absolute error of differential compound interest: formula, future value after 12 yrs

a6m0

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Hi all,

The question I'm working on is: Suppose a principal of $1000 is deposited into an account that pays annualinterest r compounded quarterly.

(a) Give the formula for the future value of the balance after 12 years.
(b) Approximate using differentials the absolute error in the future value ofthe balance after 12 years if you know the interest rate is 5% with anabsolute error of 0.1%.

For (a), I get A=1000(1+r/4)^48

and I differentiated and got 12,000(1+r/4)^47

Plugging 5% or 0.05 into r,

A=12,000(1.0125)^47
=21,515.31678

After getting here i'm kinda of stuck. How do I get the absolute error?

Do i just multiply my answer (21,515.31678) by 0.001?

Thanks
 
Suppose a principal of $1000 is deposited into an account that pays annual interest r, compounded quarterly.

(a) Give the formula for the future value of the balance after 12 years.
(b) Approximate using differentials the absolute error in the future value of the balance after 12 years if you know the interest rate is 5% with an absolute error of 0.1%.


For (a), I get A=1000(1+r/4)^48
Using the compound-interest formula, A = P(1 + r/n)^(nt), where A is the ending amount, P is the original amount (or principle), r is the annual interest rate, n is the number of compoundings per year, and t is the number of years; and assuming t is a constant (provided in the second part of this exercise), the formulation is:

. . . . .\(\displaystyle A\, =\, 1,000\,\left(1\, +\, \dfrac{r}{4}\right)^{4\cdot 12}\)

...which simplifies to agree with your equation. (For the future, note that the above, which defines the variables and states one's reasoning, is generally more useful for the volunteer helpers.) In function notation, assuming the interest rate is the variable amount, we then get:

. . . . .\(\displaystyle A(r)\, =\, 1,000\,\left(1\, +\, \dfrac{r}{4}\right)^{48}\)

and I differentiated and got 12,000(1+r/4)^47
Something is wrong here, because there is no "equals" sign and no functional name-statement on the left-hand side of that sign. Assuming that my second equation above is what was intended, and assuming that you're differentiating with respect to the variable r, then the first step of the differentiation should be:

. . . . .\(\displaystyle \dfrac{dA}{dr}\, =\, 1,000\, \big[48\, \cdot\, \left(1\, +\, \dfrac{r}{4}\right)^{47}\big]\, \cdot\, \dfrac{1}{4}\)

...by the Power, Product, and Chain Rules. Simplifying the right-hand side of the above agrees with the expression you've posted. But your expression should be an equation.

Plugging 5% or 0.05 into r,

A=12,000(1.0125)^47
=21,515.31678
How are you getting that the ending amount, A, is equal to the derivative of the formula for the ending amount?

How do I get the absolute error?
When your textbook and instructor taught this topic, did they say something like what is explained and illustrated on this web page or in this PDF? If so, where are you getting stuck in the process? If not, how is your class defining the terms, etc?

Please be complete. Thank you! ;)
 
Last edited:
Using the compound-interest formula, A = P(1 + r/n)^(nt), where A is the ending amount, P is the original amount (or principle), r is the annual interest rate, n is the number of compoundings per year, and t is the number of years; and assuming t is a constant (provided in the second part of this exercise), the formulation is:

. . . . .\(\displaystyle A\, =\, 1,000\,\left(1\, +\, \dfrac{r}{4}\right)^{4\cdot 12}\)

...which simplifies to agree with your equation. (For the future, note that the above, which defines the variables and states one's reasoning, is generally more useful for the volunteer helpers.) In function notation, assuming the interest rate is the variable amount, we then get:

. . . . .\(\displaystyle A(r)\, =\, 1,000\,\left(1\, +\, \dfrac{r}{4}\right)^{48}\)


Something is wrong here, because there is "equals" sign and no functional name-statement on the left-hand side of that sign. Assuming that my second equation above is what was intended, and assuming that you're differentiating with respect to the variable r, then the first step of the differentiation should be:

. . . . .\(\displaystyle \dfrac{dA}{dr}\, =\, 1,000\, \big[48\, \cdot\, \left(1\, +\, \dfrac{r}{4}\right)^{47}\big]\, \cdot\, \dfrac{1}{4}\)

...by the Power, Product, and Chain Rules. Simplifying the right-hand side of the above agrees with the expression you've posted. But your expression should be an equation.


How are you getting that the ending amount, A, is equal to the derivative of the formula for the ending amount?


When your textbook and instructor taught this topic, did they say something like what is explained and illustrated on this web page or in this PDF? If so, where are you getting stuck in the process? If not, how is your class defining the terms, etc?

Please be complete. Thank you! ;)

Hi thanks for the reply, and clearing things up. I dont think we went over absolute errors in class nor the textbook, and after reading the pdfs Im still kind of unsure on my next steps. From the PDFS I get that absolute error is the Exact value − Approximate value. Im guessing that the approximate value is 21,515.31678 x (0.001), but what would be the Exact value? Sorry if I'm not making much sense, Im just really confused

Thank You
 
Hi thanks for the reply, and clearing things up. I dont think we went over absolute errors in class nor the textbook, and after reading the pdfs Im still kind of unsure on my next steps. From the PDFS I get that absolute error is the Exact value − Approximate value. Im guessing that the approximate value is 21,515.31678 x (0.001), but what would be the Exact value? Sorry if I'm not making much sense, Im just really confused
The links explain that "absolute" error is the error in strictly numerical terms. The "relative" error is the error in percentage terms. ;)
 
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