1. Forecasting FX Rates Fundamental and Technical Models

2. Forecasting Exchange Rates Model Needed A forecast needs a model, which specifies a function for S t : S t = f (X t ) The model can be based on - Economic Theory (say, PPP: X t = (I d,t - I f,t )  f (X t ) = I d,t - I f,t ) - Technical Analysis (say, past trends) - Statistics - Experience of forecaster - Combination of all of the above

3. Forecasting: Basics A forecast is an expectation –i.e., what we expect on average: E t [S t+T ]  Expectation of S t+T taken at time t. It is easier to predict changes. We will concentrate on E t [s t+T ]. Note: From E t [s t,t+1 ], we get E t [S t+T ]  E t [S t+T ] = S t x (1+E t [s t+1 ]) Based on a model for S t, we are able to generate E t [S t+T ]: S t = f (X t )  E t [S t+T ] = E t [f (X t+T )] Assumptions needed for X t+T Today, we do not know X t+T. We will make assumptions to get X t+T. Example: X t+T = h (Z t ),-Z t : data available today. => We’ll use Z t to forecast the future S t+T : E t [S t+T ] = g(Z t )

4. Example: What is g(Z t ) ? Suppose we are interest in forecasting USD/GBP changes using PPP: 1. Model for S t E t [s t+1 ] = s F t+1 = (S F t+1 /S t ) - 1  I d,t+1 - I f,t+1 Now, once we have s F t+1 we can forecast the level S t+1 E t [S t+1 ] = S t x [1 + s F t+1 ] = S t x [1 + (I US,t+1 - I UK,t+1 )] 2. Assumption for I t+1 => I t+1 = h(Z t ) - I US,t =  US 0 +  US 1 I US,t-1 - I UK,t =  UK 0 +  UK 1 I UK,t-1 3. E t [S t+1 ] = g(Z t ) - E t [S t+1 ] = g(I US,t-1, I UK,t-1 ) = S t x [1 +  US 0 +  US 1 I US,t-1 -  UK 0 -  UK 1 I UK,t-1 )

5. There are two forecasts: in-sample and out-of-sample. - In-sample: it uses sample info to forecast sample values. Not really forecasting, it can be used to evaluate the fit of a model. - Out-of-sample: it uses the sample info to forecast values outside the sample. In time series, it forecasts into the future. Two Pure Approaches to Forecasting Based on how we select the “driving” variables X t we have different forecasting approaches: - Fundamental (based on data considered fundamental). - Technical analysis (based on data that incorporates only past prices).

6. Fundamental Approach Economic Model We generate E t [S t+T ] = E t [f(X t+T )] = g(X t ), where X t is a dataset regarded as fundamental economic variables: - GNP growth rate, - Current Account, - Interest rates, - Inflation rates, etc. Fundamental variables: Taken from economic models (PPP, IFE, etc.)  the economic model says how the fundamental data relates to S t. That is, the economic model specifies f(X t ) -for PPP, f(X t ) = I d,t - I f,t

7. The economic model usually incorporates: - Statistical characteristics of the data (seasonality, etc.) - Experience of the forecaster (what info to use, lags, etc.)  Mixture of art and science.

8. Fundamental Forecasting: Steps (1) Selection of Model (say, PPP model) used to generate the forecasts. (2) Collection of S t, X t (for PPP: exchange rates and CPI data needed.) (3) Estimation of model, if needed (regression, other methods) (4) Generation of forecasts based on estimated model. Assumptions about X t+T may be needed. (5) Evaluation. Forecasts are evaluated. If forecasts are very bad, model must be changed.  MSE (Mean Square Error) and MAE (Mean Absolute Error) are measures used to asses forecasting models.

9. Fundamental Forecasting: Process Model Data Estimation Forecast Evaluation Modify/Change Model Test Model Theory Pass? Practice

10. Estimation Period Out-of- Sample Forecasts Validation Forecasts Fundamental Forecasting: Usual Estimation Process 1) Select a (long) part of the sample to select a model and estimate the parameters of the selected model. (You get in-sample forecasts.) 2) Keep a (short) part of the sample to check the model’s forecasting skills. This is the validation step. You can calculate true MSE or MAE 3) Forecast out-of-sample.

