Optimal Rotation Age Research Articles

Managing ecosystem services in oleaginous forests for bioenergy provision and climate change mitigation

Oleaginous forests provide diverse ecosystem services, including timber, seed yield (a vital feedstock for biodiesel production), and substantial carbon savings. These carbon savings encompass carbon sequestration related to timber growth and carbon savings resulting from substituting fossil fuel with biodiesel. However, oleaginous forests are vulnerable to seed wasp attacks (disservice), which significantly threaten both seed yield and carbon savings. Using an integrated ecological-economic model that includes Faustmann's Land Expectation Value model and a pest damage control model, we aim to understand the intricate relationship among multiple ecosystem services and disservices of oleaginous forests. The results reveal four distinct phases contingent on varying pesticide application rates: the pesticide under-use phase, substitution phase, complementary phase, and over-use phase. Notably, a potential avenue to minimize pest damage is identified during the complementary phase by reducing the optimal rotation age at the expense of decreased carbon sequestration and bioenergy provision, posing a challenge to climate change mitigation. These findings have implications for formulating policies to manage conflicting ecosystem services of energy forests, offering valuable insights into the intersection of sustainable forest management and climate policy.

Optimal Rotation Age in Fast Growing Plantations: A Dynamical Optimization Problem.

Forest plantations are economically and environmentally relevant, as they play a key role in timber production and carbon capture. It is expected that the future climate change scenario affects forest growth and modify the rotation age for timber production. However, mathematical models on the effect of climate change on the rotation age for timber production remain still limited. We aim to determine the optimal rotation age that maximizes the net economic benefit of timber volume in a negative scenario from the climatic point of view. For this purpose, a bioeconomic optimal control problem was formulated from a system of Ordinary Differential Equations (ODEs) governed by the state variables live biomass volume, intrinsic growth rate, and area affected by fire. Then, four control variables were associated to the system, representing forest management activities, which are felling, thinning, reforestation, and fire prevention. The existence of optimal control solutions was demonstrated, and the solutions of the optimal control problem were also characterized using Pontryagin's Maximum Principle. The solutions of the model were approximated numerically by the Forward-Backward Sweep method. To validate the model, two scenarios were considered: a realistic scenario that represents current forestry activities for the exotic species Pinus radiata D. Don, and a pessimistic scenario, which considers environmental conditions conducive to a higher occurrence of forest fires. The optimal solution that maximizes the net benefit of timber volume consists of a strategy that considers all four control variables simultaneously. For felling and thinning, regardless of the scenario considered, the optimal strategy is to spend on both activities depending on the amount of biomass in the field. Similarly, for reforestation, the optimal strategy is to spend as the forest is harvested. In the case of fire prevention, in the realistic scenario, the optimal strategy consists of reducing the expenses in fire prevention because the incidence of fires is lower, whereas in the pessimistic scenario, the opposite is true. It is concluded that the optimal rotation age that maximizes the net economic benefit of timber volume in P. radiata plantations is 24 and 19 years for the realistic and pessimistic scenarios, respectively. This corroborates that the presence of fires influences the determination of the optimal rotation age, and as a consequence, the net economic benefit.

Effect of climate change on the land rent of radiata pine plantations in Chile: Site productivity and forest fires

Climate change can severely affect forest plantation productivity and fire regimes in temperate regions. The economic evaluation of these impacts is challenging because of the spatially differentiated effects expected to occur across planted areas. We calculated the combined effect of projected climate change on the stochastic land expectation value (SLEV) for a random sample of stands covering the Pinus radiata plantations in central-southern Chile. To simplify the stochastic evaluation problem with changing productivity and fire frequencies over time, we divided rotations into stages, before and after commercial thinning, and assumed that reforestation is at the end of each stage. The SLEV was calculated through Monte Carlo simulations using the coefficients from previously estimated productivity and fire risk statistical models. The predicted combined effect on the SLEV is negative in most regions except those in the southern limit of the specie distribution. In the economically crucial coastal area of the central regions, the negative effect of more frequent fires outweighs faster growth. In the northern drier areas, the SLEV becomes negative due to a large drop in site productivity. Anticipatory reassignments of management regimes have a large adaptation value in these low-productivity sites. Our optimization-simulation results suggest that relative to prescribed management regimes, the optimal rotation age in most low-productivity sites is longer and in most high-productivity sites shorter. Shortening the rotation age by one to two years has an important adaptation value in highly productive coastal sites.

