# Are Delay and Interval Effects the Same Anomaly in the Context of Intertemporal Choice in Finance?

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Delay Effect

**Example**

**1.**

**Example**

**2.**

## 3. The Interval Effect

**Example**

**3.**

**Definition**

**1.**

**Proposition**

**1.**

- 1.
- $\overline{f}(t,a)=\mathrm{exp}\{-\overline{\delta}(t,a)\}$, where$\overline{\delta}(t,a)$is the mean discount rate in the interval$[t,t+a]$;
- 2.
- $\underset{a\to 0}{\mathrm{lim}}\overline{f}(t,a)=\mathrm{exp}\{-\delta (t)\}$, where$\delta (t):=\underset{a\to 0}{\mathrm{lim}}\frac{F(t+a)-F(t)}{aF(t)}$is the instantaneous discount rate at time t.

**Proof.**

- The general expression of a discount function, according to its instantaneous discount rate, leads to $F(t+a)=\mathrm{exp}\left\{-{\displaystyle {\int}_{0}^{t+a}\delta (x)dx}\right\}$ and $F(t)=\mathrm{exp}\left\{-{\displaystyle {\int}_{0}^{t}\delta (x)dx}\right\}.$ Therefore, as $\frac{1}{a}{\displaystyle {\int}_{t}^{t+a}\delta (x)\mathrm{d}x}$ is the average of function $\delta $ in the interval $[t,t+a]$, one has$$\overline{f}(t,a)={\left[\frac{\mathrm{exp}\left\{-{\displaystyle {\int}_{0}^{t+a}\delta (x)\mathrm{d}x}\right\}}{\mathrm{exp}\left\{-{\displaystyle {\int}_{0}^{t}\delta (x)\mathrm{d}x}\right\}}\right]}^{\frac{1}{a}}=\mathrm{exp}\left\{-\frac{1}{a}{\displaystyle {\int}_{t}^{t+a}\delta (x)\mathrm{d}x}\right\}=\mathrm{exp}\{-\overline{\delta}(t,a)\},$$
- $\underset{a\to 0}{\mathrm{lim}}\overline{f}(t,a)=\underset{a\to 0}{\mathrm{lim}}{\left[\frac{F(t+a)}{F(t)}\right]}^{1/a}={1}^{\infty}$ , which is an indetermination. Let us solve this indetermination by using the well-known formula to solve this type of indetermination:$$\underset{a\to 0}{\mathrm{lim}}\overline{f}(t,a)=\mathrm{exp}\left\{\underset{a\to 0}{\mathrm{lim}}\frac{1}{a}\left[\frac{F(t+a)}{F(t)}-1\right]\right\}=\mathrm{exp}\left\{\underset{a\to 0}{\mathrm{lim}}\frac{F(t+a)-F(t)}{aF(t)}\right\}=\mathrm{exp}\{-\delta (t)\}.$$

## 4. Mathematical Analysis of the Delay and Interval Effects

#### 4.1. Assessment at a Given Benchmark (Time 0)

**Definition**

**2.**

**Theorem**

**1.**

- (i)
- If$s=t$, then$\overline{\delta}(s,a)>\overline{\delta}(t,b)$;
- (ii)
- The instantaneous discount rate is strictly decreasing;
- (iii)
- If$s\le t$, then$\overline{\delta}(s,a)>\overline{\delta}(t,b)$;
- (iv)
- The delay effect holds;
- (v)
- The subadditivity of the second order holds.

**Proof.**

- The instantaneous discount rate is constant in the interval $[r,s]$. This is not possible because by taking $a=\frac{s-r}{2}$ and $b=s-r$, one has $\overline{\delta}(t,a)=\overline{\delta}(t,b)$, in contradiction with (i);
- The instantaneous discount rate is not constant in the interval $[r,s]$. In this case, there is a subinterval of $[r,s]$, where the instantaneous discount rate is increasing and, as such, the reasoning is the same as the case in which $\delta (r)<\delta (s)$.

**Corollary**

**1.**

**Proof.**

**Definition**

**3.**

**Theorem**

**2.**

- (i)
- If$s+a=t+b$, then$\overline{\delta}(s,a)>\overline{\delta}(t,b)$;
- (ii)
- The instantaneous discount rate is strictly increasing;
- (iii)
- If$s+a\ge t+b$, then$\overline{\delta}(s,a)>\overline{\delta}(t,b)$;
- (iv)
- The reverse delay effect holds;
- (v)
- The superadditivity of the second order holds.

**Corollary**

**2.**

**Proof.**

#### 4.2. Assesment at Variable Reference (at the Front-End Delay of the Interval)

## 5. Conclusions and Future Research

- The decreasing interval effect, wherein the discount rate decreases (the FED of the short interval is less than or equal to the FED of the larger interval);
- The increasing interval effect, wherein the discount rate increases (the FED of the larger interval is less than the FED of the shorter interval).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Reference | Term Used | Definition | Experimental Work? | Mathematical Definition? |
---|---|---|---|---|

[20] | Subadditive discounting | Yes | Yes | No |

[28] | Interval effect and subadditive discounting | Yes | Yes | No |

[23] | Interval effect | [28] | No | No |

[21] | Interval effect | Yes | Yes | No |

[35] | Interval effect | [20] | No | No |

[26] | The effect of interval length | [20] | Yes | No |

[30] | Interval effect | Yes | No | No |

[27] | Interval effect | [20,23] | No | No |

[36] | Interval effect | [23] | Yes | No |

[37] | Interval effect | [20,28] | Yes | No |

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Delay | Interval | |
---|---|---|

Delay effect | Different | Equal |

Interval effect | Equal | Different |

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**MDPI and ACS Style**

Cruz Rambaud, S.; Ortiz Fernández, P.
Are Delay and Interval Effects the Same Anomaly in the Context of Intertemporal Choice in Finance? *Symmetry* **2021**, *13*, 41.
https://doi.org/10.3390/sym13010041

**AMA Style**

Cruz Rambaud S, Ortiz Fernández P.
Are Delay and Interval Effects the Same Anomaly in the Context of Intertemporal Choice in Finance? *Symmetry*. 2021; 13(1):41.
https://doi.org/10.3390/sym13010041

**Chicago/Turabian Style**

Cruz Rambaud, Salvador, and Piedad Ortiz Fernández.
2021. "Are Delay and Interval Effects the Same Anomaly in the Context of Intertemporal Choice in Finance?" *Symmetry* 13, no. 1: 41.
https://doi.org/10.3390/sym13010041