Risk Aversion

Let us very briefly comment on the inclusion of risk aversion in the Dothan model. For the log-normal process *f*(*r*) = *αr* and *g*(*r*) = *kr* and

$$f^\*(r) = [a + kq(r)]r.$$

Assuming a constant market price of risk, *q*(*r*) = *q* ≥ 0, we have

$$f^\*(r) = \mathfrak{a}^\* r, \qquad \mathfrak{a}^\* = \mathfrak{a} + q.$$

Again, *f* <sup>∗</sup>(*r*) has the same form than *f*(*r*), and all previous results will apply after making the replacement *α* → *α* + *q*.

#### **5. Some Empirical Results**

In order to choose an appropriate model for rates that would allow us to obtain realistic long-run discount functions, we performed a rather complete empirical study on interest rates combined with inflation. Our study follows the line partly initiated by Newell and Pizer [50] (see also [51]). To our knowledge, there are few empirical studies on real rates with some exceptions. We remark here the recent and excellent survey by Giglio et al. on the housing market in London and Singapore [52–54], which allowed for a rather realistic estimation of long-run discount rates.

Our first concern was knowing how the discount process depended on the underlying random process that characterizes interest rates. To this end, we collected data for the nominal interest rates and inflation of fourteen countries over time spans ranging from 87 to 318 years [26]. The countries in our sample are Argentina (ARG, 1864–1960), Australia (AUS, 1861–2012), Chile (CHL, 1925–2012), Germany (DEU, 1820–2012), Denmark (DNK, 1821–2012), Spain (ESP, 1821–2012), United Kingdom (GBR, 1694–2012), Italy (ITA, 1861– 2012), Japan (JPN, 1921–2012), Netherlands (NLD, 1813–2012), Sweden (SWE, 1868–2012),

the United States (USA, 1820–2012) and South Africa (ZAF, 1920–2012). The data are summarized in Table 2.

**Table 2.** List of the countries analyzed. CPI stand for Consumer Price Index. Data has different specificities, particularly in terms of empty records as has been reported elsewhere [26,29]. \*, We have taken the discount (ID) rate since the governmen<sup>t</sup> bond yield data was not available.


Since all but two of our nominal interest rate processes are for 10-year governmen<sup>t</sup> bonds, which pay out over a 10-year period, we smoothed out inflation rates with a 10-year moving average and subtracted the annualized inflation index from the annualized nominal rate to compute the real interest rate, as explained in the previous section by means of the Fisher's procedure (cf. Equation (61)),

$$r(t) = n(t) - i(t)\_{\prime}$$

where *n*(*t*) is the nominal rate and *<sup>i</sup>*(*t*) is the inflation rate. The particular case of the United States is plotted in Figure 2.

**Figure 2.** The construction of real interest rates *r*(*t*) in terms of the nominal rates *n*(*t*) and inflation *<sup>i</sup>*(*t*) (Fisher's procedure). Large fluctuations and negative rates are shown here for the United States (USA).

In our empirical analysis, the nominal rates are determined by IG rates constructed from the 10-year Government Bond Yield *y*(*<sup>t</sup>*, *τ*) with *τ*= 10 years. Thus, looking at Equations (63) and (64), we estimate the nominal rates by

$$m(t) \sim y(t, \tau = 10 \text{ years}).$$

Let us recall that denoting, by *<sup>B</sup>*(*<sup>t</sup>*, *t* + *<sup>τ</sup>*), the governmen<sup>t</sup> bond issued at time *t* and maturing at time *t* + *τ* with unit maturity, *<sup>B</sup>*(*<sup>t</sup>*, *t*) = 1, the yield *y*(*<sup>t</sup>*, *τ*) is defined as (cf. Equation (38))

$$y(t,\tau) \equiv -\frac{1}{\tau} \ln B(t, t+\tau) \qquad \Longrightarrow \qquad B(t, t+\tau) = e^{-\tau y(t|\tau)}.$$

One can argue that *τ* = 10 years is not a short period of time in order to consider *y*(*<sup>t</sup>*, *τ* = 10 years) a very accurate estimator of *n*(*t*) (cf. Equations (63) and (64)). Although this may be true, we must bear in mind that 10 year bonds are the shortest bonds available for most of the countries analyzed.

The inflation rate is estimated through the Consumer Price Index (CPI) as

$$i(t) \sim \frac{1}{\tau} \ln \left[ I(t+\tau) / I(t) \right],$$

where *I*(*t*) is the aggregated inflation up to time *t*, and *τ* =10 years. The relation between *I*(*t*) and the Consumer Price Index (CPI) is

$$I(t+\tau) = I(t) \prod\_{j=0}^{\tau-1} \left[ 1 + \mathcal{C}(t+j) \right].$$

where *C*(*t*) is the time series of the empirical CPI. The instantaneous rate of inflation *<sup>i</sup>*(*t*) is, therefore, estimated by the quantity *i*(*t* + *<sup>τ</sup>*), which is written in terms of the CPI reads

$$i(t) \sim i(t+\tau) = \frac{1}{\tau} \sum\_{j=0}^{\tau -1} \ln\left[1 + \mathcal{C}(t+j)\right].$$

A remarkable characteristic observed for all countries is that real interest rates frequently become negative as the real interest rates are mostly dominated by inflation *<sup>i</sup>*(*t*) > 0

(see Figure 2). In some cases, as we can see in Table 3 (see also Figure 2), negative real rates show high frequency and long periods of time, and, on average, real interest rates are negative one quarter of the time.

