basis of one voyage, possibly including a ballast leg, and is, in that sense, ... positioning decisions for single vessels operationg in the tramp voyage market. The.
Research Center Athens University of Economics and Business Report No. E194
POSITIONING AND DIVERSIFICATION IN SHIPPING
Evangelos F. Magirou 1 Harilaos N. Psaraftis 2 Lambros Babilis 3 Athanasios Denissis 4
March 1997
1 2 3 4
Professor, Athens University of Economics and Business Professor, National Technical University of Athens Ph.D. Candidate, National Technical University of Athens National Technical University of Athens.
0. EXECUTIVE SUMMARY
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0.1. ΕΥΡΕΙΑ ΠΕΡΙΛΗΨΗ
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0.2. EXECUTIVE SUMMARY
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1. INTRODUCTION 1.1. SCOPE OF THE REPORT
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1.2. ACKNOWLEDGMENTS
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2. POSITIONING TRAMP VESSELS 2.1. INTRODUCTION
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2.2. A POSITIONING MODEL
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2.3. AN APPLICATION TO CAPESIZE BULK CARRIERS
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2.4. CASE STUDY RESULTS
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2.5. CONCLUSIONS AND EXTENSIONS
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2.6. APPENDIX 2.6.1. Dynamic Programming Formulation 2.6.2. Solution methods 2.6.3. Using the Bulkpos spreadsheet
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2.7. REFERENCES
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2.8. TABLES AND FIGURES
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3. DIVERSIFICATION IN SHIPPING 3.1. INTRODUCTION
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3.2. CASE STUDY DESCRIPTION 3.2.1. Vessel Types 3.2.2. Economic performance
30 30 30
3.3. PORTFOLIO ANALYSIS IN SHIPPING 3.3.1. Review of the relevant theory 3.3.2. Portfolio analysis results 3.3.3. The Beta of Shipping 3.3.4. Conclusions
31 31 32 33 34
3.4. REFERENCES
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3.5. TABLES AND FIGURES
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2
0.
EXECUTIVE SUMMARY
0.1.
ΕΥΡΕΙΑ ΠΕΡΙΛΗΨΗ
Η έρευνα αυτή αποτελεί συνέχει προηγούµενης σχετικά µε την χρήση Ποσοτικών Μεθόδων στην Ναυτιλία5. Στην έκθεση αυτή προέκυψε το συµπέρασµα ότι η χρήση Ποσοτικών Μεθόδων γιά την λήψη στρατηγικών αποφάσεων στην Ναυτιλία ήταν περιορισµένη αλλά παρουσίαζε δυνατότητες ευρύτερης διάδοσης. Από τότε (1992) η πρόβλεψη αυτή δεν επαληθεύθηκε λόγω των εξής παραγόντων: • Οι χαµηλές τιµές καυσίµου σε συνδυασµό µε τις µικρότερες καταναλώσεις στα νεώτερα πλοία έκαναν τα υποδείγµατα βέλτιστης ταχύτητας µικρής σηµασίας. • Η µεγάλη αστάθεια της ναυλαγοράς οδήγησε σε απώλεια εµπιστοσύνης προς τα ποσοτικά υποδείγµατα. Κύριος στόχος της έρευνας αυτής είναι να παρουσιάσει υλοποιήσεις ωρισµένων υποδείγµάτων που παρουσιάσθηκαν στην προηγούµενη έκθεση. Αναπτύχθηκαν έτσι δύο υποδείγµατα, το πρώτο σχετικά µε την Στρατηγική Ναυλώσεων και το δεύτερο σχετικά µε την ∆ιαφοροποίηση των Επενδύσεων στην Ναυτιλία. Το υπόδειγµα της Στρατηγικής Ναυλώσεων εξετάζει το πρόβληµα της επιλογής ναύλου όπως αντιµετωπίζεται από ένα ιδιοκτήτη σε µία γνωστή και ενδεχοµένως περιοδική αγορά. Αναπτύσσεται η σχετική θεωρία µε έµφαση στις µεθόδους επίλυσης των εξισώσεων που προκύπτουν, και εφαρµόζονται στην κατηγορία των Πλοίων Χύδην Φορτίου Capesize µε στοιχεία του 1995. Οι µέθοδοι που αναπτύσσονται είναι άµεσα εφαρµόσιµοι στο πλαίσιο των τρεχόντων αποφάσεων ναυλώσεων. Σχετικά µε την ∆ιαφοροποίηση σε Ναυτιλιακές Επενδύσεις η βασική παρατήρηση είναι ότι ναι µεν οι συσχετίσεις ανάµεσα στις αποδόσεις πλοίων διαφόρων κατηγοριών είναι µεγάλες αλλά οπωσδήποτε όχι τέλειες. Ετσι γεννάται κάποια περιορισµένη δυνατότης διαφοροποίησης στην επιλογή του στόλου. Με επιλογές µεταξύ χαρακτηριστικών τύπων πλοίων εφαρµόζεται η θεωρία χαρτοφυλακίων Markowitz γιά να υπολογισθούν οι δοµές των αποτελεσµατικών στόλων και το αποτελεσµατικό σύνορο αποδοσης κινδύνου. Η δοµή του στόλου στην περίπτωση της ύπαρξης περιουσιακού στοιχείου βέβαιας απόδοσης έχει ιδιαίτερη σηµασία, καθώς η δοµή αυτή είναι ενιαία γιά οποιαδήποτε προτίµηση του επενδυτή ανάµεσα σε απόδοση - κίνδυνο. Τέλος, υπολογίζεται στην έκθεση η συσχέτιση των αποδόσεων µεταξύ των παραπάνω κατηγοριών πλοίων και της γενικώτερης αγοράς (στην συγκεκριµένη περίπτωση του Χρηµατιστηρίου Νέας Υόρκης). Έτσι εκτιµώνται γιά τις ναυτιλιακές επενδύσεις οι συντελεστές βήτα καθώς και θεωρητικά αναπροσαρµοσµένες αποδόσεις που λαµβάνουν υπόψη τον σχετικό βαθµό κινδύνου σύµφωνα µε την θεωρία Αποτίµησης των Περιουσιακών Στοιχείων. Τα παραπάνω υποδείγµατα αναπτύχθησαν σε εµπορικά φύλλα λογισµικού µε δυνατότητες µαθηµατικού προγραµµατισµού. Το δεύτερο υπόδειγµα µπορεί να υλοποιηθεί χωρίς καµµία δυσκολία. Το πρώτο υπόδειγµα είναι δυσκολώτερο και είναι διαθέσιµο από το Κέντρο µετά από αίτηση των τυχόν ενδιαφεροµένων.
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0.2.
EXECUTIVE SUMMARY
This report is a continuation of a previous one on the use of Quantitative Methods in Shipping5. It was observed in that report that the use for strategic purposes of Quantitative Methods in shipping is limited, but with prospects of wider use. Since then (1992), quantitative methods have rather lost ground, under the influence of the following factors • Low oil prices and decreased fuel consumption of modern vessels have made optimal speed models of little importance. • Wide charter market volatility has decreased the confidence in quantitative assessments. The main objective of this research is to provide practical implementations of some of the models appearing in the first report. Two models were thus developed, one relating to Positioning and the other concerning Diversification in Shipping. The Positioning model considers the problem of charter selection faced by an operator of a single vessel in a deterministic, periodic market. The relevant theory and solution methods (suitable for spreadsheet implementation) are developed and applied to the Capesize Bulk Carriers with data for 1995. It is evident that the positioning methodology suggested are readily applicable to the everyday chartering environment. The Diversification model starts with the observation that the correlation between the returns from operationg various types of vessels is high but not perfect, and thus there exists some limited possibility for diversification in the structure of fleets. The standard Markowitz theory is applied to obtain the risk - return tradeoffs between six representative types of vessels. The portfolia corresponding to the efficient set are also determined. Of particular interest is the single, "optimal" portfolio that is obtained in the presence of a riskless asset, which is invariant with the risk preference of the investor. Finally, the rates of return from the vessel types are compared to that of the "overall" market (in particular the NYSE). We thus obtain an estimate of the beta coefficients of various shipping investments, and hence the risk adjusted required rates of return from shipping investments. Both models were calculated in a commercial spreadsheet environment with mathematical programming features (for the Diversification application). The second model is quite straightforward. The first spreadsheet is complicated and is available from the Research Center upon request.
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Magirou V., H. Psaraftis and N. Christodoulakis [1992], Quantitative Methods in Shipping: A Survey of Current Use and Future Trends, Report No. E115, Center for Economic Research, Athens University of Economics and Business.
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1.
INTRODUCTION
1.1.
SCOPE OF THE REPORT
The terms of reference for this report were to follow up on a previous report on Quantitative Methods in Shipping (Magirou et al [1992] ) and in particular to: • Survey the use of quantitative models in shipping. • Implement some of the models appearing in the previous report. The first task was carried out through personal contacts with several practitioners. The results of these contacts can be summarized as follows: • Due to the improved ship technology and low fuel prices, the previously important optimal speed models are of lesser impact • Due to the increased charter market volatility since the end of 1994, operators have lost confidence in quantitative models • Increased availability of information systems has created the misconception that a computerized system is equivalent to the use of quantitative methods • The increased reliance on hedging mechanisms (BIFFEX Futures trading, Forward Freight Agreements, Contracts of Affreightment) have given rise to the need for more sophisticated hedging strategies Overall there is no increase in the use of Quantitative methods since the last report [1992]. Hence, the second task of this report is perhaps more important: to devolop practical implementations of particularly interesting models. We examine in the sequel of this report two such problems: • Vessel Positioning. • Diversification and Risk - Return Tradeoffs. These models were outlined in the earlier report. In the present report we provide a fairly comprehensive implementation of both in a commercial spreadsheet environment, thus easy to implement in everyday decision making. The first spreadsheet which is relatively complicated is also available from the Reseach Center upon request. It is thus hoped that this report will actually contrubute to the wider use of Quantitative Methods in Shipping.
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1.2.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the discussions they had with • P. Pappas, W. Kuijpers and B. Tsakonas of Oceanbulk Maritime S.A. (Athens, Greece), • A. Polemis of Polembros Shipping Ltd. (London, UK) • Professor C. Grammenos, Director, International Centre for Shipping, Trade and Finance of the City University Business School, London. • M. Kokkinis and S. Galanos of Golden Destiny S.A., Shipbrokers (Piraeus, Greece). Of course the authors are solely responsible for all errors and omissions.
