Assessment of investment decision in the dry bulk shipping market based on real options thinking and the shipping cycle perspective

ABSTRACT

As the dry bulk shipping market seems to have been stuck in a trough period for a long time, investors need to pay more attention to their investment strategies to survive during this period. This study aimed to find a suitable model to assess dry bulk ship investment decisions in the tough and peak periods based on real options theories. Two options, involving an abandonment option and a deferrable option, were used to define investors’ responses to the uncertainty in investment processes such as stopping or selling vessels. The option valuation was solved by using a binomial valuation model, due to data limitations. In accordance with shipping cycle theories, different volatility parameters for the tough and peak periods were calculated using a generalized autoregressive conditional heteroskedasticity (GARCH) model. The application of the real options model to a case study involving secondhand ship trading indicated its viability. According to the results of the case study, the new model has advantages over the traditional net present value (NPV) method in uncertain investment environments. Thus, the results demonstrate that the real options model is a more suitable method for use in the current dry bulk shipping market.

KEYWORDS:

• Ship investment
• real options
• shipping cycle
• deferrable option
• abandonment option
• binary model

1. Introduction

The financial crisis in 2008 caused considerable changes to the maritime environment, with a particular impact on the dry bulk shipping market. The Baltic Dry Index (BDI), which reflects the freight rate of the dry bulk shipping market, dropped from 10 245 in June 2008 to 743 in December 2008. Since that time, the BDI has been maintained at a low level and the dry bulk shipping market has been stuck in the trough period. There is still no indication that the shipping markets will enter a recovery period in the near future, and hundreds of shipping companies went bankrupt during this depressed period (Dinwoodie, Landamore, and Rigot-Muller 2014).

In the dry bulk shipping market, there are many small-scale shipowners, no market barriers for participants, and all transactions are highly transparent. Simply put, the dry bulk shipping market can be seen as an example of a perfectly competitive market (Yijie, Yin, and Sheng 2018). Companies and shipowners need to be familiar with the market and select the most suitable investment decisions to survive in such a market. For this reason, investment decisions in the dry bulk shipping market have been researched by many shipping scholars.

The traditional discounted cash flow (DCF) methodology is commonly used in general project management investment research. Bendall and Evans were the first to apply this method to ship investment, and since then, it has been put forward as an appropriate theoretical method for analyzing maritime investments (Bendall 1980; Evans 1984). The most widely used of the DCF methods is the net present value (NPV) method. The NPV method takes into account the time value of money as a basis to accept or reject proposed investment, that is the NPV is the difference between the present value of cash inflows and the present value of cash outflows that could occur as a result of undertaking an investment project. Frankel (1982) and Sloggett (1984) discussed the use of NPV for analyzing maritime investment strategies in different situations, and they highlighted several NPV calculation methods. From then on, NPV became the most widely used method for maritime investment analyses, and companies operating in the dry bulk shipping market still use it to this day.

However, the limitations of the NPV approach started to be revealed after deeper research. Hodder and Riggs (1985) identified three major NPV limitations. Firstly, it does not take the influence of inflation into account, especially for long-term investments. Secondly, a single discount rate cannot reflect the risk involved in a complex project. Finally, flexibility of management is a key factor in investment but it is disregarded in the NPV approach. For these reasons, NPV is not a suitable method for investments involving uncertainty.

Due to the limitations of traditional NPV techniques, many new methods have been developed to resolve the limitations. Real options analysis (ROA) is one of them. The term ‘real option’ was coined by Myers (1977). A real option is the right, but not the obligation, to undertake certain business initiatives, such as deferring, abandoning, expanding, staging, or contracting a capital investment project. A key factor in real options theory is option valuation. Black and Scholes (1973) defined an initial option pricing formula. Based on their seminal works, Merton (1973) published a paper on the Black and Scholes (B-S) option pricing model, which further promoted the development of real options theory. Another major option pricing method (besides the B-S model) is the binomial approach developed by Cox et al. (1979). Based on the two option valuation methods, ROA has become a useful method in investment research.

