Bio-based and Applied Economics 9(1): 53-83, 2020

ISSN 2280-6180 (print) © Firenze University Press 
ISSN 2280-6172 (online) www.fupress.com/bae

Full Research Article

DOI: 10.13128/bae-7961

Step-by-step development of a model simulating returns 
on farm from investments: the example of hazelnut 
plantation in Italy

AlisA spiegel1,*, simone severini2, WolfgAng Britz3, Attilio ColettA2

1 Business Economics Group, Wageningen University, the Netherlands
2 Department DAFNE, Università della Tuscia, Italy
3 Institute for Food and Resource Economics, Bonn University, Germany

Abstract. Recent literature reviews of empirical models optimizing long-term invest-
ments in agriculture see gaps with regard to (i) separating investment and financing 
decisions, (ii) considering explicitly risk and temporal flexibility, and (iii) accounting 
for farm-level resource endowments and other constraints. Inspired by real options 
approaches, this paper therefore stepwise develops a model extending a simple net 
present value calculation to a farm-scale simulation model based on mathematical 
programming, which considers time flexibility, different financing options and down-
side risk aversion. We empirically assess the different model variants by analysing 
investments into hazelnut orchards in Italy outside of traditional producing regions. 
The variants suggest quite different optimal results with respect to scale and timing 
of the investment, its financing and the expected NPV. The stepwise approach reveals 
which aspects drive these differences and underlines that considering temporal flex-
ibility, different financing options and riskiness can considerably improve traditional 
NPV analysis. 

Keywords. Perennial crop; real options; stochastic dynamic modelling; stochastic 
optimization.

JEL codes. C61, Q12.

1. Introduction

Recent literature reviews on empirical models for long-term investment analysis see 
gaps with regard to separating investment and financing decisions (e.g., Trigeorgis and 
Tsekrekos, 2018) and explicit consideration of associated risk and temporal flexibility (e.g., 
Shresta et al., 2016). Furthermore, opportunity costs, farm-level resource endowments, 
multiple risk sources and risk preferences are also rarely taken into account. This paper 

*Corresponding author. E-mail: alisa.spiegel@wur.nl

Editor: Fabio Gaetano Santeramo.



54 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

illustrates how to include all these aspects into farm-level investment analysis and high-
lights resulting differences based on an empirical example of investing into hazelnut trees.

The vast majority of research modelling farm-level investment behaviour opts for 
the classical investment theory, which maximizes the net present value (NPV) or alterna-
tively the internal rate of return (IRR), or minimizes the pay-off period, subject to tech-
nological and resource constraints (e.g., Schweier and Becker, 2013; Shresta et al., 2016; 
Bett and Ayieko, 2017). Two major limitations of this approach are well known (among 
others see Freixa et al., 2011; Robinson et al., 2013; Badiu et al., 2015; Sgroi et al., 2015; 
Stillitano et al., 2016). First, the risk underlying the investment project is not explicit-
ly represented and can be reflected only by increasing the discount rate above market 
levels. Other data determining cash-flow changes of the operation related to the invest-
ment enter with their expected values, only, neglecting their riskiness including potential 
correlations. Second, the classical investment theory depicts a “now exactly as defined 
or never” decision problem where neither future adjustments to the investment project 
under, for instance, changing market, policy or technological environments, nor its post-
ponement are considered. This easily overestimates necessary investment triggers and 
thus suggests a lower investment scale (Wolbert-Haverkamp and Musshoff, 2014). The 
new investment theory aims to overcome these limitations. In particular, the applica-
tion of the real options approach to agricultural investment projects has gained interest 
(e.g., Wossink and Gardebroek, 2006; Hinrichs et al., 2008; Maart-Noelck and Musshoff, 
2013; Spiegel et al., 2020). But its empirical application is still limited, for instance, in 
the domain of perennial crops. While quantitative analysis of investments into perennial 
crops has a long history (e.g., Jackson, 1985), it mainly sticks to the classical investment 
theory. Despite considerable market and production risk in orchard production, only a 
few recent studies, such as Sojkova and Adamickova (2011), consider risk. Not astonish-
ingly, they find substantial differences in optimal investment levels compared to the clas-
sical NPV approach and suggest that deterministic models may provide flawed estima-
tion of investment dynamics and scale. 

Consideration of risks in investments is also beneficial for their social and behavioural 
analysis. Social analysis mainly focuses on social networks and their effects on investment 
decision, for instance, via learning experience (Marra et al., 2003; Ghadim et al., 2005). 
Dynamic social analysis is more promising and benefits from explicit consideration of 
risks, as learning and social interactions usually affect not only expectations, but also asso-
ciated subjective risk; and optimal behaviour was found to be sensitive to strategic uncer-
tainty (Morreale et al., 2019). Behavioural investment analysis studies subjective factors, 
including irrationality, subjective beliefs, and risk attitude (see e.g., Chavas and Nauges, 
2020; Weersink and Fulton, 2020). Also here, explicit consideration of risks in dynamics is 
beneficial as it allows adjusting risk perception and risk preferences (Coelho et al., 2012). 
As for optimal financing behaviour, many studies investigate with other methods different 
aspects and determinants of farm-level demand for credits, such as present risk manage-
ment strategies (Katchova, 2005), credit source (Farley and Ellinger, 2007), interest rate 
(Turvey et al., 2012; Fecke et al., 2016), farmer’s personal characteristics and farm struc-
tural variables (Howley and Dillon, 2012). While financing behaviour is found to affect 
farm performance, financial risk, resilience, and their links to investment behaviour is still 
understudied. 



55Development of a model simulating returns on farm from investments

Building on this literature, we develop models for valuing and analysing long-term 
investment decision on farm, starting with a simple net present value calculation. We 
stepwise expand this model to a final dynamic stochastic farm-scale simulation mod-
el inspired by real options approaches, which considers different financing options and 
downside risk aversion in the form of minimum household withdrawals. To this end, the 
paper focuses on economic analysis of farm-level investment and financing options, while 
some social and behavioural aspects might be incorporated in follow-up research as dis-
cussed in the concluding chapter. Accordingly, the objectives of the paper are twofold. 
First, we aim to illustrate how additional investment drivers can be stepwise incorporated 
into models of increasing complexity, and second, we aim to demonstrate sensitivity of 
results across the model variants to underline their relevance. The novelty of the paper is 
threefold. First, we explicitly consider factors that are still widely ignored when modelling 
farm-level investment decision, namely temporal flexibility, flexibility in terms of financ-
ing options, and downside risk aversion of the farm household. Second, we stepwise intro-
duce these factors to quantify their impact on optimal scale and timing of investments 
in a case study. Third, the case study refers to perennial crops, a domain where advanced 
quantitative assessments are lacking.

Hazelnut production was chosen for the empirical application. It presents an interest-
ing case study as it requires long-lasting expensive investments in form of a plantation, 
specialized machinery and irrigation. The different models are all set up for the same 
case study farm located in Viterbo, a central Italian region, where hazelnut production is 
not traditional, but becomes an increasingly important agricultural activity. The farm is 
assumed to currently manage rainfed annual crops. It is representative by its size and farm 
program for farms that are investing into new hazelnut plantations in the region. Since 
hazelnut production was found to be characterized by a relatively high level of risks (Zin-
nanti et al., 2019), we explicitly quantify considerable market (Pelagalli, 2018), weather, 
and other production risks affecting product quality and quantity.

