Find the maximum amount of the equal annual withdrawal

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In summary, P25,000 is deposited in a savings account that pays 5% interest, compounded semi-annually. Equal annual withdrawals are to be made from the account, beginning one year from now and continuing forever. The maximum amount of the equal annual withdrawal is closest to P1625.
  • #1
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P25,000 is deposited in a savings account that pays 5% interest, compounded semi-annually. Equal annual withdrawals are to be made from the account, beginning one year from now and continuing forever. The maximum amount of the equal annual withdrawal is closest to
A. P625
B. P1000
C. P1250
D. P1625
 
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  • #2
Hello, and welcome to MHB! :)

I would let \(A_n\) be the amount in the account at the end of year \(n\), and \(W\) be the amount withdrawn at the end of year \(n\). Then we may model this problem with the following recursion:

\(\displaystyle A_{n+1}=1.05A_{n}-W\)

Can you give the homogeneous solution (based on the root of the characteristic equation)?
 
  • #3
I made an error in my above post, where I used annual compounding. We want:

\(\displaystyle A_{n+1}=1.025^2A_{n}-W=\left(\frac{41}{40}\right)^2A_{n}-W\)

And so our homogeneous solution will take the form:

\(\displaystyle h_n=c_1\left(\frac{41}{40}\right)^{2n}\)

Our particular solution will take the form:

\(\displaystyle p_n=B\)

Substitution into the recursion yields:

\(\displaystyle B-\left(\frac{41}{40}\right)^2B=-W\)

\(\displaystyle \left(\frac{9}{40}\right)^2B=W\implies B=\left(\frac{40}{9}\right)^2W\)

And so by the principle of superposition, we obtain:

\(\displaystyle A_{n}=h_n+p_n=c_1\left(\frac{41}{40}\right)^{2n}+\left(\frac{40}{9}\right)^2W\)

\(\displaystyle A_0=c_1+\left(\frac{40}{9}\right)^2W\implies c_1=A_0-\left(\frac{40}{9}\right)^2W\)

And so our closed form is:

\(\displaystyle A_{n}=\left(A_0-\left(\frac{40}{9}\right)^2W\right)\left(\frac{41}{40}\right)^{2n}+\left(\frac{40}{9}\right)^2W\)

To finish the problem, we want to set:

\(\displaystyle \lim_{n\to\infty}A_n=0\)

In order for that to happen (for the limit to be finite), we need:

\(\displaystyle W=\left(\frac{9}{40}\right)^2A_0\)

Using the value \(A_0=25000\), we then find:

\(\displaystyle W=\frac{10125}{8}=1265.625\)
 
  • #4
MarkFL said:
I made an error in my above post, where I used annual compounding. We want:

\(\displaystyle A_{n+1}=1.025^2A_{n}-W=\left(\frac{41}{40}\right)^2A_{n}-W\)

And so our homogeneous solution will take the form:

\(\displaystyle h_n=c_1\left(\frac{41}{40}\right)^{2n}\)

Our particular solution will take the form:

\(\displaystyle p_n=B\)

Substitution into the recursion yields:

\(\displaystyle B-\left(\frac{41}{40}\right)^2B=-W\)

\(\displaystyle \left(\frac{9}{40}\right)^2B=W\implies B=\left(\frac{40}{9}\right)^2W\)

And so by the principle of superposition, we obtain:

\(\displaystyle A_{n}=h_n+p_n=c_1\left(\frac{41}{40}\right)^{2n}+\left(\frac{40}{9}\right)^2W\)

\(\displaystyle A_0=c_1+\left(\frac{40}{9}\right)^2W\implies c_1=A_0-\left(\frac{40}{9}\right)^2W\)

And so our closed form is:

\(\displaystyle A_{n}=\left(A_0-\left(\frac{40}{9}\right)^2W\right)\left(\frac{41}{40}\right)^{2n}+\left(\frac{40}{9}\right)^2W\)

To finish the problem, we want to set:

\(\displaystyle \lim_{n\to\infty}A_n=0\)

In order for that to happen (for the limit to be finite), we need:

\(\displaystyle W=\left(\frac{9}{40}\right)^2A_0\)

Using the value \(A_0=25000\), we then find:

\(\displaystyle W=\frac{10125}{8}=1265.625\)
Tnx Sir
 

1. What is the concept of "Find the maximum amount of the equal annual withdrawal"?

"Find the maximum amount of the equal annual withdrawal" is a financial concept that refers to determining the largest possible amount of money that can be withdrawn each year from a given investment or retirement account while still maintaining the account's balance. This calculation takes into account factors such as the initial investment, interest rates, and the desired withdrawal period.

2. How is the maximum amount of the equal annual withdrawal calculated?

The maximum amount of the equal annual withdrawal is calculated using a formula that takes into account the present value of the investment, the expected interest rate, and the desired withdrawal period. This formula is known as the "annuity payment formula" and is commonly used in financial planning and retirement calculations.

3. What factors affect the maximum amount of the equal annual withdrawal?

The maximum amount of the equal annual withdrawal is affected by several factors, including the initial investment amount, the interest rate, the length of the withdrawal period, and any fees or taxes associated with the investment. Additionally, the type of investment or retirement account may also impact the calculation.

4. How can I use the concept of "Find the maximum amount of the equal annual withdrawal" in my financial planning?

The concept of "Find the maximum amount of the equal annual withdrawal" can be used in financial planning to help determine how much money can be safely withdrawn from an investment or retirement account each year without depleting the account balance. This can be useful in creating a sustainable retirement income plan or in determining the feasibility of early retirement.

5. Are there any limitations to using the maximum amount of the equal annual withdrawal calculation?

Yes, there are some limitations to using the maximum amount of the equal annual withdrawal calculation. This calculation assumes a consistent interest rate and does not account for fluctuations in the market or unexpected expenses. Additionally, it may not be applicable to all types of investments or retirement accounts. It is important to consult with a financial advisor and regularly review and adjust your financial plan to ensure its effectiveness.

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