Many studies began to use ROA for researching maritime investments since 2000. Bendall analyzed the adaptability of the use of ROA in ship investment analyses under conditions of uncertainty, and he compared the results of ROA and NPV applied to case studies (Bendall and Stent 2003, 2005, 2007). The continuity of the ship price index and the competitiveness of the shipping markets ensure the adaptability of ROA. The studies by Bendall indicate the advantages of ROA over NPV based on actual case studies. In recent years, ROA are commonly used in all fields of the shipping industry: Dikos (2008) and Adland, Hansson, and von der Wense (2017) used ROA for tanker market research and oil transportation research; Kyriakou et al. (2017) used ROA in dry bulk shipping market; Philipp and Spinler (2017) used ROA for liner shipping market research; A time charter extension option was defined by Yun, Lim and Lee (2018); Zheng and Negenborn (2017) adopt the expanded NPV rule and uses a real option approach to analyze terminal port investment timing decisions. In these studies, options were commonly used to present the actions taken by the investors during the processes of ship purchasing and chartering, such as trading or chartering out, to reduce their loss when facing uncertainties of the market (Bendall and Stent 2007). When making a ship investment decision, like ordering a newbuilding ship, the influences of policies, and the activities of operators should be taken into consideration (Xu and Yip 2012). In order to define the possible actions including stopping operation and terminating the investment when facing a depressed market, multiple options, involving an abandonment option, and a deferrable option, were firstly used in this study.

One limitation of the existing ROA studies of maritime investment is the partitioning of time series data of the shipping market. Today, almost all shipping scholars agree that the shipping market should be divided into two periods: the period before the financial crisis and the period after the financial crisis. Moreover, analyses of market relationships and volatility should not be the same for the two periods and the occurrence of the financial crisis could be regarded as an interference factor of dry bulk shipping market (Chou 2014). For this reason, ROA studies of maritime investment decisions before 2008 may not be suitable for understanding the current market and the period after 2008 should regard as an independent time period in ship investment research.

On the other hand, shipping cycle theory developed by Stopford (1997) indicates that cyclical fluctuations exist within the dry bulk shipping market. The shipping cycle can be divided into four periods: trough, recovery, peak, and recession. A longer 7-year cycle is well-known and appears to characterize the dry bulk shipping market cycle and a shorter 4-year cycle was raised according to the market data before the financial crisis by Chistè and Gary (2014). Obviously, the level of volatility and the relationships that exist among the shipping markets are different in each period (Yin and Shi 2018; Fan and Luo 2013), which means that different investment strategies should be applied by the individual shipping companies for each period (Greenwood and Hanson 2014). This paper used the ROA method to assess investment decisions in the dry bulk shipping market in different shipping cycle periods. Almost all companies will expand their investments in the recovery periods and reduce them in the recession periods. Therefore, the ROA model of investment decisions in the dry bulk shipping market used in this study focused on both the trough period and the peak period.

Simply put, the ROA method, combined with the shipping cycle theory, is used in this paper to assess an investment decision for the dry bulk shipping market. Compared with the traditional ROA method, the improved method contains the impact of the fluctuation of the market in different cycle stages to be more suitable for different periods of shipping market; applies multiple options to define the influences of the possible actions when facing uncertainties; raises specific calculation models and judging criteria for both deferrable options and abandonment options.

The rest of the paper is organized as follows. In Section 2, the ROA methodology and the associated valuation method for the trough period and the peak period are described. The cyclical parameter calculations for the ROA model are discussed in Section 3. The application of the ROA and traditional NPV models to a specific case study is presented in Section 4. Section 5 contains a comparison of the results of the two methods and a deeper discussion and Section 6 concludes the paper.

2. Methodology

Before presenting the methodology, the notations were introduced firstly (Table 1):

Table 1. The notations and definition.