Taking hazelnut production in the Viterbo region as an example is motivated by fur-
ther facts. Firstly, with 13% of global hazelnut production, Italy is the second largest pro-
ducer worldwide after Turkey with ca. 65% (FAO, 2019). Global demand for hazelnut and 
derived products increased over the last decades and is projected to expand further. This 
triggers new investments in different producing countries, partially initiated by interna-
tional food industry companies, of which a major one is located nearby our case study 
region. In Italy, further expansion of hazelnut orchards in the traditional hilly production 
districts under rainfed systems is not possible. New plantations are now set-up in sur-
rounding lower areas where irrigation is necessary to ensure relatively stable production 
and quality levels. Over the years 2016-2019, the Italian National Institute for Statistics 
(ISTAT) recorded a 15% increase in the total area devoted to hazelnut cultivations. Fur-
ther investments are likely in coming years, according to major companies involved in 
hazelnut-based food production which foresee and foster the cultivation of 90.000 hec-
tares in Italy alone. The trend of investing into hazelnuts as an alternative land use option 
also reflects decreased profitability of so far dominating annual crops such as grains and 
oilseed. Both socio-economic and environmental consequences of this ongoing land use 
change are lively debated (Boubaker et al., 2014; UTZ, 2016). So far, economic assess-
ments of investments into hazelnuts at farm level draw on data from specialized producers 



56 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

in the traditional districts, only. Several authors therefore stress the need to better evaluate 
investments in new producing regions (Bobic et al., 2016; Pirazzoli and Palmieri, 2017; 
Frascarelli, 2017). The empirical analysis conducted in this paper closes the gap.

The paper is organized as follows. Section 2 step-by-step develops four models where 
each one expands the previous one by relaxing some assumptions to further improve the 
analysis. Section 3 introduces data and assumptions used in our case study which also 
shows the additional data required for the model expansions. Section 4 presents main 
empirical results to highlight differences across the model variants. Section 5 concludes 
with a discussion of pros and cons of the different model variants and provides sugges-
tions for further research on farm investments.

2. Building-up a stochastic dynamic farm-level model

2.1 Farm-level endowments, economy-of-scale and alternative crop (ClassNPV)

We start with simulating discounted cash flows at farm level for either investing now 
or never – the still dominant approach in literature. In the case of hazelnuts, the nominal 
cash flows in each year depend on the age of the plantation (Fig. 1). 

A newly set-up hazelnut orchard can be first harvested in its seventh year. From 
there to the tenth year, yields increase linearly from zero to a maximum yield level (max-
Yields) which is maintained until the trees are thirty years old. Afterwards, there is a linear 
decrease in annual yields to 50% of the maximum up to the year 35. The resulting formula 
for the yields in year y is:

 (1)

Figure 1. Evolution of a new hazelnut orchard, with related investments points. Source: Own elabora-
tion based on Liso et al. (2017) and Frascarelli (2017).



57Development of a model simulating returns on farm from investments

where y depicts the year after the initial set-up and thus the age of the plantation; 
yieldhazel,y the hazelnut yields at age y in tonnes per hectare [t ha-1]; maxYields refers to the 
maximum hazelnut yield [t ha-1].

Multiplying hazelnut yields with their price and deducting variable costs defines the 
gross margin per hectare. We capture the difference between the farm-gate and the aver-
age regional market price marketPrice by so-called quality index qi, which reflects specific 
quality of hazelnuts, farmer’s negotiation power, and other related factors. Both the quality 
index and the market price are represented in the NPV calculation by their expectations. 
We also distinguish between harvesting costs  per tonne harvested, and other costs  per 
hectare, which include irrigation and fertilization costs. At each age of the plantation y, 
the cash flow per hectare equal to the gross margin is thus defined as:

E[gmhazel,y] = yieldhazel,y * E[qihazel,y] * E[marketPricehazel,y] – yieldhazel,y * harvCost – 
otherCost ∀ y ≤ 35 

(2)

where E[∙] is the expectation operator; gmhazel,y stays for the gross margin of hazelnuts 
[€ ha-1]; qihazel,y for the hazelnuts quality index; marketPricehazel,y for the average market 
price of hazelnuts [€ t-1]; harvCost for the variable harvesting costs [€ t-1]; otherCost for 
the other quasi-fixed costs related to hazelnut cultivation, including irrigation and fertili-
zation costs [€ ha-1]. Furthermore, we consider two (quasi-)fixed resources endowments: 
land and labour. Additional demand for labour can be satisfied via hired labour. The 
farm resources are distributed between hazelnuts and durum wheat - an alternative crop 
to hazelnuts. The acreages of hazelnut and durum wheat can jointly not exceed the given 
land endowment:

areahazel + areawheat ≤ endland (3)

where areahazel depicts land under hazelnuts [ha] and areawheat land devoted to durum 
wheat [ha]; endland stays for the total fixed and given land endowment [ha].

Labour requirement for the crops are expressed per hectare; for hazelnuts, additional 
labour hours per harvested tonne are considered. Total labour requirement can be covered 
by on-farm or hired labour:

areawheat * labwheat + areahazel * labhazel + areahazel * yieldhazel,y * labhm ≤ endlab +  
hiredLaby ∀y 

(4)

where labwheat stays for labour requirements for durum wheat [hours per hectare, h ha-1]; 
labhazel for quasi-fixed (i.e., independent of yields) labour requirements for hazelnuts [h 
ha-1]; labhm for variable labour requirements for hazelnuts [hours per tonne, h t-1]; endlab 
for on-farm labour use [hours, h]; hiredLaby for additionally required labour that can be 
hired [h]. The gross margin of the alternative crop is defined in a similar way as the one of 
hazelnuts, namely, based on expected yield, quality index, market price and variable costs:

E[gmwheat] = E[yieldwheat] * E[qiwheat] * E[marketPricewheat] – E[costwheat] (5)



58 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

where gmwheat stays for gross margin of durum wheat [€ ha-1]; yieldwheat for its yields [t 
ha-1]; qiwheat for its quality index; marketPricewheat for its average market price wheat [€ t-1]; 
and costwheat for its quasi-fixed costs [€ ha-1].

While durum wheat is rain-fed, hazelnuts require irrigation water, such that farm-
ers have to invest into a well and irrigation equipment in addition to the establishment 
costs of the plantation (Fig. 1). Furthermore, harvesting machinery for hazelnuts must 
be available prior to the first harvesting of hazelnuts. Harvesting machinery is physically 
depreciated while other machinery is depreciated by lifetime. The formula for NPV then 
becomes:

E[NPV] =  * areahazel +  areawheat *

* E[gmwheat] – invCostWell – 

 (6)

where NPV stays for the net present value over the overall planning horizon ∑y [€]; iniCost 
for the costs associated with initial establishment of hazelnut plantation [€ ha-1]; dr for 
the discount rate [%]; reconvCost for the costs associated with final clear-cut of hazelnut 
plantation [€ ha-1]; invCostWell for costs of well and irrigation equipment for hazelnut [€]; 
invCostMachm,y for investment costs of machinery m{smaller;standalone;irrigation;tractor;
operating} [€]; E[wage] for expected costs of hired labour [€ h-1]. We optimize the farm-
level NPV under endowment constraints (Eq. 3 and 4) by solving for the following deci-
sion variables: area of hazelnuts, area of durum wheat and investments into machinery m 
at each age of the plantation y.

