Compound interest questions -- Annual rate of 3.11% which is compounded 3 times each year

  • #1

Homework Statement:

The formula for compound interest is given by
๐ด(๐‘ก)=๐ผ(1+ (0.01ร—๐‘Ÿ/n))^๐‘›๐‘ก,
where ๐ผ is the initial amount invested in dollars, ๐‘Ÿ is the annual interest rate (expressed as a percent % rate), ๐‘› is the number of times interest is compounded per year, ๐‘ก is the time in years since the initial investment and ๐ด(๐‘ก) is the amount, or balance, in dollars after ๐‘ก years have passed.

Suppose you invest $350 today into an account with an annual rate of 3.11% which is compounded 3 times each year. Answer the following questions about the growth of your investment.

Relevant Equations:

1. What is the instantaneous rate at which your investment grows (in dollars per year) as a function of ๐‘ก ?

2. What is the rate at which the investment is growing (in dollars per year) after 17 years?

3. What is the true percent increase of the investment at the end of the first year?
1. 350*(1+(0.0311)/3)*ln(1+(0.0311)/3)

2 and 3. Not sure how to answer these questions.
 

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  • #3
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It is not clear what the instantaneous rate that the investment would grow, since the compounding is incremental. Maybe what they mean is the equivalent rate of growth if the compounding were continuous, but this would not be a function of time.
 
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  • #4
pbuk
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1. What is the instantaneous rate at which your investment grows (in dollars per year) as a function of ๐‘ก
1. 350*(1+(0.0311)/3)*ln(1+(0.0311)/3)
This is not a function of ## t ##.

You are given
The formula for compound interest is given by ๐ด(๐‘ก)=๐ผ(1+ (0.01ร—๐‘Ÿ/n))^๐‘›๐‘ก
which is a function of ## t ##, what do you think you need to do with this? It will be easier if we work in ## \LaTeX ##, I'll give you a head start by rewriting the equation in the question:
$$ A(t) = I \left(1 + \left( 0.01 \frac{r}{n} \right) \right) ^ {nt} $$
Edit: probably best if we define ## a = I \left(1 + \left( 0.01 \frac{r}{n} \right) \right) ## and write
$$ A(t) = a ^ {nt} $$
 
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  • #5
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This is not a function of ## t ##.

You are given

which is a function of ## t ##, what do you think you need to do with this? It will be easier if we work in ## \LaTeX ##, I'll give you a head start by rewriting the equation in the question:
$$ A(t) = I \left(1 + \left( 0.01 \frac{r}{n} \right) \right) ^ {nt} $$
Edit: probably best if we define ## a = I \left(1 + \left( 0.01 \frac{r}{n} \right) \right) ## and write
$$ A(t) = a ^ {nt} $$
That last equation makes no sense to me.
 
  • #6
pbuk
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That last equation makes no sense to me.
Am I missing the point here - surely it's just exponential growth?
 
  • #7
etotheipi
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I think @Chestermiller's point is that the model suggested in the question is for discrete values of ##t##, so a derivative is not well defined. For purposes of the question it is probably required to assume that the given equation holds over a continuous interval of ##t##.
 
  • #8
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The equivalent continuous compounding interest rate R would satisfy $$A=Ie^{Rt}= I \left(1 + \left( 0.01 \frac{r}{n} \right) \right)^{nt}$$or$$R=n\ln{\left(1 + \left( 0.01 \frac{r}{n} \right)\right)}$$
 
  • #9
pbuk
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I think @Chestermiller's point is that the model suggested in the question is for discrete values of ##t##, so a derivative is not well defined. For purposes of the question it is probably required to assume that the given equation holds over a continuous interval of ##t##.
Economics is a social science so you have to accommodate a certain lack of rigour :wink:

However where does it say in the question that ## t ## is discrete?
 
  • #10
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Economics is a social science so you have to accommodate a certain lack of rigour :wink:

However where does it say in the question that ## t ## is discrete?
When they say that interest is compounded 3 times a year, what does that mean to you?
 
  • #11
etotheipi
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Economics is a social science so you have to accommodate a certain lack of rigour :wink:

However where does it say in the question that ## t ## is discrete?
If ##n## is finite, e.g. if I compound it 12 times per year, then the actual ##A(t)## would be a step function. The formula given in the question would only really hold for integer values of ##t## (or at least those that coincide with the time of compounding).

However, if we make the assumption that we can convert it into an analogous interest model over a continuous interval of ##t##, then we can of course recast it in terms of ##A(t) = Ie^{Rt}##.
 
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  • #12
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If ##n## is finite, e.g. if I compound it 12 times per year, then the actual ##A(t)## would be a step function. The formula given in the question would only really hold for integer values of ##t## (or at least those that coincide with the time of compounding)..
Yes, at integral multiples of 1/n.
 
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  • #13
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If ##n## is finite, e.g. if I compound it 12 times per year, then the actual ##A(t)## would be a step function. The formula given in the question would only really hold for integer values of ##t##.
But it doesn't say any of that in the question. You are bringing extraneous information about your knowledge of how a deposit account works in the real world that is not in the question. The question presents a continuous, differentiable model of compound interest.

And as it happens the discrete model is not how it happens anyway - interest is normally accrued on a daily basis (using some conventional calculation) increasing the value of the deposit (almost) continuously. The accrued amount is not included in the amount that is used to calculate interest.
 
  • #14
pbuk
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Maybe what they mean is the equivalent rate of growth if the compounding were continuous, but this would not be a function of time.
The rate of growth in dollars per year (which is what the question asks for) surely would be a function of time.
 