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2.
POSITIONING TRAMP VESSELS
2.1.
INTRODUCTION
Tramp vessels operate in a market (the spot charter market) which, although notoriously volatile can show various regularities, mainly seasonal. One of the aims of the operator of a tramp vessel is to take advantage of these regularities by selecting charters which place the vessel at the right place at the right time. In other words, the vessel should be near regions at times when the demand for charters originating in these regions is high. This aim is referred to as the positioning goal. A positioning strategy should in principle specify the charter to be selected in every possible circumstance, i.e. at every location and time, and in such a way that if followed, it will lead the vessel at the profitable locations at the appropriate time. Thus positioning is an important aspect of chartering decisions. There are many ways for coming up with reasonable chartering decisions, the most prevalent one being selecting the charter that maximizes the daily revenue net of voyage costs (fuel, port and canal costs). This is referred to as maximizing the TCE (Time Charter Equivalent). However the TCE calculation is usually performed on the basis of one voyage, possibly including a ballast leg, and is, in that sense, a myopic criterion. Future charters can be partially taken into account by adjusting the TCE for the desirability or the undesirability of the destination. It is straightforward to perform the TCE maximization not only for a single voyage but for many future ones provided that the future charter rates can be quantified so as to reflect the market projections of the vessel operator. These projections are usually about the charter market as a whole (is it rising, falling or stable, how strong are the seasonal effects going to be, are there any impending exogenous effects?), and may take into account peculiarities such as a glut of vessels in a particular area. In principle the maximization should be performed in a discounted fashion, incorporating thus time value of money considerations. Other considerations should in principle be taken into account in positioning: Is the vessel due for drydocking in some particular shipyard and thus must be located near that yard in the near future? If the vessel is operated jointly with other vessels as a fleet, positioning should be done for the fleet as a whole rather than for each vessel separately. Finally, if the vessel is a combined carrier, the cost of switching among cargos should be taken into account. In this report we develop a deterministic dynamic model whose aim is to facilitate positioning decisions for single vessels operationg in the tramp voyage market. The model was formulated in Magirou, Psaraftis and Christodoulakis [1992], and is in the spirit of Devanney [1970] but simpler in a sense since it does not take into account layup decisions and other features appearing there. On the basis of exogenous predictions about the charter market, the model determines the positioning policy that maximizes the revenue (discounted or average) over an infinite horizon. The model implementation incorporates yearly seasonalities, and can also take into account overall market trends. The calculations required are easily performed on a spreadsheet, and in fact the results presented here have been computed in a standard commercial spreadsheet environment.
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Tramp market operators are extremely wary of any complex models, and they have a point. Uncertainties in the charter market are so marked that any kind of quantitative analysis can prove dangerous, since it provides a false sense of objectiveness. Furthermore, the working method of most decision aids is not transparent and hence not credible to practitioners. Decision making in practice involves so many considerations about the market that translating them into mathematical model input might prove of questionable value: by the time one has input all features required by a quantitative model, the market opportunity may be gone. We feel however that the model presented here is so transparent and straightforward that it can provide important support in everyday chartering decision making. In the next section we describe the modelling approach, present the fundamental formulation, discuss the solution methods. In Sections 2.3 and 2.4 we present a Case Study for Capesize Bulk Carriers. Finally, Section 2.5 presents directions for further work. The theoretical justifications of the results are presented in the Appendix, Section 2.6.
2.2.
A POSITIONING MODEL
In deciding about vessel positioning, each tramp vessel operator acts on the basis of some implicit predictions about the charter market as a whole. The overall market is usually characterized as rising, falling or stable. Furthermore, seasonal fluctuations reflecting yearly trade seasonalities are also to be taken into account. These seasonalities are assumed to be superimposed on the overall market trend. An obvious question is how operators assess overall market prospects. Numerous specialized publications and even software are available. Although predictions can prove unreliable, especially in the long term, we will assume that operators can quantify their predictions and be prepared to act upon analysis based on these quantifications. Given an operator's ex ante expectations one can try to determine the policy that will maximize some profit indicator. By policy we mean a rule that specifies for each relevant world location and each time period the charter to undertake. We will assume that the market forecasts are deterministic, i.e. the operator acts as if he had perfect foresight about the market. Thus the proposed policy's success will depend upon whether forecasts prove correct ex post. Concerning profit indicators, the usual criterion for chartering decisions is the Time Charter Equivalent as discussed in Evans and Marlow [1990]. This is essentially the average daily revenue net of costs directly related to the voyage (fuel, canal dues, port costs). To take an example from Evans and Marlow, a ship located in Rotterdam is fixed from New Orleans to Yokohama for 25,000 ±5% tons Scrap at $25 / ton minus a commission of 3,75%. The ship loads (due to draft considerations) 25,575 tons, obtaining a Net Revenue of $591,432 (net of commission). The Voyage Expenses (Fuel, Port Costs, Canal Dues) are calculated at $293,360, leading to a Voyage Surplus of $298,072. Since the total days of the voyage from Rotterdam to New Orleans and then to Yokohama are 62, the Daily Surplus is $ 4,808 per day. The TCE (Time Charter Equivalent) is defined as the Daily Surplus plus the Time Chartering commission. Assuming a 5% commission, the TCE is calculated at 8
$5,061 per day. Whether the ship makes a profit will depend on Daily Running Costs (Crew, Victuals, Administration, Financial Charges). The TCE is the main indicator for chartering decisions since the Running Costs are usually independent of the charter selection. As evident in the above example, the TCE depends crucially on the initial location: if the ship in the previous example had been located in the US East Coast, the total voyage days would be fewer and the TCE correspondingly higher. Thus the TCE indicator will depend on the ballast leg considered. Furthemore, when the planning horizon is longer, the time value of money should be taken into account. Our approach will be based on maximizing the discounted sum of future profits. It is well known, see Ross [1970], that the outcome of this optimization for small discount rates also maximizes the time average value of profits, thus maximizing TCE as well. We will also present a direct method for determining the policy that maximizes the time average value of profits. In order to avoid the dependence of the TCE calculation on the ballast leg, we consider in this paper the total surplus involved in a single voyage leg from location i to location j starting at time t. If the rate per ton is r(i,j,t), the vessel loads DWT tons (its deadweight) and the total cost is c(i,j,t) then the surplus is p(i,j,t) = r(i,j,t)DWT - c(i,j,t) The charter rate r is of course zero for a ballast leg, so the surplus p is negative and it must be counterbalanced by a positive surplus in laden legs. Without loss of generality we will assume that the surplus accrues at the loading time t and not at the discharge time, which is t+d(i,j), d(i,j) denoting the time distance between ports i and j. We assume that the overall charter market changes are manifested in changes of the surplus p, which is the usual assumption among operators. In particular, if the overall market is expected to be changing in an exponential rate π and there is a periodicity with period T then p will be of the form p(i,j,t) = (1+π)t p'(i,j,t)
(1)
with p'(i,j,t) periodic in period T, that is p'(i,j,t+T)=p'(i,j,t). This periodicity assumption about p would be true if both rates and costs are of the same functional form as (1). It is reasonable to assume that charter rates are of that form but not for too long a horizon unless the rate of increase π is zero. It is also reasonable to assume that port and fuel costs change in accordance to the overall market, since in times of good markets the cost of marine supplies is likely to increase as well. We first consider the maximization of the sum of the discounted profits for an infinite horizon, i.e. the expression ∑k p(i,j,tk) (1+ρ)tk where ρ is the discount rate, the index k ranges over all voyages and tk is the starting time of voyage k. Although the planning horizons in shipping are finite, they can be considered long enough so that the infinite horizon policy is also optimal for the initial stages of a sufficiently long finite horizon one. Thus the infinite horizon optimal policy is a reasonable one if it is recomputed at every decision period.
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In the spirit of dynamic programming let V'(i,t) be the optimal discounted profit ∑k p(i,j,tk) (1+ρ)tk when the vessel is presently located at port i at time t. It is reasonable to expect that V'(i,t+T) = (1+π)T V'(i,t), so it is simpler consider the πdiscounted value of V' i.e. V(i,t) = V'(i,t)/(1+π)t , which is obviously periodic in T, and so we just need to determine its values for t = 1,2,…T. Obviously, the knowledge of V suffices for the solution of the positioning problem. We can obtain a dynamic programming relation for V as follows: If at location i and time t the voyage to location j is undertaken, the immediate surplus of the leg is p(i,j,t). The vessel will be at location j at time t+d(i,j), from which it expects a profit to go equal to V'[ j,t+d(i,j)], which in present value terms is V'[j,t+d(i,j)] /(1+ρ)d(i,j) . The total surplus is thus p(i,j,t) + V'[ j,t+d(i,j)] (1+π)t+d(i,j) /(1+ρ)d(i,j) = (1+π)t { p'(i,j,t) + V[ j,t+d(i,j)] [(1+π)/(1+ρ)]d(i,j) } By the principle of optimality, V'(i,t) equals the maximum with respect to the destination j of the above expression, and taking into account that V'(i,t) = V(i,t)/(1+π)t we obtain the following dynamic programming relation for V(i,t) { where 1+δ = (1+ρ) / (1+π) }: V(i,t) = max locations j { p'(i,j,t) + V[ j, t+d(i,j)] / (1+δ)d(i,j) } Since p'(i,j,t) is periodic in t with period T it is obvious that the V(i,t) defined above is periodic in T, i.e. V(i,t) = V(i,t+T) and hence the above dynamic programming equation simplifies to V(i,t) = max locations j { p'(i,j,t) + V(j,t') / (1+δ)d(i,j) } and t' = ( t+d(i,j)-1) modT+1
(2)
The above dynamic programming equation has a finite solution provided that the dscount rate δ is positive, or equivalently the discount rate ρ is greater than the expected market growth π. Determining the actual solution can be done by several methods as described in the Appendix. The proofs of the above claims are straightforward and are also given in the Appendix. Another important optimization criterion is the average value of profits over time, under the assumption of steady rates up to periodicity and no discounting. A dynamic programming approach to this problem is as follows: Assume that an optimal positioning policy has been determined and leads to a "best" average profit rate α. If a vessel is located at location i at time t, its profits for a long horizon H are assumed to be of the form hit+αH, the term hit denoting the benefit resulting from being at location i at time t relative to some standard location and time (say location 1 at time 1). If the choice is to go to location j, the resulting profit will be the voyage profit p(i,j,t) plus the profit for the rest of the horizon. But the remaining horizon is Hd(i,j), while the starting location is j at time t+d(i,j), and thus the profit to go is hj,t+d(i,j)+α[H-d(i,j)]. Therefore the optimal destination is determined by maximizing over j the expression p(i,j,t) + hj,t+d(i,j) + α[H-d(i,j)]. The outcome of this maximization should equal the optimal value, which is by definition hit+αH. Therefore we get the equation
10
hit + αH = max locations j { p(i,j,t) + hj,t+d(i,j) + α[H-d(i,j)] } or simplifying hit =max locations j { p(i,j,t) + hj,t+d(i,j) - αd(i,j)} . It is customary in shipping to deal with the daily rates, say r(i,j,t) = p(i,j,t)/d(i,j), and thus the previous equation becomes hit = max locations j { (r(i,j,t) - α)d(i,j)+ hj,t+d(i,j) }. If we assume furthermore that the rates are periodic, i.e. r(i,j,t) = r(i,j,t+T), then it is easy to verify that the h's should also be periodic and of the same periodicity. We finally get the analog of (2) as hit =max locations j { (r(i,j,t) - α)d(i,j)+ hj,t' }
t' = ( t+d(i,j)-1) modT+1
(3)
A rigorous derivation of (3) follows the standard analysis for Markovian Decision Processes (Ross [1970]) where it is also rigorously proven that solving (3) leads to a solution of the original maximization problem. It is instructive to interpret (3) in terms of the vessel positioning problem. When located at location i, the operator computes for each available charter its benefit relative to the optimal average return α, i.e. (r(i,j,t) - α)d(i,j). To this value he adds the expected profit of being at location j at time t+d(i,j), and chooses the charter maximizing this sum. This decision procedure is close to actual chartering practice: If a vessel is located at an unfavorable spot from where cargos are available at low rates (r< α), it will prefer to go to a better location even in ballast (with r=0), rather than take up a charter with positive r which ties him up for a long period in the same unfavorable location.