The model advances by accounting for all required investments as well as resource 
endowments. It also captures the associated economy-of-scale; in our example, via life-
time and capacities of machines and via fixed costs for a well and irrigation equipment. 
In another case study, the gross margin of the alternative land use option could also rep-
resent average returns from a portfolio of alternative crops instead of one crop, only, as 
in here durum wheat. As the result, we simulate the maximum possible farm-level NPV 
under given conditions and constraints.

This model still suffers from limitations as seen in literature. First, it operates with 
expected variables, ignoring their underlying riskiness when maximizing the NPV. Sec-
ond, it implies investing into hazelnuts now or never. Yet, in the case of uncertainty and 
high sunk costs of an investment project, investors might prefer to wait for new infor-
mation before making a decision. Here, sunk costs relate to setting up the plantation and 
investments into a well, irrigation equipment and specialized machines while future pric-
es, yields, and costs are uncertain, and the first yield is generated only seven years after 
the investment. These circumstances might create an additional value of waiting and of 
getting more information, such as on price developments of hazelnut, and motivate using 
the real options instead of a classical NPV approach.



59Development of a model simulating returns on farm from investments

2.2 Risk and flexibility in timing (RealOpt)

Spiegel et al. (2018; 2020) demonstrate the advantages of stochastic-dynamic program-
ming for farm-level investment analysis, since it also considers risks besides (quasi-fixed) 
assets, such as land and on-farm labour already found in the model ClassNPV above, and 
addresses both time and scale flexibility as elements of a real-options approach. Spiegel et 
al. (2018; 2020) overcome the curse of dimensionality found in binary lattices or similar 
scenario tree approaches by employing a scenario tree reduction technique. Building on 
their work, we transform the ClassNPV model developed in the section above into a sto-
chastic-dynamic farm-level model. In contrast to Spiegel et al. (2018; 2020), we consider 
a second replantation period in order to expand the finite planning horizon so far in the 
future that differences to an infinite one become marginal from a numerical perspective.

We assume the following aspects of management flexibility. The farmer can decide 
during the first five years to introduce hazelnut or to continue cultivating durum wheat 
as an alternative annual crop (time flexibility 1). After reaching an age between thirty-
two and thirty-five years, the hazelnut trees must be removed; afterwards the land can be 
either planted again with new hazelnut trees or cropped with durum wheat (time flexibil-
ity 2). The subsequent plantation must be closed down again after thirty-two to thirty-
five years (time flexibility 3). This results in a finite planning horizon of seventy-five years 
such that differences between an infinite and this finite planning horizon should be neg-
ligible for any reasonable private discount rate. In order to increase computational speed, 
we divide the total land endowment into distinct plots of sizes 2n with n = 0,1,2… which 
in combination allow any integer plantation size between 0 and the land endowment (scale 
flexibility). Using fixed plot sizes instead of a continuous fractional plantation size allows 
for a mixed integer program instead of a mixed non-linear integer one. Integers are need-
ed anyhow to capture indivisibilities in investment (well, machinery). Time flexibility is 
considered separately for each plot.

Differences compared to the previous model ClassNPV are threefold. First, we con-
sider now not only the expected values of stochastic variables, but also the associated 
riskiness. More specifically, all expected values are replaced by probability distributions 
or stochastic processes, represented by a scenario tree. Each node of the tree contains a 
vector of stochastic variables’ realizations. Second, we now distinguish between the time 
period and the age of the plantation. In the previous simpler model, hazelnuts could only 
be planted in the first year such that the plantation’s age was equal to the year. Due to the 
time flexibility in RealOpt model, time and plantation age become two different dimen-
sions as the time flies regardless of the farmer’s decision to introduce hazelnuts or not. 
Accordingly, a plantation now can consist of plots of different age. As a consequence, in 
the expanded model, decision variables and risky parameters carry now both a time and 
node indices, such that the gross margins of both crops are defined as follows:

gmhazel,p,t,n = hahazel,p,t,n *[ yieldhazel,p,t,n * qihazel,t,n * marketPricehazel,t,n – yieldhazel,p,t,n * 
harvCost – otherCost] 

(7)

gmwheat,t,n = yieldwheat,t,n * qiwheat,t,n * MarketPricewheat,t,n – costwheat,t,n (8)



60 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

where gmhazel,p,t,n stays for the gross margin of hazelnuts [€ ha-1] on plot p in time period 
t{t1,t2,…,T} and node of the scenario tree n; yieldhazel,p,t,n for hazelnut yields [t ha-1] on plot 
p in time period t and node of the scenario tree n. The hazelnut yield depends on the 
difference between current year t and the year  when they were planted on this plot on 
the same path from the root to the current node n according to Eq.(1). hahazel,p,t,n stays 
for a binary variable of devoting a plot p into hazelnuts in time period t and node of the 
scenario tree n (1 = the plot is cultivated with hazelnuts; 0 = otherwise); qihazel,t,n for the 
hazelnuts quality index in time period t and node of the scenario tree n; marketPricehazel,t,n 
for the market price of hazelnuts in time period t and node of the scenario tree n [€ t-1]; 
gmwheat,t,n for gross margin of durum wheat [€ ha-1] in time period t and node of the sce-
nario tree n; yieldwheat,t,n for yields of durum wheat [t ha-1] in time period t and node of 
the scenario tree n; qiwheat,t,n for quality index of durum wheat in time period t and node 
of the scenario tree n; MarketPricewheat,t,n for the market price of durum wheat [€ t-1] in 
time period t and node of the scenario tree n; costwheat,t,n for quasi-fixed costs for durum 
wheat [€ ha-1] in time period t and node of the scenario tree n.

The farm’s operating income is thus defined as follows:

operIncfarm,t,n = areawheat,t,n * gmwheat,t,n +  sizep * gmhazel,p,t,n –  inip,t,n * iniCost * 

sizep–  reconvp,t,n * reconvCost * sizep – invWellt,n * invCostWell –  invMachm,t,n * 

invCostMachm – hiredLabt,n * waget,n ∀t,n 

(9)

where operIncfarm,t,n stays for farm’s operating income in time period t and node of the sce-
nario tree n [€]; areawheat,t,n for land area under durum wheat in time period t and node of 
the scenario tree n [ha]; sizep for the size of the plot p [ha]; inip,t,n for a binary variable of 
exercising the initial establishment of a hazelnut plantation on plot p in time period t and 
node of the scenario tree n (1 = hazelnuts are introduced; 0 = otherwise); reconvp,t,n for a 
binary variable of exercising clear-cut of hazelnuts plantation onto a plot p in time period 
t and node of the scenario tree n (1 = hazelnuts are clear-cut; 0 = otherwise); invWellt,n for 
a binary variable of exercising investments into a well and irrigation equipment in time 
period t and node of the scenario tree n (1 = investments into a well and irrigation equip-
ment are exercised; 0 = otherwise); invMachm,t,n for a binary variable of exercising invest-
ments into required machinery m in time period t and node of the scenario tree n (1 = 
investments into machinery are exercised; 0 = otherwise).