Laurel Regeneration Management by Smallholders to Generate Agroforestry Systems in the Ecuadorian Amazon Upper Basin: Growth and Yield Models

Laurel (Cordia alliodora Ruiz & Pav. Oken) is a Neotropical native tree that is easily regenerated in the secondary forest within the Amazon region. Amazonian smallholders use this tree regeneration to obtain a homogeneous forest cover when developing local agroforestry systems, which do not depend on nursery seedling production for tree planting. The objective of the present investigation was to develop growth and yield models for Laurel within the local agroforestry systems. A total of 226 sampling plots were measured between 2010–2011 and 2014–2015. Chapman- Richard, Hosslfeld II, and the generalized algebraic difference approach (GADA) form of the Chapman-Richard’s function was used for modeling height-age and diameter-age relationships. Eight volume models were tested to describe total stem volume. The GADA method was suited to describe the Laurel height and diameter-age growth. The cutting cycle for agroforestry systems with a density of 300 trees ha−1 at the best site index (SI) (22 m) produced 13.9 m3 ha−1 year−1 and a total wood yield of 195.1 m3 ha−1 at age 14. In the worst SI (14 m), the average annual yield was 3.5 m3 ha−1, with a total yield of 83.3 m3 ha−1 at age 24 years. The Spurr potential model was the best fit to describe the volume of the Laurel according to the Akaike information criteria. The Laurel biological (optimal) rotation age suggests that the minimum cutting diameter should be lowered from 30 cm of DBH in the research zone. Management of the natural regeneration of secondary forests by smallholders is a local agroforestry practice that should be given greater attention, especially within protected forest areas.

Forest Land Expectation Value or Maximum Sustained Yield? Resolving A Long-Standing Paradox

The Faustmann formula, equivalent to the land expectation value (LEV), yields the present value, starting with bare land, of an infinite series of future timber rotations for a stand. If this formula is used to find the rotation age that maximizes the present value, a lower annual harvest will result when compared to a maximum sustained yield (MSY) regime for an ownership with many stands. However, the LEV is strongly preferred by economists. This is the LEV vs. rents paradox. Herein, this paper demonstrates that an infinite series of annual harvests for an ownership under an even flow regime, discounted to the present using any interest rate, will provide the same optimal rotation age as the time of the peak MAI for a single stand, though with different estimates for the profitability, depending on the interest rate. Thus, rotations producing the MSY and the maximum financial return are identical when analyzed at the ownership scale but are not the same when based on the analysis at the stand scale using the Faustmann formula (LEV). This is the solution to the paradox. The formulas for internal rate of return (IRR) and the land valuation for transaction purposes are also presented.

Determining optimal forest rotation ages and carbon offset credits: Accounting for post‐harvest carbon storehouses

AbstractSequestering carbon in forest ecosystems is important for mitigating climate change. A major policy concern is whether forests should be left unharvested to avoid carbon dioxide (CO2) emissions and store carbon, or harvested to take advantage of potential carbon storage in post‐harvest wood product sinks and removal of CO2 from the atmosphere by new growth. The issue is addressed in this paper by examining carbon rotation ages that consider commercial timber as well as carbon values. A discrete‐time optimal rotation age model is developed that employs data on carbon fluxes stored in both living and dead biomass as opposed to carbon as a function of timber growth. Carbon is allocated to several ecosystem and post‐harvest product pools that decay over time at different rates. In addition, the timing of carbon fluxes is taken into account by weighting future carbon fluxes as less important than current ones. Using simple formulae for determining optimal rotation ages, we find that: (1) Reducing the price of timber while increasing the price of carbon will increase rotation age, perhaps to infinity (stand remains unharvested). (2) An increase in the rate used to discount physical carbon generally reduces the rotation age, but not in all cases. (3) As a corollary, an increase in the price of carbon increases or reduces rotation age depending on the weight chosen to discount future carbon fluxes. (4) Site characteristics and the mix of species on the site affect conclusions (2) and (3). (5) A large variety of carbon offset credits from forestry activities could be justified, which makes it difficult to accept any.