Notation	Definition	Unit
$PV$	the total present value of the investment revenue	Million dollars
$T_0$	the capital investment costs of the vessel	Million dollars
$n$	the investment horizon	N/A
$r$	the risk-adjusted discount rate	N/A
$C{F}_t$	the operating income in period t	Million dollars
$T_n$	the residual value of the vessel after depreciation	Million dollars
$C_{i,j}$	the option value during period i for situation j	Million dollars
P	the rising probability of the option value	N/A
$P^{\ast }$	the subjective rising probability judgments of the investors	N/A
u	the rising volatility	N/A
d	the falling volatility	N/A
$C{F}_i$	the sum present value of the options during period i	Million dollars
$F_s$	the freight rate of the stopping node	Dollars per ton
$F_r$	the freight rate of the restarting node	Dollars per ton
AOC	the average operating cost	Dollars per ton
SC	the stopping cost	Million dollars
RC	the restarting cost	Million dollars
Q	the vessel capacity under the operation state	Million tons
$F_t$	the operation revenue in time t	Dollars per ton
FC	the fixed cost of the vessel whether in the operation or stopping state	Million dollars
$F_{ij}$	the freight rate during period i for situation j	Dollars per ton
$T_{A\left(i,j\right)}$	the abandonment trigger during period i for situation j	Million dollars
ABC	the vessel abandonment cost	Million dollars
$A_{ij}$	the abandonment option value	Million dollars
$T_i$	the residual value of the vessel in period i	Million dollars
$R_{i,j}$	the operation revenue if investors do not exercise the abandonment option	Million dollars
p	the order of the GARCH ${\sigma }^2$ terms	N/A
q	the order of the ARCH ${ϵ}^2$ terms	N/A
${\sigma }^2$	the variance in the GARCH model	N/A
$\omega$	the constant term which means the long-term fluctuation of the sequence	N/A
R_SHP	the first-order difference SHP data	N/A

The revenue of a dry bulk ship investment can be defined as the sum of the net operating income and the variation in ship price during the investment project.

(1) $PV=-T_0+\sum _{t=1}^n\frac{C{F}_t}{{\left(1+r\right)}^t}+\frac{T_n}{{\left(1+r\right)}^n}$(1)

Multiple options, involving an abandonment option and a deferrable option, were used in this study to define $C{F}_t$ and $T_n$ in the calculation of the total present value of the investment revenue. The main risk in the operating period is the fluctuation of the dry bulk freight rate. Shipowners may have to take measures such as Forward Freight Agreement (FFA) or adjusting operation strategy to reduce their losses when the freight rate becomes very low (Yin, Luo, and Fan 2017). They can choose to stop operating the vessel if the market is depressed. When the market recovers to the normal level, shipowners can choose to restart operating the vessel. This right to make the above-mentioned choices is defined as the deferrable option. In a worse situation, when the market is very depressed, shipowners may choose to sell the vessel or send it to the demolition market. This means that the shipowners abandon their above-mentioned right, and is defined as the abandonment option.

2.1. Option valuation method

The B-S and binomial valuation models are the most commonly used valuation methods in ROA. The B-S model is a continuous valuation method with strict requirements regarding the data. One assumption is that the data should obey the normal distribution in any limited time period; however, dry bulk shipping market data do not fit this assumption. Therefore, the binomial valuation model was selected as the option valuation method. Given the risk of the investment, the future cash flow is uncertain. To apply the binomial valuation model, two possible outcomes are defined for each period and the following three assumptions (Brandão, Dyer, and Hahn 2008) must be satisfied:

(1) The market is perfectly competitive.

(2) The risk-adjusted discount rate and the probability of the rise and fall of the option value will not change.

(3) Shipowners are risk-neutral investors.

The abandonment and deferrable options in this study are based on European options. The binomial valuation model of the European options is shown in Figure 1.

Figure 1. Binomial valuation model.

In the binomial valuation model in Figure 1, each node $C_{i,j}$ represents the option value during period i for situation j. There are total of i + 1 situations during period i. During each period, the option value will rise with a probability of P and fall with a probability of 1-P, and u and d are the volatility range of rising and falling, respectively. (Parameters u, d, and P are discussed in more detail in Section 2.4.) Each option value during period i can be calculated based on the option value during period i + 1:

(2) $C_{i,j}=P\ast C_{i+1,j}+\left(1-P\right)\ast C_{i+1,j+1}$(2)

(3) $C{F}_i=\frac{1}{{\left(1+r\right)}^i}\ast \sum _{j=0}^i\frac{i!}{j!\left(i-j\right)!}{P}^{i-j}{\left(1-P\right)}^j{C}_{i,j}$(3)

In this case, Eq. (1) can be redefined as:

(4) $PV=-T_0+\sum _{i=1}^{n}C{F}_i+\frac{T_n}{{\left(1+r\right)}^n}$(4)

2.2. Deferrable option

One assumption of the deferrable option is that investors consider stopping or restarting vessel operation only due to the influence of the freight rate but not that of other risks in the dry bulk shipping market. Two freight rate nodes, i.e., the stopping and restarting nodes, are defined.