11. Example: In-sample PPP forecasting of USD/GBP PPP equation for USD/GBP changes: E t [s t+1 ] = s F t+1  I US,t+1 - I UK,t+1 => E t [S t+1 ] = S F t+1 = S t x [1+ s F t+1 ] Data: Quarterly CPI series for U.S. and U.K. from 1996:1 to 1997:3. US-CPI: 149.4, 150.2, 151.3 UK-CPI: 167.4, 170.0, 170.4 S 1996:1 = 1.5262 USD/GBP. S 1996:2 = 1.5529 USD/GBP. 1. Forecast S F 1996:2 I US,1996:2 = (USCPI 1996:2 /USCPI 1996:1 ) - 1 = (150.2/149.4) - 1 =.00535. I UK,1996:2 = (UKCPI 1996:2 /UKCPI 1996:1 ) - 1 = (170.0/167.4) - 1 =.01553 s F 1996:2 = I US,1996:2 - I UK,1996:2 =.00535 -.01553 = -.01018. S F 1996:2 = S F 1996:1 x [1 + s F 1996:2 ] = 1.5262 USD/GBP x [1 + (-.01018)] = = 1.51066 USD/GBP.

12. Example (continuation): S F 1996:2 = 1.51066 USD/GBP. 2. Forecast evaluation (Forecast error: S F 1996:2 -S 1996:2 )  1996:2 = S F 1996:2 -S 1996:2 = 1.51066 – 1.5529 = -0.0422. For the whole sample: MSE: [(-0.0422) 2 + (-0.0471) 2 +.... + (-0.1259) 2 ]/6 = 0.017063278

13. Note: Not a true forecasting model. Example: Out-of-sample Forecast: E t [S t+T ] Simple forecasting model: Naive forecast (E t [I t+1 ]=I t ) E t [s t+1 ] = s F t+1 = (E t [S t+1 ]/S t ) – 1  I d,t - I f,t. Using the above information we can predict S 1996:3 : 1. Forecast S F 1996:3 s F 1996:3 = I US,1996:2 - I UK,1996:2 =.00535 -.01553 = -.01018. S F 1996:3 = S 1996:2 x [1 + s F 1996:3 ] = 1.5529 x [1 + (-.01018)] = 1.53709 2. Forecast evaluation  1996:3 = S F 1996:3 -S 1996:3 = 1.53709 – 1.5653 = -0.028210.

14. More sophisticated out-of-sample forecasts can be achieved by estimating regression models, survey data on expectations of inflation, etc. For example, consider the following regression model: I US,t =  US 0 +  US 1 I US,t-1 +  US.t. I UK,t =  UK 0 +  UK 1 I UK,t-1 +  UK,t. Suppose we estimate both equations. The estimated coefficients (a’s) are: a US 0 =.0036, a US 1 =.64, a UK 0 =.0069, and a UK 1 =.43. Therefore, I F US,1996:3 =.0036 +.64 x (.00535) =.007024 I F UK,1996:3 =.0069 +.43 x (.01553) =.013578. s F 1996:3 = I F US,1996:3 - I F UK,1996:3 =.007024 - 013578 = -.00655. S F 1996:3 = 1.5529 USD/GBP x [1 + (-.00655)] = 1.5427 USD/GBP.  1996:3 = S F 1996:3 -S 1996:3 = 1.5427 – 1.5653 = -0.0226.

15. Example: Exchange Rate Forecasts US Excel Regression Results: How much variability of Y t is explained by X t t-stat tests H o : a i =0 t a1 =a 1 /SE(a 1 )= 0.640707/0.071274=8.9893 SUMMARY OUTPUT Regression Statistics Multiple R0.629674 R Square0.396489 Adjusted R Square0.391583 Standard Error0.006811 Observations125 ANOVA dfSSMSFSignificance F Regression10.003748 80.807523.66E-15 Residual1230.0057054.64E-05 Total1240.009454 CoefficientsStandard Errort StatP-value Intercept0.003660.0009233.9658670.000123 X Variable 10.6407070.0712748.98933.66E-15

