QuestionsCategory: Business MathematicsA 52-year-old client asks an accountant how to plan for his future retirement at age 62. He expects income from Social Security in the amount of \$21,600 per year and a retirement pension of \$40,500 per year from his employer.
A 52-year-old client asks an accountant how to plan for his future retirement at age 62. He expects income from Social Security in the amount of \$21,600 per year and a retirement pension of \$40,500 per year from his employer. He wants to make monthly contributions to an investment plan that pays 8%, compounded monthly, for 10 years so that he will have a total income of \$83,700 per year for 30 years. What will the size of the monthly contributions have to be to accomplish this goal, if it is assumed that money will be worth 8%, compounded continuously, throughout the period after he is 62? To help you answer this question, complete the following.
1. How much money must the client withdraw annually from his investment plan during his retirement so that his total income goal is met?

1. How much money S must the client's account contain when he is 62 so that it will generate this annual amount for 30 years? (Him: S can be considered the present value over 30 years of a continuous income stream with the amount you found in Question 1 as its annual rate of flow.)

1. The monthly contribution R that would, after 10 years, amount to the present value S found in Question 2 can be obtained from the formula:
2. R = S / ((1 + i/12)^(12*n) - 1) / (i/12)
where i represents the monthly interest rate and n the number of months. Find the client's monthly contribution, R.

1.  To determine how much the client must withdraw annually from his investment plan during retirement, we need to subtract his expected income from Social Security and retirement pension from his total income goal of \$83,700 per year:

Total income goal - Social Security income - Retirement pension

= \$83,700 - \$21,600 - \$40,500 = \$21,600 per year

Therefore, the client must withdraw \$21,600 per year from his investment plan during retirement to meet his total                           income goal.

1. To calculate the amount S that the client's account must contain when he is 62 so that it will generate an annual income of \$21,600 for 30 years, we can use the present value formula for a continuous income stream:

S = R/i * (1 - e^(-i*n))

Where R is the annual income, i is the continuous interest rate, and n is the number of years. In this case,

R = \$21,600, i = 0.08 (the annual interest rate of 8% compounded continuously), and n = 30. Substituting in these                       values,       we get:

S = \$21,600 / 0.08 * (1 - e^(-0.08*30))

= \$240,909.09

Therefore, the client's account must contain \$240,909.09 when he is 62 in order to generate an annual income of \$21,600 for 30 years.

1. To find the monthly contribution R that would, after 10 years, amount to the present value S found in Question 2, we can use the formula:

R = S / ((1 + i/12)^(12*n) - 1) / (i/12)

where i is the monthly interest rate (0.08/12) and n is the number of years (10). Substituting in these values, we get:

R = \$240,909.09 / ((1 + 0.08/12)^(12*10) - 1) / (0.08/12)

= \$1,326.98

Therefore, the client must make monthly contributions of \$1,326.98 to his investment plan for 10 years in order to have \$240,909.09 when he is 62 and generate an annual income of \$21,600 for 30 years.

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