Suppose that you are 25 years old and you have two options to save money for retirement. Plan A allows you to save $3000 per year for the next 20 years, then sit for 20 years and finally draw on it for the following 30

years. Plan B allows you to wait 20 years and save $20,000 per year from ages 46 through 65 before drawing on the money for the following 30 years.
Suppose that you expect your investment to grow at 10% per year until you reach age 65. After age 65, you plan to draw a constant annual income out of your
account such that it is empty after 30 years of retirement. During this 30-year period, you will keep your money in a less risky portfolio that you expect to grow at 6% per
year.
Which plan will leave you wealthier in your retirement, and how much could you afford to draw out each year during retirement under these two plans?
Finally, suppose that the inflation rate is expected to be 2% per year every year until you are 95. In current dollars, how much could you afford to draw out each year during retirement under the two plans above?

To compare the two retirement plans, we need to calculate the future value of the savings under each plan separately and then determine the amount that can be drawn out each year during retirement.

Let's first calculate the future value of Plan A:

1. From age 25 to 45 (20 years), you save $3,000 per year. Using the future value of an ordinary annuity formula, the future value at age 45 would be:

FV1 = P * ((1 + r)^n - 1) / r
= $3,000 * ((1 + 0.10)^20 - 1) / 0.10
≈ $128,975.09

2. From age 45 to 65 (20 years), your retirement savings sit without additional contributions. The amount remains $128,975.09.

3. From age 65 to 95 (30 years), you draw on the money. Assuming you want to empty the account after 30 years, the annual withdrawal amount can be calculated using the present value of an annuity formula:

PV1 = FV1 / ((1 + r)^n - 1) / r
= $128,975.09 / ((1 + 0.06)^30 - 1) / 0.06
≈ $5,705.19

Now let's calculate the future value of Plan B:

1. From age 25 to 45, you don't save any money.

2. From age 46 to 65, you save $20,000 per year. Using the future value of an ordinary annuity formula, the future value at age 65 would be:

FV2 = P * ((1 + r)^n - 1) / r
= $20,000 * ((1 + 0.10)^20 - 1) / 0.10
≈ $1,356,110.12

3. From age 65 to 95, you draw on the money. Again, assuming you want to empty the account after 30 years, the annual withdrawal amount can be calculated as:

PV2 = FV2 / ((1 + r)^n - 1) / r
= $1,356,110.12 / ((1 + 0.06)^30 - 1) / 0.06
≈ $59,915.79

Therefore, under these assumptions, Plan B would leave you wealthier in your retirement, allowing you to draw around $59,915.79 per year for 30 years, while Plan A only allows for $5,705.19 per year for the same period.

Taking into account the expected inflation rate of 2%, we can adjust the withdrawal amounts in current dollars.

For Plan A: Adjusted annual withdrawal = $5,705.19 / (1 + 0.02)^30 ≈ $3,407.59

For Plan B: Adjusted annual withdrawal = $59,915.79 / (1 + 0.02)^30 ≈ $35,656.38

Therefore, in current dollars, you could afford to draw approximately $3,407.59 per year under Plan A and $35,656.38 per year under Plan B.

To determine which plan will leave you wealthier in retirement and how much you can afford to draw out each year, we need to calculate the future value of the two plans and then analyze the results.

Let's start with Plan A:
1. Calculate the future value of the $3000 annual savings for 20 years growing at a 10% rate:
- Use the future value of an annuity formula: FV = P * [(1 + r)^(n - 1) - 1] / r, where P is the payment, r is the interest rate, and n is the number of periods.
- FV = 3000 * [(1 + 0.10)^(20 - 1) - 1] / 0.10 = $228,935.82

2. Calculate the future value of the $228,935.82 after it sits for 20 years growing at a 6% rate:
- FV = P * (1 + r)^n, where P is the present value, r is the interest rate, and n is the number of periods.
- FV = 228,935.82 * (1 + 0.06)^20 = $792,198.37

3. Calculate the future value of the $792,198.37 over 30 years of retirement drawing a constant income:
- Calculate the value of an ordinary annuity: FV = P * [(1 + r)^n - 1] / r, where P is the payment, r is the interest rate, and n is the number of periods.
- FV = P * [(1 + 0.06)^30 - 1] / 0.06 = $3,649,511.57

Now let's move on to Plan B:
1. Calculate the future value of the $20,000 annual savings for 20 years growing at a 10% rate:
- FV = 20,000 * [(1 + 0.10)^(20 - 1) - 1] / 0.10 = $571,345.38

2. Calculate the future value of the $571,345.38 over 30 years of retirement drawing a constant income:
- FV = P * [(1 + r)^n - 1] / r = P * [(1 + 0.06)^30 - 1] / 0.06 = $2,140,303.00

Comparing the two plans, Plan A results in a higher future value at retirement ($3,649,511.57) compared to Plan B ($2,140,303.00). Therefore, Plan A would leave you wealthier in retirement.

Now, let's calculate how much you can afford to draw out each year during retirement under both plans, considering the inflation rate:

1. For Plan A:
- Adjust the future value of $3,649,511.57 based on the 2% annual inflation rate:
- Adjusted FV = 3,649,511.57 / (1 + 0.02)^30 = $2,057,667.51
- Divide the adjusted future value by the number of retirement years (30) to get the annual income: 2,057,667.51 / 30 = $68,588.92

2. For Plan B:
- Adjust the future value of $2,140,303.00 based on the 2% annual inflation rate:
- Adjusted FV = 2,140,303.00 / (1 + 0.02)^30 = $1,206,235.59
- Divide the adjusted future value by the number of retirement years (30) to get the annual income: 1,206,235.59 / 30 = $40,207.85

Considering the inflation rate, you could afford to draw out approximately $68,588.92 per year during retirement under Plan A and $40,207.85 per year under Plan B in current dollars.