The discounted operating income is the objective variable to be maximized, defined as 
follows:

NPV =  probn *  (10)

where probn stays for the probability of the node  to occur [percentage points].
At each node of the constructed scenario tree, the model takes into account available 

time and scale flexibility, the state of the stochastic variables, as well as resources endow-
ments, and provides the following output:



61Development of a model simulating returns on farm from investments

- Land distribution between hazelnuts and durum wheat. Observing changes in land 
distribution between different nodes of the tree allows to observe (re)planting deci-
sions, as well as decisions to expand or clear-cut hazelnut plantations;

- Investments into a well and harvesting and other machinery for hazelnuts, the latter 
differentiated by size;

- Related economic variables such as costs and revenues.
Although the RealOpt model is fairly complex and presumably closer to real world 

decisions on investments, it has still two major drawbacks considering gaps found in 
literature. First, due to high costs related to the initial investments, the farmer will face 
considerable negative cash flows during the first years after a plantation is set up. Relat-
ed costs for financing are most probably underestimated by the average discount rate in 
the model. Second, the model neglects downside risk aversion, while the production cycle 
of hazelnuts implies significant negative cash flows in several time periods and related 
financing costs. We address these drawbacks stepwise in the two final models.

2.3 Costs of financing (RealOptFin)

The RealOptFin model introduces a current account of the farm operation. It serves 
as the source to cover variable and investment costs and receives subsidies and the operat-
ing income from selling products. In order to finance investments beyond accumulated 
cash, the model considers different types of loans with fixed repayment times and interest 
rates. The benefit for the farmer from the farm operation is represented now by annual 
profit withdrawals from the current account of the farm, discounted by his private dis-
count rate. Accordingly, the private discount rate now does not longer need to reflect the 
costs of financing. Instead, the market based discount rate is implicit and endogenously 
determined depending on the financing decisions.

The farmer now optimizes the expected net present value of future profit withdrawals 
from the farm operation, considering simultaneously investment and financing decisions. 
Farm operating income operInc enters the current account as follows:

curAcct,n =  curAcct-1,n1 + operIncfarm,t,n – withdrawt,n +  newLoansloans,t,n – 

 repaymloans,t,n –  intpaymloans,t,n  ∀t,

 (11)

where curAcct,n stays for the current account in the year t and node n [€]; withdrawt,n for 
annual farm household withdrawals [€]; newLoansloans,t,n for the loans acquired in the year 
t and node n [€]; repaymloans,t,n for the debt to-be-paid in the year t and node n [€]; and 
intpaymloans,t,n for the interest to-be-paid in the year t and node n [€]. The household with-
drawals are defined for each combination {t,n} based on investment and financing decisions.

The reader should note that introducing endogenous financing decisions implies and 
requires a more accurate simulation of cash flows. In particular, if the previous two mod-
els could omit cash flows independent of investment decisions, such as decoupled subsi-
dies under the Common Agricultural Policy’s (CAP) first pillar, all cash flows related to 



62 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

the farm operation have to be included now, since they affect the required financing. The 
operating income is hence defined as:

operIncfarm,t,n = areawheat,t,n * gmwheat,t,n +  sizep * gmhazel,p,t,n –  inip,t,n * iniCost  

* sizep –  reconvp,t,n * reconvCost * sizep – invWellt,n * invCostWell  

–  invMachm,t,n * invCostMachm – hiredLabt,n * waget,n + endland * prem ∀t,n (12)

where prem stays for the Common Agricultural Policy (CAP) first pillar direct payments 
[€ ha-1].

The discounted household withdrawals are now the objective variable to be maxi-
mized and defined as follows:

NPV =  probn *  (13)

2.4 Downside risk aversion (RealOptFinRisk)

Explicitly considering profit withdrawals allows introducing a lower limit of 
income from the farm operation such as to ensure household survival. This limit also 
acts as risk floor. The previous RealOptFin model assumes such minimum withdrawals 
to be zero, i.e. there are combinations of years and nodes possible where the household 
will not receive any income from the farm. This is likely to occur especially in the first 
years after setting up the plantation where high investment costs coincide with zero or 
low yields of hazelnuts. Our final RealOptFinRisk model instead assumes a minimum 
withdrawal level in each year and each node of the scenario tree. It is calculated by 
multiplying the level of the farm resource endowments with assumed minimum risk-
free returns: 

withdrawt,n ≥ endlab * minWage + endland * prem (14)

where minWage is a minimum risk-free off-farm wage [€ h-1]. Similar, the minimum with-
drawal limits above assumes that the farmer would be able to receive at least the premium 
of the first pillar of CAP as returns to its land, for instance, by renting it out. Different 
assumptions to define minimum withdrawals could clearly be chosen.

Financing and deciding on the annual withdrawals are hence also measures of risk 
management. While we ensure that the amount of new long-term loans cannot exceed 
investment costs in a year – assuming that bank will link such loans to a business plan 
– short-run loan and postponed withdrawals allow flattening the impact of stochastic 
operational cash flows from the farm on household withdrawals, i.e. income. The reader 
should note further that we assume that the quality indices, yields and prices of hazelnut 
and durum wheat are not correlated. Combining arable farming and a hazelnut plantation 
thus by itself reduces risk due to natural hedging.



63Development of a model simulating returns on farm from investments

We consider a lower limit on annual household withdrawals as a rather transparent 
and easy to communicate measure of risk aversion. Changing the limit in sensitivity anal-
ysis can help to inform a decision taker on the trade-offs between ensuring a minimum 
income level under any potential future development and his expected discounted income 
level. It does not require to introduce explicitly a risk-utility function in the framework 
above which is another avenue to develop the model further, for instance, to introduce 
behavioural aspects.

Figure 2 graphically represents the model variant and its major components. Each node 
of the scenario tree contains a vector of realizations of the seven stochastic variables. These 
realizations enter the calculations of net revenues in each node of the tree, which also reflect 
set-up and removal decisions with respect to hazelnuts made in this one and its ancestor 
nodes. These decisions translate into the future according to the production cycle and deter-
mine required future financing, as well as future costs of adjusting these production deci-
sions. Financing decisions need to ensure minimum household withdrawals and a non-
negative current account. The model simultaneously solves for optimal behaviour in all its 
nodes, maximizing the net present value (Eq. 13) under endowment and other constraints.

Figure 2. Graphical representation of the RealOptFinRisk model’s major components and relations 
between them.

Note: H stays for hazelnuts; DW stays for durum wheat.