Determining spatially varying profit‐maximizing management practices for miscanthus and switchgrass production in the rainfed United States

AbstractDetermining optimal management practices for the profitable production of perennial energy crops is critical for scaling up production beyond experimental levels. Although many experimental field studies have examined the effects of management practices on the performance of miscanthus and switchgrass, there are no recommendations for economically optimal nitrogen (N) application rates and how they should vary spatially and with the age of the energy crop as well as on optimal rotation age of the energy crop to maximize profits. We develop a modeling framework to determine economically optimal crop management decisions and simulate the variability under various scenarios for miscanthus and switchgrass production across 2287 counties in the rainfed United States. We find that profit‐maximizing N recommendations for these crops vary across maturity stages and regions and can increase the landowner's profits compared with a uniform N rate across ages and regions. We also find that the optimal rotation for these crops is shorter than the productive physical lifespan (15–20 and 10 years for miscanthus and switchgrass, respectively). Specifically, the N rate that maximizes the economic returns is negligible for miscanthus and 111 kg ha−1 for switchgrass production at age 2. The mean profit‐maximizing N rate increases with age for miscanthus, peaking at 151 kg ha−1 at age 11 before declining to 114 kg ha−1 at the optimal rotation age of 13 years while that for switchgrass is 150 kg ha−1 for middle‐aged stands and declines to 114 kg ha−1 at the optimal rotation of 8–9 years. We find that miscanthus is the most profitable energy crop in the northern region of the rainfed United States while switchgrass is most profitable in the south of the rainfed United States. Our findings are useful for improving assessments of the profitability of energy crops and guiding future management decisions by landowners.

Stochastic Dynamic Optimization for Forest Rotation with Uncertain Stumpage Prices

Abstract This study investigates the effect of uncertain stumpage prices on the optimal forest rotation decision and related profitability under various silvicultural scenarios. The study applies loblolly pine (Pinus taeda L.) growth and yield models with distinct silvicultural scenarios in the Piedmont and Upper Coastal Plain region of the southeastern United States. It then applies chance-constrained dynamic optimization to derive optimal rotation age and related profitability on per acre basis. The growth and yield models show that using a combination of herbicide and fertilizer silvicultural treatment leads to a higher timber yield and profit at a lower optimal rotation age than other treatment scenarios. Furthermore, the stochastic optimization model shows that low risk tolerance results in low returns. Within a particular risk tolerance level, silvicultural treatment options with higher standard deviations almost always produce higher returns. The findings of this article contribute to the literature on optimizing forest management when considering uncertain stumpage prices over time, given different risk tolerance levels.

Responsiveness of miscanthus and switchgrass yields to stand age and nitrogen fertilization: A meta‐regression analysis

AbstractOptimal management of the perennial bioenergy crops, miscanthus and switchgrass, requires an understanding of their responsiveness to nitrogen (N) fertilizer at different maturity stages across locations and growing conditions. Earlier studies that have examined the yield response of these crops to N and stand age using field experiments or meta‐analysis techniques provide mixed evidence. We extend earlier studies by applying a multi‐level mixed‐effects (MLME) meta‐regression model to conduct a more extensive multivariate regression of yield response of these crops to N and stand age, while controlling for climate and location conditions and unobserved factors related to study design. Our findings are based on 1403 and 2811 yield observations for miscanthus and switchgrass, respectively, from experiments conducted between 2002 and 2019 across the rainfed region in the United States. We find statistically significant evidence that an additional year of maturity increases miscanthus and switchgrass yields but at a decreasing rate; yields peak at the 7th and 6th year respectively, for the observed range of applied N rates and stands. We also find that an increase in N application increases yield by a statistically significant level, but at a declining rate; the magnitude of the yield response to N is, however, small and varies with the age of the crop. The impact of N is larger on older compared to younger and middle‐aged stands of miscanthus. In contrast, the impact of N on switchgrass is larger on middle‐aged compared to younger and older stands of switchgrass. We do not find a statistically significant effect of soil productivity on yield for either crop. This analysis provides a basis for developing N application recommendations and optimal rotation age for miscanthus and switchgrass and shows that these energy crops can grow just as productively on low productivity land as on high productivity land.