16. Example (continuation): Inflation Forecasts - UK Regression Results: SUMMARY OUTPUT Regression Statistics Multiple R0.425407 R Square0.180971 Adjusted R Square0.174312 Standard Error0.011305 Observations125 ANOVA dfSSMSFSignificance F Regression10.003473 27.177847.6E-07 Residual1230.0157190.000128 Total1240.019192 CoefficientsStandard Errort StatP-value Intercept0.0069180.0014034.9326372.57E-06 X Variable 10.4281320.0821245.2132377.6E-07 I UK,t-1 explains 18.10% of the variability of I UK,t t-stat is significant at the 5% level (|t|>1.96) => Lagged Inflation explains current Inflation

17. Example: Out-of-sample Forecasting FX with an Ad-hoc Model Forecast monthly MYR/USD changes with the following model: s MYR/USD,t = a 0 + a 1 (I MYR – I USD ) t + a 2 (y MYR – y USD ) t + ε t Excel Regression Results: SUMMARY OUTPUT Regression Statistics Multiple R0.092703 R Square0.018594 Adjusted R Square-0.0087 Standard Error0.051729 Observations112 ANOVA dfSSMSFSignificance F Regression20.0025280.0012640.472420.624762 Residual1090.2916660.002676 Total1110.294195 CoefficientsStandard Errort StatP-value Intercept0.0069340.0051751.3399030.180277 X Variable 1 (I MYR – I USD ) t 0.2159270.1058242.0404350.041307 X Variable 2 (y MYR – y USD ) t 0.0915920.0516761.7724280.076326 X t explains 1.86% of the variability of s t t-stat tests H o : a i =0 t a1 =a 1 /SE(a 1 )= 0.215927/0.105824=2.040435

18. Example (continuation): Out-of-sample Forecasting w/Ad-hoc Model s MYR/USD,t = a 0 + a 1 (I MYR – I USD ) t + a 2 (y MYR – y USD ) t + ε t 0. Model Evaluation Estimated coefficient: a 0 =.0069, a 1 =.2159, and a 2 =.0915. t-stats: t a1 =|2.040435|>1.96 (reject H 0 ) ; t a2 =|1. 772428|<1.96 (can’t reject H 0 ) Do the signs make sense? a 1 =.2159 > 0 => PPP a 2 =.0915 > 0 => Trade Balance 1. Forecast S F t+1 E[s MYR/USD,t ] =.0069 +.2159 (I MYR – I USD ) t +.0915 (y MYR – y USD ) t Forecasts for next month (t+1): E t [INF t+1 ]= 3% and E t [INC t+1 ]= 2%. E t [s MYR/USD,t+1 ] =.0069 +.2159 x (.03) +.09157 x (.02) =.0152. The MYR is predicted to depreciate 1.52% against the USD next month.

19. Example (continuation): Out-of-sample Forecasting w/Ad-hoc Model 1. Forecast S F t+1 (continuation) E t [s MYR/USD,t+1 ] =.0152. Suppose S t =3.1021 MYR/USD S F t+1 = 3.1021 USD/MYR x (1+.0152) = 3.1493 USD/MYR. 2. Forecast evaluation Suppose S t+1 = 3.0670  t+1 = S F t+1 -S t+1 = 3.1493 - 3.0670 = 0.0823. ¶

20. Practical Issues in Fundamental Forecasting Issues: - Are we using the "right model?" - Estimation of the model. - Some explanatory variables (Z t+T ) are contemporaneous. => We also need a model to forecast the Z t+T variables. Does Forecasting Work? RW models beat structural (and other) models: Lower MSE, MAE. Richard Levich compared forecasting services to the free forward rate. He found that forecasting services may have some ability to predict direction (appreciation or depreciation). For some investors, the direction is what really matters, not the error.