64 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

2.5 Comparison of the models

Fig. 3 and Table 1 below give an overview on the four model variants. ClassNPV 
calculates discounted annual cash flows at farm level under the assumption to convert a 
part of land into hazelnuts now or never, i.e. it considers scale flexibility under endow-
ment constraints. Consequently, it also considers that additional labour might be needed 
depending on available farm family labour and the chosen investment program. RealOpt 
adds time flexibility, i.e., it captures and optimizes returns from investments at different 
time points, drawing on a real options approach. That model is next expanded to RealOpt-
Fin by introducing a difference between the private discount rate, used by the farmer to 
discount cash flows, and the costs of financing investments, i.e. it also optimizes financing 
decisions. RealOptFinRisk finally ensures that the farm household can withdraw in each 
year a minimum sum of money from the farming operation. It is also worth to mention 
that ClassNPV does not require a scenario tree as only the expected realizations are need-
ed in each time period. However, the tree realizations can be used post-model to report 
on the riskiness of the NPV optimized without considering risk. 

Figure 3. Comparison of components of the four model variants.



65Development of a model simulating returns on farm from investments

2.6 Solution approach

We use the solution approach suggested by Spiegel et al. (2018; 2020), which com-
bines Monte-Carlo simulation, a scenario tree reduction technique, and stochastic-
dynamic programming (Fig. 4). First, 5’000 Monte-Carlo draws are obtained for all the 
stochastic variables, using empirically predefined stochastic processes and distributions. 
Jointly this results in a huge scenario tree with 5’000 equally probable independent paths 
and a realization vector for the seven stochastic variables in each node. This step is done 
in Java based on standard libraries and own developed code to overcome speed limita-
tions in GAMS. The GAMS-package SCENRED2 by Heitsch and Römisch (2009) reduces 
this scenario tree in the second step. The underlying scenario reduction technique merges 
selected paths and nodes and provides new outcomes (i.e., the expected mean of merged 
outcomes) and the respective probabilities (i.e., the thickness of merged paths). The rela-
tion between nodes across time in a scenario tree is captured by an ancestor matrix, gen-
erated by SCENRED2. The final step combines the obtained scenario tree with the farm-
level model and solves for the optimal investment behaviour using stochastic program-
ming. Due to manifold dynamic relations between endogenous variables, all nodes on the 
same path to a final leave are interrelated. As all paths start with from the same root node, 
that implies that all nodes need to be simultaneously solved. The code of scenario tree 
composition and the farm-level model is available online.

3. Data and parameters

The parameters of the model draw on multiple data sources, including the Italian 
Farm Accountancy Data Network (FADN-CREA), Eurostat, World Bank, Census data 
(ISTAT, 2010), agricultural output prices (ISTAT, 2018) and the Italian Central Bank, 
as well as available literature (Frascarelli, 2017; Liso, 2017; Ribaudo, 2011) and expert 
judgement. The FADN data are only available for the period 2008-2016; the data from 
ISTAT, Eurostat, and the World Bank were selected for the period 2000-2016. All mon-
etary values were deflated using the GDP deflator for Italy provided by the World Bank 
(2015=100) to ensure comparability over time. 

Traditionally, hazelnut orchards were found in a specific district of the Viterbo prov-
ince, only, which is specifically suitable for hazelnut cultivation but nowadays doesn’t offer 

Table 1. Comparison of the four model variants.

ClassNPV RealOpt RealOptFin RealOptFinRisk

(i) Production cycle Yes Yes Yes Yes
(ii) Spatial flexibility Yes Yes Yes Yes
(iii) Economy-of-scale Yes Yes Yes Yes
(iv) Resources endowments Yes Yes Yes Yes
(v) Time flexibility No Yes Yes Yes
(vi) Optimising financing costs No No Yes Yes
(vii) Downside risk preferences No No No Yes



66 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

any additional space for new hazelnut cultivation. Therefore, new investments are located 
in municipalities close by, following a gradient of falling hazelnut yields depending on soil 
characteristics, climate conditions and often higher irrigation requirements, which mostly 
depends on the distance to the traditional growing zone. Data have been retrieved from the 
individual farm FADN database (2008-2016) considering only 21 municipalities of the prov-
ince of Viterbo1 (Lazio region) where hazelnut represents a limited share of the Utilized Agri-
cultural Area according to 2010 Census data (less than 5%), but which have recently experi-
enced relevant relative increases due to new plantations. We furthermore filter FADN data  to 
account for two factors. First, observations referring to years at or close after the establish-
ment of hazelnut plantations were excluded to reflect that no yields occur in the first six years 

1 Arlena di Castro, Bassano in Teverina, Blera, Castel Sant’Elia, Celleno, Civita Castellana, Gradoli, Graffignano, 
Marta, Monte Romano, Montefiascone, Monterosi, Oriolo Romano, Orte, Piansano, Tuscania, Vejano, Vetralla, 
Villa San Giovanni, Vitorchiano in Tuscia, Viterbo. 

Figure 4. Graphical representation of the solution process. (Source: based on Spiegel et al., 2018; 
2020).



67Development of a model simulating returns on farm from investments

after planting (Frascarelli, 2017). Second, only observations above 1 ha are included to neglect 
non-commercial activities in form of “hobby farms”. The regional focus and the two filters 
led to 62 observations in total. Census data suggest a representative farm size of 30 ha, and, 
for the considered municipalities, cropping of rain-fed arable crops with durum wheat as the 
dominant one as the benchmark before considering a hazelnut plantation.

Table 2 provides an overview of the parameter values and underlying data sources. 
For durum wheat and hazelnuts, expected yields are derived from the FADN sample based 
on total production and area. Since there is no information on the age of the respective 
plantations, we corrected the resulting average hazelnut yields by a coefficient of 1.25 and 
assumed it to be the maximum hazelnut yields. That coefficient reflects the average rela-
tion between the maximal yield and the yield developments depicted in Eq.(1). Also, due to 
limited data on hazelnut yields, we assume no riskiness in maximum hazelnut yields max-
Yields and the yields derived thereof yieldhazel,p,t,n. Instead, stochasticity in hazelnut produc-
tion is captured by a stochastic quality index and market price. In order to estimate the 
expected market prices of unshelled hazelnuts and durum wheat, the market prices in Italy 
provided by ISTAT (for hazelnuts) and Eurostat (for durum wheat) were used.

Furthermore, we correct the expected hazelnut price derived from historical observa-
tions by a multiplicative coefficient of 1.18. Assuming higher future prices seems appro-
priate due to increasing global demand of hazelnut, while production is expanding into 
less suitable production areas with a lower yield potential and higher costs, such caused 
by the required irrigation. Furthermore, all four models suggest no investments at all into 
hazelnuts under the expected historical price level. This contradicts observed farmers’ 
behavior, and suggests that farmer expect higher future prices. We used sensitivity analysis 
to find a suitable future expected mean price level where some but not all land was devot-
ed to hazelnut in at least one of the models, reflected by the factor of 1.18.

For quality indices, the FADN data were used to derive annual per unit farm specific 
prices of hazelnuts and durum wheat by dividing crop revenues by sold quantities. These 
calculated farm-gate prices were normalized by the market prices in Italy provided by ISTAT 
(2018) for hazelnuts and durum wheat to define samples of farm specific quality indices. 