Price Premium Requirements for Growing Higher Quality Pine Sawtimber in Even-Aged Systems in the Southeastern United States

Abstract Intensive pine silviculture has become the dominant management paradigm in the southeastern United States. Although productivity has been substantially increased by the combination of cultural, silvicultural, and genetic advancements, wood quality is sometimes sacrificed in intensive silviculture. Extending the optimal rotation allows trees to grow more timber, which may result in the production of better quality sawtimber; however, landowners may require incentives to do so. We simulated loblolly, slash, shortleaf, and longleaf pine for growth and yield using the Forest Vegetation Simulator (FVS) to determine sawtimber price premiums landowners would require to offset the costs associated with delaying the final harvest by 10 to 30 years in even-aged systems. Required incentives increased with the length of harvesting delay beyond the financially optimal rotation age. On medium productivity sites, landowners would be willing to delay the final harvest by 10 years for sawtimber price premiums of $5.06/ton (20.47%) for loblolly, $5.34/ton (21.6%) for slash, $4.56/ton (18.45%) for longleaf, and $6.71/ton (27.14%) shortleaf pine, respectively. Harvest delays of 10 to 20 years were financially justifiable, whereas extensions exceeding 30 years were prohibitively costly for all species. Delaying the optimal harvest could benefit landowners by generating a premium price for their sawtimber while providing important ecosystem services.

Age trends in genetic parameters for growth performance across country-wide provenances of the iconic conifer tree Araucaria angustifolia show strong prospects for systematic breeding and early selection

Understanding the growth patterns of long-lived conifer tree species is important to devise breeding and early selection strategies, predict future biomass productivity and assess adaptive tree fitness for long term conservation efforts. We investigated the genetic variation for growth traits of Araucaria angustifolia, the grandiose renowned “Paraná pine” tree, in a trial involving 122 families across 15 provenances covering the entire natural range of the species in Brazil. Measurements at ages 7, 24, 32, 33 and 35 were used to adjust continuous growth curves based on nonlinear mixed-effect models for all 2158 trees, providing annual estimates for unmeasured ages in the 7-to-35-year interval. Estimated values closely matched observed ones and a reduction of the coefficient of residual variation was observed in the estimated data, possibly due to removal of random error in the observed measurements, making the estimated curves more reliable to predict growth patterns. Genetic variation for growth within provenances was greater than between, with a trend of increasing heritabilities over time for most provenances. Substantial genetic variation found both within and between families could drive efficient early selection at both levels. All provenances included individual trees and families with good potential to be selected for shorter rotations. Growth curves show that trees invest first in height and later in diameter growth. Considerable variation was observed across provenances for the optimal age and optimal tree volume at which annual growth increment peaks, a tipping point that could be used as a predictor of the optimal rotation age and expected tree volume. The data clearly indicate potential for early selection for growth at age 7–10 with an 85% prediction accuracy of growth at age 35. Additionally, growth data indicate potential of shortening harvest age from 30–35 to 15–20 years by selecting the best individuals and families. These results underscore the potential of expanding investments in breeding and plantation forestry of A. angustifolia, which in parallel could contribute to enhancing conservation efforts of this iconic subtropical conifer.

Developing a growth and yield model for planted big-leaf mahogany (Swietenia macrophylla King) at advanced age in subtropical moist forest in Puerto Rico

The ability to predict growth and potential yield is essential for planning forest management. Here we developed a narrow size-class diameter model to simulate growth in a mature plantation (age 42 years) of big-leaf mahogany (Swietenia macrophylla King) in a subtropical moist forest in Puerto Rico. We calculated optimal rotation age by means of the biological and economic criteria. The analyses involved diameter measurements over 31 years and a variety of contrasting economic scenarios, which were evaluated by means of the Faustman formula and sensitivity analyses. The optimal biological rotation was 90 years while the optimal economic rotation was 42 years in most of the scenarios, regardless of wood prices and costs of reforestation. Conspicuous changes in the optimal economic rotation were only observed under low interest rates, in which it was prolonged to around 80 years, close to the optimal biological rotation. Abundant natural regeneration and a prolonged biological rotation suggest that other silvicultural systems, rather than traditional clear-cutting and replanting could be adopted. However, to evaluate alternative silvicultural strategies we would need to improve our ability to predict growth. This would require expansion of our database to include information on growth conditions of individual trees and young tree dynamics, as well as size and growth of other species, to model natural regeneration and competition.

