Forecasting spot and forward prices in t (1)

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Forecasting spot and forward prices in the

international freight market

Roy Batchelor

a,

⁎

, Amir Alizadeh

a,1

, Ilias Visvikis

b,2

a_{Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, UK}

b_{Athens Laboratory of Business Administration (ALBA), Athinas Ave and 2A Areos Str, 166 71 Vouliagmeni, Athens, Greece}

Abstract

This paper tests the performance of popular time series models in predicting spot and forward rates on major seaborne freight routes. Shipping is a nonstorable service, so the forward price is not tied to the spot by any arbitrage relationship. The developing forward market is dominated by hedgers, and it is an empirical question whether forward rates contain information about future spot rates. We find that vector equilibrium correction (VECM) models give the best in-sample fit, but implausibly suggest that forward rates converge strongly on spot rates. In out-of-sample forecasting all models easily outperform a random walk benchmark. Forward rates do help to forecast spot rates, suggesting some degree of speculative efficiency. However, in predicting forward rates, the VECM is unhelpful, and ARIMA or VAR models forecast better. The exercise illustrates the dangers of forecasting with equilibrium correction models when the underlying market structure is evolving, and coefficient estimates conflict with sensible priors.

JEL classification:G13; G14

Keywords:Forecasting; Freight market; Commodity market: Vector equilibrium correction model; ARIMA model

1. Introduction

In this paper we investigate the performance of alternative univariate and bivariate linear time-series models in generating short-term forecasts of spot

freight rates in the international dry bulk shipping market, and corresponding rates fixed in the Forward Freight Agreement (FFA) market. The FFA market is interesting for several reasons. First, it is relatively new and under-researched, and our findings come from a unique and specially constructed database of forward freight rates. There is practical value to users of the market–ship owners and shipping agents–in know-ing whether and how forward rates can best be used to predict spot rates. Second, the underlying asset traded in the FFA market is a service rather than a storable commodity. This means that arbitrage between spot

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⁎Corresponding author. Tel.: +44 207 040 8733; fax: +44 207 040 8881.

E-mail addresses:[email protected](R. Batchelor), [email protected](A. Alizadeh),[email protected] (I. Visvikis).

1

Tel.: +44 207 040 0199; fax: +44 207 040 8881.

2

Tel.: +30 210 896 4531 8; fax: +30 210 896 4737.

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and forward markets is not possible, so spot and forward prices are not linked by the rigid cost-of-carry relationships observed in most commodity markets. Third, unlike the large established markets in com-modities and financial futures, the FFA market is small and dominated by the activities of hedgers rather than speculators. It cannot therefore be taken for granted that all information relevant to future freight rates is automatically incorporated into the forward price.

In these circumstances we should expect to observe certain characteristics in the time series of spot and forward freight rates. In speculatively efficient markets for nonstorable commodities, forward prices are un-biased forecasts of future spot prices, and changes in forward prices for fixed target dates are close to ran-dom, reflecting the arrival of news. The thinness of the FFA market and the absence of a strong speculative interest mean that forward freight rates may exhibit neither of these properties. Similarly, in arbitrage-dominated markets, forward rates are tied to spot rates, and both tend to move to ensure convergence at the expiry of contracts, as the cost of carry falls. The absence of arbitrage in the FFA market means that the spot rate may converge on the forward rate, provided that the forward rate embodies some expectations about future spot rates. However, there is no reason why the forward rate should converge on the spot rate. The most general model we use is a vector error-correction (VECM) model linking spot and forward rates for four major shipping routes. This model is used to make inferences about the efficiency and usefulness of FFA rates. For example, if forward rates are expec-tations of spot rates we would expect (a) there to be a cointegrating vector linking spot and forward rates, and (b) the cointegrating vector to be the basis (that is, spot rate−forward rate = 0), and (c) this equilibrium to

be established by spot rates converging on forward rates, but not vice versa.