(5) $\left\{\begin{array}{c}{F}_s=AOC-SC/Q\\ F_r=AOC+RC/Q\end{array}\right.$(5)

Investors will stop operating the vessel if the sum of the operation revenue and cost of stopping the vessel is lower than the average operating cost ($F_t<F_s$, where $F_t$ is the operation revenue in time t). Also, if the operation revenue is higher than the sum of the average operating cost and the cost of restarting the vessel ($F_t>F_r$), investors will restart operating the vessel. In this case, the value of the deferrable option can be calculated as follows:

(6) $C_{i,j}=\left\{\begin{array}{c}\left(F_{i,j}-AOC\right)\ast Q-FC,F_{i,j}>F_r\\ dependson{F}_{i-1,j},F_s<F_{i,j}<F_r\\ -SC-FC,F_{i,j}<F_s.\end{array}\right.$(6)

When the freight rate is between $F_s$ and $F_r$ ($F_s<F_{i,j}<F_r$), $F_{i,j}$ should be assessed regardless of whether the vessel is operating. The operation state is assumed to be maintained at the state in the previous period.

2.3. Abandonment option

In the abandonment option, investors have the right to decide when to finish their investment. This means shipowners may choose to sell the vessel or send it to the demolition market when the dry bulk shipping market is extremely depressed. In this case, an abandonment trigger ($T_A$) can be defined as follows:

(7) $T_{A\left(i,j\right)}=\sum _{u=i+1}^n\left[\frac{1}{{\left(1+r\right)}^{\left(u-i\right)}}\ast \sum _{v=j}^u\frac{u!}{v!\left(u-v\right)!}{P}^{u-v}{\left(1-P\right)}^v{C}_{u,v}\right]+ABC$(7)

The abandonment trigger, $T_{A\left(i,j\right)}$, is used to assess the sum of the cash flows after $C{F}_{i,j}$ (Figure 2). If the trigger $T_{A\left(i,j\right)}$ is less than zero, investors will not have any revenue if they keep the vessel. In this case, investors will sell the vessel or send it to the demolition market. Once the abandonment option is exercised, the residual value of the vessel has to be recalculated. The abandonment option value $A_{ij}$ equals the difference between the vessel residual values minus the vessel abandonment cost:

(8) $A_{i,j}=\left\{\begin{array}{c}{T}_i-R_{i,j}-\frac{T_n}{{\left(1+r\right)}^{\left(n-i\right)}}-ABC,T_{A\left(i,j\right)}<0\\ \\ 0,T_{A\left(i,j\right)}\ge 0.\end{array}\right.$(8)

Figure 2. Abandonment trigger, $T_{A\left(i,j\right)}$. # stopping node; * restarting node.

2.4. Shipping cycle factors

In this paper, only two shipping cycle periods were considered in the ROA investment model (the trough period and the peak period). According to shipping cycle theories, the volatility of the freight rate and of the secondhand ship price in the trough period and the peak period is totally different. For this reason, the rising and falling volatility (u and d) of the freight rate, ship price, and option value should be calculated for the trough period and peak period, respectively, as well as the rising probability (P).

Freight rate, ship price, and their movements over time are of great importance to shipping markets research and almost all maritime studies consider the volatility of the freight rate and of the ship price.

Since Engle (1982) introduced the autoregressive conditional heteroskedasticity (ARCH) model, several extensions and variations of it have appeared in the literature. Later on, Kavussanos (1996, 1997) argued that generalized autoregressive conditional heteroskedasticity (GARCH) models are more suitable than ARCH models for defining the volatility of the freight rate and ship price. The GARCH model has been proved to be suitable for dry bulk shipping market volatility analyses. Therefore, the GARCH model was used in the ROA method to define the rising and falling volatility range of the option value. The basic GARCH (p, q) specification can be expressed as follows:

(9) $GARCH\left(p,q\right)={\sigma }_t^2=\omega +\sum _{i=1}^q{\alpha }_i{ϵ}_{t-i}^2+\sum _{j=1}^p{\beta }_j{\sigma }_{t-j}^2$(9)

In the GARCH model, if the coefficient condition, $\sum _{i=1}^q{\alpha }_i+\sum _{j=1}^p{\beta }_j<1$, is satisfied, the long-term volatility will be stable:

