Difference equation for interest and bank account

In summary: There are some other issues with your solution as well, but I won't go into them here.In summary, we discussed the interest rate and annual service charge of a bank account and the relation between deposits and balances using difference equations. We also calculated the balance after 25 years with and without monthly deposits, taking into account the compounding frequency and possible errors in the equations.
  • #1
xortan
78
1

Homework Statement



Consider your bank offers a 2% annual interest rate, and has no annual service charge. Let y[n] represent your account balance at the beginning of the month n and x[n] represent the amount of money you deposit during the month n. Assume that deposits during month n are credited to the balance at the end of that month but earn no interest until the following month.

a. Use difference equations to express the relation between deposits and balances.
b. Assume that you deposit $1000 in the bank and make no further deposits. Solve your difference equations to determine your balance after 25 years.
c. Determine your balance after 25 years if you deposit $100 each month

I think I have figured out the answer.

Homework Equations



r = interest rate
y[0] = 1000

The Attempt at a Solution



For part a I found the equation to be:

y[n] - (1+r)y[n-1] = x[n]

For part b the equation will become:

y[n] = (1 + r)n*y[0]

This gave an answer of $280 234.51

For part c I am kind of unsure about but after finding the zero-input and zero-state response and performing convolution I found the equation to be

y[n] = 1000*(1 + r)n + 100*((1 - (1+r)n) / (1 - (1 + r))

This gave an answer of $2 276 407.05

Please let me know if I am on the right track here!

Thanks!
 
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  • #2
Firstly, the interest will depend on the compounding frequency. Usually (by convention), this is per year. In other words, AER. You have assumed not AER. So your answer might not be correct for this reason.

Also, I don't think your equation in part a) is correct. You wrote: y[n] - (1+r)y[n-1] = x[n] But this means that the amount of money in your bank account at the start of the month depends on the amount of money you put in the bank account during that month. Which would be very useful, but unfortunately violates causality :)
 

1. What is a difference equation for interest and bank account?

A difference equation for interest and bank account is a mathematical equation that describes the relationship between the amount of money in a bank account and the interest earned on that money over time. It is used to track the growth of a bank account, taking into account the initial deposit, interest rate, and any additional deposits or withdrawals.

2. How is a difference equation used to calculate interest?

A difference equation is used to calculate interest by taking the previous balance in a bank account, multiplying it by the interest rate, and adding that amount to the balance. This process is repeated for each time period (e.g. month, year) to track the growth of the account over time.

3. What factors affect the growth of a bank account according to a difference equation?

The growth of a bank account according to a difference equation is affected by several factors, including the initial deposit, interest rate, compounding frequency, and any additional deposits or withdrawals. These variables can be adjusted to see how they impact the growth of the account over time.

4. Can a difference equation accurately predict the growth of a bank account?

A difference equation can provide a good estimate of the growth of a bank account, but it may not be completely accurate. Factors such as changes in interest rates or unexpected deposits or withdrawals can affect the actual growth of the account. However, a difference equation can give a good approximation of the growth over a certain period of time.

5. How can understanding a difference equation be useful for managing personal finances?

Understanding a difference equation can be useful for managing personal finances because it allows individuals to see how different factors, such as interest rates and compounding frequency, can impact the growth of their bank account. This knowledge can help individuals make informed decisions about where to save their money and how to maximize their savings over time.

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