<blockquote>
<ol>
<li>I'm aware that A is a constant that is the amount repaid in each time period t
i.e. it is independent of t. But what does it mean by &quot;continuous&quot;</li>
</ol>
</blockquote>
<p>Continuous means that it happens in every 'instant' over some interval as opposed to having discrete repayments where the repayment would occur only at some specific discrete moments (e.g. every month, year etc). Continuous time is usually because it is more convenient to work with and usually model predictions won't qualitatively change as result.</p>
<blockquote>
<ol start="2">
<li>What is intuition for A/r
what is this actually telling us? How would the model be different if magically somehow we had just A rather than A/r
on the RHS.</li>
</ol>
</blockquote>
<p>The <span class="math-container">$A/r$</span> would be the value if debt would be perpetuity. Recall that in discrete time perpetuity is a special case of annuity where formula for value of annuity is <span class="math-container">$$P=\frac{A}{r}\left(1-\frac{1}{(1-r)^t}\right) \tag{*}$$</span>. When <span class="math-container">$t \to \infty$</span> you end up with perpetuity <span class="math-container">$A/r$</span>.</p>
<p>The whole term:</p>
<p><span class="math-container">$$\frac{A}{r}(1−e^{rt})$$</span></p>
<p>is just discrete time equivalent of <span class="math-container">$*$</span>.  However, this being said you should not simply interpret it separately since you are not dealing with perpetuity.</p>
<p>If <span class="math-container">$r$</span> would 'magically' disappear the equation would no longer make sense.  To be clear <span class="math-container">$r$</span> could be set to 1 which would make it 'disappear' but then you need to set it to 1 everywhere. If you want to deal with debt that literally has no interest you need to derive and solve new ODE.</p>

CC BY-SA 4.0
2023-04-22T10:36:27.270Z
Thanks for this detailed answer. 1) Regarding the continuous point then can I assume t is any positive real number. So I shouldn’t think about it as time period 0,1,2…etc. in fact the only useful partitions are t=0 t=T and then all the real number in between, would you say that is correct? Hence why we are using a differential equation, rather than a difference equation which we could use for a relationship based on discrete time intervals?
CormJack
CC BY-SA 4.0
2023-04-22T10:38:58.973Z
2) Regarding the perpetuity point, I looked this up, and it said “the state or quality of lasting forever.” so I assume it’s like the debt never maturing. As if it were an interest only  loan. But would you be able to go into slightly more detail on this point? Thanks!
CormJack
CC BY-SA 4.0
2023-04-22T11:21:27.790Z
@CormJack 1) yes, any real number but your examples are natural numbers, it could be 0.1 or 0.00001 etc. 2) yes perpetuity is loan that never expires such as consol bonds in britain, what more detail would you like about it?
1muflon1
CC BY-SA 4.0
2023-04-22T18:19:10.790Z
Yes completely agree, I wrote "Shouldn't" think about it as 0,1,2...but as you rightly say 0.0001 etc. Thanks again for clarifying. What I think I don't understand about perpetuity.... Do you literally mean the term A/r would be the value of the debt if it never matured? I don't understand how that works? If you have time...if you could just break down the relationship/intuition between A/r and perpetuity. That would be awesome?
CormJack
CC BY-SA 4.0
2023-04-22T18:21:55.647Z
I'm looking for the kind of intuition where for example I can explain why we have a term $D(0)e^rt$ because this shows the growth of the original debt over time. and I can understand that $\frac{A}{r}(1 - e^{rt})$ as an overall term get's more negative as more payments are made over time, counteracting the debt growth. And I understand that we can think of A/r as a ration marking the effectiveness of repayments vs the size of interest...If you can add that kind of intuition around your perpetuity point (if  you have time), would be awesome!
CormJack
CC BY-SA 4.0
2023-04-22T19:10:02.817Z
@CormJack even though perpetuity never expires you can sell it on market for debt, even if you do not sell it value of it is not zero
1muflon1
CC BY-SA 4.0
2023-04-22T09:27:40.097Z
2023-04-22T09:27:40.097Z
1muflon1
<p>Given the model of debt and repayments given by the first-order linear ordinary differential equation (ODE):</p>
<p><span class="math-container">$\frac{dD}{dt} - rD = -A$</span></p>
<p>In this model, <span class="math-container">$D(t)$</span> represents the consumer's debt at time <span class="math-container">$t$</span>, <span class="math-container">$r$</span> is the constant interest rate paid on the debt, and <span class="math-container">$A$</span> is the <em>continuous rate of debt repayment</em>. The equation describes how the debt evolves over time, considering the interest rate and repayment rate.</p>
<p>The solution to this ODE is:</p>
<p><span class="math-container">$D(t) = D(0)e^{rt} + \frac{A}{r}(1 - e^{rt})$</span></p>
<ol>
<li>I'm aware that A is a constant that is the amount repaid in each time period <span class="math-container">$t$</span> i.e. it is independent of t. But what does it mean by &quot;continuous&quot;</li>
<li>What is intuition for <span class="math-container">$\frac{A}{r}$</span> what is this actually telling us? How would the model be different if magically somehow we had just A rather than <span class="math-container">$\frac{A}{r}$</span> on the RHS.</li>
</ol>