(10) ${\sigma }^2=\frac{\omega }{1-{\sum }_{i=1}^q{\alpha }_i-{\sum }_{j=1}^p{\beta }_j}$(10)

Base on the long-term volatility calculated using the GARCH model, the rising and falling volatility ranges (u and d) in different shipping cycle periods can be calculated as follows:

(11) $u=e^{\sigma };d=e^{-\sigma }=\frac{1}{u}$(11)

In contrast, the rising probability, P, is very hard to quantify because investors have different opinions based on different trends in the dry bulk shipping market. According to assumption 3 (Brandão, Dyer, and Hahn 2008) of the binomial valuation model, the investors are risk neutral, which means that risk appetite for investment would not have an impact on P. In this case, P mainly depends on the subjective judgments of the investors. Therefore, P can be defined as (50% + $P^{\ast }$), where $P^{\ast }$ is the subjective judgments of the investors.

3. Volatility calculation

In this section, the volatility parameters (u and d) for different shipping cycle periods are presented. Companies and investors will pay more attentions on the trough period and the peak period because the other two periods, the recovery and recession periods, usually last for a short time and the dry bulk shipping market has a significant fluctuation trend in both periods. Therefore, the volatility calculation only focuses on the trough period and the peak period. The financial crisis in 2008 caused considerable changes in the field of shipping cycles. For this reason, the tough period consisted of only the period after 2008 and the peak period was from 2004 to 2008. The two major kinds of data used in this paper are the BDI and the Bulk Carrier Secondhand Price Index (SHP). The BDI shows the freight rate of the dry bulk shipping market and the vessel residual value is defined by the SHP. The details of the data used are shown in Table 2.

Table 2. Volatility data for tough and peak periods.

Shipping Cycle	Time	Time span	Index	Frequency	Volume
With Shipping Cycle	Tough period	2010.07.16–2017.05.05	BDI	Weekly	347
			SHP	Weekly	347
	Peak period	2004.07.09–2008.09.12	BDI	Weekly	219
			SHP	Monthly	51
Without Shipping Cycle	Recent 10 years	2007.05.06–2017.05.05	BDI	Monthly	120
			SHP	Monthly	120
	Recent 5 years	2012.05.11–2017.05.05	BDI	Weekly	259
			SHP	Weekly	259

Source: https://sin.clarksons.net

Obviously, the more frequent the data, the more accurate the volatility rate estimates. But the SHP source did not provide weekly data until 2010.07.16. For this reason, monthly data for the peak period was used instead.

In addition, the recent 10 years data and recent 5 years data are involved to compare the ROA method with and without considering the shipping cycle.

As each volatility rate calculation involves the same procedure, only the procedure for SHP in the tough period is provided here, as an example. The results of the other volatility rate calculations are given at the end of this section.

Before using the GARCH model, several basic tests of the SHP data for the tough period needed to be carried out. The first-order difference SHP data in logarithm form, designated R_SHP, was found to be stationary according to the unit root test. Other descriptive statistics were also acquired from the basic set of tests: skewness value: −1.080127; kurtosis value: 10.91914; and Jarque-Bera value: 971.3882. These results reject the hypotheses of normality of the R_SHP data, which meant that the B-S valuation model was not suitable. An auto-correlation test indicated the presence of auto-correlation, based on the fat tail and spiked peak of the graph of the R_SHP data. By changing the time lag coefficient, the best autoregressive (AR) model for the R_SHP data was found to be AR(3), according to the Akaike information criterion (AIC). The ARCH Lagrange multiplier test results (Table 3) indicated the presence of autocorrelated conditional heteroskedasticity in the R_SHP residuals. The AR(3) model is as follows:

(12) $R\mathrm{\_}SH{P}_t=0.21365R\mathrm{\_}SH{P}_{t-1}+0.13098\mathrm{R}\mathrm{\_}SH{P}_{t-2}+0.15407R\mathrm{\_}SH{P}_{t-3}+{\epsilon }_t$(12)

Table 3. The statistics results and ARCH lagrange multiplier test results of the AR and GARCH models.