2.3.
AN APPLICATION TO CAPESIZE BULK CARRIERS
The model was applied to Capesize Bulk Carriers, with data typical of the market conditions at the end of 1995 - beginning 1996. These carriers are of 125 -150,000 DWT and their main routes are carrying coal, iron ore and sometimes grains to Japan and North West Europe from Australia (Coal, Grains), South Africa (Coal), South America (Iron Ore), North America (Coal) or the Caribbean (Iron Ore). For the purposes of this case study, the world is divided in six major locations, although a more detailed division would be desirable if one were to carry out a case study for smaller vessels. The locations considered were: US East Coast (USEC), South Africa (S. Africa), South America (S. America), North West Europe (NWE), Australia and Japan. The rates per ton were obtained from specialized publications like the Shipping Intelligence Weekly published by Clarkson Research Studies Ltd. The rates relevant for the end of 1995 - beginning 1996 are shown in Table 1. The voyage S. Africa to Japan is not really practiced as vessels usually load coal in Hampton Roads, Virginia up to their maximum draft and then proceed to Richards Bay, S. Africa to load extra cargo. Thus the rate USEC to Japan includes an implicit haul from S. Africa, and there are few cargoes exclusively from S. Africa to Japan. The S. Africa to Japan rate was calculated implicitly from TCE considerations, but as will be shown in the next section it does not play any role in charter selection.
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Of the above routes the ones corresponding to backhauls from Japan to NWE (Australia or S. Africa to NWE) are usually at discount and show smaller TCE's, since many vessels are available for these routes after discharging in Japan. These routes show less marked seasonal patterns from the remaining, standard routes. Seasonal fluctuations were estimated informally by operators and are given in Table 2. Voyage costs consist of fuel, port and canal costs. The only relevant canal is Suez and its toll is at $100,000 per passage. We assume that ships will always use the canal, although it is possible to incorporate this choice in the model, at the cost of higher dimensionality. Port costs are given in Table 3. We consider a Capesize Vessel of 125,000 DWT with a nominal speed of 12 knots and fuel consumption of 42 tons per day. One can modify the speed within small intervals provided daily consumption is modified by say the cube law {see Evans and Marlow [1990] }. Since diesel oil consumption for electricity generation is almost constant per day, we do not take it into account. The time step is one week, with a periodicity of one year. The year is conventionally set at 48 weeks, meaning that a time step in the model consists of 365/48 = 7.6 days. The time index t ranges from 1 to 48. We assume that the ship takes 7 days to load or discharge. To illustrate the time calculations, consider a vessel that is loaded and ready to sail from the USEC to Japan, a distance of 14,776 miles at the beginning of period 5. It will need 14,776/(24*12)=51.3 days to sail and 7 days to discharge, a total 58.3/7.6 = 7.7 time steps, which we round to the nearest integer, i.e. 8 steps. Since the leg from Japan is in ballast, the vessel will be ready to sail immediately at the beginning of period 13. If the ship were in the USEC at period say 46, it would be ready to sail from Japan at time step 5 of the new year, which is consistent with the time t' specification in (2) and (3). The discount and the market growth rate are input in yearly form, and are adjusted to weekly rates by the compound interest method, namely ρweek= (1+ρyear)1/48 -1, and similarly for the market growth rate π. Naturally, no discount or growth parameters need to be specified for the average profit case.
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2.4.
CASE STUDY RESULTS
To determine optimal positioning policies one has to actually solve equations (2) and (3) for the discounted and average case repectively. The methods used to solve these equations are presented in the appendix. All calculations wre carried out in a commercial spreadsheet environment, described in detail in Magirou et al [1997]. We calculated positioning policies for several combinations of discount vs market growth parameters (scenaria) as well as the policy corresponding to a steady market - no discounting. Scenaria 1,2 and 3 assume a 15% rate of discount, which is reasonable in view of the risk involved in capesize vessel investments. Market change is assumed at 0%, 10% and -10% p.a. respectively. All scenaria assume a 12 knot vessel. Scenario 4 examines the situation in a rapidly falling market while Scenaria 5 and 6 consider faster ships with the same fuel consumption, i.e. incorporating more modern technology. Finally, Scenario 7 refers to average profit optimization. These scenaria are summarized in Table 4. Table 5 gives the positioning strategy for Scenario 1 (Stable market, 15% discount). This table displays the charter specified by the policy when the ship is in a specific location and time. Thus, if the ship is in NWE in the 2nd week of May, the corresponding charter is found in the column corresponding to NWE, and the second of the May rows. The table entry is USEC, and thus the policy suggests that the ship should be fixed for the USEC. Table 6 gives the sequence of charters of a vessel following the suggested policy for 10 voyages (20 legs), assuming that actual rates match the predicted ones. An examination of Tables 5 and 6 shows that the optimal voyage is usually the USEC - NW Europe one. Vessels load in the USEC, unload in Europe and back again. However, when in the summer rates for the route fall to 60% (August), it is best to try to get a charter in a route that does not fall by that much, ie either the Australia - Europe or the S. Africa - Europe. Thus voyages USEC - Japan and NWE - S. Africa are selected. Note that if a vessel is initially located in a different part of the world, the optimal strategy will lead it towards the USEC - NWE route. Also note that the voyage S. Africa - Japan is never undertaken. The above positioning strategy is insensitive to the small market changes (±10%) examined in Scenaria 2 and 3. Furthermore, it remains essentially the same for falling discount rates (calculations not shown here) and more important, for the average case of Scenario 7. Scenario 4 examines what happens in a rapidly falling market, which decreases by 40% annually. The results in Table 7 show that the preferred route in the second half of the year is USEC - Japan - Australia - NWE - USEC instead of the USEC NWE round trip. This is reasonable because in a falling market longer routes are more favoured, since they lock on a profit for a longer time. Thus the Japan route which was not profitable for a stable market is at a premium in a falling one, and thus the optimal strategy selects it for several summer trips as well as for the winter period. Scenaria 5 and 6 correspond to a stable market and 15% discount for faster ships doing 13 and 14 knots at the same fuel consumption. The optimal strategy for a 13 knot ship is very similar to that of a 12 knot ship except for the observation that it 13
calls for waiting (idle) in the USEC or the NWE in August, expecting an improvement in charter rates in the fall. For a 14 knot ship though (Table 8), the preferred routes are either USEC - NWE - USEC or the USEC - Japan - S.Africa - NWE - USEC instead of the route USEC - Japan - Australia - NWE - USEC. This can be explained by noting that a faster ship is better suited for longer distance trips, and the Japan to NWE via S. Africa route is longer than via Australia. The results for ships of different speeds show that vessels of different characteristics will follow different routes. Obviously, different routes will be used by operators who have differing projections about the market. In general, one would expect that the aggregate behaviour of operators will not lead to market imbalances, namely all ships aiming to do the same route, since such a behaviour would of course alter the charter rates offered. It is an interesting question to incorporate positioning questions in overall models for the charter market, like the one outlined in Stopford [1993]. Although the positioning strategy in the average case is similar to that of the discounted case and thus gives nothing new, there is a useful side product of the calculation, namely the optimal average profit α and the location values hit. These are shown in Table 9 and graphically in Figure 1, where the effects of seasonalities can be directly observed. One notes that the high values are all located in the Autumn, while the low points fall in two categories: The Pacific locations (Japan, Australia, S. Africa) show an earlier low point (April or May) than the remaining locations (US, NWE, S. America) for which the low values are in the summer. The α and hit parameters are all that is needed in order to calculate the optimal positioning strategy6, using the dynamic programming relation (3). As an example of the dynamic programming approach to positioning, consider a vessel of 125,000 DWT located in Australia in the first week of May. It is offered two charters, not necessarily compatible with the nominal rates shown in Table 1, and for durations again not compatible with the vessel's speed. In particular, it is offered two time charters, one to Japan for 4 weeks at $170,000 per week, and another to Europe at $145,000 per week for 6 weeks. The operator can accept one of the charters or turn down both and wait for a better offer. To evaluate the charters one should consult Table 9, under the implicit assumption that the overall market is still consistent with the parameters assumed in the case study. Table 9 shows an average revenue of $101,457 per week, which we round to $100,000 for ease of calculation. Furthermore, the value of being in Japan in the first week of June (when the charter ends) is -$593, 853 (rounded to -$590,000) and the value of being in Europe in the third week of June is -$316,798 (rounded to $315,000). To evaluate the Japan charter, its profit relative to the optimal is 4 * (170,000 -100,000) - 590,000 = - $310,000. Similarly, for the Europe charter the value is 6 * (145,000-100,000) - 315,000 = - $45,000. Thus the Europe charter leads to a smaller loss relative to the optimal and is preferable to the Japan charter, Concerning whether the operator should refuse both charters and wait for some time, we note that the option of staying idle for one week has a value of about 6
In the discounted case it is the solution to (2), V(i,t), that is required to compute the optimal policies using (2). However, this solution lacks intuitive appeal. The advantage of the α, hit parameters is that they are easy to interpret in the setting of everyday chartering decisions.