21. Example: Two forecasts: Forward Rate and Forecasting Service (FS) F t,1-month =.7335 USD/CAD E FS,t [S,t+1-month ]=.7342 USD/CAD. (Sternin’s stragey: buy CAD forward if FS forecasts CAD appreciation.) Based on the FS forecast, Ms. Sternin decides to buy CAD forward at. (A) Suppose that the CAD appreciates to.7390 USD/CAD. MAE FS =.7390 -.7342 =.0052 USD/CAD. Sternin makes a profit of.7390 -.7335 =.055 USD/CAD. (B) Suppose that the CAD depreciates to.7315 USD/CAD. MAE FS = .7315 -.7342  =.0027 USD/CAD. (smaller!) Sternin takes a loss of.7315 -.7335 = -.0020 USD/CAD. ¶

22. Technical Analysis Approach Based on a small set of the available data: past price information. TA looks for the repetition of specific price patterns.  Discovering these patterns is an art, not a science. TA attempts to generate signals: trends and turning points. Popular models: - Moving Averages (MA) - Filters - Momentum indicators.

23. (1) MA models The goal of MA models is to smooth the erratic daily swings of FX to signal major trends. The double MA system uses two MA: Long-run MA and Short-run MA. LRMA will always lag a SRMA (gives smaller weights to recent S t ). Buy FC signal: When SRMA crosses LRMA from below. Sell FC signal: When SRMA crosses LRMA from above. Buy FC signal

24. Example: S t (USD/GBP) Double MA (red=30 days; green=150 days)

25. (2) Filter models The filter, X, is a percentage that helps a trader forecasts a trend. Buy signal: when S t rises X% above its most recent trough. Sell signal: when S t falls X% below the previous peak. Idea: When S t reaches a peak  Sell FC When S t reaches a trough  Buy FC. Key: Identifying the peak or trough. We use the filter to do it: When S t moves X% above (below) its most recent peak (trough), we have a trading signal.

26. Example: X = 1%, S t (CHF/USD) Peak = 1.486 CHF/USD (X = CHF.01486)  When S t crosses 1.47114 CHF/USD, Sell USD Trough = 1.349 CHF/USD (X = CHF.01349)  When S t crosses 1.36249 CHF/USD, Buy USD

27. Example: X = 1%, S t (CHF/USD) Peak = 1.486 CHF/USD (X = CHF.01486)  When S t crosses 1.47114 CHF/USD, Sell USD 1.47114 Sell USD signal 1.36249 Peak = 1.486 CHF/USD

28. (3) Momentum models They determine the strength of an asset by examining the change in velocity of asset prices’ movements. We are looking at the second derivative (a change in the slope). Buy signal: When S t climbs at increasing speed. Sell signal: When S t decreases at increasing speed. (USD/GBP) time StSt Buy GBP signal

29. TA Summary: TA models monitor the derivative (slope) of a time series graph. Signals are generated when the slope varies significantly. Technical Approach: Evidence RW model: seems to be a very good forecasting model. Many economists have a negative view of TA:  TA runs against market efficiency (see RC). Informal empirical evidence for TA: The marketplace is full of newsletters and consultants selling technical analysis forecasts and predictions. Some academic support: Bilson points out that linear comparisons -i.e., based on correlations- are meaningless, since TA rely on non-linearities. Bilson, based on non-linear models, finds weak support for TA.

30. Exchange Rate Volatility Many agents are interested in forecasting the variance of returns. Models where variance is important:CAPM Black-Scholes Value-at-Risk (VaR) The assumption of a constant variance is called homoscedasticity. Variances of asset returns, however, tend to be time-varying:  returns of financial assets are heteroscedastic. S t is quite volatile in some periods, while during other periods S t does not move very much. Robert Engle won a Nobel Prize modeling volatility.

31. Autoregressive Conditional Heteroscedasticity (ARCH) Engle introduced the ARCH(q) model: σ 2 t = α 0 + Σ i q α i e 2 t-i. The variance at time t depends on past conditional errors. Engle's model incorporates two empirical facts observed in returns: (1)large (small) changes in returns are likely to be followed by large (small) changes in returns. When e t-1 is big,  σ 2 t tends to be big. (volatility clustering). (2) The unconditional distribution of financial assets' returns has thicker (fatter) tails than the normal distribution. Example: Volatility clustering and fat tails is observed in time series plots and histograms of the USD/GBP. ¶