	AR(1)	AR(2)	AR(3)
AIC	−5.70466	−5.74064	−5.83255*
	GARCH(1,1)	GARCH(1,2)	GARCH(2,1)
AIC	−5.68208	−5.77977*	−5.65927
	AR(3) model		GARCH(1,2) model
Time lag coefficient	F-statistic	Prob.	F-statistic	Prob.
1	6.079950	0.0436	0.201109	0.6541
1	4.274157	0.0147	0.099914	0.9049
3	3.694993	0.0121	0.065330	0.9782
4	2.763627	0.0276	0.067943	0.9915
5	2.235471	0.0106	0.410995	0.8411
10	2.044006	0.0285	0.523848	0.8731

*Null hypothesis rejected (under 5% level of significance)

The results of the ARCH Lagrange multiplier test for the GARCH models are also presented in Table 3. According to the AIC, the best GARCH model is the GARCH(1,2) model, which is as follows:

(13) ${\sigma }_{shp}^2=2.15\times {10}^{-5}+0.012936{\epsilon }_{t-1}^2+0.722000{\sigma }_{t-1}^2+0.0636771{\sigma }_{t-2}^2$(13)

The results of the ARCH Lagrange multiplier test indicate that the ARCH effect was eliminated in the GARCH model. According to Eq. (10), the weekly volatility, ${\sigma }_w^2$, was calculated as $1.11\times {10}^{-4}$. Assuming that there are 52 business weeks in a year, the yearly volatility of SHP, ${\sigma }_{SHP\left(T\right)}$, was calculated as 0.0760. Thus, the rising range u is 1.0789 and the falling range d is 0.9268.

The other volatility results are given in Table 4. And the rising and falling volatility ranges of recent 10 years are 2.3665 and 1.4563, respectively, which is evidently not suitable for the comparison.

Table 4. Volatility parameter calculation results.

Shipping Cycle	Time	Index	Volatility ${\sigma }^2$	Rising rang u	Falling range d
With Shipping Cycle	Tough period	BDI	$6.276\times {10}^{-4}$	1.1972	0.8353
		SHP	$1.110\times {10}^{-4}$	1.0789	0.9268
	Peak period	BDI	$3.823\times {10}^{-4}$	1.1514	0.8685
		SHP	$2.816\times {10}^{-3}$	1.2018	0.8321
Without Shipping Cycle	Recent 10 years	BDI	$6.183\times {10}^{-2}$	2.3665	0.4226
		SHP	$1.177\times {10}^{-2}$	1.4563	0.6867
	Recent 5 years	BDI	$1.417\times {10}^{-3}$	1.3119	0.7623
		SHP	$2.210\times {10}^{-4}$	1.1132	0.8983

4. Case study

In this study, a real case¹ was selected to compare the investment results using the traditional NPV method and the ROA method introduced in this paper. A Panamax dry bulk secondhand ship, called ‘GRAIN PEARL’, was sold to Greece in December, 2015. The vessel is 81600DWT and was made in China in 2013. The final trading transaction price was $16.8 million. In that time, the SHP was 79 and the BDI was 718. The initial freight rate was $20/ton. As for the investment duration, Bulut (2013) raised that the duration of a general ship investment project should more than four years. On the other hand, one cycle in the shipping market is usually completed itself every ten or less than ten years (Stopford 1997). As our volatility parameters are all calculated based on each specific cycle stage, the duration had better last no longer than a cycle stage. Therefore, we assume that the investment duration in this case study is 5 years and each period last 1 year. Other parameters of this case study are presented in Table 5.

Table 5. Case study parameters.

Parameter	Value	Unit
Annual Fixed Cost (FC)	2	Million dollars
Annual Freight Volume (Q)	0.8	Million tons
Annual Stopping Cost (SC)	0.8	Million dollars
Annual Restarting Cost (RC)	0.5	Million dollars
Annual Average Operating Cost (AOC)	15	Dollars per ton
Risk-adjusted Discount Rate (r)	8%	N/A
Annual Depreciation Rate (DR)	10%	N/A
Abandonment Cost (ABC)	5	Million dollars

Table 6. Results of ROA and NPV methods.