The validity of the VECM model is tested by benchmarking forecasts from it against forecasts from a number of alternative linear time series models, and against the random walk. Even if two price series are cointegrated, incorporating the information contained in the cointegrating relationship in the model is not guaranteed to improve predictability. Moreover, as discussed by Clements and Hendry (1995, 1998, 2001), the VECM is not robust to structural change. The equilibrium correction term forces variables to

their average historical relationship, so long-term fore-casts in particular may be inaccurate if the underlying relationship has shifted. Stock and Watson (1996)

show that most mainstream macroeconomic variables have been subject to significant structural change in recent decades. As a consequence, Allen and Fildes

(2001) find that in practice, VECM models have a

mixed track record in forecasting such time series. In commodity markets, the balance of evidence seems to favour the VECM approach. For example,Zeng and

Swanson (1998)estimate VECM and other models for

spot and futures prices of the S&P500 index, the US 30-year T-bond, gold and oil. They find that the VECM predicts better than all simpler models, and also the random walk. However, this is not surprising since in their chosen markets the possibility of arbitrage ensures that the basis (futures price−spot price) is

equal to the cost of carry (borrowing cost less own rate of return on the spot asset), and their best-performing models use this as the cointegrating vector. Cash-and-carry arbitrage (borrow funds, buy spot, sell forward) is not feasible in the freight market since the underlying asset cannot be stored.Cullinane (1992)andCullinane, Mason, and Cape (1999)report success in forecasting spot freight rates using simpler univariate ARIMA models. Kavussanos and Nomikos (2003) compare joint VECM forecasts of spot freight rates and the now defunct exchange traded BIFFEX futures freight rates with forecasts from ARIMA, VAR and Random Walk models. They find that the VECM generates the most accurate forecasts of spot prices but not of futures prices. Their tests use overlapping forecast intervals, which Tashman (2000) argues may bias forecast evaluation, and we have been careful to design our forecast evaluation procedures to avoid this problem.

Alizadeh and Nomikos (2003)examine the directional

forecast accuracy of FFAs and Freight Futures contracts (BIFFEX) in four routes and concluded that FFAs do not seem to be very accurate in revealing the direction of future freight rates. They report that the directional accuracy of FFAs in forecasting freight rates varies between 46% and 74%, and, in general, forecasting accuracy declines as maturity increases.

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performance of the alternative models. Finally, Section 6 summarises our conclusions.

2. Spot and forward freight rate data

The aim of the creation of the Freight Forward Market in 1992 was to provide a mechanism for hedging freight rate risk in the dry-bulk and wet-bulk sectors of the shipping industry. FFAs are principal-to-principal contracts between a seller and a buyer to settle a freight or hire rate for a specified quantity of cargo or type of vessel, for one (usually) or more major trade routes. One party–the charterer–is concerned that in some future month the spot price of a standard freight may be higher than expected, and buys FFA contracts. The other party–the shipowner–is concerned about rates falling, takes the opposite position, and sells FFA contracts. Settlement is made on the difference between the contracted forward price and the average price for the route selected in the index over the last seven working days of the settlement month. The market is intermediated by a small number of specialist brokers. The standard routes underlying the FFA contracts have changed since its inception. Currently, FFA con-tracts are written on freight rates in routes of the Baltic Panamax Index (BPI), the Baltic Handymax Index (BHMI), the Baltic Capesize Index (BCI), the Baltic Dirty Tanker Index (BDTI), and Baltic Clean Tanker Index (BCTI). The most liquid routes, those in the BPI, are described inTable 1. Our data set consists of daily spot and FFA prices in Panamax Atlantic routes 1 and 1A from 16 January 1997 to 31 July 2000, and daily spot and FFA prices in Panamax Pacific routes 2 and 2A from 16 January 1997 to 30 April 2001. The difference in the sample periods between the Atlantic and Pacific routes arises because the Atlantic routes are characterised by modest FFA trading and FFA brokers stopped publishing FFA quotes for those routes after July 2000. FFA trading is concentrated mostly on the Pacific routes 2 and 2A, since most market agents operate their Panamax vessels in the Pacific region.