We differentiate two sizes of a specialized harvester for hazelnuts between which the 
model can chose endogenously. The cheaper harvester is drawn by a tractor ordinarily 
used for other activities. The more expensive self-driving harvester reduces per ha labour 
needs and has a longer lifetime measured in harvested area. 

Compared to the ClassNPV model, the other models require converting expectations 
of stochastic variables  into stochastic processes or distributions. All the stochastic variables 
are assumed to be mutually independent, i.e. a correlation coefficient between any two sto-
chastic variables of zero is chosen. In particular, the market prices of hazelnuts and durum 
wheat are captured by uncorrelated mean-reverting stochastic processes defined as follows:

dprHt = μhazel (θhazel – prHt)dt + σhazel dWthazel (15)

dprDWt = μwheat (θwheat – prDWt)dt + σwheat dWtwheat (16)

where t is the time period; hazel indicates hazelnuts and wheat durum wheat; prHt is the 
natural logarithm of hazelnuts price; prDWt the natural logarithm of durum wheat price; 



68 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

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69Development of a model simulating returns on farm from investments

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70 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

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71Development of a model simulating returns on farm from investments

μ the speed of reversion; θ the long-term logarithmic average level of price; σ the standard 
deviation; and dWthazel the standard Brownian motion independent from dWtwheat. Other 
stochastic variables, namely a quality index of hazelnuts and a quality index, yield and 
variable costs of durum wheat are captured by distributions that were selected based on 
Akaike Information Criteria (AIC) (Akaike 1998) using @RISK software. More details on 
deriving the stochastic processes and distributions based on historical data are presented 
in the Appendix.

4. Results and discussion

We focus in this section on differences between the models with respect to key 
results: scale and timing of optimal hazelnuts introduction, expected NPV, as well as 
financing decision (Table 3). In particular, according to the ClassNPV model, hazelnuts 
cannot compete with the representative alternative arable crop durum wheat. According-
ly, the expected NPV of ClassNPV (rows 4-5 in Table 3) reflects returns from cultivating 
durum wheat only and hazelnuts are never introduced. In contrast, a hazelnut plantation 
might be set-up in later years in the RealOpt model which considers temporal flexibility. 
Specifically, that model suggests that a land share of about 48% of hazelnuts in the sec-
ond year or later is optimal. This does not imply that in any future stochastic scenario 
hazelnuts are cultivated. Temporal flexibility means that the farmer can wait, observe how 
the stochastic environment evolves, and take an investment decision depending on which 
node of the scenario tree is realized in the future. The 48% is hence an expected share. 
Row 2 in Table 3 reports the earliest time point where any hazelnuts are introduced (if at 
all). While both RealOpt and RealOptFin imply waiting at least for two years before setting 
up the first time a plantation, RealOptFinRisk suggests even longer postponement as the 
minimal year profit withdrawal is increased from zero in RealOptFin to opportunity costs 
reflecting off-farm wages and renting out land. Durum wheat exceeds these opportunity 
costs in any year and node, but hazelnuts do not. Accordingly, the RealOptFinRisk model 
has to postpone investments until hazelnuts are only introduced on such nodes where the 
minimal income of farming exceeds opportunity costs. For the remainder of the stochastic 
tree, only durum wheat is cropped. Compared to RealOpt or RealOptFin, this implies a 
lower average discounted household income at however reduced downside risk.

The temporal flexibility introduced in RealOpt allows increasing the expected NPV 
by 9.5% compared with the ClassNPV model. Note that generally the NPV can nev-
er decrease when additional flexibility is considered if all other assumptions are equal. 
Explicitly considering the costs of financing in RealOptFin slightly decreases the competi-
tiveness of hazelnuts and reduces the NPV by 4.4% compared with the RealOpt model. 
That means that the discount rate used in RealOpt underestimates the true costs of financ-
ing under assumed loan conditions. Yet, considering downside risk aversion in the Real-
OptFinRisk model has an even stronger effect: only around 6% of the total land is con-
verted to hazelnut in the third year or later. The expected NPV drops by 8.2% compared 
with the RealOpt model and by 3.9% compared with the RealOptFin model. However, the 
expected NPV under RealOptFinRisk still slightly exceeds the one of the ClassNPV model 
by 0.6%. Fig. 5 compares the riskiness of the resulting NPV in the four models described 
above, plus the forceHazel variant which forces immediate conversion of the whole farm 



72 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

Table 3. Comparison of empirical results of different models.

ClassNPV RealOpt RealOptFin RealOptFinRisk

(1)
Expected area under hazelnuts, % of total 
farm land endowment

- 48.07 40.80 6.03

(2)
Time period when introducing hazelnuts for 
the first time

-
(in 2 years) (in 2 years) (in 3 years)

(3) Is earlier reconversion applied? yes yes yes
(4) Expected NPV at farm-level, € 541,740.32 593,267.05 567,052.33 544,800.89

(5)
Expected NPV per hectare, € 
[calculated as (4) divided over the total farm 
land endowment]

18,058.01 19,775.57 18,901.74 18,160.03

(6) Used harvesting machine(s) - Large Large Large

(7)

Total expected amount of new loans over the 
planning horizon, €

Short: 
110,534.94

Middle:
2,602.33

Long:
 1,720,204.63

Short:
140,573.06

Middle:
11,792.54

Long:
432,047.90

(8) Total expected amount of interest paid, € 481,262.14 130,775.94

Figure 5. Distributions of maximized net present values in the five model variants, incl. forceHazel – an 
additional model variant that forces immediate conversion of the whole farm into hazelnuts. The forceHa-
zel model assumes no financing constraint, as it has no feasible solution otherwise. Both forceHazel and 
classNPV models ignore the associated risk and treat all the stochastic variables as their expectations, yet 
we recovered the riskiness of resulting NPVs based on the optimal behaviour that the models suggest.



73Development of a model simulating returns on farm from investments

into hazelnuts. The forceHazel model considers no financing options, as otherwise it has 
no feasible solution. The forceHazel model is therefore similar to the classNPV model 
except for having no scale flexibility. The models forceHazel and classNPV hence represent 
the two corner solutions: the former suggests devoting all resources to hazelnuts, the lat-
ter to the alternative crop. Both deterministic models forceHazel and classNPV ignore any 
risk by using expected values, only, for any stochastic variable related to both hazelnuts 
and the alternative crop durum wheat. We however recovered the associated riskiness in 
resulting NPVs post-model by applying the optimal behaviour in both models to the con-
structed scenario tree (Fig. 5). One can observe that hazelnuts imply much more risk of 
the resulting NPV, while also leading to a slightly lower expected NPV (compare force-
Hazel and classNPV in Fig.5). In contrast, the other three models directly report the riski-
ness of the NPV and consider it when searching for the optimal investment and financing 
behaviour. While realOpt and realOptFin are quite similar in terms of the spread of the 
NPV, the model realOptFinRisk clearly outputs a less risky NPV due to its lower limit on 
annual household withdrawals, however as noted already above, at the costs of a lower 
expected NPV (Fig. 5). The realOpt and realOptFin show some outliers (indicated as dots 
in the box-and-whisker charts) with quite low NPVs that are removed at the RealOptFin-
Risk model, which however also considerably reduces upside risk.