14
$500,000 (since the value of being in Australia in the second week of May is about $400,000 while the weekly optimal rate is $100,000). Thus the Australia - Europe time charter should be accepted. On the other hand, consider a case where the only available choice is a time charter to Japan for 4 weeks at $110,000 a week, a very bad situation indeed. Its value would be 4 * (110,000-100,000) - 590,000 = -$550,000. This figure is even less than the value of the alternative of waiting idle in Australia for a week. Thus the Japan charter should be turned down. A further application of the location values is in the choice of delivery location of vessels that are candidate for purchase. Assume that a perspective owner considers the purchase of one of two identical 125,000 DWT vessels, one that can be delivered in the first week of September in Japan, while the other is to be delivered again in the first week of September in the US East Coast. Of course the second vessel is at a premium since it can be chartered immediately. The premium can be calculated by examining the difference between the two location values which is about $700,000 (the USEC value for September is about $350,000 while that for Japan is -$350,000). A similar estimate can be obtained by looking at the optimal discounted revenue which gives similar figures (for a discount rate of 15% the value is $500,000).
2.5.
CONCLUSIONS AND EXTENSIONS
The case study presented above showed that the variations in discount rates do not seriously affect positioning strategy, which is more sensitive to variations in the overall market. Changes in the vessel characteristics seriously affect the choice of charters, and so does variations in the relative level of rates and their seasonalities. The model presented above can be extended in various ways. We already noted that the decision of using a canal or not can be incorporated in the model by including the choice between two types of voyages, one using the canal and the other not using it. Incorporating combined carriers is more difficult as it requires the expansion of the state space. It is not clear what is the best way to incorporate uncertainty in a positioning model, something which is required in case one intends to apply such models to positioning a whole fleet. A finite horizon positioning model can be easily implemented and would be useful in case an operator is not satisfied with the infinite horizon assumptions employed here. Of course, unless the horizon is reasonably long such a model would depend on the specification of the profit at the end of the horizon. One can use the values of this model as end of horizon values in a finite time model. We note that such finite horizon models would be useful in case an operator must incorporate drydocking decisions in the overall chartering framework. Finally, as an extension of the classical optimal speed problem (See again Evans and Marlow), the model can be run for various speeds with consumption following a cube law, and the speed for which maximizes the time average return is the generalized optimal speed.
15
2.6.
APPENDIX
2.6.1. Dynamic Programming Formulation It is straightforward to show using standard dynaminc programming techniques as in Ross [1970] that equation (2) describes completely the solution of the problem in the sense that if an optimal policy exists, it gives rise to a value function satisfying (2) and conversely the (unique) solution of (2) determines the optimal policy. We provide a sketch of the proof of these claims. Let V(i,t) be a solution of V(i,t) = max locations j { p'(i,j,t) + V(j,t') /(1+δ)d(i,j) }
(2)
and let dmax(i,t) be defined by the inversion of (2), i.e dmax(i,t) = arg [ max locations j { p'(i,j,t) + V(j,t') /(1+δ)d(i,j) } ] We claim dmax(i,t) is optimal. N
To justify this claim consider JN =
∑
p(ik,ik+1,tk)/(1+ρ)tk. Consider also
k =1
N
LN= ∑ p(ik, ik+1,tk)/(1+ρ)tk + V(ik+1,tk+1)/ (1+δ)tk+1 - V(ik,tk)/(1+δ)tk k =0
It is easy to see by telescoping sums that LN = JN + V(iN+1,tN+1)/(1+δ)tN+1 - V(i1,t1)/( 1+δ)t1 Since LN can be written as N
LN= ∑ [ p'(ik, ik+1,tk) + V(ik+1,tk+1)/ (1+δ)tk+1-tk -V(ik,tk) ] /(1+δ)tk k =1
it follows from the dynamic programming equation (2) that each term in the sum in nonpositive, and so is LN. Therefore JN ≤ V(i1,t1)/(1+δ)t1 - V(iN+1,tN+1)/ (1+δ)tN+1 The expression becomes an equality if a policy satisfying (2) is chosen. Furthermore, the term V(iN+1,tN+1)/(1+δ)tN+1 approaches zero for large N, by the finiteness of V. Based on these observations, the correctness of the dynamic programming approach is evident. A similar approach works to show that (3) suffices to determine the optimal average profit maximizing policy. It will be further explained in the next section. 2.6.2. Solution methods The dynamic programming equation can be considered as one corresponding to a Markovian Decision Process (MDP) in states (i,t), where the decision to take a charter to location j corresponds to the deterministic transition to (j,t') with t' defined consistently with (2). Hence we can apply he standard theory for the solution of the 16
dynamic programming equations in MDP's. We refer to the comprehensive survey of Puterman [1990]. Although the policy iteration algorithm would be the natural choice for the solution of (2), the problem size does not allow a spreadsheet implementation. We use instead the modified policy iteration method, which consists essentially of two steps: The matrix inversion step consists of a number of Gauss-Seidel iterations to solve for the value function corresponding to a certain policy. The policy iteration step adjusts the candidate optimal policy through the updating of the k-th iterate of the value function Vk by the expression Vk+1(i,t) = max locations j { p'(i,j,t) + Vk(j,t') /(1+δ)d(i,j) }
(2')
The method does not guarantee to converge in a finite number of steps as does policy iteration. However, it can be shown that if the successive iterates Vk are close enough, then the policy maximizing the right hand side of (2') gives rise to a value function that is ε - close to the optimal. Proposition 6.23 in Puterman [1990] gives a precise and effective stopping criterion in terms of ε,δ. In our examples we use an accuracy of 1%, i.e. we stop when ε is guaranteed to be less than 1% of the current value function. Solving in the case of average profit can be accomplished either by considering the problem as a Markovian one or as a Semi-Markovian one. Again the appropriate value iteration methods are surveyed in Puterman [1990]. However these methods are indirect in the sense that they proceed by reducing the problem to a Markovian one. We used instead a solution procedure which is outlined in the following paragraphs. The approach relies in two straightforward lemmas, similar to the standard ones appearing in Puterman. For convenience, we will drop the time subscript t and use only a location subscript, i or j. Lemma 1. Assume that there exist constants hi i=1,..,N and α such that hi ≥ (cij - α)dij + hj
∀ i,j.
Then the average return is less than α. Lemma 2. Assume that there exist constants hi i=1,..,N and α, ε≥ 0 such that hi - ε ≤ max j {(cij - α)dij + hj }
∀i
Assume also that dij ≥ 1. Then the policy defined by going from location i to the location specified by arg max j {(cij - α)dij + hj } achieves an average profit greater than α-ε. A direct algorithm to determine the solution of (3) can be obtained as follows: Select arbitrary h's and try to find a α such that the condition of Lemma 1 is satisfied. The smallest such α is given by α(h) = max jj {cij+ (hi-hj)/dij}, and by Lemma 1 provides an upper bound to the average profit. If we now update the h's by using α(h), i.e. h'i =max j {(cij - α(h))dij + hj}, one can verify that h'j≤hj. Furthermore if we repeat 17
the procedure with the new h' we get an updated α=α(h') which is smaller than before, i.e. α(h) ≥ α(h'), giving thus a better upper bound to the average profit. In case α(h) = α(h') it is not true that the α's are optimal. However due to the convexity of the function α(h), it can be shown that α([h'+h]/2) < α(h'), thus again obtaining a better upper bound with h'' = [h'+h]/2 . In either case we get a decreasing sequence of α's and h's which will converge to the solution of (3). Lemma 2 can be used to assess whether the procedure has converged to the required accuracy. 2.6.3. Using the Bulkpos spreadsheet The implementation of the discounted case of the model described earlier is in the EXCEL 5 spreadsheet BULKPOS (BULKPOS.XLS). It has been written for an experienced EXCEL user. Its operation is described in this section, which should be read after the rudiments of the model have been understood. Bulkpos consists of the following sheets: • Distances • Rates • Revenues • Parameters • DP • Iterations • Routes • DP-Macro • Aux1 and Aux2 We describe the function of each sheet. 2.6.3.1. Distances This sheet has two matrices. The first (A14;G20) contains the distances among the various geographical areas in nautical miles. The second (A22;G28) contains 0 for routes that do not involve a Suez Canal passage and 1 otherwise. This data is used for adding one day to voyage duration and Suez canal tolls expenses to the voyage costs. The rows of each table are origins/departures and the columns destinations/arrivals. 2.6.3.2. Rates The first table (B4;H11) of this sheet contains the freight rates in $/ton, assumed to hold in the beginning of a year, as well as a row of bunker prices at each location. The second table (Row 15 up to 97) contains the multiplicative seasonality indices for each route on a monthly basis (with 100 denoting the January rate). The last column contains seasonality indices for bunkers at each location. The remainder of the sheet (below row 101) is reserved for internal calculations. 2.6.3.3. Revenues The table of this sheet contains the gross profit per trip (revenues minus voyage cost) in USD. There are 72 rows (6 origins × 12 months) and 6 destinations. The gross profit of a voyage depends on the starting month due to seasonal freight rates 18
and bunker prices' fluctuation. This table is computed by the sheet, but it should be consulted by the user to check for possible errors, since it is the most important input to the algorithm. In interpreting it, note that negative entries stand for ballast legs (zero freight rate). 2.6.3.4. Parameters This sheet contains the necessary parameters affecting the economic performance of the vessel: DWT, Speed and Consumption, Laytime as well as the relevant interest rates. There are two interest rate parameters, the discount rate (B13) and the expected rate of market appreciation (B12). Both rates are to be input in yearly form. The program calculates the overall discount to be used in the calculations (i.e. the δ parameter introduced in the text). The calculations produce a table called TIME DISTANCE containing the total voyage duration (in days) for each leg, which is discretized to the period used in the model (area dist_wk). Among the parameters to be used is nominal speed and nominal consumption at that speed. Another input is the actual speed (cell C8), and based on the actual speed the sheet computes the actual consumption (cell C9) using the cube law. Nominal speed, consumption figures can also be changed by the user (Cells B5,B6). 2.6.3.5. DP The sheet contains the cells required to implement the basic policy iteration step as shown in equation A.2. The table from row 9 and on has rows corresponding to each time-location combination (a total of 6*48 rows), while the columns correspond to the locations to be visited next. The cells contain the revenue corresponding to the given column-row combination, while the last column (Column K) computes the maximum over the candidate locations. The entries in the main part of the table are computed using the relevant data for the immediate revenues (Revenue sheet) and the discounted value of future revenues. The latter are computed iteratively inside the DP-Macro loop. The previous iterate appears in Column M of this same sheet. Finally, the current optimal policy appears in the sheet in Column O, and is repeated in more readable form in the sheet Routes. 2.6.3.6. Iterations In this sheet the user must specify • The desired percentage error in the value function - Cell C2 • The maximum number or policy improvement iterations - Cell C3 • The number of policy evaluation iterations - Cell C4 • Whether it is desired to start with a zero initial value function of with the previous one - Cell H2 The Visual Basic Program implementing the Modified Policy Iteration Algorithm is invoked as a Macro (Menu Tools - Macro) under the name Running. The results of the macro calculation appear (to some extent) in this sheet. Upon the end of the calculation the sheet contains the elapsed time and the guaranteed percentage error that was achieved after the number of iterations specified by the user. In case the desired error is not below the desired one the program must be rerun, keeping the same initial value (ie set H2 at 0). The program execution will stop if specified error reached earlier.