Spot price data are from the Baltic Exchange. FFA rates are not available from any data vendor. Our data on daily bid and offer quotes for every trading route of the BPI, and for the nearby FFA contracts for the four Panamax routes, have been collected manually from the records of Clarkson's Securities Ltd., a leading broker in the FFA market. FFA prices are always those

of the nearby contract because it is highly liquid and is the most active contract. However, to avoid thin market and expiration effects (when futures and forward contracts approach their settlement day, the trading volume decreases sharply), we roll over to the next nearest contract 1 week before the nearby contract expires. In 1999, the total FFA volume was about 1200 contracts, a figure similar to 1998. According to Clarkson's Securities Ltd., in 2000 the total FFA volume was about 2000 contracts, while more than 8000 contracts were traded in 2004 and about 8500 in 2005, with a market value of around US$29 billion. In practice, most of the trading concentrates in the nearby (one month) and first-out (2 months) contracts, with the second-out (3 months) contracts characterised by

Table 1

Baltic Panamax Index (BPI)—route definitions

Routes Descriptions Size of vessel (dwt, tonnes)

Weight in index (%)

1 1–2 safe berths/anchorages Mississippi River not above Baton Rouge/Antwerp, Rotterdam, Amsterdam

55,000 10

1A Transatlantic (including ESCA) round of 45/60 days on the basis of delivery and redelivery Skaw–Gibraltar range

70,000 20

2 1–2 safe berths/anchorages Mississippi River not above Baton Rouge/1 no combo port South Japan

54,000 12.5

2A Basis delivery Skaw–Gibraltar range, for a trip via Gulf to the Far East, redelivery Taiwan– Japan range, duration 50/ 60 days

70,000 12.5

3 1 port US North Pacific/1 no combo port South Japan

54,000 10

3A Transpacific round of 35/50 days either via Australia or Pacific (but not including short rounds such as Vostochy/Japan), delivery and redelivery Japan/ South Korea range

70,000 20

4 Delivery Japan/South Korea range for a trip via US West Coast–British Columbia range, redelivery Skaw–Gibraltar range, duration 50/60 days

70,000 15

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very low volume figures. More recently, quarter and calendar contracts have also been traded in certain routes and on time-charter rates or daily earnings of different types of vessels.

Combining information from FFA contracts with different times to maturity may create breaks in the series at the date of the forward rollover, since FFA returns for that day are calculated between the price of the expiring contract and the price of the next nearest contract. To address this issue, we experimented with a series of synthetic prices for a “perpetual” 22-day ahead FFA contract. The prices are calculated as a weighted average of near and distant FFA contracts, weighted according to their respective number of days from maturity. This procedure generates a series of FFA prices with constant maturity and avoids the problem of price-jumps caused by the expiration of a particular FFA contract. However, use of these data yields empirical results which are qualitatively the

same as those reported below, so there is no evidence that the FFA contract rollover biases our findings.

Figs. 1–4 present co-movements and fluctuating

patterns of spot and FFA rates for routes 1, 1A, 2 and 2A, respectively. Freight rates weakened after the economic crises in Russia and the Far East in 1997–

1998, but recovered through 1999 to their starting levels by 2000–2001. Since then, as world trade–and especially trade with China–has accelerated, freight rates have risen very sharply, and in 2004–2005 were some three to four times higher than in 2001. Forward rates are typically close to spot rates, but occasionally (for example, in 1999–2000) drift away.Kavussanos and Visvikis (2004)investigate the lead–lag relation-ships between forward and spot markets, and find bidirectional causality in price movements in all routes, but find less clear evidence on the direction of volatility spillovers between spot and forward prices across different routes. In another study, Kavussanos,

Fig. 1. FFA and Spot Prices in Route 1 (16/01/97–31/07/00).

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Visvikis, and Menachof (2004)show that FFA prices 1 and 2 months prior to maturity are generally unbiased predictors of the realised spot prices, while Kavussa-nos, Visvikis, and Batchelor (2004)argue that the onset of FFA trading did have a general stabilising influence on the spot price volatility.

For the purpose of analysis, all prices are transformed to natural logarithms. Summary statistics of logarithmic first differences (“log returns”) of daily spot and FFA prices are presented inTable 2for the whole period, in the four Panamax routes. The results indicate excess kurtosis in all series, and excess skewness in most, and the Jarque–Bera tests indicate departures from normality for both spot and FFA prices in all routes. The Ljung–BoxQ(36) andQ2(36) statistics on the first 36 lags of the sample autocorrela-tion funcautocorrela-tion of the level series and of the log-squared series indicate significant serial correlation and existence of heteroskedasticity, respectively. In contrast to speculative storable commodities, such as

stock prices and exchange rates, there is no reason to expect changes in spot freight rates to be serially uncorrelated. Demand and supply for freight services are determined by the needs of trade. Scope for intertemporal substitution and substitution across routes and vessel types is severely limited. Serial correlation in spot prices may also be exaggerated by the way shipbroking companies calculate freight rates. These rates are mostly based on actual fixtures, but in the absence of an actual fixture they depend on the shipbroker's estimate of what the rate would be if there was a fixture; discussions with brokers suggest that their estimates are typically a mark-up over the previous day's rate.