Figure 6 visualizes the riskiness of the four models in greater details. ClassNPV 
implies no hazelnuts and reflects the moderate riskiness of durum wheat cultivation, only. 
The upper panel shows that quite clearly, as the cloud with the points showing the dif-
ferent outcomes for the farm income is quite dense. In contrast, RealOpt implies much 
more risky withdrawals, including considerable positive and negative outliers. Moreover, 
annual withdrawals implied by RealOpt echo the production cycle of hazelnuts: negative 
withdrawals in the beginning of the time horizon (establishment of the first plantation) 
and between time periods 35 and 40 (establishment of the second plantation), combined 
with high positive withdrawals that are associated with periods of maximum yields of the 
hazelnut plantation.

Both models with financing (the lower part of Fig. 6) cut off the negative withdrawals 
by covering them with short-term credits or by not withdrawing all profits in some years, 
i.e. using a retained profit position. Without these internal and external financing options, 
a lower limit of household withdrawals of zero or above in any year under all potentially 
considered futures cannot be achieved. This is visible from the upper panel as even under 
the classNPV where only durum wheat is grown, there are some years where farm prof-
its become negative. These last two models differ mainly in financing behaviour. The 
RealOptFin model only needs to maintain a positive current account of the firm but can 
reduce household withdrawals in certain years down to zero. As a consequence, it uses 
almost solely long-term credits (Table 3, row (7)) to finance the initial investment costs 
of plantation set-up and the well, as well as in some later years investment in a harvest-
er. The costs relate to an expected 41% land share under hazelnuts (Table 3, row (1)). In 
contrast, the RealOptFinRisk model ensures minimum annual withdrawals above opportu-
nity costs and has to use also short- and especially middle-term credits to balance annual 
fluctuations in withdrawals (Table 3, row (7)). These reflect foremost the production cycle, 
i.e. plantation ages of no or low hazelnuts yields, but also relate to nodes in scenario tree 
with lower than average prices and/or quality indices. Since only 6% compared to 41 % of 



74 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

total land is in the expected mean devoted to hazelnuts, the required investment costs are 
considerably lower such that the amount of long-term credits decreases substantially com-
pared with the RealOptFin model. 

The empirical results are in line with the available literature. Comparison of the 
results of ClassNPV and RealOpt models indicate that uncertainty and time flexibil-
ity leads to later investments at a higher expected scale. Trigeorgis and Reuer (2017) and 
Musshoff (2012) confirm that managerial flexibility usually increases the value of waiting, 
hence leading to postponement of investments; the reader should note here that relatively 
small uncertainty might lead to no value of waiting and hence immediate investments. As 
for investment scale, Hassett and Metcalf (1993) confirm that if immediate investment is 
worthless, uncertainty could create its value in the future. However, the effect of uncer-
tainty might be the opposite if immediate exercising of investment is profitable in a risk-
free environment. In this case, considering temporal flexibility might lead to lower expect-
ed investment scale depending on how the stochastic environment evolves. The resulting 
effect would depend on the underlying uncertainty, as well as on relationship between 
stochastic variables and the optimal investment scale. In this regard, our empirical results 
stating that uncertainty leads to larger expected area under hazelnuts shall be treated as 
a special case. Trigeorgis and Reuer (2017) also report that managerial flexibility reduc-
es downside riskiness of investment, which is confirmed by our results, in particular the 

Figure 6. Distributions of annual withdrawals across the planning horizon in the four models.



75Development of a model simulating returns on farm from investments

upper part of Fig. 6. Comparison of the results of RealOpt and RealOptFin models suggest 
that explicit consideration of financing behaviour reduces investment scales, yet does not 
affect the earliest time of investments. The results can indirectly be confirmed by Chen 
(2003) and Lin (2009), both claiming that a higher debt ratio leads to a higher investment 
threshold. However, we explicitly highlight here that the literature focusing on financ-
ing of investment under uncertainty is extremely limited. Finally, comparing the results 
of RealOptFin and RealOptFinRisk, one can conclude that consideration of downside risk 
aversion leads to later investment at a lower scale, as well as lower resulting riskiness. Pre-
vious studies confirm that risk aversion, and downside risk aversion in particular, reduces 
incentives to invest (e.g., Chronopoulos et al., 2011).

5. Discussion and conclusion

Our case study results highlight that the assumptions underlying the different model 
variants can considerably affect key results. The comparison confirms that more advanced 
models are more informative: they provide additional insights and can provide more 
detailed advice to decision takers, such as on how to best finance an investment and how 
to buffer income fluctuations from production and market risks. The step-by-step devel-
opment of the advanced farm-level models allows to identify the relative importance of 
the additional elements considered and to illustrate their value added. For instance, the 
simple NPV calculation suggests not planting hazelnut at all while all the other more 
complex models suggest doing so, however at varying time periods and scales. Constrain-
ing the downside risk of income from the farm operation in the most advanced models 
not only highlights the trade-off between mean income and reduced downside risk, but 
also shows the resulting consequences on the scale and timing of investments, as well as 
on financing behaviour.

Clearly, there is a trade-off between additional insights and potentially more realis-
tic results on the one hand, and increased data demands (Table 2) and model complex-
ity on the other one. Additionally, higher data requirements imply typically also higher 
uncertainty. For instance, the more advanced model with explicit financing costs does not 
simply require one average interest rate, but interest rates for different finance instruments 
which depend on a number of factors, such as credit amount or farmer’s credit scores. The 
results – both additional ones and the ones also found in simpler models – are sensitive 
to what is assumed here in detail on top of the parameter found also in simpler models. 
Compared to sensitivity to one average discount rate only, the more advanced model dis-
tinguishes between different components of discount rate, i.e., time preferences, risk pref-
erences, costs of financing, etc., which all can be subject to sensitivity analysis to inform 
on their importance individually. Furthermore, such sensitivity analysis could also help to 
find a set of parameters which best fits the observed behaviour (e.g. Troost and Berger, 
2014). In our case, expected hazelnuts yields and market prices as well as their riskiness 
would be obvious first candidates for such an analysis.

As a word of caution, we remind the reader that using more advanced methods such as 
real options does not necessary imply a better fit to observed behaviour. Indeed, especially 
the full rationality assumption inherent in optimization approaches might be questioned. 
A potential promising avenue here is to complement the optimization model with other 



76 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

methodologies (Colen et al., 2016), for instance, to expose farmers facing investment deci-
sions to results of such models in order to learn more, e.g., on how they frame the decision 
problem including which results matter to them most, or to contrast subjective perceptions 
of market developments and related risk with findings from statistical analysis. The detailed 
what, how and when view of dynamic programming approaches might ease that kind of 
dialogue as it might be similar to the one used by the farmer itself. Alternatively, results 
obtained with other methodologies, e.g., econometric or experimental techniques for objec-
tive or constraint functions, can serve as input for optimization model and allow introduc-
ing behavioural aspects, for instance in form of a risk utility function (Chronopoulos et 
al., 2014). Finally, further research might put greater focus on how learning affects future 
expectations, for instance, how experiences of rare but catastrophic events shape expecta-
tions, and how this can be reflected, for instance, in a scenario tree.