19
2.6.3.7. Routes This sheet contains the results of the Running Macro execution. Of course, for the results to be reliable the desired error must have been reached. The first part of this sheet contains a route table with 10 consecutive voyages. It extracts the result from the positioning strategy that appears from line 32 and on in this same sheet. The user should fill in the ORIGIN and the DEPARTURE WEEK and then press Shift-F9 to perform the required calculations. The program will find out the details such as origin, departure week, destination, arrival week, gross revenue, Time Charter Equivalent per day and voyage duration in days for 10 consecutive voyages. It also contains the mean Time Charter Equivalent per day for these consecutive voyages. The resulting Time Charter Equivalent for the whole period is compared with the theoretically computed value from dynamic programming considerations. The two figures should be close only for small values of the discount rate and a stable market. The second table (Line 32 and on) shows the positioning policy calculated by the macro. The first column, titled TIME, contains the week of the year (1-48). The other columns contain the origin/departure locations. The cell at the intersection of a week number row with the origin/departure column shows the suggested destination of a ship leaving from the specific origin/departure location at the specific week of the year. If the destination is the same with the origin/departure column, the ship should stay for one more week at port waiting for improved freight rates. The user may go to the policy matrix by choosing the range POLICY from the range name list. 2.6.3.8. DP-Macro This sheet contains the visual basic program used to solve the dynamic programming equation. The program is a straightforward application of the Modified Policy Iteration method. The number of Policy Evaluation iterations is taken from Iterations!(C4) while the total number of Policy Improvement iterations is again from Iterations!(C3). The running time of each Policy Evaluation iteration is about 1 sec on a 486-33MHz, and about 100 policy iterations would usually suffice for a 1% accuracy. 2.6.3.9. Aux1 and Aux2 These sheets repeat some mainly from the DP sheet but are significantly shorter. They are thus used to speed up the DP-Macro operation.
20
2.7.
REFERENCES
Devanney J. [1973], A model of the tanker charter market and a related dynamic program, in Shipping Management, P. Lorange and V. Norman editors, Institute for Shipping Research, Bergen Norway and Maritime Research Center, the Hague Netherlands. Evans J. and P. Marlow [1990], Quantitative Methods in Maritime Economics, Second Edition, Fairplay Publications, Coulsdon, Surrey, UK. Magirou V., H. Psaraftis and N. Christodoulakis [1992], Quantitative Methods in Shipping: A Survey of Current Use and Future Trends, Report No. E115, Center for Economic Research, Athens University of Economics and Business. Magirou V., H. Psaraftis, L. Babilis and A. Denisis [1997], Positioning and Diversification in Shipping, Center for Economic Research, Report No. E194, Athens University of Economics and Business. Puterman M. [1990], Markov Decision Processes, in Stochastic Models, D. Heyman and M. Sobel Editors, Handbooks in Operations Research and Management Science Volume 2, North Holland, Amsterdam, Netherlands. Ross S. [1970], Applied Probability Models with Optimization Applications, Holden Day, San Francisco. Stopford M. [1993], Maritime Economics, Routledge, London UK
21
2.8.
TABLES AND FIGURES R at es in $ /t o n Destination Origin USEC S. AFRICA S. AMERICA AUSTRALIA
NWE 6.1 7.1 5.8 8.0
Japan 12.9 7.8 10.8 4.7
Table 1: December 95 - January 96 Capesize Rates in $/ton Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Standard route 100 100 110 120 110 100 80 60 80 90 100 100
Discount route 100 100 105 110 105 100 95 90 95 100 100 100
Table 2: Rate seasonalities, January = 100 Region
Port Costs (in ,000 USD)
USEC S. Africa S. America NEW Australia Japan
45 30 55 125 90 100
Table 3: Port costs
Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7
Discount rate % p.a. 15 15 15 15 15 15 0
Market change % p.a. -0 10 -10 -40 0 0 0
Table 4: Scenario description
22
Speed (Knots) 12 12 12 12 13 14 12
Origins JAN_1
WEEK
USEC NWE
S.AFRICA NWE
S.AMERICA NWE
NWE USEC
AUSTRALIA NWE
JAPAN AUSTRALIA
JAN_2 JAN_3
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
JAN_4
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
FEB_1 FEB_2
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
FEB_3
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
FEB_4 MAR_1
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
MAR_2
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
MAR_3 MAR_4
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
APR_1
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
APR_2 APR_3
NWE NWE
NWE NWE
NWE JAPAN
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
APR_4
NWE
NWE
JAPAN
USEC
NWE
AUSTRALIA
MAY_1
NWE
NWE
JAPAN
USEC
NWE
AUSTRALIA
MAY_2 MAY_3
JAPAN JAPAN
NWE NWE
JAPAN JAPAN
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
MAY_4
JAPAN
NWE
JAPAN
USEC
NWE
AUSTRALIA
JUN_1 JUN_2
JAPAN JAPAN
NWE NWE
JAPAN JAPAN
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
JUN_3
NWE
NWE
JAPAN
USEC
NWE
AUSTRALIA
JUN_4 JUL_1
NWE NWE
NWE NWE
S.AFRICA S.AFRICA
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
JUL_2
NWE
NWE
S.AFRICA
USEC
NWE
AUSTRALIA
JUL_3 JUL_4
NWE NWE
NWE NWE
S.AFRICA S.AFRICA
S.AFRICA USEC
NWE NWE
AUSTRALIA AUSTRALIA
AUG_1
NWE
NWE
S.AFRICA
USEC
NWE
AUSTRALIA
AUG_2 AUG_3
NWE NWE
NWE NWE
S.AFRICA S.AFRICA
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
AUG_4
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
SEP_1
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
SEP_2 SEP_3
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
SEP_4
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
OCT_1 OCT_2
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
OCT_3
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
OCT_4 NOV_1
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
NOV_2
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
NOV_3 NOV_4
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
DEC_1
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
DEC_2 DEC_3
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
AUSTRALIA AUSTRALIA
DEC_4
NWE
NWE
NWE
USEC
NWE
AUSTRALIA
Table 5: Positioning Strategy for Stable Market 23
T C E / D ay f o r t h e p e r i o d : $ 1 2 , 1 5 1 ORIGIN
DEPAR.