  • #15
pbuk
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When they say that interest is compounded 3 times a year, what does that mean to you?
Sorry, missed this. It means that 3 times a year the balance of accrued interest is added to the principal balance of the account.

It doesn't matter though, the question does not ask you to write an equation for the amount of money in the account based on the information that interest is compounded 3 times a year, it gives you that equation.
 
  • #16
etotheipi
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But it doesn't say any of that in the question. You are bringing extraneous information about your knowledge of how a deposit account works in the real world that is not in the question. The question presents a continuous, differentiable model of compound interest.

And as it happens the discrete model is not how it happens anyway - interest is normally accrued on a daily basis (using some conventional calculation) increasing the value of the deposit (almost) continuously. The accrued amount is not included in the amount that is used to calculate interest.
I think you are right in practice, I cannot imagine financial analysts sitting around wasting their time with discrete models when ##Pe^{rt}## is a very good approximation anyway for finite ##n##.

But in the context of this question, the wording is pretty ambiguous. It doesn't specify the domain of ##t##, and given that we compound only thrice per year, that would seem to imply a very pronounced step function, so we cannot infer immediately that the model is continuous and differentiable...

In any case, the only way to answer the question is to assume it holds for all ##t## and differentiate.
 
  • #17
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Poor OP, he must think we are mad. To recap: the question tells you that at time ## t ##, your investment is worth ## A(t) = a ^ {nt} ## dollars. What is the rate of change of your investment at time ## t ##?
 
  • #18
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I think you are right in practice, I cannot imagine financial analysts sitting around wasting their time with discrete models when ##Pe^{rt}## is a very good approximation anyway for finite ##n##.

But in the context of this question, the wording is pretty ambiguous. It doesn't specify the domain of ##t##, and given that we compound only thrice per year, that would seem to imply a very pronounced step function, so we cannot infer immediately that the model is continuous and differentiable...

In any case, the only way to answer the question is to assume it holds for all ##t## and differentiate.
I think you give financial analysts too much credit. I don't think most financial analysts would know what continuous compounding is if it jumped up and bit them on the butt (or what an exponential function is).
 
  • #19
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Poor OP, he must think we are mad. To recap: the question tells you that at time ## t ##, your investment is worth ## A(t) = a ^ {nt} ## dollars. What is the rate of change of your investment at time ## t ##?
I see what you are saying, I think. The math teacher that dreamt up this problem did not fully understand the implications of discrete compounding and imagined that all the student needed to do was differentiate the equation that was given with respect to t.
 
  • #20
pbuk
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I think you are right in practice
I know I am right in practice, I do this for a living. But I am also right in theory.

, I cannot imagine financial analysts sitting around wasting their time with discrete models when ##Pe^{rt}## is a very good approximation anyway for finite ##n##.
I think you mean discrete ## n ##. And it's not a 'very good approximation': at every time ## t_i ## that the discrete model is defined it is equal to the value of the continuous model at ## t_i ##.

But in the context of this question, the wording is pretty ambiguous.
Not to me, and not to the examiner. The question presents you with an explicit model and asks you about its rate of change. The question would be meaningless if its rate of change were not defined.
 
  • #21
Articulating and defending my answers:

1. Instantaneous rate as a function of t: FIND THE DERIVATIVE (d/dx a^x=a^x*ln(a). Derivative A'(t)=1050*(3.0311/3)^(3t)*ln(3.0311/3)
2. Total amount invested after 17 years: A(t)=350*(3.0311/3)^3t. A(17)=$592.24.
3. Rate at which the investment IS growing after 17 years: USE DERIVATIVE. Thus, A'(t)=1050*(3.0311/3)^(3t)*ln(3.0311/3). A'(17)=18.32%.
4. True percent increase at the end of the first year: (361-350)/350=3.14%.

I think I got it after some thought and DESMOS. Thank you!
 
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  • #22
pbuk
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Articulating and defending my answers:

1. Instantaneous rate as a function of t: FIND THE DERIVATIVE (d/dx a^x=a^x*ln(a). Derivative A'(t)=1050*(3.0311/3)^(3t)*ln(3.0311/3)
2. Total amount invested after 17 years: A(t)=350*(3.0311/3)^3t. A(17)=$592.24.
3. Rate at which the investment IS growing after 17 years: USE DERIVATIVE. Thus, A'(t)=1050*(3.0311/3)^(3t)*ln(3.0311/3). A'(17)=18.32%.
4. True percent increase at the end of the first year: (361-350)/350=3.14%.

I think I got it after some thought and DESMOS. Thank you!
I haven't checked your arithmetic but I'm sure its right - as you have discovered, this is actually a pretty easy question when you realise that you have to throw away some extraneous information.
 
  • #23
pbuk
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I haven't checked your arithmetic but I'm sure its right - as you have discovered, this is actually a pretty easy question when you realise that you have to throw away some extraneous information.
Edit: as pointed out in #24 and #30 I am not very good at adding fractions.

Oops - almost right. You had the term 1+(0.0311)/3 in your original answer which was correct but now you have 3.0311/3 which is not.
 
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  • #24
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Oops - almost right. You had the term 1+(0.0311)/3 in your original answer which was correct but now you have 3.0311/3 which is not.
Check his algebra again. It is right.
 
  • #25
pbuk
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I think you give financial analysts too much credit. I don't think most financial analysts would know what continuous compounding is if it jumped up and bit them on the butt (or what an exponential function is).
Having revealed my profession in #20 (it's on my profile anyway) I am going to ignore that remark.
 

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