Overall, our paper underlines that the conceptual and technical elements are read-
ily available to build farm-scale models based on dynamic stochastic optimization. This 
allows to determine scale and timing of long-term investments under production and 
market risk and endowment constraints, drawing on real options. We also highlight that 
such models are extensions of the widely used farm programming approaches and show 
the additional insights which can be gained from their application. We demonstrated the 
different models using the example of hazelnut production in an Italian region. An appli-
cation to, for instance, other perennials can draw on the basic model structure and solu-
tion approach. But it will clearly require other data, and potentially also adjustments in 
some model detail, for instance introducing variables and equations to reflect additionally 
required investments such as relating to storage or post-harvest treatment.

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Appendix. Capturing stochastic variables with stochastic processes and distributions 
based on historical data

Market price of hazelnuts and durum wheat

In order to estimate the stochastic processes for market prices of unshelled hazelnuts 
and durum wheat, the market prices in Italy provided by ISTAT (for hazelnuts) and Euro-
stat (for durum wheat) were used (Fig.A1).

We omit the observations from the years 2008 and 2014-2016 for hazelnuts, as they 
do not fit the general trend and hence should be excluded when estimating stochastic pro-
cesses. We ran the following stationarity tests: Augmented Dickey-Fuller (ADF) test; Phil-
lips–Perron (PP) Unit Root test; and Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test. For 
both data samples, non-stationarity hypothesis cannot be rejected based on the ADF and 
PP tests, while the KPSS test concludes that stationarity hypothesis cannot be rejected. In 
light of the conflicting results of these tests, we decide on the appropriate method based 
on economic reasoning and therefore apply an MRP estimation. This assumes stationarity 



80 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

reflecting that the market price likely fluctuates around a constant long-term per unit pro-
duction cost. The result of the MRP estimations are summarized in the Table A1.

Furthermore, as above, we correct every price draw by a multiplicative coefficient of 
1.18 in order to account for expected increase in hazelnut price due to increasing demand. 
This price level also leads to introduction of hazelnut in some but not all model variants 
and also to highlight differences.

Quality index for hazelnut and durum wheat

The FADN data were used to derive annual per unit farm specific prices of hazelnuts 
and durum wheat by dividing crop revenues by sold quantities. These calculated farm-gate 

Figure A1. Real durum wheat (DW) and hazelnut (H) prices, € 100kg-1. Source: ISTAT and Eurostat; the 
prices are deflated (2015=100) using the GDP deflator in Italy provided by the World Bank.

0,00

5,00

10,00

15,00

20,00

25,00

30,00

35,00

40,00

0,00

50,00

100,00

150,00

200,00

250,00

300,00

350,00

400,00

450,00

500,00

2000 2002 2004 2006 2008 2010 2012 2014 2016

H DW

Table A1. Estimated parameters of mean-reverting processes for hazelnut and durum wheat prices. 
Source: own estimation based on the ISTAT (for hazelnuts, years 2000-2013) and Eurostat (for durum 
wheat, years 2000-2016, excl. 2008) data. The prices were deflated (2015=100) using the GDP deflator 
for Italy provided by the World Bank.

Natural logarithm of hazelnut price Natural logarithm of durum wheat price

Long-term mean 7.6782 5.4690
Speed of reversion 0.9219 3.1053
Standard deviation 0.1933 0.4808
Starting value 8.0669 5.4036



81Development of a model simulating returns on farm from investments

Figure A2. Distribution fitting for the quality index of hazelnut. Source: own elaboration based on 
FADN and ISTAT data.

5.0% 90.0% 5.0%

4.4% 91.9% 3.8%

0.511 1.363

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

1.
2

1.
4

1.
6

1.
8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Fit Comparison for QI 15

RiskLaplace(0.92467,0.23981)

Input

Minimum 0.1766

Maximum 1.6130

Mean 0.9236

Std Dev 0.2497

Values 56

Laplace

Minimum −∞

Maximum +∞

Mean 0.9247

Std Dev 0.2398

@RISK Course Version

Wageningen University

Figure A3. Distribution fitting for the quality index of durum wheat. Source: own elaboration based 
on FADN and Eurostat data.

5.0% 90.0% 5.0%

6.3% 88.9% 4.9%

0.602 1.406

0.
0

0.
5

1.
0

1.
5

2.
0

2.
5

3.
0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fit Comparison for QI 4

RiskLaplace(0.98170,0.25802)

Input

Minimum 0.0555

Maximum 2.5309

Mean 0.9795

Std Dev 0.2660

Values 653

Laplace

Minimum −∞

Maximum +∞

Mean 0.9817

Std Dev 0.2580

@RISK Course Version

Wageningen University



82 Alisa Spiegel, Simone Severini, Wolfgang Britz, Attilio Coletta

prices were normalized by the market prices in Italy provided by ISTAT for both hazel-
nuts and durum wheat to define samples of farm specific quality indices. These observa-
tions for quality indices were fitted to a Laplace distribution with a mean of 0.9247 and 
standard deviation of 0.2398 (Fig.A2) for hazelnut, and mean of 0.9817 and standard devi-
ation of 0.2580 (Fig.A3) for durum wheat.

Yields and variable costs for durum wheat

For durum wheat, yields derived from the FADN sample based on total production 
and area were fitted to a Laplace distribution with a mean of 3.9120 and standard devia-
tion of 1.1984 (Fig.A4). The observations for durum wheat costs were fitted to a Gamma 
distribution with a shape parameter of 3.8286 and a scale parameter of 97.098 (Fig.A5).

Figure A4. Distribution fitting for the yields of durum wheat. Source: own elaboration based on FADN 
data.

5.0% 90.0% 5.0%

5.1% 90.8% 4.1%

1.98 6.04

0 1 2 3 4 5 6 7 8 9

10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fit Comparison for Yields_DW_t_per_ha

RiskLaplace(3.9120,1.1984)

Input

Minimum 0.4427

Maximum 9.8519

Mean 3.9127

Std Dev 1.2456

Values 653

Laplace

Minimum −∞

Maximum +∞

Mean 3.9120

Std Dev 1.1984

@RISK Course Version

Wageningen University



83Development of a model simulating returns on farm from investments

Figure A5. Distribution fitting for the variable costs of durum wheat. Source: own elaboration based 
on FADN data.

5.0% 90.0% 5.0%

4.0% 88.8% 7.2%

114 676

-2
00

0

20
0

40
0

60
0

80
0

1,
00

0

1,
20

0

1,
40

0

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

Fit Comparison for Dataset 3

RiskGamma(3.8286,97.098)

Input

Minimum 13.30

Maximum 1,363.66

Mean 371.75

Std Dev 190.45

Values 647

Gamma

Minimum 0.00

Maximum +∞

Mean 371.75

Std Dev 189.99

@RISK Course Version

Wageningen University


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