DESTINATION
STATUS
REVENU E
WEEK
DAYS
NWE
6
USEC
BALLAST
-74025
18
USEC
8
NWE
LADEN
507538
18
NWE
10
USEC
BALLAST
-74025
18
USEC
12
NWE
LADEN
582694
18
NWE
14
USEC
BALLAST
-74025
18
USEC
16
NWE
LADEN
657850
18
NWE
18
USEC
BALLAST
-74025
18
USEC
20
JAPAN
LADEN
1389023
58
JAPAN
28
AUSTRALIA
BALLAST
-111055
22
AUSTRALIA
31
NWE
LADEN
379931
48
NWE
37
USEC
BALLAST
-74025
18
USEC
39
NWE
LADEN
432381
18
NWE
41
USEC
BALLAST
-74025
18
USEC
43
NWE
LADEN
507538
18
NWE
45
USEC
BALLAST
-74025
18
USEC
47
NWE
LADEN
507538
18
NWE
1
USEC
BALLAST
-74025
18
USEC
3
NWE
LADEN
507538
18
NWE
5
USEC
BALLAST
-74025
18
USEC
7
NWE
LADEN
507538
18
Table 6: Voyages following positioning policy
24
Origins WEEK JAN_1 JAN_2 JAN_3 JAN_4 FEB_1 FEB_2 FEB_3 FEB_4 MAR_1 MAR_2 MAR_3 MAR_4 APR_1 APR_2 APR_3 APR_4 MAY_1 MAY_2 MAY_3 MAY_4 JUN_1 JUN_2 JUN_3 JUN_4 JUL_1 JUL_2 JUL_3 JUL_4 AUG_1 AUG_2 AUG_3 AUG_4 SEP_1 SEP_2 SEP_3 SEP_4 OCT_1 OCT_2 OCT_3 OCT_4 NOV_1 NOV_2 NOV_3 NOV_4 DEC_1 DEC_2 DEC_3 DEC_4
USEC JAPAN JAPAN NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN
S.AFRICA NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE
S.AMERICA NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN S.AFRICA S.AFRICA S.AFRICA S.AFRICA S.AFRICA S.AFRICA S.AFRICA NWE NWE NWE NWE NWE NWE NWE JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN JAPAN NWE
NWE USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC S.AFRICA S.AFRICA S.AFRICA USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC USEC
AUSTRALIA NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE NWE
JAPAN AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA AUSTRALIA
Table 7: Positioning Strategy for a Rapidly Falling Market
25
Origins WEEK JAN_1
USEC NWE
S.AFRICA NWE
S.AMERICA NWE
NWE USEC
AUSTRALIA S.AFRICA
JAPAN S.AFRICA
JAN_2
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
JAN_3 JAN_4
NWE NWE
NWE NWE
NWE NWE
USEC USEC
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
FEB_1
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
FEB_2 FEB_3
NWE NWE
NWE NWE
NWE NWE
USEC USEC
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
FEB_4
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
MAR_1
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
MAR_2 MAR_3
NWE NWE
NWE NWE
NWE NWE
USEC USEC
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
MAR_4
NWE
NWE
NWE
USEC
NWE
S.AFRICA
APR_1 APR_2
NWE NWE
NWE NWE
NWE NWE
USEC USEC
NWE NWE
S.AFRICA S.AFRICA
APR_3
NWE
NWE
NWE
USEC
NWE
S.AFRICA
APR_4 MAY_1
NWE NWE
NWE NWE
JAPAN JAPAN
USEC USEC
NWE NWE
S.AMERICA S.AFRICA
MAY_2
NWE
NWE
JAPAN
USEC
NWE
S.AFRICA
MAY_3 MAY_4
JAPAN JAPAN
NWE NWE
JAPAN JAPAN
USEC USEC
NWE NWE
S.AFRICA S.AFRICA
JUN_1
JAPAN
NWE
JAPAN
USEC
NWE
S.AFRICA
JUN_2 JUN_3
JAPAN JAPAN
NWE NWE
JAPAN JAPAN
S.AFRICA S.AFRICA
NWE NWE
S.AFRICA S.AFRICA
JUN_4
NWE
NWE
JAPAN
S.AFRICA
NWE
S.AFRICA
JUL_1
NWE
NWE
JAPAN
S.AFRICA
NWE
S.AFRICA
JUL_2 JUL_3
NWE NWE
NWE NWE
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
NWE S.AFRICA
S.AFRICA S.AFRICA
JUL_4
NWE
NWE
S.AFRICA
S.AFRICA
S.AFRICA
S.AFRICA
AUG_1 AUG_2
NWE NWE
NWE NWE
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
AUG_3
NWE
NWE
S.AFRICA
S.AFRICA
S.AFRICA
S.AFRICA
AUG_4 SEP_1
NWE NWE
NWE NWE
S.AFRICA S.AFRICA
S.AFRICA USEC
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
SEP_2
NWE
NWE
S.AFRICA
USEC
S.AFRICA
S.AFRICA
SEP_3 SEP_4
NWE NWE
NWE NWE
NWE NWE
USEC USEC
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
OCT_1
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
OCT_2 OCT_3
NWE NWE
NWE NWE
NWE NWE
USEC USEC
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
OCT_4
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
NOV_1
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
NOV_2 NOV_3
NWE NWE
NWE NWE
NWE NWE
USEC USEC
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
NOV_4
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
DEC_1 DEC_2
NWE NWE
NWE NWE
NWE NWE
USEC USEC
S.AFRICA S.AFRICA
S.AFRICA S.AFRICA
DEC_3
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
DEC_4
NWE
NWE
NWE
USEC
S.AFRICA
S.AFRICA
Table 8: Positioning for a 14 knot vessel 26
O P T IM A L A VER A G E P RO F IT S IN U S$ WEEKLY PROFIT
101,457
DAILY
13,350
L O C A T IO N VA L U ES IN U SD Origins WEEK JAN_1 JAN_2 JAN_3 JAN_4 FEB_1 FEB_2 FEB_3 FEB_4 MAR_1 MAR_2 MAR_3 MAR_4 APR_1 APR_2 APR_3 APR_4 MAY_1 MAY_2 MAY_3 MAY_4 JUN_1 JUN_2 JUN_3 JUN_4 JUL_1 JUL_2 JUL_3 JUL_4 AUG_1 AUG_2 AUG_3 AUG_4 SEP_1 SEP_2 SEP_3 SEP_4 OCT_1 OCT_2 OCT_3 OCT_4 NOV_1 NOV_2 NOV_3 NOV_4 DEC_1 DEC_2 DEC_3 DEC_4
USEC 462,392 441,057 429,331 427,214 434,707 413,371 401,645 399,529 407,021 366,897 336,382 315,476 304,180 245,266 195,962 156,267 126,182 86,057 55,542 34,637 23,340 -18,776 -53,278 -55,395 -39,860 -31,659 -5,807 29,654 74,725 128,546 191,976 265,016 347,665 363,907 389,759 425,221 470,292 467,745 474,809 491,481 517,763 496,427 484,701 482,585 490,077 468,742 457,016 454,899
S.AFRICA 329,636 327,519 335,012 294,887 264,372 254,541 254,319 206,479 168,249 139,629 120,618 91,567 72,127 40,147 17,776 -35,415 -80,991 -94,182 -89,721 -92,595 -77,817 -53,430 -11,391 23,314 75,670 137,635 209,210 214,378 229,156 275,692 331,837 340,365 358,502 386,249 423,605 413,344 412,692 410,575 418,068 396,732 385,006 382,890 390,383 369,047 357,321 355,204 362,697 341,362
S.AMERICA 261,933 250,207 248,090 255,583 215,459 202,912 199,975 206,648 165,703 134,368 112,642 100,525 78,369 29,885 -8,733 27,861 -1,951 -22,154 -32,747 -33,731 -57,410 -105,073 -143,127 -144,262 -72,687 -67,519 -52,741 -6,205 49,940 58,468 76,605 139,166 172,557 197,589 232,230 276,481 273,114 279,357 295,209 320,671 317,304 305,578 303,461 310,954 289,619 277,892 275,776 283,269
NWE 152,392 150,275 157,768 136,433 124,707 122,590 130,083 89,958 59,443 38,537 27,241 -31,672 -80,977 -120,671 -150,757 -190,881 -221,396 -242,302 -253,598 -295,715 -330,217 -332,333 -316,798 -308,598 -282,746 -247,284 -194,171 -148,393 -84,962 -11,923 70,726 86,969 112,821 148,282 193,353 190,807 197,870 214,542 240,824 219,489 207,763 205,646 213,139 191,803 180,077 177,961 185,453 164,118
Table 9: Average Case Optimisation Results
27
AUSTRALIA 961 -39,164 -69,679 -90,585 -101,881 -148,333 -185,177 - 212,410 -230,035 -257,698 -275,753 -284,197 -283,033 -337,610 -384,573 -399,151 -396,077 -400,337 -386,946 -363,945 -323,293 -289,976 -239,006 -178,428 -108,240 -104,458 -91,067 -68,066 -35,456 -25,542 -6,018 23,116 61,858 52,984 53,719 64,063 84,017 62,681 50,955 48,839 56,331 34,996 23,270 21,153 28,646 7,311 -4,415 -6,532
JAPAN -506,010 -517,306 -563,759 -600,602 -627,836 -645,460 -673,124 -691,178 -699,623 -698,458 -753,036 -799,998 -814,576 -811,502 -815,762 -802,371 -779,371 -738,719 -705,401 -654,432 -593,853 -523,665 -519,883 -506,492 -483,491 -450,882 -440,967 -421,443 -392,310 -353,567 -362,441 -361,706 -351,362 -331,409 -352,744 -364,470 -366,587 -359,094 -380,429 -392,155 -394,272 -386,779 -408,115 -419,841 -421,957 -414,465 -454,589 -485,104
Location values 600,000
400,000
200,000
-
USEC
Value
S.AFRICA S.AMERICA -200,000
NWE AUSTRALIA JAPAN
-400,000
-600,000
-800,000
Time
Figure 1: Location values with seasonality effects
28
DEC_1
NOV_1
OCT_1
SEP_1
AUG_1
JUL_1
JUN_1
MAY_1
APR_1
MAR_1
FEB_1
JAN_1
-1,000,000
3.
DIVERSIFICATION IN SHIPPING
3.1.
INTRODUCTION
Investing in shipping has been traditionally a risky business. In the past, the dangers of shipping were related to weather, trade risk (since trading was undertaken by shippers themselves) and even piracy. It's interesting to note that in the turn of the century shipping loans were not due to be repaid in case the ship sank7. In modern times most of the risk is related to risks of the shipping business itself, which is considerable, given the wild fluctuations in the shipping market. Relatively recently several mechanisms have been introduced aimed at reducing risk, apart from traditional marine insurance. Noteworthy are steps towards the deepening of the shipping market, as manifest in the institutions related to the Baltic Exchange Futures trading. However, fundamental risk is still an important factor, as evidenced in the post 1995 dramatic drop in rates. A traditional approach to risk reduction is that of diversification. In shipping, an obvious question is to what extent this is applicable if one wants to remain within the confines of shipping. Then diversification can be accomplished if a shipping concern is not exclusively active in one shipping sector, but operates several types of vessels. The benefit of diversification is to be weighed against possible increases in operating costs that result from running several types of vessels, and the loss in expected earnings that goes hand in hand with reduced risk. It is evident that if one remains within the confines of shipping, the opportunities arising from diversification are limited, since it is a common observation that ALL shipping rates show high positive correlations. The sources for these correlations are obvious: If say the tanker market is high, this means that economic activity is high (except in crisis situations) and thus demand for bulk carriers will also be high. Also, at a more technical level, if the tanker market is high the combined carriers will switch from bulk to oil, thus reducing the supply of bulkers and increasing the corresponding rates. Sources of negative correlations are less evident. Lack of complete correlation between the performance of several types of ships will definetely lead to some possibilities for diversification, but not much should be expected from it. It is the purpose of this research to provide the basis for such an analysis. Such ideas appeared among other sources in Lorange and Norman [1973], and reiterated in Magirou et al [1992]. To our knowledge it is the first time that actual data collection was carried out to assess in a quantitative fashion these matters. A related question to that of diversification is that of actually assessing the risk of shipping in terms of the rate of return required from shipping investments. Again, we are not aware of any such quantitative estimates, referring to vessel types per se8. 7 8
See Polemis [1995]. There are numerous studies relating to the behaviour of stocks of shipping companies in various stock exchanges, as for instance the one by Grammenos and Markoulis [1996].