33. The GARCH(q,p) model: σ 2 t = α 0 + Σ i q α i e 2 t-i + Σ i p ß i σ 2 t-i. The variance at time t depends on past errors and past variances. A variance has to be positive, we need to impose restrictions: α 0  0, α i  0 (i=1,...,q), and ß i  0 (i=1,...,p) After some algebra, we get the unconditional (or average) variance, σ 2 : σ 2 = α 0 /(1 - Σ i q α i - Σ i p ß i ). For the unconditional variance to be well defined, we need: Σ i q α i + Σ i p ß i < 1. GARCH models have been successfully employed for exchange rates. Empirical regularity: A GARCH(1,1) model tends to work well: σ 2 t = α 0 + α 1 e 2 t-1 + ß 1 σ 2 t-1.  σ 2 is well defined when λ=α 1 +ß 1 <1.

34. GARCH models are usually estimated using a method called maximum likelihood (ML). Once the parameters α i 's and ß i 's are estimated, equation (V.4) can be used to forecast the next-period variance. Example: Sietes Swiss uses an AR(1)-GARCH(1,1) for S t - USD/CHF: s t = [log(S t ) - log(S t-1 )]x100 = a 0 + a 1 s t-1 + e Ft,e t  Ψ t-1 ~ N(0,σ 2 t ). σ 2 t = α 0 + α 1 e 2 t-1 + ß 1 σ 2 t-1. Sample size: 1548 observations (January 6, 1977 - January 30, 2010). The estimated model for s t is given by: s t =-0.055 +0.059 s t-1, (1.10)(1.78) σ 2 t =0.140 +0.079 e 2 t-1 +0.876 σ 2 t-1. (2.19)(4.16)(25.95) Note: λ=.955 < 1.

35. The model estimates: e 1548 = -0.551 σ 2 1548 = 2.554. Now, Sietes Swiss forecasts σ 2 1549 : σ 2 1549 = 0.140 + 0.079 (-0.551) 2 + 0.876 (2.554) = 2.401. ¶

36. Multiperiod variance forecasts using a GARCH(1,1) model: E t [σ 2 t+k ]=(α 1 + ß 1 ) k [σ 2 t - α 0 /(1-α 1 -ß 1 )]+ α 0 /(1-α 1 -ß 1 ) =λ k [σ 2 t - σ 2 ] + σ 2 If λ=(α 1 +ß 1 )<1, as k , λ k  0.  as we go more into the future, the forecasts converges to σ 2. Note: λ represents the rate of convergence (decay) of σ 2 t towards σ 2. Example: We estimate a GARCH(1,1) model using 300 daily observation for the GBP/USD exchange rate. We obtain these estimates: α 0 =.175, α 1 =.123, ß 1 =.847, and σ 2 300 = 1.959. We want to forecast the variance 10 days from now. The unconditional variance is σ 2 =.175/(1 -.123 -.847) = 5.833. The 10-day-ahead variance forecast is: E 300 [σ 2 310 ] = (.97) 10 [ 1.959 - 5.833] + 5.833 = 2.976.  the volatility forecast 10 days ahead is σ 310 = 1.725%. ¶

37. J.P. Morgan's RiskMetrics Approach In October 1994, J.P. Morgan unveiled its risk management software. RiskMetrics is a method that estimates the market risk of a portfolio, using the Value-at-Risk (VAR) approach. RiskMetrics uses a simple specification to model σ 2 t : σ 2 t = (1-γ) e 2 t-1 + γ σ 2 t-1.  next period's variance is a weighted average of this period's squared forecast error and this period's variance. RiskMetrics approach involves an IGARCH model:α 0 =0 γ = (1-α 1 ) = ß 1. J.P. Morgan set:γ=.94, for daily data. γ=.97, for monthly volatility. Given the large number of series that J.P. Morgan uses (more than 500), RiskMetrics uses the same γ for all the series.

38. Example: Suppose we estimate a time-varying variance with the RiskMetrics model using 300 daily observations for GBP/USD. Estimates σ 2 300 = 1.959 and e 300 = 0.663. Forecast for the variance tomorrow: E 300 [σ 2 301 ] = (.94) 1.959 +.06 (.633) 2 = 1.8655 That is, the volatility forecast for tomorrow is E 300 [σ 301 ] = 1.366%. ¶