# The Financial Valuation Risk in Pepper Production: The Use of Decoupled Net Present Value

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}year

^{−1}) and structural risks (EUR 1622 € ha

^{−1}year

^{−1}). The use of decreasing discount functions has permitted a more realistic investment estimation. Finally, the sensitivity analysis shows that decoupled net present value (DNPV) is little affected by changes in interest rates in contrast to traditional net present value (NPV).

## 1. Introduction

## 2. Materials and Methods

- (a)
- Discounting function $\mathrm{d}\left({\mathrm{t}}_{\mathrm{i}}\right)$. Traditionally, exponential functions are used; however, their use means penalizing projects whose profits occur in the final period of their useful life. Some authors such as Chichilnisky [69] or Weitzman [52] use hyperbolic functions. In particular, Almansa and Martínez [46] use decreasing discounting functions for the valuation of future environmental impacts. By using a discount function as follows:$$\mathrm{d}\left({\mathrm{t}}_{\mathrm{i}}\right)={\mathrm{e}}^{-{\mathrm{r}}_{\mathrm{i}}\mathrm{t}}$$$$\mathrm{f}\left(\mathrm{r}\right)=\frac{{\mathsf{\beta}}^{\mathsf{\alpha}}}{\Gamma \left(\mathsf{\alpha}\right)}{\mathrm{h}}^{\mathsf{\alpha}-1}{\mathrm{e}}^{-\mathsf{\beta}\mathrm{r}}$$

- (b)
- Cash flows. They are obtained as the difference between income and foreseen payments. In most cases, future net cash flows (${\tilde{\mathrm{c}}}_{\mathrm{i}}$) can be considered as uncertain. In any case, they can be defined as the difference between cash inflows ${\tilde{\mathrm{I}}}_{\mathrm{i}}$ and cash outflows ${\tilde{\mathrm{H}}}_{\mathrm{i}}$, which are also considered uncertain$${\tilde{\mathrm{c}}}_{\mathrm{i}}={\tilde{\mathrm{I}}}_{\mathrm{i}}-{\tilde{\mathrm{H}}}_{\mathrm{i}},\mathrm{i}=1,\dots ,\mathrm{n}$$

- (a)
- Certainty equivalent. In this case, the uncertain value ${\tilde{\mathrm{c}}}_{\mathrm{i}}$ is multiplied by a factor ${\mathsf{\epsilon}}_{\mathrm{t}}$ which fluctuates between 0 and 1 and represents the reduction in value required by the investor to transform a risk investment into a risk-free investment.$$\mathrm{P}\left(\mathrm{C}\left(\mathrm{t}\right)\right)=\sum {\mathsf{\epsilon}}_{\mathrm{t}}{\tilde{\mathrm{c}}}_{\mathrm{i}}\mathrm{d}\left(\mathrm{t}\right)\mathrm{P}\left({\mathrm{L}}_{{\mathrm{t}}_{\mathrm{i}}}\left(\mathrm{t}\right)\right)$$
- (b)
- Decoupled net present value. In general, the problem with risk investments is due to the fact that the real value of future cash flows tends to be worse than the foreseen value. For this reason, future income and expense flow is reduced with the value of the corresponding synthetic insurance, which protects the investor from future risks of loss in value of future cash flows. In this sense, the value of the synthetic insurance of inflows is ${\tilde{\mathrm{S}}}_{{\mathrm{I}}_{\mathrm{i}}}$, and that of outflows is ${\tilde{\mathrm{S}}}_{{\mathrm{H}}_{\mathrm{i}}}$.$$\mathrm{P}\left[\mathrm{C}\left(\mathrm{t}\right)\right]={\displaystyle \sum _{\mathrm{i}}{\tilde{\mathrm{c}}}_{\mathrm{i}}}\mathrm{P}\left[{\mathrm{L}}_{{\mathrm{t}}_{\mathrm{i}}}\left(\mathrm{t}\right)\right]={\displaystyle \sum _{\mathrm{i}}\left[{\tilde{\mathrm{I}}}_{\mathrm{i}}-{\tilde{\mathrm{H}}}_{\mathrm{i}}-\left({\tilde{\mathrm{S}}}_{{\mathrm{I}}_{\mathrm{i}}}-{\tilde{\mathrm{S}}}_{{\mathrm{H}}_{\mathrm{i}}}\right)\right]}\mathrm{P}\left[{\mathrm{L}}_{{\mathrm{t}}_{\mathrm{i}}}\left(\mathrm{t}\right)\right]$$