Augmented Dickey Fuller (ADF) and Phillips–Perron (PP) unit root tests on the log levels and log first differences of the daily spot and FFA price series indicate that all variables are log-first difference stationary, all having a unit root on the log levels representation. ADF and PP tests are sometimes criticised for their lack of

Fig. 3. FFA and Spot Prices in Route 2 (16/01/97–10/08/01).

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power in rejecting the null hypothesis of a unit root when it is false. This lack of power is addressed by the KPSS test proposed byKwiatkowski et al. (1992), which has stationarity as the null hypothesis, and the results of the KPSS tests confirm our inferences.

3. Forecasting models

To identify the model that provides the most accurate short-term forecasts of spot and FFA prices in the market, four time series models are considered. These are theBox

and Jenkins (1970) ARIMA model, the Vector

Auto-regression (VAR) model of Sims (1980) on price changes, a general Vector Equilibrium Correction model (VECM) suggested for cointegrated variables by

Engle and Granger (1987), and a restricted VECM

model. The VAR model here can be considered as a restricted version of the VECM in which the equilibrium correction term is dropped. For evaluation purposes, the data are split into an in-sample estimation set and an out-of-sample forecast set. The various time-series models are initially estimated over the period 16 January 1997 to 30 June 1998 for all routes. The period from 1 July 1998 to 31 July 2000 for the Atlantic routes and the period from 1 July 1998 to 30 April 2001 for the Pacific routes are then

used to evaluate independent out-of-sample N-days ahead forecasts.

The ARIMA(p,1,q) model for spot and forward freight rates is

DSt¼a10þ

Xp

i−1

a1iDSt−iþ

Xq

j−1

b1ie1t−j

þe1t ; e1tfiidNð0;r

2 1Þ

DFt¼a20þ

Xp

i−1

a2iDFt−iþ

Xq

j−1

b_2ie2t−j

þe2t ; e2tfiidNð0;r

2

2Þ ð1Þ

whereΔFtandΔStare changes in log futures and spot

prices, respectively, and theεtare random error terms.

The corresponding bivariate VAR(p) model is:

DSt¼a10þ

Xp

i−1

a1iDSt−1þ

Xp

j−1

b_1iFt−jþe1t

DFt¼a20þ

Xp

i−1

a2iDSt−iþ

Xp

j−1

b_2iFt−jþe2t: ð2Þ Table 2

Descriptive statistics for spot and FFA prices

Statistics Route 1 Route 1A Route 2 Route 2A

Spot FFA Spot FFA Spot FFA Spot FFA

N 891 891 891 891 1078 1078 1078 1078

Skew −0.17 −0.15 1.81 −0.04 0.63 0.29 −1.28 0.10

Kurtosis 13.35 5.43 30.78 4.71 522.25 5.04 31.35 6.23

Jarque–Bera 6609.83 1096.71 35,637.27 822.28 12,271.08 1158.85 44,392.48 1747.41

Q(36) 216.43 304.47 186.29 258.59 507.73 285.55 157.80 292.49

QSQ(36) 264.77 283.47 290.45 221.35 215.81 276.36 246.01 404.80

ADF (levels) −1.53 −1.65 −1.89 −1.61 −1.85 −1.52 −2.12 −1.89

PP (levels) −1.32 −1.57 −1.67 −1.78 −1.83 −1.63 −1.89 −1.95

ADF (differences) −9.17 −31.72 −10.34 −29.55 −14.00 −30.42 −12.48 −30.01 PP (differences) −15.75 −32.07 −14.05 −29.71 −32.77 −30.46 −15.22 −30.03

KPSS 0.93 0.86 0.74 0.75 0.83 0.86 0.74 0.75

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The potential advantage of the bivariate VAR model over the univariate ARIMA model is that it takes into account the information content in the spot price movement in determining the forward price and vice versa.