<mathematical-economics><finance><debt>
Meaning of $\frac{A}{r}$ in standard debt model:
CormJack
# Meaning of $\frac{A}{r}$ in standard debt model:

(Asked by: username_0 on 2023-04-21)

Given the model of debt and repayments given by the first-order linear ordinary differential equation (ODE):

$\frac{dD}{dt} - rD = -A$

In this model, $D(t)$ represents the consumer's debt at time $t$, $r$ is the constant interest rate paid on the debt, and $A$ is the *continuous rate of debt repayment*. The equation describes how the debt evolves over time, considering the interest rate and repayment rate.

The solution to this ODE is:

$D(t) = D(0)e^{rt} + \frac{A}{r}(1 - e^{rt})$

1. I'm aware that A is a constant that is the amount repaid in each time period $t$ i.e. it is independent of t. But what does it mean by "continuous"
2. What is intuition for $\frac{A}{r}$ what is this actually telling us? How would the model be different if magically somehow we had just A rather than $\frac{A}{r}$ on the RHS.

**Tags:** mathematical-economics, finance, debt

---

## 1 answers

### Answer by username_1 on 2023-04-22. Score: 1 | Accepted answer

> 1. I'm aware that A is a constant that is the amount repaid in each time period t
>    i.e. it is independent of t. But what does it mean by "continuous"

Continuous means that it happens in every 'instant' over some interval as opposed to having discrete repayments where the repayment would occur only at some specific discrete moments (e.g. every month, year etc). Continuous time is usually because it is more convenient to work with and usually model predictions won't qualitatively change as result.

> 2. What is intuition for A/r
>    what is this actually telling us? How would the model be different if magically somehow we had just A rather than A/r
>    on the RHS.

The $A/r$ would be the value if debt would be perpetuity. Recall that in discrete time perpetuity is a special case of annuity where formula for value of annuity is $$P=\frac{A}{r}\left(1-\frac{1}{(1-r)^t}\right) \tag{\*}$$. When $t \to \infty$ you end up with perpetuity $A/r$.

The whole term:

$$\frac{A}{r}(1−e^{rt})$$

is just discrete time equivalent of $\*$. However, this being said you should not simply interpret it separately since you are not dealing with perpetuity.

If $r$ would 'magically' disappear the equation would no longer make sense. To be clear $r$ could be set to 1 which would make it 'disappear' but then you need to set it to 1 everywhere. If you want to deal with debt that literally has no interest you need to derive and solve new ODE.

> **Comments:**
> **username_0**: Thanks for this detailed answer. 1) Regarding the continuous point then can I assume t is any positive real number. So I shouldn’t think about it as time period 0,1,2…etc. in fact the only useful partitions are t=0 t=T and then all the real number in between, would you say that is correct? Hence why we are using a differential equation, rather than a difference equation which we could use for a relationship based on discrete time intervals?  
> **username_0**: 2) Regarding the perpetuity point, I looked this up, and it said “the state or quality of lasting forever.” so I assume it’s like the debt never maturing. As if it were an interest only  loan. But would you be able to go into slightly more detail on this point? Thanks!  
> **username_1**: @CormJack 1) yes, any real number but your examples are natural numbers, it could be 0.1 or 0.00001 etc. 2) yes perpetuity is loan that never expires such as consol bonds in britain, what more detail would you like about it?  
> **username_0**: Yes completely agree, I wrote "Shouldn't" think about it as 0,1,2...but as you rightly say 0.0001 etc. Thanks again for clarifying. What I think I don't understand about perpetuity.... Do you literally mean the term A/r would be the value of the debt if it never matured? I don't understand how that works? If you have time...if you could just break down the relationship/intuition between A/r and perpetuity. That would be awesome?  
> **username_0**: I'm looking for the kind of intuition where for example I can explain why we have a term $D(0)e^rt$ because this shows the growth of the original debt over time. and I can understand that $\frac{A}{r}(1 - e^{rt})$ as an overall term get's more negative as more payments are made over time, counteracting the debt growth. And I understand that we can think of A/r as a ration marking the effectiveness of repayments vs the size of interest...If you can add that kind of intuition around your perpetuity point (if  you have time), would be awesome!  
> **username_1**: @CormJack even though perpetuity never expires you can sell it on market for debt, even if you do not sell it value of it is not zero  

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