The effects of cultivar and rotation length (5 vs. 10 years) on biomass production and sustainability of poplar (Populus spp.) bioenergy plantation

AbstractPoplars (Populus spp.) managed in short rotation woody crop (SRWC) systems are considered very promising in terms of biomass production for energy purposes in a temperate zone. In this study, several gaps in knowledge regarding the selection of the plant material, cultivar‐specific responses to coppicing and rotation length are addressed. Five‐ and 10‐year rotation lengths were considered, with a focus on the sustainability of the plantation and the quantification of the biomass production of 10 commercially available hybrid poplar cultivars during two consecutive 5‐year rotation cycles versus one 10‐year rotation in northern Poland. The biomass production varied considerably among cultivars, ranging from 1.6 (‘AF8’) to 7.5 (‘NE42’) Mg ha−1 year−1 at the end of the first 5‐year rotation, and 0.0 (‘AF8’)–5.4 (‘Degrosso’) Mg ha−1 year−1 in the second 5‐year rotation, while in a single 10‐year rotation, the mean annual increment (MAI) ranged from 1.0 (‘AF8’) to 15.1 (‘NE42’) Mg ha−1 yr−1. The 5‐year rotation cycles negatively influenced the sustainability of the plantation due to the high mortality of stems and sprouts in the second 5‐year coppice rotation. In the case of 5‐year and longer rotations, replanting (not resprouting) after harvest may be a more advantageous practice for maximizing the regeneration. The 10‐year rotation cycle allowed for enough development of the crown and root systems to produce sufficient basal area and height to achieve maximal MAI in the trees. The study revealed that an age ≥10 is a biologically optimal rotation age as the slope of MAI was equal/close to zero. This study further showed the crucial importance of testing cultivars before introducing them on a commercial scale and provided valuable information on poplar cultivars available to stakeholders in the biomass market.

Diversity of Carbon Storage Economics in Fertile Boreal Spruce (Picea Abies) Estates

A “normal forest”, an idealized estate with a uniform distribution of stand ages, can be used in the study of sustainable management practices. As the normal forest contains a variety of stand ages, the characteristics of the stands can be represented in terms of a “normal stand”, with properties known as a function of age. This paper takes seven never-thinned stands as seven “normal stands”, which describe seven estates of normal forest. The intention is to study the robustness of carbon storage microeconomics to varying estate characteristics. It was found that the economically optimal rotation ages vary. The state sums of volume and capitalization, corresponding to any optimal rotation, also vary significantly. Growth rates vary more than the optimal expected stand volumes. Consequently, any excess volume related to carbon storage adds on to an almost unified basic volume. For all seven normal estates, the most economical way of increasing carbon storage is to increase the size of trees retained in thinning from above.

Management of spontaneously regenerated mixed stands of birch and Norway spruce in Sweden

Timber production and profitability were evaluated for spontaneously-regenerated mixtures on two formerly clearcut areas. The abandoned areas developed into birch-dominated ( Roth and Ehrh.) stands with successional ingrowth of Norway spruce ( (L.) H. Karst.). An experiment with randomized treatments within blocks was established, using three management strategies and one unthinned control, resulting in variation in optimal rotation age, merchantable volume and species composition. The management strategies were evaluated based on total production (volume) by using measured growth data 42 years after clearcutting and the modelled future stand development. The long-term effects of spontaneous regeneration and management strategies were evaluated based on land expectation value (LEV) and compared with a fifth management strategy using artificial regeneration and intense thinnings. 12 years after treatment, at a stand age of 42 years, the unthinned control had produced the highest total stem volume. At interest rates of 2% or higher, the unmanaged forest was an economically viable strategy, even compared to an intensive management strategy with a preferred merchantable timber species. Interest rates clearly impacted the profitability of the different management strategies. This study shows that when spontaneous regeneration is successful and dense, the first competition release can have a high impact on the development of future crop trees and on the species mixture.Betula pendulaBetula pubescensPicea abies

Productivity and profitability of Larix principis-rupprechtii and Pinus tabuliformis plantation forests in Northeast China

This study analyzes the productivity and profitability of Larix principis-rupprechtii and Pinus tabuliformis plantation forests in Northeast China. Three management regimes with different thinning intensities (heavy thinning, light thinning, and control) were considered. It is found that high initial planting densities without thinning cause high mortality and slow growth rates, leading to an unprofitable outcome. Thinning operations increase both diameter and height growth of the retained trees, boosting the merchantable volume of large-diameter trees and thus the profitability and productivity. The optimal rotation ages with heavy thinning of Larix principis-rupprechtii and Pinus tabuliformis plantations are 48 and 49 years, 7–8 years longer than what the current management regulation recommends. But the productivity per ha per year of Larix principis-rupprechtii in the heavy thinning regime doubled to 8.21 m3. For Pinus tabuliformis, the thinning treatments cause the productivity gains to increase from 5.1 to 7.9%. The constraints of government regulation without allowance for thinning to the near-mature stands and increasing the intensity could severely reduce the profitability and productivity.