Our bivariate VECM (p) models are of the form

DSt¼a10þ

The term in brackets represents the cointegrating (long-run) relationship between the spot and forward prices. The VECM model is argued to be more

appro-priate than the univariate ARIMA and bivariate VAR models in modelling the spot and forward prices as it takes into account both the short-run dynamics and the long-run relationship between variables. In Eq. (3) the coefficients γ1and γ2 measure the speed of

adjust-ment of spot and forward prices to their long-run equilibrium.

The VAR model is itself a restricted version of the VECM, where the two equilibrium correction terms are omitted (γ1=γ2= 0). The VAR model therefore

may require a larger number of parameters than the VECM to capture the dynamic behaviour of the vari-ables, and this lack of parsimony may cause problems when the model is used for forecasting. One problem is that the collinearity between the different lagged variables may lead to imprecise coefficient estimates. Another problem is that the large number of para-meters may lead us to model some sample-specific noise in finite samples, and this overfitting can give the

Table 3.1

Estimates of the forecasting models for route 1

Regressor ARIMA VAR VECM S-VECM

ΔSt ΔFt ΔSt ΔFt ΔSt ΔFt ΔSt ΔFt

R−2 _0.3536 _0.0047 _0.4215 _0.0258 _0.4216 _0.0385 _0.4219 _0.0355

Q(12) 2.245

Estimated coefficients of ARIMA, VAR, VECM and restricted VECM (S-VECM) estimated using daily data on spot and FFA freight rates over the period 16/01/97 to 30/06/98. The cointegrating vector is restricted to be the lagged basisSt−1−Ft−1in route 1. Figures in parentheses under

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appearance of a good within-sample fit, but poor out-of-sample forecasts.

We estimate both an unrestricted VECM and a restricted version which is simply the parsimonious version of Eq. (3) derived by successively eliminat-ing statistically insignificant coefficients. In general this model has different regressors in the two equations, and is therefore estimated as a system of Seemingly Unrelated Regression Equations (SURE), since this method yields more efficient estimates than Ordinary Least Squares in these circumstances (Zellner, 1962). The restricted model is denoted S-VECM.

4. Estimation results

As noted above, the results of the unit root tests on the log levels and log first differences of the daily spot and FFA price series indicate that all variables are log-first difference stationary, all having a unit root on the log-levels representation. This means that the first differences of spot and forward series should be used in the ARMA and VAR models, while cointegration tests should be performed to ascertain the long-run relationship between the series if the VECM model is going to be used.

Johansen's (1988)multivariate cointegration test results indicate that spot and FFA prices are cointegrated in all

Table 3.1A

Estimates of the forecasting models for route 1A

0.545 0.0096 0.5551 0.0425 0.5578 0.0641 0.5574 0.0626

Q(12) 15.663

See notes toTable 3.1.

For route 1A, the cointegrating vector isSt−1−1.0143⁎Ft−1+ 0.1585.

Table 3.2

Estimates of the forecasting models for route 2

0.443 0.0161 0.4592 0.0417 0.4872 0.0417 0.4871 0.0442

Q(12) 8.717

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routes. The cointegrating vectorzt−1= (St−1+δ1Ft−1+δ0)

is restricted to be the lagged basis (St−1−Ft−1) in routes 1

and 2A; that is,δ1=−1 andδ0= 0. The other

cointegrat-ing vectors are zt−1=St−1−1.0143⁎Ft−1+ 0.1585 in

route 1A andzt−1=St−1−1.0067⁎Ft−1+ 0.0324 in route

2. The results of the likelihood ratio tests for the over-identifying restrictions applied to the cointegrating vectors are 4.901 [0.086] for route 1; 8.011 [0.018] for route 1A; 8.481 [0.014] for route 2; and 3.581 [0.167] for route 2A, where figures in square brackets arep-values. This discrepancy in the results might be result of the different economic and trading conditions that prevailed in each trading route in our sample period.

The results of ARIMA models for spot and forward rates for the four routes are presented in the first columns of Tables 3.1, 3.1A, 3.2 and 3.2A. The lag length for the autoregressive and moving average parts are chosen to minimise the Schwartz Bayesian Criterion (SBC;Schwartz, 1978). All ARIMA models seem to be well specified, as is indicated by relevant diagnostic tests for autocorrelation and heteroskedas-ticity. It can be noted that across all routes, the adjusted coefficient of determination for changes in spot rates

are higher than those of forward rates, indicating potentially higher predictability in spot rates than forward rates.

