Calculating accumulated profits when profit increases at an increasing rate

[Akanksha]

Hello guys,
I am struggling with understanding the concept of “accumulated profits“ in the context of this graph given.


For context: The graph plots the increase in profits (and not profit earned per year) for successive years, so profits increase at an increasing rate for this company, and we need to know if the investment worth 2 M is worth it, considering the client wants his money back in 2 years max.


• What I did not understand: when we can already see that for Y2, increase in profits from year 1 is 2M, which is equal to the value of our investment; why are we calculating the area under the line graph?
What is the significance? While I understand what this means conceptually, such as calculating the distance in speed-time graph or work done in Force-time graphs, I don’t know what accumulated profits actually mean in this context.

I am at loss of words, I‘ve been trying to understand this for over 8 hours now and I feel dumb.
What do accumulated profits even mean here?

 

[BigBeachBanana]

Accumulated Profit Year (2) = Profit in Year (1) + Profit in Year (2).

In the example,
Profit Year (1) i.e the area of the triangle from Year(0) to Year(1): $\frac{1\times 1}{1}=1$

Profit Year (2) is the area between Year(1) and Year(2), which is also $1.$

Then the Accumulated Profit up to Year(2) is $1 + 1 =2$, which is the equivalent of the area of the triangle from Year(0) to Year(2).

 

[JeffM]

Why are we calculating the area under the curve? It represents accumulated profit.

You say why is it necessary to worry about accumulated profit when determining whether to pay 2 million for an opportunity that generates an increase in profits of 2 million in just one year?

Well, what if the profit in year 1 is - 4 million and the increase in profits is 2 million in year 2. That means accumulated profits are minus 6 million. Would you pay 2 million to incur a loss of 6 million?

Are you still telling me that you do not see the relevance of the concept of accumulated profit?

Or is your question that you do not see the relevance in this specific case?

I must say that the relevance is not obvious in this case because the problem is so artificial. If I am given the opportunity to invest 2 million now, get 1 million back in a year with certainty, and get, with certainty, another 4 million back (principal of 2 million and profit of 2 million) at the end of two years, I do not need anything more than arithmetic to make me say yes.

They may be trying to show in a super simple case where the ideas behind integral calculus come in useful (though I must admit that I never needed integral calculus in a long career in finance).

Or they may be leading up to the idea that investing is always about choosing between alternatives, and differences in accumulated profit, which is just a different name for total profit over the period of investment, are always relevant although not always dispositive.

 

[Deleted member 4993]

Why are we calculating the area under the curve? It represents accumulated profit.

You say why is it necessary to worry about accumulated profit when determining whether to pay 2 million for an opportunity that generates an increase in profits of 2 million in just one year?

Well, what if the profit in year 1 is - 4 million and the increase in profits is 2 million in year 2. That means accumulated profits are minus 6 million. Would you pay 2 million to incur a loss of 6 million?

Are you still telling me that you do not see the relevance of the concept of accumulated profit?

Or is your question that you do not see the relevance in this specific case?

I must say that the relevance is not obvious in this case because the problem is so artificial. If I am given the opportunity to invest 2 million now, get 1 million back in a year with certainty, and get, with certainty, another 4 million back (principal of 2 million and profit of 2 million) at the end of two years, I do not need anything more than arithmetic to make me say yes.

They may be trying to show in a super simple case where the ideas behind integral calculus come in useful (though I must admit that I never needed integral calculus in a long career in finance).

Or they may be leading up to the idea that investing is always about choosing between alternatives, and differences in accumulated profit, which is just a different name for total profit over the period of investment, are always relevant although not always dispositive.

In Introduction to Engineering classes, we often choose "monetary" examples - because everybody thinks they understand money. The problem above can be easily transformed into "weight gain of a plate" due to "deposition".