**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 *m*<sup>ˆ</sup> , ˆ *k* (in %) and *α*ˆ are estimates from the country time series; *<sup>r</sup>*<sup>ˆ</sup>∞ (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 *α* 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 *<sup>r</sup>*<sup>ˆ</sup>∞ > 0.


This makes the Feller and log-normal models—as well as any other model assuming positive interest rates [13]—less interesting or at least less appropriate to model real interest *r*(*t*) = *n*(*t*) − *<sup>i</sup>*(*t*) instead of solely considering nominal rates *<sup>n</sup>*(*t*). It is, however, necessary to remember the fact that nominal rates can indeed become negative as has recently been observed in Western economies. We, therefore, confined the empirical work to the OU (Vasicek) model and then assumed the Local Expectation Hypothesis [36–38], according to which, we live in a risk neutral world, and the market price of risk is zero. Let us recall, as explained in Section 3, that the market price of risk *q* = *q*(*<sup>r</sup>*, *t*) may be any function of the rate and time. There is, hence, no unique expression for it. Thus, in Section 4, we presented several expressions of the long-run rate, which include risk in different forms for all market models analyzed. The usual assumption in the literature [33,38] is that the price of risk is a constant that is independent of time and the value of the rate but without any empirical justification. This is a sensitive issue since data is quite scarce, particularly in environmental applications, for obtaining a credible estimation of *q*. Moreover, to our knowledge, in environmental problems, the estimations of the long-run rates do not take into account, nor even mention, the market price of risk [14–16,50,54]. In any case, we do not lessen the importance of taking into account some kind of risk in estimating log-run rates; however, unfortunately, with the data available to us, we cannot make any reliable estimation of *q*. For this reason, we have not taken into account the market price of risk, assuming risk-neutral investors and following the Local Expectation Hypothesis. In any case, the question is under consideration).

We can estimate the parameters *m*, *α* and *k* of the Vasicek model to each of the data series. There are several possible procedures. One of the possible methods is to deal with stationary averages. The parameter *m* can be estimated through the stationary mean value of the rate (cf. Equation (74))

$$m = \mathbb{E}[r(t)].$$

Parameters *α* and *k* can be estimated via the correlation function of the Ornstein–Uhlenbeck process. Thus, from Equation (75), we have

$$C(t - t') = \frac{k^2}{2\alpha} e^{-\alpha|t - t'|}.$$

The empirical correlation can then be fitted by an exponential, which in turn allows us to estimate *α* (measured in 1/year units) for each country. The parameter *k* is obtained from the empirical standard deviation *σ*<sup>2</sup> = <sup>E</sup>[|*r*(*t*) − *<sup>m</sup>*|<sup>2</sup>], and for the Vasicek model, it is given by

$$k = \sigma \sqrt{2a}.$$

The resulting parameters are shown in Table 3. The minimum and maximum values for each country allows us to show that parameters may indeed fluctuate over different periods of time.

Finally, the long-run discount rate can be evaluated from Equation (79),

$$r\_{\infty} = m - k^2 / 2\alpha^2.$$

For this calculation, we neglected the market price of risk as mentioned above.

The countries studied can be divided into two groups. Nine countries have long-run positive rates (boldface in Table 3). The average historical rate for these nine countries is *m* = 2.7 % while the average long-run rate is *r*∞ = 2.1 %, which, on average, is 29 % lower than *m*. Five countries with less stable behavior have long-run negative rates and an exponentially increasing discount.

Four cases of this group have a negative average rate *m* due to at least one period of runaway inflation; the exception is Spain, which has a (highly positive) mean real interest rate but still has a long-run negative rate. Convergence in this case to the long-run rate happens within 30 years and typically within less than a decade. This contrasts with other treatments, which assume that short term rates are always (or nearly always) positive and predict that the decrease in the discounting rate happens over a much longer timescale, which can be measured in hundreds of years [50,51,55–58].

Alternatively, we can estimate parameters using the well-established maximum likelihood procedure. For the Vasicek model, the maximum likelihood estimation is extensively documented in the financial mathematics literature (see for instance [13]). The approach differs from the previous one as it focuses attention on two consecutive steps of our time series (generally consecutive years) and takes the conditional probability to perform the estimation. Table 4 shows that the most inaccurate estimator is *α*ˆ, an unsurprising fact since the estimation of *α* is known to be quite difficult for the Vasicek model [59]. The last two columns in Table 4 include the long-run interest rate estimator *<sup>r</sup>*<sup>ˆ</sup>∞ and its error calculated through error propagation.

