# How to create a mathematical model of investment?

• Eclair_de_XII
In summary, the problem involves someone with no initial capital investing at a constant annual rate of return, with investments made and compounded continuously. The solution involves using the differential equation ##\frac{dS}{dt}=r(S+\frac{k}{r})## to model the scenario.
Eclair_de_XII

## Homework Statement

"Someone with no initial capital ##S_0=0## invests ##k \frac{dollar}{year}## at an annual rate of return ##r##. Assume that investments are made continuously and that the return is compounded continuously."

Find the solution ##S## that solves the differential equation modeling this scenario.

## Homework Equations

Let ##S## be the amount of capital at any time.

## The Attempt at a Solution

I'm trying to make sure that the units of all my quantities are the same, but the main problem I'm having is the discontinuity at ##t=0## when I divide ##S## by ##t##.

##\frac{dS}{dt}=\frac{S}{t}+k(r+1)##
##\frac{dS}{dt}-\frac{S}{t}=k(r+1)##
##\frac{d}{dt}(t^{-1}S)=\frac{k(r+1)}{t}##
##S(t)=(tlnt)(k(r+1))##

I frankly do not know how to model this problem. Can anyone help me understand what is happening in this problem? It's hard to insert ##S## into the equation because the ##\frac{dS}{dt}## does not rely on ##S##; it relies on the rate of return ##r## which only applies to the yearly investments ##k##. Then it's compounded continually...? I do not understand this.

Eclair_de_XII said:

## Homework Statement

"Someone with no initial capital ##S_0=0## invests ##k \frac{dollar}{year}## at an annual rate of return ##r##. Assume that investments are made continuously and that the return is compounded continuously."

Find the solution ##S## that solves the differential equation modeling this scenario.

## Homework Equations

Let ##S## be the amount of capital at any time.

## The Attempt at a Solution

I'm trying to make sure that the units of all my quantities are the same, but the main problem I'm having is the discontinuity at ##t=0## when I divide ##S## by ##t##.

##\frac{dS}{dt}=\frac{S}{t}+k(r+1)##
##\frac{dS}{dt}-\frac{S}{t}=k(r+1)##
##\frac{d}{dt}(t^{-1}S)=\frac{k(r+1)}{t}##
##S(t)=(tlnt)(k(r+1))##

I frankly do not know how to model this problem. Can anyone help me understand what is happening in this problem? It's hard to insert ##S## into the equation because the ##\frac{dS}{dt}## does not rely on ##S##; it relies on the rate of return ##r## which only applies to the yearly investments ##k##. Then it's compounded continually...? I do not understand this.

For a very small time increment of ##\Delta t## years, the amount invested between times ##t## and ##t + \Delta t## is ##k \Delta t ## dollars; that is what is meant by a continuous investment at rate ##k##.

If S(t) is the amount in the bank at time ##t## (years) we have
$$S(t+\Delta t) = S(t) (1 + r \Delta t) + k \Delta t \hspace{2cm}(1)$$
because the balance ##\$S(t)## at ##t## would grow by the interest-rate factor factor ##1 + r \Delta t## in the short time interval of length ##\Delta t##, even if we did not invest any more---but additional investment makes it grow more. That is what continuous compounding means: a dollar in the account at time zero grows to ##\$e^{rt}## by time ##t > 0##, and so the growth in value from ##t## to ##t + \Delta t## is
$$e^{r(t +\Delta t)} - e^{rt} = e^{rt} \left( e^{r \Delta t}-1 \right) \doteq e^{rt} (1 + \Delta t),$$
which is a growth rate of ##1 + r \Delta t.##

We get the differential equation for ##S(t)## from eq.(1) above.

For more on continuous compounding, see, eg.,
http://www-stat.wharton.upenn.edu/~waterman/Teaching/IntroMath99/Class04/Notes/node11.htm
http://people.duke.edu/~charvey/Classes/ba350_1997/preassignment/proof1.htm
http://math.stackexchange.com/questions/539115/proof-of-continuous-compounding-formula

Last edited:
Thanks. Should I worry about the equation being continuous at ##t=0##? Yes, right? So I can't have any of the variables be divided by ##t## for the ##\frac{dollar}{year}## unit.

Can anyone tell me what is wrong with my current model?

##\frac{dS}{dt}=r(S+kt)=rS+rkt##
##\frac{dS}{dt}-rS=rkt##
##\frac{d}{dt}(Se^{-rt})=(rkt)(e^{-rt})##

Basically, I get an equation with an extra linear term: ##S(t)=-kt+\frac{k}{r}(e^{rt}-1)##. It's another equation that satisfies the IVP at ##S(0)=0##. I'm not sure what to make of that, but at least I got something done tonight.

Last edited:
Eclair_de_XII said:
Can anyone tell me what is wrong with my current model?

##\frac{dS}{dt}=r(S+kt)=rS+rkt##
##\frac{dS}{dt}-rS=rkt##
##\frac{d}{dt}(Se^{-rt})=(rkt)(e^{-rt})##

Basically, I get an equation with an extra linear term: ##S(t)=-kt+\frac{k}{r}(e^{rt}-1)##. It's another equation that satisfies the IVP at ##S(0)=0##. I'm not sure what to make of that, but at least I got something done tonight.

*************************************

The model above assumes that the person invests at rate ##kt## at time t, so is a variable investment rate that increases over time. The problem told you the investment rate is constant, not variable.

************************************

Eclair_de_XII said:
Thanks. Should I worry about the equation being continuous at ##t=0##? Yes, right? So I can't have any of the variables be divided by ##t## for the ##\frac{dollar}{year}## unit.

I have no idea what you are talking about. There is no issue of discontinuity anywhere, even at ##t = 0##. You do not divide by ##t##---that is not what rates are about. I explained exactly what was meant by rates, so all you need to do is re-read the post.

Last edited:
Oh, I got it: ##\frac{dS}{dt}=r(S+\frac{k}{r})##.

Thanks.

## 1. How do I choose the variables for my investment model?

Choosing variables for an investment model can be a complex process and will depend on the specific goals and parameters of your model. Some common variables to consider include the initial investment amount, expected return rate, inflation rate, and time horizon. It may also be helpful to gather data on historical market performance and economic indicators to inform your variable selection.

## 2. What mathematical equations should I use for my investment model?

The equations used in an investment model will also vary depending on the specific goals and parameters of your model. Some commonly used equations include the compound interest formula, the Black-Scholes model for option pricing, and the capital asset pricing model (CAPM). It is important to carefully consider which equations are most appropriate for your model and to ensure they are applied correctly.

## 3. How do I account for risk in my investment model?

Risk is a crucial factor to consider in any investment model. One approach is to incorporate a risk premium into your expected return rate to account for the higher potential returns associated with riskier investments. Another option is to use a risk-adjusted discount rate, which discounts future cash flows based on their level of risk. There are also more advanced statistical models available for measuring and predicting risk.

## 4. Can I use historical data to create my investment model?

Historical data can be a valuable resource for creating an investment model, but it should not be relied on exclusively. Economic and market conditions are constantly changing, and using outdated data can lead to inaccurate predictions. It is important to regularly update and reassess your model using current data to ensure its accuracy and relevance.

## 5. How can I test the accuracy of my investment model?

There are several methods for testing the accuracy of an investment model. One approach is to compare the model's predictions to actual market performance over a specific time period. Another option is to use backtesting, which involves applying the model to past market data to see how well it would have performed. It is also helpful to continually monitor and adjust the model as needed to improve its accuracy.

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