#### 2.1. Risks to Be Evaluated

#### 2.2. Sensitivity Analysis

## 3. Results

#### 3.1. Risk of Temporary Fall in Prices

^{−1}. There is a risk of a temporary fall in prices down to threshold profitability (0.378 € kg

^{−1}). In accordance with (13), the center of gravity has been obtained from part of the curve between 0.378 and 0.53 € kg

^{−1}, obtaining a value of 0.458 € kg

^{−1}. In this way, we can summarize all the prices in this range into a single price, 0.458 €/kg. For this price, the loss is 0.07 € kg

^{−1}. In addition, the probability of the price being between 0.378 and 0.53 € kg

^{−1}is 0.31. Consequently, the product of loss through its probability permits obtaining risk as a percentage, which multiplied by mean production of the given week (12.473 € kg

^{−1}) involves a risk of EUR 122.36 (Table 2) for the given week. This methodology has also been used for the analysis of investments in the energy sector, where Espinoza and Rojo [73] study the variation of solar radiation.

^{−1}and the weekly mean decreases as the mean prices decrease. This happens until week 21 (third week of May), when it remains more or less stable until week 35 (end of August), when this probability begins to increase. The result obtained is EUR 3720.27. That is to say, the annual risk that the grower assumes due to a temporary price fall is EUR 3720.27, which was 3.08 % of the production value. It should be remembered that most of the risk is due to precocity in the first weeks of harvest (the first three weeks present a risk of EUR 2542.34).

#### 3.2. Probability of Permanent Drop in the Price of Peppers

#### 3.3. Risk due to Nematodes and Aphids

^{−1}. The product of loss per probability gives a risk occurrence of 3.41% for nematodes and 0.28% for aphids.

^{−1}) minus production for each case. For nematode risk, the loss risk profile is 30,000 kg ha

^{−1}, being the difference between 88,000 kg ha

^{−1}and production obtained without treatment (58,000 kg ha

^{−1}). Espinoza and Rojo [73] make a similar analysis to evaluate the tariff risk profile in renewable energies.

#### 3.4. Risk Due to Bad Quality Irrigation Waters

#### 3.5. Risk due to Increase in Costs

#### 3.6. Obtaining Present Value

#### 3.7. Obtaining the DNPV

^{−1}for year 1. Traditional calculation involves determining some annual yields of EUR 25,814, but with this methodology we are able to identify the crop risks, focusing predominantly on problems related to pests and variations in prices. Therefore, the risk-free net yield is reduced to EUR 10,727.01 in the first year and to EUR 14,936.76 in the last year.

- DNPV is not affected by the use of decreasing discount functions;
- NPV with decreasing rate is higher than traditional NPV in 58% of occasions;
- In the comparison between NPV and DNPV, the latter is higher in 68% of the cases and in 75% when decreasing discount functions are used.

## 4. Conclusions

^{−1}year

^{−1}and the latter at 1622 € ha

^{−1}for the first year, a risk which decreases as the greenhouse’s end of useful life approaches. Other risks considered are the loss caused by pests, valued at −3248.70 € ha

^{−1}per year and risks from loss in water quality (−5168.25 € ha

^{−1}per year). Finally, although the amount is small, risks from the increase in costs of pluriannual projection elements that need to be replaced have also been considered.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Probability of obtaining the indicated net present value (NPV) or a lower NPV (V = NPV, D = decoupled net present value (DNPV), g = use of gamma discount function).

**Figure 7.**Comparisons per pairs of updated values obtained (V = NPV, D = DNPV, g = gamma discount function).

Date | Production | Date | Production |
---|---|---|---|

April 12, 2016 | 12,112.50 | April 11, 2017 | 13,986.00 |

April 19, 2016 | 8075.00 | April 18, 2017 | 9324.00 |

April 28, 2016 | 6729.17 | April 30, 2017 | 9324.00 |

May 18, 2016 | 10,266.67 | May 20, 2017 | 16,090.00 |

June 07, 2016 | 12,473.33 | June 14, 2017 | 30,808.88 |

July 06, 2016 | 37,700.00 | July 05, 2017 | 11,383.33 |

July 29, 2016 | 24,100.00 | August 02, 2017 | 19,025.36 |

August 11, 2016 | 10,216.67 | August 16, 2017 | 9800.00 |

Total | 121,673.34 | 119,741.57 |

**Table 2.**Obtaining weekly risk due to a temporary fall in prices based on the mean and standard deviation.

Week | Mean | Standard Deviation | p (µ > price > 0.378) | Expected Value (EUR) | Loss (EUR) | Risk per Kilo | Production (%) | Risk (EUR) |
---|---|---|---|---|---|---|---|---|