Financial analysis of potential Pinus patula plantations in Antioquia, Colombia

The establishment of commercial forest plantations requires the selection of sites where reasonable profitability can be attained. A financial analysis was made for the identification of the most suitable areas for the establishment of new Pinus patula plantations in the central region of Antioquia, Colombia. The analysis was performed assuming basic silvicultural treatments at the establishment but no management during the entire rotation period. A volume yield data at the stand level was obtained from a previously fitted model that uses biophysical variables and stand density as predictors. The estimated stand volume, a detailed cash flow, and a derived stumpage price were combined to perform a financial analysis. The Land Expectation Value (LEV) and Internal Rate of Return (IRR) at the optimal rotation age, along with their spatial variation, were calculated in this study. Results suggest that the estimated volume and the current stumpage price are not sufficient to guarantee reasonable profitability for new timberland investments. While the LEV was negative, the IRR was in the range 4.1±1.5%, which is less than the discount rate of 6.8% used in the financial analysis. However, a positive LEV and an IRR at 8% would be achieved if forest productivity increases by 20% because of silvicultural practices or costs reduction in a similar proportion (obtaining IRRs up to 8.4%). Moreover, if the government provide subsidies, the IRR would increase up to 10.3% (without requiring an increase in productivity or a decrease in costs) on sites with high growth potential (mean annual increment greater than 16 m3 ha-1 year-1), and close to the mills (less than 45 km radii).

Joint production of wood, resin, and carbon from pine plantation forest in Java

Pine (Pinus merkusii) plantation forest has been developed in Java Island under Perhutani management with clear-cutting and artificial regeneration silviculture system. The main products of pine plantation forests are wood and resin. Although the Indonesian government has committed to reduce the greenhouse gases (GHGs) emission for 26% by 2020 with voluntary efforts, pine plantation forest may contribute to reduce GHGs emission through enhancing carbon stock. The additional income from carbon sales might or might not change the optimal rotation of the pine forest. This study aims to analyze the effects of additional income from resin and carbon benefits on the optimal rotation of pine plantation forest in Java. The modified Faustmann model is used to determine the optimal rotation. The results of this study show that (1) the inclusion of resin benefits may lengthen the optimal rotation age, (2) the optimal rotation of timber and resin production of pine forest in Java for size class III, IV, V, and VI are 34, 33 32, and 30 years respectively, (3) additional income from carbon sales may increase carbon stock by 8%, and lengthen optimal rotation for 3 years. Perhutani may change the optimal rotation due to additional benefits from resin and carbon.

Cut or keep: What should a forest owner do after a windthrow?

After a windstorm hits a forest stand the forest owner has two possibilities: clear cut and start a new rotation or let the standing trees grow until maturity. We compute the Land Expectation Value (LEV) under the risk of windthrow and introduce an endogenous rule to account for this decision. We compare the result with the default alternative of systematic clear felling and replanting (commonly assumed in existing literature). We have calibrated our model to represent maritime pine plantations in south-western France. Results show that by following this rule, payoffs may increase by up to 90% when the economic risk (i.e. probability of having a windstorm) is high. About 75% of the time it is profitable to keep the standing trees until maturity. Although payoffs are sensitive to economic risk, the optimal rotation age does not change much with significant increases in the risk probability.

Debunking the forest rent model fallacy in a fully regulated forest

In this paper, we first show both mathematically and empirically that the forest rent model is the wrong model to determine the optimal rotation age of a fully regulated forest as long as the standing timber has value. We then demonstrate that even if the forest is acquired for free, a fully regulated forest based on the forest rent model is actually worth more than the present value of its annual income. By converting to a fully regulated forest with a shorter rotation under the land expectation model, a higher present value of over 10% can be realized.