# Valuing the Future and Discounting in Random Environments: A Review

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## Abstract

**:**

## 1. Introduction

## 2. The Process of Discounting—Fundamentals

#### 2.1. Definitions and General Setting

#### 2.2. The Feynman–Kac Approach

#### 2.3. The Fourier Transform Approach

#### 2.4. Adding Risk Aversion

## 3. Pricing Bonds—The Term Structure of Interest Rates

#### 3.1. Dynamics of the Bond Price

#### 3.2. The Market Price of Risk

#### 3.3. The Term Structure Equation and the Risk-Neutral Measure

## 4. Standard Models

#### 4.1. Bonds and Real Rates

#### 4.2. The Vasicek (Ornstein–Uhlenbeck) Model

#### Risk Aversion

#### 4.3. The Cox–Ingersoll–Ross (Feller) Model

#### Risk Aversion

#### 4.4. The Log-Normal Model

#### Risk Aversion

## 5. Some Empirical Results

## 6. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The Vasicek and Cox–Ingersoll–Ross discount functions. The parameters used are those corresponding to the United States and are provided by Table 5 of Ref. [29] (see Section 5). In the top figure, we plot the discount function $D\left(t\right)$, while in the bottom figure, we plot the log ratio $-lnD\left(t\right)/t$. In the top figure, we observe the asymptotic exponential decay of the discount after more than a hundred years, while in the bottom figure, we clearly see the existence of a long-run discount rate for the Vasicek model (cf. Equation (80)). The initial rate ${r}_{0}$ is arbitrarily taken to be $1\%$. In both models, we assume no market price of risk $q\left(r\right)=0$ (the Local Expectation Hypothesis).

**Figure 2.**The construction of real interest rates $r\left(t\right)$ in terms of the nominal rates $n\left(t\right)$ and inflation $i\left(t\right)$ (Fisher’s procedure). Large fluctuations and negative rates are shown here for the United States (USA).

**Table 1.**Key statistical features for three standard models: the Vasicek (Ornstein–Uhlenbeck), the Cox–Ingersoll–Ross (Feller) and the log-normal models. The average and variance are provided in terms of the model parameters to better compare the asymptotic behavior of $D\left(t\right)$. The asymptotic discount is provided by showing an exponential decay with a long-run rate of discount ${r}_{\infty}$ for the Vasicek and the Cox–Ingersoll–Ross models and also in the log-normal case for a specific combination of parameters (${k}^{2}/2<\alpha $, mild fluctuations). The parameter $\delta $ is defined in Equation (111).

Model | $\mathbb{E}\left[\mathit{r}\right(\mathit{t}\left)\right]$ | Var$\left[\mathit{r}\right(\mathit{t}\left)\right]$ | $\mathit{D}(\mathit{t}\to \mathbf{\infty})$ | ${\mathit{r}}_{\mathbf{\infty}}$ | |
---|---|---|---|---|---|

Vasicek | m | ${k}^{2}/\alpha $ | $exp(-{r}_{\infty}t)$ | $m-{k}^{2}/2{\alpha}^{2}$ | |

Feller | m | $m{k}^{2}/\left(2\alpha \right)$ | $exp(-{r}_{\infty}t)$ | $\frac{2m}{1+\sqrt{1+2{k}^{2}/\alpha}}$ | |

Log-normal | ${r}_{0}{e}^{\alpha t}$ | ${r}_{0}^{2}{e}^{2\alpha t}[{e}^{{k}^{2}t}-1]$ | constant | (${k}^{2}/2>\alpha $) | $--$ |

$exp(-{r}_{\infty}t)$ | (${k}^{2}/2<\alpha $) | $(\alpha -{k}^{2}/2)/\delta $ | |||

