Question

40.  Problem 5.40 (Required Annuity Payments)

A father is now planning a savings program to put his daughter through college. She is 13, plans to enroll at the university in 5 years, and she should graduate 4 years later. Currently, the annual cost (for everything - food, clothing, tuition, books, transportation, and so forth) is $12,000, but these costs are expected to increase by 5% annually. The college requires total payment at the start of the year. She now has $7,000 in a college savings account that pays 7% annually. Her father will make six equal annual deposits into her account; the first deposit today and sixth on the day she starts college. How large must each of the six payments be? (Hint: Calculate the cost (inflated at 5%) for each year of college and find the total present value of those costs, discounted at 7%, as of the day she enters college. Then find the compounded value of her initial $7,000 on that same day. The difference between the PV of costs and the amount that would be in the savings account must be made up by the father's deposits, so find the six equal payments that will compound to the required amount.) Do not round intermediate calculations. Round your answer to the nearest dollar. 39.  Problem 5.39 (Required Annuity Payments Your father is 50 years old and will retire in 10 years. He expects to live for 25 years after he retires, until he is 85. He wants a fixed retirement income that has the same purchasing power at the time he retires as $45,000 has today. (The real value of his retirement income will decline annually after he retires.) His retirement income will begin the day he retires, 10 years from today, at which time he will receive 24 additional annual payments. Annual inflation is expected to be 6%. He currently has $130,000 saved, and he expects to earn 9% annually on his savings. How much must he save during each of the next 10 years (end-of-year deposits) to meet his retirement goal? Do not round intermediate calculations. Round your answer to the nearest dollar. $

Answer 1

1) Current Annual Cost = 12,000 | Savings Account Balance = 7,000 | Inflation rate or Growth rate of costs = 5%

Rate of return on savings account = 7%

At Age 13, Annual Cost = 12,000

At Age 18, First year of College, Annual cost = Current Annual cost * (1+G)⁵ = 12000 * (1+5%)⁵ = 15,315.38

At Age 19, Second year of College, Annual cost = 15,315.38 * (1+5%) = 16,081.15

At Age 20, Third year of College, Annual cost = 16,081.15 * (1+5%) = 16,885.21

At Age 21, Fourth year of College, Annual cost = 16,885.21 * (1+5%) = 17,729.47

Now we will calculate the Present Value all annual costs during college years at the time of joining the college, that is, at Age 18. Therefore, cost determined at First year of college will be Year 0 and will not be discounted. Rate will be 7% for discounting.

PV of First year cost at Joining Year = 15,315.38

PV of Second year cost at Joining Year = 16,081.15 / (1+7%) = 15,029.11

PV of 3rd Year Cost at Joining Year = 16,885.21 / (1+7%)² = 14,748.19

PV of 4th Year cost at Joining Year = 17,729.47 / (1+7%)³ = 14,472.52

Total PV of Annual Cost at the Time of Joining = 15,315.38 + 15,029.11 + 14,748.19 + 14,472.52

Total PV of Annual Cost at the Time of Joining = $ 59,565.21

Now we will find the Value of 7,000 account balance at the time of joining, that is, 5 years from now, using 7% rate of return.

Value of account balance at Joining Year = 7,000 * (1+7%)⁵ = $ 9,817.86

Remaining amount required at the Joining year = 59,565.21 - 9,817.86 = $ 49,747.34

Since her father wants to make equal payment starting from today, it becomes an Annuity-Due. Hence, we will use Future Value of Annuity-Due formula for calculation of equal payments

FV of Annuity Due formula = (PMT / R)*((1+R)^T - 1) * (1+R)

FV = 49,747.34 | PMT = P | R = 7% | T = 5

=> 49,747.34 = (P / 7%) * ((1+7%)⁵ - 1)*(1+7%)

=> P = (49,747.34 * 7%) / (((1+7%)⁵ - )*(1+7%))

Equal Payment = $ 8,084.67 or $ 8,085

Hence, her father should deposit an equal payment of $ 8,085 starting from today to reach the goal.

2) Current Income = 45,000 | Inflation rate = 6% | Rate of return = 9% | Current Savings = 130,000

As father wants same purchasing power as today, then, first cashflow in 10 years should be compounded by the inflation rate for 10 years.

Cashflow at retirement = CF*(1+Inflation rate)^T = 45,000 * (1+6%)¹⁰ = 80,588.15

Since cashflow will continue to increase at inflation rate each year, it will be a growing annuity and as the first payment will be at the time he retires, therefore, it will be a Growing Annuity-Due.

PV of Growing Annuity-Due formula = (PMT/(R-Inflation))*(1-((1+Inflation)/(1+R))^T)*(1+R)

First PMT = Cashflow at retirement = 80,588.15 | R = 9% | Inflation = 6% | T = 25 (as he want to receive 25 paymenst)

PV of Retirement Income at the time of retirement = (80,588.15/(9%-6%))*(1-((1+6%)/(1+9%))²⁵)*(1+9%)

Solving the above equation, we will get the Present Value of all retirement incomes at the time of retirement.

PV of Retirement Income at the time of retirement = $ 1,470,696.97

Now we will calculate the Value of his current savings at the time of retirement to find the remaining amount needed to meet his goal.

Value of current savings at Retirement = 130,000 * (1+9%)¹⁰ = $ 307,757.28

Remaining Amount needed at Retirement = PV of Retirement Income at retirement - Value of savings at Retirement

Remaining Amount needed at Retirement = 1,470,696.97 - 307,757.28 = $ 1,162,939.69

Now that we know the remaining requirement, we can calculate the Year-end equal payments which would be a normal annuity, hence, the formula used. The remaining amount is Future Value, therefore, FV of Annuity will be used.

FV of Annuity formula = (PMT/R)*((1+R)^T - 1)

FV = 1,162,939.69 | R = 9% | T = 10

=> 1,162,939.69 = (PMT/9%)*((1+9%)¹⁰ - 1)

=> PMT = (1,162,939.69 * 9%) / ((1+9%)¹⁰ - 1)

Solving the equation, we will get the equal year-end payments over 10 years which should be made by father.

Year-end Payment = $ 76,544.79 or $ 76,545

Hence, Year-end payment required to meet father's goal is $ 76,545