, we set out to analyze
the implications of the different modeling strategies in a practical application,
namely Value-at-Risk calculations.
The results of the study indicate that the RV based models consistently produce
superior forecasts to those of the ARCH-models and offer a forecasting accuracy
improvement of up to 50 percent. This seems to confirm the notion that there
may be valuable information contained within intraday price data when
forecasting short horizons. Therefore, practitioners interested in forecasting FX
rate volatility ought to regard using RV as a highly pertinent alternative to
ARCH. As for the practicality of either modeling framework, the high-frequency
data is admittedly more complicated to attain and handle. However, the
EUR/SEK +23%
EUR/JPY +3%
EUR/GBP +4%
EUR/USD +56%
Average VaR difference in percent between LOG-ARMA and GARCH with 1% VaR calculated with 100’000 Euro as portfolio. A positive number indicates that Log-ARMA has a larger VaR-value.
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convenience of using standard time series ARMA methodology should not be
overlooked. It is of course also possible that the potential gains in forecasting
precision afforded by the RV methodology alone can outweigh the cost and
inconvenience of using intraday data.
As for the Value-at-Risk, we found that the RV models generally predicated
higher volatilities and thus estimated greater values at risk. That is, the results
indicate that the frequently used ARCH models tend to seriously underestimate
the risk associated with holding a given portfolio of assets. If so, our results
bears the implication that risk managers may justifiably need to upgrade their
VaR numbers, in order to accurately portray their real risk.
On a final note, it is worth noting that the ARMA models we used to model the
RV, were the most simplistic available. For completeness and potential further
gains in predictive accuracy, higher orders and more flexible models ought to be
evaluated. In a further study, it would be interesting to see if the residual serial
dependence of the ARMA models could be regularly reduced to near zero.
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6. REFERENCES
Articles:
Andersen, T.G. & Bollerslev, T. & Diebold F.X. & Labys, P. (2001), ”The
Distribution of Realized Exchange Rate Volatility”, Journal of the American
Statistical Association, No. 96, pp. 42-55.
Andersen, T.G. & Bollerslev, T. & Diebold F.X. & Labys, P. (2000), ”Exchange rate
returns standardized by Realized Volatility are (nearly) Gaussian”,
Multinational Finance Journal, No. 4, pp. 159-179.
Andersen, T.G. & Bollerslev, T. & Diebold F.X. & Labys, P. (2002), ”Modeling and
forecasting Realized Volatility”, Econometrica, No. 71, pp. 529-626.
Frinjs, B. & Margaritis, D. (2008), “Forecasting daily volatility with intraday
data”, The European Journal of Finance, Vol. 14, No. 6, pp. 523-540.
Gardner, Jr. E.S. (2006), “Exponential smoothing: The state of Art – Part ll”,
International Journal of Forecasting, vol. 22, issue 4, pp. 637-666.
Teräsvirta, T. (2006), “An Introduction to Univariate GARCH models”, SSE/EFI
Working Papers in Economics and Finance, No. 646.
Books:
Alexander, C. (2008), Practical Financial Econometrics, 1st ed., John Wiley &
Sons Inc, Hoboken, USA.
Jäckel, P. (2002), Monte Carlo – Methods in finance, 1st ed., John Wiley & Sons
Inc, Hoboken, USA.
Philippe, J. (2007), Value At Risk: The New Benchmark for Managing Financial
Risk, 3rd ed., The McGraw-Hill Companies Inc, NY, USA.