16 | 1.04 | 0.33 | 0.48 | 0.865 | 0.17 | 0.082 | 10.8% | 1.067.43 |

17 | 0.89 | 0.30 | 0.46 | 0.689 | 0.20 | 0.092 | 7.2% | 800.72 |

18 | 0.85 | 0.30 | 0.44 | 0.658 | 0.19 | 0.084 | 6.6% | 674.19 |

21 | 0.52 | 0.21 | 0.25 | 0.452 | 0.07 | 0.016 | 10.9% | 212.05 |

24 | 0.53 | 0.19 | 0.28 | 0.458 | 0.07 | 0.020 | 5.2% | 122.36 |

25 | 0.52 | 0.16 | 0.31 | 0.454 | 0.06 | 0.019 | 12.8% | 296.04 |

28 | 0.45 | 0.07 | 0.33 | 0.417 | 0.03 | 0.010 | 20.3% | 244.58 |

31 | 0.46 | 0.11 | 0.28 | 0.416 | 0.04 | 0.012 | 10.0% | 141.83 |

32 | 0.45 | 0.12 | 0.24 | 0.415 | 0.04 | 0.008 | 7.9% | 80.20 |

33 | 0.46 | 0.12 | 0.25 | 0.421 | 0.04 | 0.010 | 4.2% | 50.98 |

34 | 0.44 | 0.12 | 0.20 | 0.410 | 0.03 | 0.006 | 4.1% | 29.88 |

Total | 100.0% | 3720.27 |

**Table 3.**Obtaining weekly risk of pepper production as product of weekly production per estimated loss and its probability.

Week | Mean | Standard Deviation | p (price < 0.378) | Expected Value | Loss | Loss × Probability | Production | Risk (EUR) |
---|---|---|---|---|---|---|---|---|

16 | 1.04 | 0.33 | 0.02 | 0.268 | 0.11 | 0.002 | 10.8% | 30.73 |

17 | 0.89 | 0.30 | 0.04 | 0.267 | 0.11 | 0.005 | 7.2% | 41.11 |

18 | 0.85 | 0.30 | 0.06 | 0.264 | 0.11 | 0.007 | 6.6% | 55.81 |

21 | 0.52 | 0.21 | 0.25 | 0.262 | 0.12 | 0.030 | 10.9% | 390.86 |

24 | 0.53 | 0.19 | 0.22 | 0.274 | 0.10 | 0.022 | 5.2% | 140.27 |

25 | 0.52 | 0.16 | 0.19 | 0.291 | 0.09 | 0.016 | 12.8% | 253.31 |

28 | 0.45 | 0.07 | 0.17 | 0.342 | 0.04 | 0.006 | 20.3% | 151.94 |

31 | 0.46 | 0.11 | 0.22 | 0.314 | 0.06 | 0.014 | 10.0% | 171.70 |

32 | 0.45 | 0.12 | 0.26 | 0.304 | 0.07 | 0.020 | 7.9% | 185.62 |

33 | 0.46 | 0.12 | 0.25 | 0.307 | 0.07 | 0.018 | 4.2% | 89.87 |

34 | 0.44 | 0.12 | 0.30 | 0.302 | 0.08 | 0.023 | 4.1% | 111.66 |

Total | 100.0% | 1622.90 |

Risks | Production Treated Plants | Production Untreated Plants | Loss (%) | Probability (%) | Risk (%) |
---|---|---|---|---|---|

Nematodes | 88,000 | 58,000 | 34 | 10 | 3.41 |

Aphids (early maturation) | 92,000 | 66,000 | 28 | 1 | 0.28 |

Salinity (dSm^{−1}) | Production | Loss (%) | Probability | Risk |
---|---|---|---|---|

Quality | 63,000 | 0% | 0.50 | 0.00% |

2.5 | 60,000 | −5% | 0.30 | −1.43% |

3.5 | 51,000 | −19% | 0.15 | −2.86% |

6.0 | 43,000 | −32% | 0.05 | −1.59% |

Salinity risk | −5.87% |

Initial Price (EUR) | Increase (EUR) | Probability | Increase in Costs (EUR) | Costs Risk (%) | |
---|---|---|---|---|---|