29
An obvious method is to apply the Capital Asset Pricing Model, obtain the beta's for shipping as a whole, or the corresponding values for particular types of ships and thus obtain the rates of return stipulated by the CAPM. The structure of this report is as follows: We first state briefly the data used in the study. Then, following a brief introduction to Markowitz portfolio theory we review the results obtained. We next compare returns from shipping and a representative capital market (the NYSE) to obtain some estimates for the beta of shipping.
3.2.
CASE STUDY DESCRIPTION
3.2.1. Vessel Types The vessel types that could be examined are quite numerous. For convenience, we restricted our analysis to the following vessels types, as listed in Table 1. Other vessel categories (Ro-ro's, Reefers, Passenger) or ages (including newbuildings) were not used mainly for lack of published data. 3.2.2. Economic performance To assess the overall return from owning and operating a ship of a certain type, we need roughly the following information for each type: • Rates • Operating Costs • Resale value Although data is not transparent, several specialized publications regularly list such information. In Table 2 we provide a compilation of the relevant Charter rates, in Table 3 the operating costs, while in Table 4 the resale prices One is mainly interested in the economic outcome from owning and operating a ship. In the spirit of porfolio theory, we examine the yearly results from owning an operating a ship for a year, i.e. purchasing it in the beginning of the year, operating for the duration and selling it at the end of the year. Then the return on the original investment, denoted by ri,t is given by the expression ri,t = (Sale pricei, t+1 ∗(1-α)+ Yearly ratei,t -Op. Cost i,t)/ (Purchase pricei,t) - 1 where the index i refers to the vessel type, while the index t refers to the time index. The parameter α refers to the effect of ageing plus the sale commission. In practice the value of the parameter is about 3%. The results are given in Table 5. Note that the average return is slightly larger than the long term return9 (LT Return in Table 5). It is interesting to compare the average yearly returns with their standard deviation. Overall, the vessel types of higher average returns show increased risk, as 9
The expected long term return equals µ-σ2/2, where µ is the time average rate of return, while σ2 is the variance of the return (with respect to time). See Hull [1993] Ch. 10.3
30
expressed by the standard deviation. The only possible exception is type 3, (Capesize bulk carriers) which shows a much higher risk with just a slight return advantage over Panamax Bulk carriers. Graphically, the risk-return tradeoff is shown in Table 6 and Figure 1. One would also expect that returns would be highly correlated with rates. It turns out that this is not so, since the main component in the return computation is capital appreciation, and this is imperfectly, and even negatively correlated with rates. In order to apply portfolio theory to shipping we require estimates of the relevant correlations between the returns of the various types of vessels. These are shown in Table 7. We note that as expected the correlation coefficients are quite high. Still the correlations are less than perfect. They are lower between tanker and bulk carriers as a group, and higher in the bulk - tanker categories (with the exception of the correlation between small and medium tankers which is as low as 0.420).
3.3.
PORTFOLIO ANALYSIS IN SHIPPING
3.3.1. Review of the relevant theory The standard approach to portfolio analysis hinges on assuming that risk is measured by standard deviations. In this sense Figure 1 gives the risk-return tradeoffs. From the algebraic viewpoint, porfolio selection can be stated as follows: Assume that the portfolio share in vessel type i is Xi, the expected return is Ri, the standard deviation is σi while the correlation between the returns of type i,j is ρij, the portfolio variance is ∑i,j ρijσi σj Xi Xi, the expected return is ∑iriXi, and furthermore ∑i Xi = 1. Consider now the case where the investor wants to an achieva an average return greater of equal than a parameter R, while keeping the risk, i.e. the portfolio variance, as low as possible. Then this objective can be stated in terms of mathematical programming as follows min ∑i,j ρijσi σj Xi Xi = σPortfolio2(R) subject to ∑iri Xi ≥ R ∑i Xi = 1 Xi ≥ 0 i=1,..,N (but not for i=0) Note that the outcome of the minimization is a function of the relevant parameters, and by writing it as σPortfolio2(R) we stress its dependence on the required rate of return R. This dependence is known as the Capital Market Line (See Sharpe [1970]). In case the investor's choice is expanded to include a riskless asset, i.e. investing in bonds etc., the above formulation is essentially the same with the addition of an asset indexed by i=0 with constant return r, and σ0, ρ0j equal to zero. In that case the function σPortfolio2(R) is linear in R, and furthermore the investor's choice is extremely easy to describe: There is a particular portfolio of vessel types, i.e. a choice Xi* with ∑iXi* = 1, called the market porfolio. Depending on his required rate of return R the investor will split his own capital as follows: he will invest a share x in the riskless 31
asset and a share (1-x) in the market portfolio, in the shares dictated by the Xi*'s. He will thus invest a share (1-x) Xi* in vessel type i. Thus the Xi*'s in a sense represent an "optimal shipping portfolio". This property is referred to a "Separation Property". The separation property holds even if an investor can borrow at a rate equal to the riskless rate r, while it is not quite valid in case the borrowing rate is different (higher), but similar properties can be derived. Furthermore, in case the borrowing and lending rate are close, the separation property holds to great accuracy. In the next section we present the results of portfolio analysis based on the riskreturn estimates of Section 2.2, Table 6. 3.3.2. Portfolio analysis results
3.3.2.1. Efficient set - no riskless asset The solution of the mathematical programming problems of the previous section was carried out in a commercial spreadsheet environment with mathematical programming facilities. The outcome of various runs is given in Table 8 and Figure 2. These results have obvious interpretations: First, for low required rates the risk minimizing portfolio consists of low risk and return vessels of types Handysize Bulk Carrier, Handysize and Aframax Tankers. With rising rates of return, Panamax Bulk Carriers quickly become attractive and replace Handys for medium rates of return. For high rates of return it is the riskiest and most lucrative type, the VLCC Tanker that comes into play. Also note (comparing Tables 6 and 8) by using the optimal portfolia an investor gets a much better risk return profile than the one provided by restricting the investement to a single vessel type.
3.3.2.2. Efficient set - riskless asset The results of the previous paragraph did not include any investment in a riskless asset. If such an asset is available, then the theory outlined in Section 2.3.1 shows that the risk - return tradeoff is linear, and shipping investment is restricted to a single shipping portfolio. These results are borne out by the relevant computations whose results are shown in Table 9 and Figure 3. Here we assumed that there is a riskless asset (deposits or loans) at a 5% interest rate. Note that positive amounts invested in the riskless asset mean deposits, while negative amounts mean borrowings. The risk return characteristic is now linear, its slope being an estimate of the price of risk (i.e. the Capital Market Line of Portfolio Theory). Furthermore, the risky portfolio, made up of various vessel types is independent of the required return, as evident in Table 9. This portfolio is made up by Panamax Bulk Carriers (37%), Handy Tankers (19%), Aframax Tankers (32%) and VLCC's (12%). Note that for high required rate of returns (25% or higher) the proposed portfolio consists of a negative investment in the riskless asset, i.e. borrowing at the available interest rate of 5%. In case the lending rate is higher than the deposit rate of 5%, the portfolio structure is somewhat 32
different but for small spreads between lending and borrowing rates these differences are not significant. 3.3.3. The Beta of Shipping The previous analysis shows that investment in shipping can gain by diversification, if owners behaviour is properly described by the assumptions of the CAPM. A larger question exists, as to the proper required rate of return from shipping. A partial answer to that question can be obtained by reference to the CAPM. An elementary look at this model gives an operative measure of an investments risk through the beta coefficient. In the CAPM we assume the existence of a market porfolio, indexed by M. Then any investment has two kinds of uncertainty, the systematic risk which can be properly viewed with reference to the market as a whole, and the non-systematic risk which can be reduced by diversification. The systematic risk is characterized by the beta coeficient: In particular, if s is the investment index the relevant beta βs is defined as βs = Cov (rs,rM)/σM2. Then the appropriate return rs for the investment s is given by rs=rf+(rM-rf)∗ βs, where rf is the return on the riskless investment, and rM is the return on the market portfolio. This condition results from considering the Lagrangian minimization conditions in the Markowitz portfolio problem. The beta coefficient can also be examined on its own: A value larger than 1 means a riskiness greater than the overall market, and vice versa for smaller values. A negative beta value shows an investment which can be useful for hedging purposes. As a preliminary application of these ideas for shipping, we examine the covariance of the returns from investments in the vessel types examined in the previous section with a reasonable market index. We choose the Dow Jones index for the NYSE, for which data is readily available, and obtain the values indicated in the following Table 10: Our data for the DJIA is yearly and covers the same time span as the shipping investments (i.e. 1980 - 95). In compiling this table we assumed that the return in the riskless asset is 5.0% while the DJIA return not including dividents had an average value of 12.5%. Thus the market return is about 15%. The entry denoted by Portfolio refers to the optimal portfolio of the previous paragraph, i.e. the one appearing in the presence of the riskless asset. There are three striking features in Table 10. First, the covariance between shipping and the NYSE is negative. This fact reflects mainly the recent trend of explosive rise in the NYSE coupled with a weak shipping market. Although one would expect a positive covariance, this is not necessarily so since shipping is mainly related to economies other than that of the US. Second, the beta's for sipping investments are quite low, many of them being in fact negative. It seems thus that shipping investments could provide a hedge with the overall NYSE. Third, compared to the CAPM rates of return, actual shipping returns seem high (again relative to the NYSE). All three observations deserve further study.
33
3.3.4. Conclusions The above preiliminary analysis was carried out using a standard commercial spreadsheet with mathematical programming capacities. The study started as an effort to apply quantitative methods in shipping and not as a study in shipping economics per se. Still, we think that the results obtained, even if they are not taken at face value give interesting insights on the risk return interplay in shipping. Further possibilities exist in assessing the risk of particular vessels, for instance by considering the beta's with respect to the world fleet (representing the shipping market portfolio), and then assessing the beta of the world fleet versus the return from a portfolio in the world economy.
34
3.4.