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APPENDIX
GRAPH 7.1: OUT-OF-SAMPLE ARCH MODELS FORECAST FOR GBP
GRAPH 7.2: OUT-OF-SAMPLE ARCH MODELS FORECAST FOR JPY
.00000
.00001
.00002
.00003
.00004
.00005
.00006
.00007
.00008
25 50 75 100 125 150 175 200
Actual ( RV ) ARCH GARCHEGARCH NAIVE
EUR / BGP
.00000
.00004
.00008
.00012
.00016
.00020
.00024
.00028
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) ARCH GARCHEGARCH NAIVE
EUR / JPY
24
GRAPH 7.3: OUT-OF-SAMPLE ARCH MODELS FORECAST FOR USD
GRAPH 7.4: OUT-OF-SAMPLE ARCH MODELS FORECAST FOR SEK
.00000
.00005
.00010
.00015
.00020
.00025
.00030
25 50 75 100 125 150 175 200
Actual ( RV ) ARCH GARCHEGARCH NAIVE
EUR / USD
.00000
.00001
.00002
.00003
.00004
.00005
.00006
.00007
.00008
25 50 75 100 125 150 175 200
Actual ( RV ) ARCH GARCHEGARCH NAIVE
EUR / SEK
25
GRAPH 7.5: REALIZED VOLATILITY FORCASTS FOR USD
GRAPH 7.6: LOG REALIZED VOLATILITY FORCASTS FOR USD
.00000
.00005
.00010
.00015
.00020
.00025
.00030
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( RV )MA ( RV ) ARMA ( RV )
EUR / USD
.00000
.00005
.00010
.00015
.00020
.00025
.00030
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( lnRV )MA ( lnRV ) ARMA ( lnRV )
EUR / USD
26
GRAPH 7.6: SQUARE ROOT REALIZED VOLATILITY FORCASTS FOR USD
GRAPH 7.7: REALIZED VOLATILITY FORCASTS FOR GBP
.00000
.00005
.00010
.00015
.00020
.00025
.00030
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( sqrt RV )MA ( sqrt RV ) ARMA ( sqrt RV )
EUR / USD
.00000
.00001
.00002
.00003
.00004
.00005
.00006
.00007
.00008
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( RV )MA ( RV) ARMA ( RV )
EUR / BGP
27
GRAPH 7.8: LOG REALIZED VOLATILITY FORCASTS FOR GBP
GRAPH 7.9: SQUARE ROOT REALIZED VOLATILITY FORCASTS FOR GBP
.00000
.00001
.00002
.00003
.00004
.00005
.00006
.00007
.00008
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( lnRV )MA ( lnRV ) ARMA ( lnRV )
EUR / BGP
.00000
.00001
.00002
.00003
.00004
.00005
.00006
.00007
.00008
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( sqrt RV )MA ( sqrt RV ) ARMA ( sqrt RV )
EUR / BGP
28
GRAPH 7.10: REALIZED VOLATILITY FORCASTS FOR SEK
GRAPH 7.11: LOG REALIZED VOLATILITY FORCASTS FOR SEK
.0000000
.0000050
.0000100
.0000150
.0000200
.0000250
.0000300
.0000350
.0000400
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( RV )MA ( RV ) ARMA ( RV )
EUR / SEK
.0000000
.0000050
.0000100
.0000150
.0000200
.0000250
.0000300
.0000350
.0000400
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( lnRV )MA ( lnRV ) ARMA ( lnRV )
EUR / SEK
29
GRAPH 7.12: SQUARE ROOT REALIZED VOLATILITY FORCASTS FOR SEK
GRAPH 7.13: REALIZED VOLATILITY FORCASTS FOR JPY
.0000000
.0000050
.0000100
.0000150
.0000200
.0000250
.0000300
.0000350
.0000400
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( sqrt RV )MA ( sqrt RV ) ARMA ( sqrt RV )
EUR / SEK
.00000
.00004
.00008
.00012
.00016
.00020
.00024
.00028
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( RV )MA ( RV ) ARMA ( RV )
EUR / JPY
30
GRAPH 7.14: LOG REALIZED VOLATILITY FORCASTS FOR JPY
GRAPH 7.15: SQUARE ROOT REALIZED VOLATILITY FORCASTS FOR JPY
.00000
.00004
.00008
.00012
.00016
.00020
.00024
.00028
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( lnRV )MA ( lnRV ) ARMA ( lnRV )
EUR / JPY
.00000
.00004
.00008
.00012
.00016
.00020
.00024
.00028
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
2010
Actual ( RV ) AR ( sqrt RV )MA ( sqrt RV ) ARMA ( sqrt RV )