Cost of whitewashing | 260 | 20 | 0.10 | 2 | 0.77% |

Plastic | 7500 | 750 | 0.10 | 75 | 1.00% |

Drip irrigation | 4600 | 460 | 0.15 | 69 | 1.50% |

Total | 3.27% |

Date | Week | Total Production | Mean Price | Amount |
---|---|---|---|---|

April 12, 2016 | 16 | 12,113 | 1.04 | 12,553 |

April 19, 2016 | 17 | 8075 | 0.89 | 7187 |

April 28, 2016 | 18 | 6729 | 0.85 | 5714 |

May 18, 2016 | 21 | 10,267 | 0.52 | 5311 |

June 07, 2016 | 24 | 12,473 | 0.53 | 6577 |

July 06, 2016 | 28 | 37,700 | 0.45 | 16,862 |

July 29, 2016 | 31 | 24,100 | 0.46 | 11,042 |

August 11, 2016 | 33 | 10,217 | 0.46 | 4700 |

Total | 121,673 | 69,945 | ||

Date | ||||

April 11, 2017 | 16 | 13,986 | 1.04 | 14,495 |

April 18, 2017 | 17 | 9324 | 0.89 | 8298 |

April 30, 2017 | 18 | 9324 | 0.85 | 7917 |

May 20, 2017 | 21 | 16,090 | 0.52 | 8323 |

June 14, 2017 | 25 | 30,809 | 0.52 | 15,881 |

July 05, 2017 | 28 | 11,383 | 0.45 | 5091 |

August 02, 2017 | 32 | 19,025 | 0.45 | 8579 |

August 16, 2017 | 34 | 9800 | 0.44 | 4312 |

Total | 119,741 | 72,896 | ||

Average | 120,707 | 71,420 |

Concept | Amount |
---|---|

1. VARIABLE COSTS | 34,896 |

1.1 Raw materials | 20,051 |

Irrigation water | 1722 |

Seed (Herminio variety) | 5000 |

Seedbed | 925 |

Agrocelhone disinfectant | 4495 |

Pesticides | 2640 |

Auxiliary insects | 2750 |

Manure | 1200 |

Compost | 1319 |

1.2. Labor | 13,145 |

1.3. Own machinery | 1700 |

2. FIXED COSTS | 2680 |

2.1. Machinery | 2680 |

3. OTHER EXPENSES | 6530 |

3.1. Social security | 3090 |

Operators | 1050 |

Owner | 2040 |

3.2. Payments to public administrations | 640 |

Tax contribution | 40 |

Income taxes | 600 |

3.3. Administration and consultancy expenses | 2800 |

ADDITIONAL EXPENSES WHITEWASHING | 1500 |

Installation (painting the greenhouse) | 1500 |

Concept | Useful Life (years) | Amount (EUR) |
---|---|---|

Structure, staking, polycarbonate, and assembly | 24 | 103,895 |

Installation drip irrigation | 10 | 4600 |

Plastic covering (thermal) | 3 | 7500 |

Pluriannual costs | 115,995 |

Risks | Year 1 | Year 24 |
---|---|---|

Nematodes | 2434.78 | 2434.78 |

Aphids | 201.84 | 201.84 |

Salinity | 4194.52 | 4194.52 |

Whitewashing | 11.54 | 23.08 |

Plastic | 0.00 | 0.00 |

Drip irrigation | 0.00 | 0.00 |

Loss due to temporary fall in prices | 2201.21 | 2201.21 |

Loss due to definitive fall in prices | 6042.98 | 321.69 |

Total risks | 15,086.87 | 9377.11 |

Net yield with risks | 25,813.87 | 24,313.87 |

Risk-free net yield | 10,727.01 | 14,936.76 |

Concept | Amount (EUR) | Useful Life in Years |
---|---|---|

Structure, staking, polycarbonate, and assembly | 103,895 | 24 |

Installation of drip irrigation | 4600 | 10 |

Plastic covering (thermal) | 7500 | 3 |

Total costs of installation and assembly | 115,995 |

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## Share and Cite

**MDPI and ACS Style**

López-Marín, J.; Gálvez, A.; del Amor, F.M.; Brotons, J.M.
The Financial Valuation Risk in Pepper Production: The Use of Decoupled Net Present Value. *Mathematics* **2021**, *9*, 13.
https://doi.org/10.3390/math9010013

**AMA Style**

López-Marín J, Gálvez A, del Amor FM, Brotons JM.
The Financial Valuation Risk in Pepper Production: The Use of Decoupled Net Present Value. *Mathematics*. 2021; 9(1):13.
https://doi.org/10.3390/math9010013

**Chicago/Turabian Style**

López-Marín, Josefa, Amparo Gálvez, Francisco M. del Amor, and Jose M. Brotons.
2021. "The Financial Valuation Risk in Pepper Production: The Use of Decoupled Net Present Value" *Mathematics* 9, no. 1: 13.
https://doi.org/10.3390/math9010013