Question

Assume that you have just turned 21, are graduating from college, and are planning for your retirement at age 55. You currently have no money Saved, but plan to make significant investments into a retirement account now that you have gotten a high-paying job. Because of moving and additional expenses associated with the start of your new job, you believe that you will only be able to invest $2,000 on your 22nd and 23rd birthdays (2 payments). You then expect to invest $10,000 each year on your 24th through your 30th birthdays (7 payments) $20,000 each year on your 31" through 40th birthdays (10 payments)and $30,000 each year on your 41st through 55 birthdays (15 payments). During this 34-year period you are willing to take some investment risks and you believe that your investment account can earn a nominal annual rate of return of 9 percent, compounded monthly. At age 55 you plan to retire and will use the money in your investment account to buy a 40-year, guaranteed annuity from an insurance company that will pay you a fixed amount on your 56th through 95th birthdays. Since this annuity is guaranteed, the insurance company uses a nominal annual rate of return of 6 percent, compounded quarterly. Given this information, determine the amount you can expect to receive each year after you retire. Enter your answer without symbols, to the nearest cent. For example, if your answer is $11.467.40, enter "11467.40"

Answer 1

The amounts saved are $2000 - two payments

$10000-next seven payments

$20000-next ten payments

$30000-next fifteen payments

9% compounded monthly is equivalent to (1+0.09/12)^12 -1 =0.0938 =9.38% compounded annually

So, the future value of your payments i..e the balance in your retirement account on 55th birthday (day of retirement) is

=(2000*1.0938^33 + 2000*1.0938^32)+ (10000*1.0938^31 +10000*1.0938^30 +....+10000*1.0938^25) + (20000*1.0938^24+.... +20000*1.0938^15)+(30000*1.0938^14+....+30000)

This can be rearranged in the form of annuities of $2000 till 34years plus annuity of $8000 till 32 years, annuity of $10000 till 25 years and an annuity of $20000 till 15 years

=(2000*1.0938^33 + ... +2000)+ (8000 * 1.0938^31+....+8000)+(10000 * 1.0938^24+....+10000)+(20000 * 1.0938^14+....+20000)

=2000/0.0938 * (1.0938^34-1) + 8000/0.0938*(1.0938^32-1) +10000/0.0938*(1.0938^25-1) +20000/0.0938*(1.0938^15-1)  

=$ 3,347,374

This amount is used to buy the 40 year annuity

As the interest rate in the post retirement period is 6% compounded quarterly,it is equivalent to (1+0.06/4)^4 -1 =0.061364=6.1364% compounded annually

The Value of withdrawals as on retirement day should be equal to the retirement balance of $ 3,347,374

3347374 = A/ 0.061364 * (1- (1/1.061364)^40)

where A is the annual withdrawal

So, A = 205406.74/ 0.907651 = $226305.91

So, an amount of 226305.91 is expected to be received each year after retirement.