The next set of columns inTables 3.1–3.2Apresent the estimates of coefficients of the VAR models. These are similar to the ARIMA model in terms of the appropriate number of lag lengths used and the diagnostic tests. As expected, the adjusted coefficients of determination for the VAR models are slightly higher than those of the ARIMA model due to the use of extra information, namely, the lagged forward rates in the spot equation and vice versa. The final columns of Tables 3.1–3.2A show the estimation results for the unrest-ricted VECM and restunrest-ricted S-VECM models. These fit the data a little better than the VAR models, and the S-VECM is preferred by the Schwartz Bayesian criterion in all cases. “Granger Causality”, as measured by the significance of lagged forward rates in the spot equation and lagged spot rates in the forward equation, seems to run both ways. In all cases, the one-period lagged change in forward rates is significant in the spot rate equation, and the one-period lagged change in spot rates is significant in the forward equation. There are also

Table 3.2A

Estimates of the forecasting models for route 2A

0.5396 0.0259 0.5808 0.0467 0.5868 0.0588 0.5892 0.0715

Q(12) 14.047

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effects from longer lags of the forward rate on the spot price in all models and all routes, but not vice versa, which suggests that forward rates lead spot rates at longer horizons.

However, the size and significance of the equilibrium correction coefficients in the VECM models are not consistent with our prior that in an efficient market for a nonstorable service, the spot rate should converge on the forward, but not vice versa. The coefficients of the

lagged equilibrium correction terms show that rates do converge, and the relative importance of movements in spot rates and movements in forward rates differs across routes. In Route 1 only the forward rate adjusts to correct disequilibrium, but in Route 2 only the spot rate moves. In Routes 1A and 2A, adjustment coefficients on both spot and forward rates are significant. In routes 1 and 1A the adjustment coefficient on the forward rate is some eight times higher than that on the spot rate, and in route

Table 4

Forecast error reduction for spot freight rate models

Route Horizon Number of Forecasts

Benchmark RW

Incremental % reduction in RMSE of % reduction from S-VECM

ARIMA VAR VECM S-VECM

1 1 520 1.64 12 −20 0 −1 −9

2 260 0.72 −2 −19 0 −1 −22

3 173 0.92 −17 −18 1 −1 −36

4 130 0.99 −23 −18 1 −1 −41

5 104 1.16 −30 −9 0 −1 −40

10 52 0.87 −44 −1 −2 0 −47

15 34 1.54 −42 −11 1 −1 −54

20 26 1.06 −47 −2 −1 0 −51

1A 1 520 1.31 0 −14 −1 0 −14

2 260 1.45 −13 −18 −1 0 −32

3 173 1.75 −23 −18 0 0 −41

4 130 1.67 −28 −19 0 0 −47

5 104 2.30 −37 −4 0 0 −42

10 52 1.92 −42 −1 0 0 −43

15 34 2.57 −38 −6 0 0 −43

20 26 1.68 −61 −1 −1 −1 −64

2 1 707 0.79 −10 −3 −2 0 −15

2 353 0.99 −28 −4 −2 0 −34

3 235 1.05 −37 −1 −2 0 −39

4 176 1.28 −42 −3 −1 0 −47

5 141 1.18 −43 −3 −1 0 −47

10 70 0.92 −45 −3 −3 0 −51

15 47 1.22 −46 −1 −1 0 −48

20 35 0.92 −36 −2 −5 0 −44

2A 1 707 1.34 −11 −7 0 −1 −19

2 353 1.76 −36 −5 −1 0 −42

3 235 1.65 −38 −2 0 −1 −41

4 176 2.35 −48 −5 −1 0 −53

5 141 1.94 −39 −2 −1 −1 −42

10 70 1.34 −47 0 −2 −1 −50

15 47 2.39 −45 −3 0 −1 −49

20 35 1.75 −55 −1 −2 −1 −60

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2A some three times higher. So in three out of the four routes, it is the forward rate rather than the spot rate that bears the burden of adjustment.