${t}^{-1/2}$ | (${k}^{2}/2=\alpha $) | $--$ |

Country | CPI | Bond Yield | From | To | Records |
---|---|---|---|---|---|

Italy | CPITAM | IGITA10 | 12/31/1861 | 09/30/2012 | 565 |

annual from 12/31/1861 | quarterly | ||||

quarterly from 12/31/1919 | |||||

Chile | CPCHLM | IDCHLM | 03/31/1925 | 09/30/2012 | 312 |

quarterly | quarterly | ||||

Canada | CPCANM | IGCAN10 | 12/31/1913 | 09/30/2012 | 357 |

quarterly | quarterly | ||||

Germany | CPDEUM | IGDEU10 | 12/31/1820 | 09/30/2012 | 729 |

annual from 12/31/1820 | quarterly | ||||

quarterly from 12/31/1869 | |||||

Spain | CPESPM | IGESP10 | 12/31/1821 | 09/30/2012 | 709 |

annual from 12/31/1821 | quarterly | ||||

quarterly from 12/31/1920 | |||||

Argentina | CPARGM | IGARGM | 12/31/1864 | 03/31/1960 | 342 |

annual from 12/31/1864 | quarterly | ||||

quarterly from 12/31/1932 | |||||

Netherlands | CPNLDM | IGNLD10D | 12/31/1813 | 12/31/2012 | 189 |

annual | annual | ||||

Japan | CPJPNM | IGJPN10D | 12/31/1921 | 12/31/2012 | 325 |

quarterly | quarterly | ||||

Australia | CPAUSM | IGAUS10 | 12/31/1861 | 09/30/2012 | 564 |

annual from 12/31/1861 | quarterly | ||||

quarterly 12/31/1991 | |||||

Denmark | CPDNKM | IGDNK10 | 12/31/1821 | 09/30/2012 | 725 |

annual from 12/31/1821 | quarterly | ||||

quarterly from 12/31/1914 | |||||

South Africa | CPZAFM | IGZAF10 | 12/31/1920 | 09/30/2012 | 329 |

quarterly | quarterly | ||||

Sweden | CPSWEM | IGSWE10 | 12/31/1868 | 09/30/2012 | 135 |

annual | annual | ||||

United Kingdom | CPGBRM | IDGBRD ${}^{*}$ | 12/31/1694 | 12/31/2012 | 309 |

annual | annual | ||||

United States | CPUSAM | TRUSG10M | 12/31/1820 | 10/30/2012 | 183 |

annual | annual |

**Table 3.**The OU (Vasicek) model parameter estimation in yearly units using stationary averages. “Neg RI” provides the time percentage and the number of years with negative real interest rates. The columns $\widehat{m}$, $\widehat{k}$ (in %) and $\widehat{\alpha}$ are estimates from the country time series; ${\widehat{r}}_{\infty}$ (in %) is evaluated from Equation (79). The Min and Max columns give reasons regarding the level of robustness of the estimation as they provide the minimum and the maximum values of the parameter estimation for four data blocks of equal length. The parameter $\alpha $ is estimated by fitting the empirical correlation function to an exponential (cf. Equation (75)) after using the whole data block. Countries in boldface are those considered historically more stable with positive long-run rates ${\widehat{r}}_{\infty}>0$.

Country | Neg RI | $\widehat{\mathit{m}}$ | Min | Max | $\widehat{\mathit{k}}$ | Min | Max | $\widehat{\mathit{\alpha}}$ | ${\widehat{\mathit{r}}}_{\mathbf{\infty}}$ |
---|---|---|---|---|---|---|---|---|---|

Italy | $28\%\phantom{\rule{0.166667em}{0ex}}\left(40y\right)$ | $-0.3$ | $-9.1$ | $5.6$ | $6.9$ | $0.8$ | $10.1$ | $0.22$ | $-5.4$ |

Chile | $56\%\phantom{\rule{0.166667em}{0ex}}\left(43y\right)$ | $-6.8$ | $-20.2$ | $12.0$ | $25.2$ | $5.6$ | $44.1$ | $0.40$ | $-26$ |

Canada | $22\%\phantom{\rule{0.166667em}{0ex}}\left(20y\right)$ | $\mathbf{2.9}$ | $0.1$ | 6 | $\mathbf{2.3}$ | $1.1$ | $2.0$ | $0.26$ | $\mathbf{2.5}$ |

Germany | $14\%\phantom{\rule{0.166667em}{0ex}}\left(25y\right)$ | $-10.7$ | $-51.0$ | $4.0$ | $33.9$ | $0.9$ | $61.4$ | $0.20$ | $-160$ |