Only four countries (the Netherlands, Sweden, the United Kingdom and the United States) show a positive long-run rate, *r*∞ > 0. This estimation procedure leads to more negative *r*<sup>∞</sup>. This feature can be attributed to the fact that, in most of the countries, estimating *α* via the maximum likelihood brings smaller values, which in turn leads to more drastic corrections to the long-run rate as *r*∞ is inversely proportional to *α* (remember that *r*∞ = *m* − *<sup>k</sup>*2/2*α*2). This effect is particularly relevant in most turbulent countries during last century (e.g., Germany) thus signaling a more intense lack of stationarity in empirical data. The averaged *r*∞ over all countries estimated via maximum likelihood is also sensitively smaller.

However, if we focus the attention on stable countries (with *r*∞ > 0) both estimation procedures bring quite similar results (see, for instance, the United States case in

Tables 3 and 4, 2.1% versus 1.8%). As in the previous estimation procedure, we also neglected the effects of risk aversion and the market price of risk.

The Vasicek model is therefore the only one among the three classic models allowing for negative rates, and for this reason, both the Feller and the log-normal models have been excluded from our analysis. However, for the Cox–Ingersoll–Ross (Feller) model, it is possible to redefine the model by shifting the process *y* = *r* − *rmin* where *rmin* < 0.

The estimation through the maximum likelihood procedure and its error analysis is then possible [59], and Figure 1 includes the shifted Cox–Ingersoll–Ross discount and compares it with the equivalent result assuming the Vasicek model. We demonstrated in Ref. [29] how to redefine the Feller process and how maximum likelihood estimation could be possible.

**Table 4.** Maximum likelihood estimation of the long-run interest rate for the Vasicek model. *m*<sup>ˆ</sup> estimates of the mean real interest rate in 1/years (in %). *α*ˆ estimates the characteristic reversion time in 1/years. The squared root of ˆ *k*2 is given in terms of 1/(year)<sup>3</sup> (multiplied by 10<sup>4</sup> to be comparable with the results in Table 3). These estimators are accompanied by the square root of the variance of each estimator. *<sup>r</sup>*<sup>ˆ</sup>∞ estimates the long-run real interest rate with 1/year (in %). Negative values of *<sup>r</sup>*<sup>ˆ</sup>∞ 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 (*<sup>r</sup>*∞ > 0) and the less stable countries (*<sup>r</sup>*∞ < 0). The error provided corresponds to the standard deviation of the *<sup>r</sup>*<sup>ˆ</sup>∞ for the different countries.


### **6. Discussion**

We reviewed one of the most important aspects of economics and finance, i.e., the problem of discount, which weights the future relative to the present. The problem is clearly very relevant in finance over relatively short time spans; however, it is even more crucial for long-run planning in addressing environmental problems on how to act now with measures to mitigate the effects of climate change.

To our knowledge, this is a rather unknown issue to the econophysics community, and this review is particularly intended for this group. We thus addressed the problem with a simple approach and ye<sup>t</sup> with a high level of rigor and generality. In this way, we also developed the traditional method used in mathematical finance to address the problem, i.e., the Feynman–Kac approach. In addition, we reviewed the bond pricing theory and its close similarity with discounting and presented a short introduction to the term structure of interest rates along with the market price of risk.

We obtained quantitative results on the problem by studying, in some detail, three standard models for the dynamical evolution of rates. These models are based on the Ornstein–Uhlenbeck process (the Vasicek model), thus, allowing for both positive and negative rates and also on the Feller and log-normal processes for positive rates. We presented the exact results for the discount function and asymptotic expressions as *t* → ∞ leading to the long-run discount rate, and we discussed the modifications of these expressions when the market price of risk is taken into account.

An important conclusion is that, for all models, the long-run discount rate is always less than the long-time average rate. This is a conclusion that necessarily has to have consequences in any long-run economic planning. Finally, we reviewed our recent empirical study on 14 different countries, which obtained numerical values for the parameters that appear in the Vasicek model. We demonstrated two different estimation procedures and briefly discussed their differences and similarities.

**Author Contributions:** Conceptualization, J.M., M.M., J.P., J.D.F. and J.G.; methodology, J.M., M.M., J.P., J.D.F. and J.G.; formal analysis, J.M., M.M., J.P., J.D.F. and J.G.; investigation, J.M., M.M., J.P., J.D.F. and J.G.; resources, J.M., M.M., J.P., J.D.F. and J.G.; writing—original draft preparation, J.M.; writing—review and editing, J.M., M.M., J.P., J.D.F. and J.G.; funding acquisition, M.M. and J.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by MINEICO (Spain), Agencia Estatal de Investigación (AEI) gran<sup>t</sup> number PID2019-106811GB-C33 (AEI/10.13039/501100011033) (JM, MM and JP); by Generalitat de Catalunya gran<sup>t</sup> number 2017 SGR 608 (JM, MM and JP); by National Science Foundation gran<sup>t</sup> 0624351 (JG); and by the Institute for New Economic Thinking (JDF).

**Data Availability Statement:** The study reports data described and analyzed in Refs. [26–30].

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript or in the decision to publish the results.