Method	Result	Implication
NPV	-$\mathrm{\$}$2.06 million	Reject
ROA	$\mathrm{\$}$2.59 million	Accept
ROA(without shipping cycle)	-$\mathrm{\$}$0.37 million	Reject

4.1. NPV method

In the traditional NPV method, the freight rate and secondhand ship price are maintained at a fixed value in the investment periods. In this case, the freight rate, $F_i$, is always $20/ton and the estimate of the ship price only takes depreciation into consideration, based on the NPV assumption. The revenue in each period and the residual value of the vessel are as follows:

$C{F}_i=\left(F_i-AOC\right)\ast Q-FC=\left(20-15\right)\ast 0.8-2=\mathrm{\$}2\mathrm{m}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{o}\mathrm{n}$

The total present value of the investment revenue is as follows:

$PV=-T_0+\sum _{t=1}^T\frac{C{F}_t}{{\left(1+r\right)}^t}+\frac{T_n}{{\left(1+r\right)}^n}=-16.8+\sum _{t=1}^5\frac{2}{{\left(1+8\mathrm{\%}\right)}^t}+\frac{9.92}{{\left(1+r\right)}^5}=-\mathrm{\$}2.06$ million

The results of the NPV method indicate that the total present value of the investment revenue is negative (Table 6). Therefore, the investment should be abandoned according to the NPV method.

4.2. ROA method

According to Eq. (5), the stopping node $F_s=AOC-SC/Q$ equals $14/ton and the restarting node $F_r=AOC+RC/Q$ equals $15.7/ton. The volatility parameters for the tough period were calculated as detailed in Section 3: $u_{BDI}$ is 1.1972 and $d_{BDI}$ is 0.8353. An assumption in this case is that the subjective judgment of the investors in the dry bulk shipping market is neutral, which means that the rising probability of the freight rate, P, equals 50%. Figure 3 shows the binomial freight rate and operating model based on these calculations and assumptions.

Figure 3. Binomial freight rate and operating model.

Decisions regarding the stopping and restarting nodes are based on comparison of $F_s$, $F_r$ and $F_{i,j}$. $C_{i,j}$ is calculated according to Eq. (6), based on the judgments regarding the stopping and restarting nodes. Regarding the restarting nodes, $C_{i,j}$ can have two values, depending on the conditions. For example, $C_{3,2}$ is -$0.6 million in normal conditions, while investors should pay an extra $0.5 million to restart operations if the previous node is a stopping node, like $N_{2,2}$, means node in Figure 3 in the period i for situation j.

At the stopping nodes, the abandonment option is considered according to the value of trigger $T_{A\left(i,j\right)}$ in Eq. (7). $T_{A\left(2,2\right)}$ = 0.53 > 0 and $T_{A\left(3,3\right)}$ = −0.03 < 0, which means that the sum of the total revenue after $N_{3,4}$ is negative. Hence, the investor should exercise the abandonment option in the third year if the freight rate becomes $11.6/ton. The deferrable option should be replaced by the abandonment option after $N_{3,4}$. In this case, $C_{4,3}$, $C_{4,4}$, $C_{5,3}$, $C_{5,4}$, $C_{5,5}$ all equal zero and $C_{3,3}$ equals $A_{3,3}$. (Also, $C_{4,3}$ and $C_{5,3}$ will have two values, depending on the conditions, like $C_{3,2}$.) The abandonment option value of $A_{3,3}$ is −1.25 according to Eq. (8). In this case, the sum of cash flow $C{F}_i$ is defined by Eq. (3).

$C{F}_1=\frac{1}{{\left(1+r\right)}^1}\ast \sum _{j=0}^1\frac{i!}{j!\left(i-j\right)!}{p}^{i-j}{\left(1-p\right)}^j{C}_{i,j}=\mathrm{\$}2.11million;$

$C{F}_2$=$\mathrm{\$}$2.20 million; $C{F}_3$=$\mathrm{\$}$2.65 million;$C{F}_4$=$\mathrm{\$}$2.72 million;$C{F}_5$=$\mathrm{\$}$2.96 million;

$\mathrm{P}\mathrm{V}=-T_0+\sum _{i=1}^{n}C{F}_i+\frac{T_n}{{\left(1+r\right)}^n}=\mathrm{\$}2.59$ million

The results of the ROA method indicate that the total present value of the investment revenue is positive. In this case, the investment is viable for the investors according to the ROA method.

By using ROA method without considering shipping cycle (recent 5 years data), the PV value is calculated as—$\mathrm{\$}$0.37 million and the investment will be abandoned according to this calculation, which is total inverse compared with the result of ROA method with considering shipping cycle.

5. Discussion

The traditional NPV method and the ROA method provided totally different results for the same case study. The ROA method indicates support for the investment decision (indicating $2.59 million revenue) while the NPV method indicates that the investment should be rejected (indicating $2.06 million in losses).