Spain | $25\%\phantom{\rule{0.166667em}{0ex}}\left(45y\right)$ | $5.7$ | $-0.5$ | $13.5$ | $2.9$ | $1.2$ | $3.6$ | $0.06$ | $-6.4$ |

Argentina | $20\%\phantom{\rule{0.166667em}{0ex}}\left(17y\right)$ | $\mathbf{2.4}$ | $-2.9$ | $6.8$ | $\mathbf{6.2}$ | $2.8$ | $6.7$ | $0.39$ | $\mathbf{1.1}$ |

Netherlands | $17\%\phantom{\rule{0.166667em}{0ex}}\left(33y\right)$ | $\mathbf{3.2}$ | $0.8$ | $5.4$ | $\mathbf{1.6}$ | $0.8$ | $2.2$ | $0.14$ | $\mathbf{2.4}$ |

Japan | $33\%\phantom{\rule{0.166667em}{0ex}}\left(26y\right)$ | $-2.2$ | $-7.8$ | $4.0$ | $9.7$ | $1.1$ | $13.2$ | $0.24$ | $-10$ |

Australia | $23\%\phantom{\rule{0.166667em}{0ex}}\left(33y\right)$ | $\mathbf{2.6}$ | $-0.7$ | $4.9$ | $\mathbf{2.3}$ | $0.7$ | $2.8$ | $0.19$ | $\mathbf{1.9}$ |

Denmark | $18\%\phantom{\rule{0.166667em}{0ex}}\left(33y\right)$ | $\mathbf{3.2}$ | $1.5$ | $4.3$ | $\mathbf{2.3}$ | $1.1$ | $2.9$ | $0.23$ | $\mathbf{2.7}$ |

South Africa | $43\%\phantom{\rule{0.166667em}{0ex}}\left(36y\right)$ | $\mathbf{1.8}$ | $-2.2$ | $5.5$ | $\mathbf{2.5}$ | $1.2$ | $2.0$ | $0.21$ | $\mathbf{1.1}$ |

Sweden | $28\%\phantom{\rule{0.166667em}{0ex}}\left(38y\right)$ | $\mathbf{2.3}$ | $-0.3$ | $3.9$ | $\mathbf{2.5}$ | $0.6$ | $3.4$ | $0.25$ | $\mathbf{1.9}$ |

United Kingdom | $14\%\phantom{\rule{0.166667em}{0ex}}\left(45y\right)$ | $\mathbf{3.3}$ | $1.4$ | $4.3$ | $\mathbf{1.9}$ | $1.0$ | $2.4$ | $0.19$ | $\mathbf{2.8}$ |

United States | $31\%\phantom{\rule{0.166667em}{0ex}}\left(36y\right)$ | $\mathbf{2.6}$ | $1.0$ | $4.0$ | $\mathbf{1.8}$ | $1.2$ | $2.1$ | $0.18$ | $\mathbf{2.1}$ |

Stable countries | $23\%\phantom{\rule{0.166667em}{0ex}}\left(33y\right)$ | $\mathbf{2.7}$ | $-0.14$ | $5.0$ | $\mathbf{2.6}$ | $1.04$ | $2.94$ | $0.23$ | $\mathbf{2.1}$ |

Unstable counntries | $31\%\phantom{\rule{0.166667em}{0ex}}\left(36y\right)$ | $-\mathbf{2.9}$ | $17.7$ | $1.8$ | $\mathbf{16}$ | $1.9$ | $26.5$ | $0.22$ | $-\mathbf{42}$ |

**Table 4.**Maximum likelihood estimation of the long-run interest rate for the Vasicek model. $\widehat{m}$ estimates of the mean real interest rate in 1/years (in %). $\widehat{\alpha}$ estimates the characteristic reversion time in 1/years. The squared root of $\widehat{{\mathit{k}}^{2}}$ is given in terms of $1/{\left(\mathrm{year}\right)}^{3}$ (multiplied by ${10}^{4}$ to be comparable with the results in Table 3). These estimators are accompanied by the square root of the variance of each estimator. ${\widehat{r}}_{\infty}$ estimates the long-run real interest rate with 1/year (in %). Negative values of ${\widehat{r}}_{\infty}$ imply that the discount function is asymptotically increasing. The standard error is obtained through error propagation. The last two rows show the average over all countries with the more stable countries (${r}_{\infty}>0$) and the less stable countries (${r}_{\infty}<0$). The error provided corresponds to the standard deviation of the ${\widehat{r}}_{\infty}$ for the different countries.