REFERENCES
Grammenos C. and S. Markoulis, A cross-section analysis of stock returns : The case of shipping firms, Maritime Policy and Management, Vol. 23, No.1 [1996] Denisis A., Optimal Fleet Structure and Portfolio Theory, MS Thesis, National Technical University of Athens [1996] Hull J., Options, Futures and other Derivative Securities, Second Edition, Prentice Hall International, London [1993] Lorange P. and V. Norman, Portfolio Planning in Bulk Shipping Companies, in Lorange P. and V. Norman, eds., Shipping Management, Proceedings from a Seminar in Bergen, Institute for Shipping Research, Bergen, Norway and Maritime Research Center, the Hague, Netherlands,[1973] Magirou V., H. Psaraftis and N. Christodoulakis [1992], Quantitative Methods in Shipping: A Survey of Current Use and Future Trends, Report No. E115, Center for Economic Research, Athens University of Economics and Business. Markovitz H., Portfolio Selection: Efficient Diversification of Investments, John Wiley and Sons, New York [1959] Polemis D.,The sailing ships of Andros, Kaireios Library Editions, Andros, Greece (in Greek) [1995] Sharpe W., Portfolio Theory and Capital Markets, McGraw Hill, New York [1970]
35
3.5.
TABLES AND FIGURES
No.
Type
Name
DWT
Age (years)
1
Bulk Carrier
Handy
30000
5
2
Bulk Carrier
Panamax
70000
5
3
Bulk Carrier
Capes
120000
5
4
Tanker
Handysize
30000
8
5
Tanker
Aframax
100000
8
6
Tanker
VLCC
250000
10-15
Table 1: Vessel Types
Year
1
2
3
4
5
6
1980
9.0
12.5
16.0
16.0
15.5
13.0
1981
9.3
14.5
17.2
12.0
15.0
12.5
1982
6.5
13.1
8.5
8.0
5.5
7.5
1983
3.8
5.1
5.8
7.5
7.0
5.8
1984
4.7
6.0
7.9
6.6
11.5
6.6
1985
4.8
7.0
8.0
7.0
10.8
9.0
1986
4.0
5.6
6.8
5.3
8.2
7.4
1987
3.8
5.8
6.5
6.5
7.6
8.2
1988
8.0
12.5
13.8
7.5
8.0
11.5
1989
9.0
14.2
18.0
8.2
14.5
17.3
1990
8.5
13.9
19.1
11.0
14.0
20.5
1991
8.5
13.0
18.5
12.6
17.6
21.2
1992
7.5
9.0
13.0
12.1
13.7
15.5
1993
8.5
12.5
18.0
9.0
12.0
15.2
1994
7.5
10.0
11.5
11.1
14.3
13.0
1995
9.5
14.0
16.6
11.6
16.8
15.3
Table 2: Timecharter Rates
36
(US$ 000/day)
Year
1
2
3
4
5
6
1980
1.100
1.300
1.525
1.350
1.800
2.000
1981
1.075
1.250
1.500
1.300
1.750
1.950
1982
1.050
1.200
1.450
1.250
1.725
1.925
1983
1.025
1.225
1.475
1.250
1.650
1.900
1984
1.025
1.240
1.500
1.275
1.700
1.950
1985
1.050
1.245
1.500
1.300
1.725
2.000
1986
1.050
1.245
1.550
1.350
1.800
2.100
1987
1.085
1.285
1.575
1.400
1.900
2.200
1988
1.130
1.330
1.600
1.450
2.100
2.300
1989
1.140
1.355
1.770
1.500
2.100
2.450
1990
1.150
1.440
1.890
1.550
2.200
2.550
1991
1.200
1.515
1.995
1.600
2.400
2.750
1992
1.275
1.615
2.120
1.650
2.600
3.000
1993
1.355
1.715
2.200
1.675
2.800
3.150
1994
1.400
1.750
2.275
1.700
3.000
3.300
1995
1.425
1.800
2.300
1.725
3.150
3.500
5
6
Table 3: Operating Costs
(US$ millions)
Sh i p T ype Year
1
1980
7.5
1981
8.0
1982
8.6
1983
2
3
4
8.0
9.0
7.8
9.0
5.5
9.0
10.0
8.0
8.5
5.0
11.0
11.3
8.5
7.0
4.8
7.8
8.3
10.4
8.2
8.7
4.0
1984
7.0
8.6
13.0
7.1
7.9
5.6
1985
5.8
7.8
12.5
6.0
7.5
6.0
1986
4.5
6.0
8.0
4.7
6.0
5.5
1987
6.0
7.8
13.0
8.0
8.5
11.8
1988
8.5
14.2
24.0
10.5
13.3
17.0
1989
12.3
17.5
28.5
15.2
20.5
23.5
1990
14.6
22.0
32.5
15.2
29.0
30.5
1991
11.5
18.0
27.0
19.0
24.0
28.0
1992
13.9
22.0
35.0
17.5
11.0
17.0
1993
14.0
19.0
28.0
14.5
14.0
20.0
1994
14.5
19.0
30.0
11.0
15.0
15.0
1995
16.0
21.0
29.0
13.0
16.0
18.0
1996
18.0
22.5
34.0
14.5
17.0
22.5
Table 4: Second-Hand Ship Prices
37
(US$ millions)
Ship Type Year
1
2
3
4
5
6
1980
30.8%
47.6%
53.1%
54.0%
31.9%
34.5%
1981
31.5%
61.1%
54.8%
39.3%
21.1%
41.6%
1982
2.2%
4.0%
2.8%
11.8%
23.4%
-4.6%
1983
-9.0%
7.3%
26.6%
0.8%
-2.7%
39.1%
1984
-10.8%
-2.0%
3.0%
-3.5%
21.5%
10.4%
1985
-13.9%
-9.9%
-27.5%
-4.9%
5.0%
8.1%
1986
37.1%
38.0%
68.0%
75.9%
55.3%
117.0%
1987
41.5%
86.1%
84.5%
38.3%
60.7%
45.4%
1988
60.0%
41.0%
28.6%
51.6%
54.8%
44.2%
1989
31.5%
42.6%
26.5%
6.0%
51.7%
41.2%
1990
-11.1%
-5.1%
-4.7%
36.4%
-10.4%
4.2%
1991
32.7%
35.4%
42.3%
4.1%
-39.9%
-24.4%
1992
7.4%
-9.3%
-15.5%
-4.9%
43.4%
28.4%
1993
12.0%
11.0%
18.6%
-16.2%
13.9%
-16.4%
1994
15.5%
16.4%
-0.4%
34.5%
16.8%
24.7%
1995
21.0%
18.7%
25.8%
26.2%
20.1%
31.6%
Mean Return
17.4%
23.9%
24.2%
21.8%
22.9%
26.6%
Std.Deviation
21.3%
26.8%
29.9%
25.5%
26.2%
31.7%
LT Return
15.4%
21.2%
20.5%
19.2%
19.7%
22.8%
Table 5: Annual Returns 0 .4 0
0 .3 5
0 .3 0
6 0 .2 5
4
5
3
2
0 .2 0
1 0 .1 5
0 .1 0
0 .0 5
0 .0 0 0 .1 5
0 .2 0
0 .2 5
S ta n d a r d
0 .3 0
D e v ia tio
Figure 1: Risk return characteristics of vessel types
38
0 .3 5
Type
DWT
r
σ
Bulk Carrier
30000
0.174
0.213
Bulk Carrier
70000
0.239
0.268
Bulk Carrier
120000
0.242
0.299
Tanker
30000
0.218
0.255
Tanker
100000
0.229
0.262
Tanker
250000
0.266
0.317
Table 6: Risk return for vessel types
1
2
3
4
5
6
1
1
0.827
0.728
0.600
0.505
0.443
2
0.827
1
0.906
0.562
0.431
0.426
3
0.728
0.906
1
0.591
0.335
0.517
4
0.600
0.562
0.591
1
0.420
0.705
5
0.505
0.431
0.335
0.420
1
0.682
6
0.443
0.426
0.517
0.705
0.682
1
Table 7: Correlation Coefficients ρ between the returns of vessel types
Vessel type
Return %
S. Dev. %
Share in optimal portfolio
1
17.4
21.3
48.2%
31.5%
15.0%
0.0%
0.0%
0.0%
0.0%
2
23.9
26.8
0.0%
10.5%
21.6%
32.6%
40.8%
45.4%
21.4%
3
24.2
29.9
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
4
21.8
25.5
23.8%
26.3%
28.3%
28.4%
10.7%
0.0%
0.0%
5
22.9
26.2
28.0%
31.7%
35.1%
37.3%
26.9%
10.1%
0.0%
6
26.6
31.7
0.0%
0.0%
0.0%
1.7%
21.6%
44.5%
78.6%
S. Dev.
19.6%
19.9%
20.4%
21.1%
22.2%
24.0%
27.8%
Return
20.0%
21.0%
22.0%
23.0%
24.0%
25.0%
26.0%
Table 8: Structure of optimal portfolia
39
Efficient Frontier
29.0% 27.0%
Return
25.0% 23.0%
Return
21.0% 19.0% 17.0% 15.0% 15.0%
20.0%
25.0%
30.0%
35.0%
Std. Deviation
Figure 2: Efficient frontier Por t f o lio St ruct ure Asset type Riskless asset
Total
Risky
73%
Total
Risky
46%
Total
%
Risky
19%
Total
Risky
-8%
Total
Risky
-35%
Bulk 30000
0%
0%
0%
0%
0%
0%
0%
0%
0%
0%
Bulk 70000
10%
37%
20%
37%
30%
37%
40%
37%
50%
37%
Bulk 120000
0%
0%
0%
0%
0%
0%
0%
0%
0%
0%
Tanker 30000
5%
19%
10%
19%
15%
19%
21%
19%
26%
19%
Tanker 100000
9%
32%
17%
32%
26%
32%
34%
32%
43%
32%
Tanker 250000
3%
12%
7%
12%
10%
12%
13%
12%
17%
12%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
Sum Return
10%
15%
20%
25%
30%
St.dev.
6%
12%
18%
24%
29%
Table 9: Risk return with riskless asset
Risk return with riskless asset
Return
30% 20%
R-portfolio
10% 0% 0%
10%
20%
30%
Std. Deviation
Figure 3: Risk return characteristic 40
Ship type
Beta for type
CAPM return
Actual return
0.218%
0.1242
6.2%
17.4%
-0.199%
-0.113
3.9%
23.9%
0.161%
0.0917
5.9%
24.2%
Tank-Handy
-0.591%
-0.337
1.6%
21.8%
Tank-Aframax
-0.143%
-0.081
4.2%
22.9%
0.447%
0.2551
7.6%
26.6%
-0.176%
-0.101
4.0%
23.5%
Bulk-Handy Bulk-Panamax Bulk-Cape
Tank-VLCC Portfolio
Covarianc e
Table 10: Relation between the DJIA and Shipping investments
41