The major reason for this conflict is due to the methods themselves. Based on one of the NPV assumptions, the freight rate, and revenue are kept at a fixed value whether or not the markets improve or worsen. On the other hand, the ROA method focuses on investors’ actions in different situations. According to the ROA approach, investors have the right to stop the operation or sell their vessel to cut their investment losses, particularly when the market gets worse. Simply put, the ROA method takes into consideration the influence of investors’ actions over the whole investment horizon while the NPV method does not take into account market uncertainty and investors’ responses. Obviously, investment strategies should be based on the combination of investment decisions during the whole investment period rather than a simple decision about whether to buy the vessel. This is the reason why the results of the two methods support different investment decisions. Taking into account the influence of investors’ actions over the whole investment horizon is the biggest advantage of the ROA method over the NPV method.

Another interesting factor regarding the ROA results relates to the present value of cash flow, $C{F}_i$, in each year: $C{F}_1$ = $\mathrm{\$}$2.11 million; $C{F}_2$ = $\mathrm{\$}$2.20 million; $C{F}_3$ = $\mathrm{\$}$2.65 million; $C{F}_4$ = $\mathrm{\$}$2.72 million; $C{F}_5$ = $\mathrm{\$}$2.96 million. The option value increases with time. The revenue in the first year is the lowest revenue while the revenue of the final year is the highest. The major reason for this phenomenon is related to the uncertainty during different periods. Once an investment has been decided on, the market uncertainty will increase with time. In the ROA model, the uncertainty is presented as the number of possible freight rates and their variation range in each year. Traditional investment theory posits that uncertainty in a project will decrease the value of the project. In contrast, real options theory regards the uncertainty as an opportunity associated with the project that increases the projected value. The different opinions underlying the two theories caused the phenomenon observed for the present case study. Uncertainty in ship market investments means more drastic fluctuations in the shipping market. In the worse situation, investors will take action to reduce their losses while in better situations, there will be higher profits. This causes the option value to increase with increases in uncertainty.

The option valuation method is the major limitation of the application of the ROA method to the dry bulk shipping market. The most commonly used method is the binomial valuation model but its accuracy limitation is well known. Other improved discrete methods such as the ternary model can improve the accuracy a little, but at the cost of a large increase in the calculated quantities. Continuous methods such as the B-S method can achieve a high accuracy but their strict requirements regarding the data means that these methods are not suitable for use in the dry bulk shipping market. Therefore, the future research on the application of ROA to the dry bulk shipping market may be focused on the option valuation method.

6. Conclusion

This study involved the development of a real options model to help companies and investors make suitable investment decisions in the dry shipping market. Compared with the traditional DCF method (i.e., the NPV method), the new model takes into account responses of investors facing uncertainty in the investment processes. The fluctuations of the freight rate and ship price are the major uncertainties in the dry bulk shipping market. When facing a depressed market, investors may choose to stop operating or sell their vessel to reduce their losses. The actions of stopping and restarting vessel operation are encompassed by the deferrable option and the action of selling the secondhand ship is encompassed by the abandonment option. What is more, different volatility parameters for the tough and peak periods (for use in the option valuation method) were calculated by using a GARCH model to consider the influence of the shipping cycle to the investment decisions.

The application of the traditional NPV method and the ROA model to a specific case study involving secondhand ship trading was carried out to examine the adaptation of the ROA model and to compare the two models. The result indicates the feasibility of using the ROA method for investigating ship market investment decisions and the advantage of the ROA method over the NPV method. Compared with the traditional NPV method, the ROA method treats uncertainty as an opportunity that increases the overall project value. In this case, the ROA method takes into account investors’ loss reduction actions and provides more accurate results. This concluded that the ROA method combined with shipping cycle theories is found to be more suitable to assess a ship investment decision than the traditional method. In future shipping market investment studies, a ROA method involving continuous valuation methods may be assessed.

Acknowledgments

The authors would like to thank the editor and three anonymous referees for their insightful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported jointly by National Social Science Fund of China [grant number 17BGL259] and National Natural Science Foundation of China [grant number 51508325].

Notes

1. The reference trade case was obtained from Shipping Exchange Bulletin, Vol. 34, 2015, pp. 54.