Country | $\widehat{\mathit{m}}$ | ${\mathit{\sigma}}_{\widehat{\mathit{m}}}$ | $\widehat{\mathit{\alpha}}$ | ${\mathit{\sigma}}_{\widehat{\mathit{\alpha}}}$ | $\widehat{{\mathit{k}}^{2}}$ | ${\mathit{\sigma}}_{\widehat{{\mathit{k}}^{2}}}$ | ${\widehat{\mathit{r}}}_{\mathbf{\infty}}$ | ${\mathit{\sigma}}_{{\widehat{\mathit{r}}}_{\mathbf{\infty}}}$ |
---|---|---|---|---|---|---|---|---|

Italy | 1.97 | 15.95 | 0.0056 | 0.0089 | 0.1146 | 0.068 | −177.8 | 19.2 |

Chile | −5.79 | 31.46 | 0.0201 | 0.0227 | 31.07 | 2.49 | −391.7 | 44.2 |

Canada | 2.66 | 3.91 | 0.0142 | 0.0178 | 0.275 | 0.021 | −4.15 | 3.94 |

Germany | −9.45 | 66.95 | 0.0071 | 0.0089 | 41.72 | 2.19 | −4094 | 228 |

Spain | 6.71 | 6.92 | 0.0167 | 0.0137 | 2.371 | 0.126 | −35.78 | 7.28 |

Argentina | 3.15 | 7.09 | 0.0228 | 0.0231 | 2.240 | 0.171 | −18.31 | 7.27 |

Netherlands | 5.99 | 0.78 | 0.1648 | 0.0550 | 1.797 | 0.243 | 5.66 | 0.78 |

Japan | 5.02 | 24.68 | 0.0053 | 0.0114 | 1.396 | 0.109 | −243.1 | 31.4 |

Australia | 3.97 | 4.50 | 0.0089 | 0.0112 | 0.223 | 0.013 | −10.29 | 4.58 |

South Africa | 2.69 | 4.72 | 0.0154 | 0.0193 | 0.435 | 0.034 | −6.49 | 4.77 |

Sweden | 2.79 | 1.66 | 0.0676 | 0.0317 | 1.692 | 0.206 | 0.95 | 1.67 |

Denmark | 4.10 | 2.59 | 0.0161 | 0.0133 | 0.315 | 0.017 | −1.97 | 2.61 |

United Kingdom | 3.42 | 0.62 | 0.1635 | 0.0326 | 3.137 | 0.253 | 2.83 | 0.62 |

United States | 3.19 | 1.23 | 0.0603 | 0.0257 | 1.003 | 0.105 | 1.81 | 1.24 |

Stable countries | 3.85 | 1.07 | 0.1140 | 0.0362 | 1.907 | 0.202 | 2.81 | 1.08 |

Unstable countries | 1.50 | 16.86 | 0.0132 | 0.0150 | 8.120 | 0.523 | −498.4 | 35.3 |

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**MDPI and ACS Style**

Masoliver, J.; Montero, M.; Perelló, J.; Farmer, J.D.; Geanakoplos, J.
Valuing the Future and Discounting in Random Environments: A Review. *Entropy* **2022**, *24*, 496.
https://doi.org/10.3390/e24040496

**AMA Style**

Masoliver J, Montero M, Perelló J, Farmer JD, Geanakoplos J.
Valuing the Future and Discounting in Random Environments: A Review. *Entropy*. 2022; 24(4):496.
https://doi.org/10.3390/e24040496

**Chicago/Turabian Style**

Masoliver, Jaume, Miquel Montero, Josep Perelló, J. Doyne Farmer, and John Geanakoplos.
2022. "Valuing the Future and Discounting in Random Environments: A Review" *Entropy* 24, no. 4: 496.
https://doi.org/10.3390/e24040496