5. Forecasting performance of the time-series models

These alternative univariate and multivariate models, estimated over the initial estimation period, are used to generate independent forecasts of the spot and FFA

prices up to 20-steps ahead. Then as recommended in

Tashman (2000), independent out-of-sampleN-period ahead forecasts are generated over the forecast period; that is, from 1 July 1998 to 31 July 2000 for the Atlantic routes and the period from 1 July 1998 to 30 April 2001 for the Pacific routes. In order to avoid the bias induced by serially correlated overlapping forecast errors, we recursively augment our estimation period byN-periods ahead every time (whereNcorresponds to the number of steps ahead). For example, in order to compute 5

steps-Table 5

Forecast error reduction for forward freight agreement models

Route Horizon Number of Forecasts

Benchmark RW

Incremental % reduction in RMSE of % reduction from S-VECM

ARIMA VAR VECM S-VECM

1 1 520 3.13 −33 −1 0 0 −34

2 260 3.18 −30 −2 0 −1 −32

3 173 2.60 −31 0 −1 0 −31

4 130 3.22 −29 −1 0 0 −31

5 104 2.41 −30 −1 1 −1 −31

10 52 1.47 −34 1 4 −3 −31

15 34 0.81 −37 12 −3 7 −20

20 26 1.35 −34 7 −1 1 −26

1A 1 520 3.83 −29 −2 0 −1 −31

2 260 4.11 −33 −1 0 0 −34

3 173 4.18 −28 −2 0 −1 −31

4 130 4.74 −35 −3 −1 0 −39

5 104 3.62 −27 −2 0 −1 −30

10 52 3.38 −29 −2 2 −2 −31

15 34 4.46 −31 −4 1 −1 −34

20 26 2.73 −31 −2 1 0 −33

2 1 707 2.28 −27 −1 0 0 −28

2 353 2.35 −35 0 0 0 −36

3 235 2.55 −28 −2 −1 0 −32

4 176 2.58 −32 −4 −1 0 −36

5 141 1.97 −28 0 0 0 −28

10 70 2.03 −31 0 1 0 −30

15 47 2.09 −27 −2 0 0 −30

20 35 2.37 −38 −2 0 0 −40

2A 1 707 3.88 −28 −2 0 −1 −30

2 353 3.39 −31 −2 0 0 −33

3 235 4.62 −29 −2 0 −1 −31

4 176 3.86 −34 −2 −1 0 −37

5 141 4.06 −20 −4 1 −1 −24

10 70 4.23 −34 −3 2 −1 −37

15 47 5.73 −31 −2 2 −1 −33

20 35 4.32 −38 −3 0 0 −41

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ahead forecasts, we augment our estimation period by

N= 5 observations each time. This method yields 104 and 141 independent non-overlapping forecasts in the Atlantic and Pacific routes, respectively. Similarly, in order to compute 10 steps-ahead forecasts, the method yields 52 and 70 independent non-overlapping forecasts in the Atlantic and Pacific routes. This methodology provides two desirable characteristics of an out-of-sample test; adequacy (enough forecasts at each forecasting horizon) and diversity (desensitising fore-cast error measures to special events).

The forecasting performance of each model for spot prices across the different forecast horizons is pre-sented in matrix form in Tables 4 and 5for spot rate forecasts and forward rate forecasts, respectively. The forecast accuracy of each model is assessed using the conventional root mean square error metric (RMSE). The RMSE is considered a fairly accurate forecast accuracy metric. It assumes a symmetric loss function for forecast users, which seems reasonable given the monetary amount of FFA contracts which can be anything over $0.5 m for a monthly contract and up to several million dollars in the case of a calendar contract. We benchmark the models against each other, and also against random walk, no-change, forecasts for spot and forward rates. To highlight the relative performance of the models, we report the RMSE of the random walk model, and the incremental percentage reduction in this RMSE achieved by moving to progressively more complicated (and statistically preferred) models. Consider the first row ofTable 4. The RMSE of the 1-day ahead forecast of the (log) spot rate from the random walk model is 0.00642, or approximately 0.64% of the spot price. The corresponding RMSEs for the ARIMA, VAR, VECM and S-VECM models are 0.00716, 0.00590, 0.00592, and 0.00585, respectively. The ARIMA model is less accurate than the random walk and increases the RMSE by 100 × (.00642−.00716)/.00642 = 12%, and this is

reported in Table 4. The VAR model is much more accurate than the ARIMA model, and improves the RMSE by 100 × (0.00590−0.00716)/0.00642 =−20%.

This represents an error reduction of about 12−20 = −8% of the RMSE of the random walk forecast, and so

on. Positive entries inTables 4 and 5indicate that the model has failed to improve on the previous model. Negative entries indicate an error reduction. The size of the entry indicates the gain from using that model rather

than the previous model, expressed as a percentage of the RMSE of the random walk model. To test the significance of any outperformance, we also apply the

Diebold and Mariano (1995)test of the hypothesis that the RMSEs from two competing models are equal, exhaustively to all pairs of models. Results indicate that the RMSEs of the S-VECM and VECM forecasts are, in general, significantly lower than other competing models, especially for shorter maturities. Full tables of these model comparisons are not shown here but are available from the authors.

The final column of the table shows the total percentage reduction in RMSE achieved by using the S-VECM model, the sum of the incremental gains from all the models. In the case of 1-day ahead spot rate forecasts for route 1, the use of the S-VECM model reduces the RMSE from the random walk forecast by 9% (as it happens, an atypically small improvement).

Several regularities stand out from the tables. First, as expected, the forward rates are much harder to forecast than the spot rates. The higher volatility of the forwards is evidenced by RMSEs from the no-change random walk model that are some three to four times higher than the corresponding spot rate RMSEs.

Second, all the S-VECM models comfortably outperform the random walk benchmark. On average over all routes and horizons they achieve a reduction of about 40% for the spot rate RMSE and 30% for the forward rate RMSE. For spot rate forecasts the improvement is less marked at very short horizons (about 15–20% for 1–4 days ahead), but about 50% for longer horizons. For the forward rate forecasts, the advantage of the model over the random walk is uniform across horizons. Not surprisingly, in all cases, the Diebold–Mariano statistic rejects the hypothesis of equal RMSE between the random walk and S-VECM models. These results are comparable to the findings of

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Third, the models with equilibrium correction fea-tures (VECM and S-VECM) perform better than VAR models for forecasts of spot rates, but not for forecasts of forward rates. Looking at the spot rate forecasts in

Table 4, in the liquid Pacific routes 2 and 2A, the use of the VECM in place of the VAR reduces the benchmark RMSE by up to 5%, and the use of the restricted S-VECM reduces it by a further 1%. In the less liquid Atlantic routes 1 and 1A, the improvements in accu-racy are less marked. Most improvements in forecast accuracy can be achieved by a VAR in place of the univariate ARIMA model, and the incremental gains from using a VECM are small. But at no horizon does the use of the VECM for spot rate forecasting reduce the forecast accuracy relative to the VAR. In contrast, the statistics in Table 5 show that for forward rate forecasting most improvements in accuracy relative to the random walk can be achieved by a simple ARIMA model. Use of the VAR actually reduces accuracy in longer term forecasts in route 1 and reduces the RMSE by up to 4% in other routes, and the use of the VECM and S-VECM models in place of the VAR leads to very small and inconsistent changes in accuracy in forecast-ing forward rates.

6. Conclusion

The forward freight market is relatively new, relatively illiquid and relatively under-researched. In this paper, we have explored the value of various standard linear time series models as forecasting tools to jointly predict the spot and forward freight rates. If the forward market is liquid enough to embody some information about futures spot rates into the prices, we should observe cointegration between spot and forward rates, and convergence of spot towards forward rates, rather than vice versa. Spot and forward freight rates are indeed cointegrated. But the models inTables 3.1–3.2A

suggest that, contrary to our expectations, forward rates adjust more strongly than spot rates to close the gap between spot and forward rates. However, out-of-sample forecasting with these equilibrium correction models show that they are not helpful in predicting forward rate behaviour, but do help predict spot rates, a finding more consistent with market efficiency.

For shipowners and charterers, the findings of our study are encouraging, in the sense that they suggest that spot freight rates are forecastable, and the rates

offered by Forward Freight Agreements to some extent help anticipate spot freight rates. For analysts of commodity markets the message is more cautionary, an illustration of the dangers of forecasting with an equilibrium correction model when the underlying market is evolving, and the parameter estimates conflict with sensible theoretical priors.

Acknowledgements

We thank Professor Michael Clements and an anonymous referee for their extremely helpful com-ments. This paper has also largely benefited from the comments of participants at the 2003 International Association of Maritime Economists Conference, Busan, Korea, and the 2003 International Symposium on Forecasting, Merida, Mexico.

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