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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 \(\$ 40,000\) 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; and he will then receive 24 additional annual payments. Annual inflation is expected to be 5 percent. He currently has \(\$ 100,000\) saved, and he expects to earn 8 percent 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?

Short Answer

Expert verified
He needs to save approximately $8,956 annually.

Step by step solution

01

Calculate the Real Value of Retirement Income

First, we need to find the amount necessary to have a future 'real value' equivalent to \(40,000 today, considering 5% inflation for 10 years. This will give us the fixed annual retirement income.The formula for future value with cumulative inflation is:\[ FV = PV \times (1 + r)^n \]Where:- \( FV \) is the future value- \( PV \) is the present value, i.e., \)40,000- \( r \) is the inflation rate (5% or 0.05)- \( n \) is the number of years (10)Plug the values into the formula:\[ FV = 40,000 \times (1 + 0.05)^{10} \]Calculate to get the required retirement income in 10 years.
02

Calculate the Present Value of the Retirement Annuity Needed

The problem specifies your father needs the annuity payments for 25 years after retirement. We need to calculate the present value of this annuity (in year 10) assuming he withdraws them over 25 years, starting from his retirement year.Use the present value of annuity formula:\[ PV_{A} = PMT \times \left(1 - \frac{1}{(1 + r)^n}\right) / r \]Assuming the future value (\( FV = 65,155.30 \)) we calculated from the previous step as annual payment \( PMT \): - \( r = 0.08 \)- \( n = 25 \)This will tell us the amount of money he needs at the start of retirement to sustain those annuity payments.
03

Calculate Future Value Required at Start of Retirement

Next, we need to find how much money your father needs at the start of retirement for the present value of the annual payments we calculated.Use the future value formula:\[ FV = PV \times (1 + r)^n \]Set \( PV \) as the amount from the present value of the annuity we calculated in Step 2, \( r = 0.08 \) and \( n = 10 \).Solve this to check how much total savings is required at the retirement date.
04

Determine Additional Savings Needed Each Year

Your father has saved $100,000 which will also grow at 8% for 10 years. Calculate its future value:\[ FV = 100,000 \times (1 + 0.08)^{10} \]Subtract this future value from the amount needed at the start of retirement (the result from Step 3) to determine the additional amount needed.
05

Calculate Annual Saving Requirement

Now we need to determine how much he needs to save each year to meet this additional amount. Use the future value of annuity formula:\[ FV_A = PMT \times \frac{(1 + r)^n - 1}{r} \]Set \( FV_A \) as the additional amount (from Step 4), \( r = 0.08 \), and \( n = 10 \).Solve for \( PMT \) to find the annual payment required to meet the goal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Retirement Planning
Retirement planning is all about preparing financially for the years when you will no longer be earning a regular income. It's like setting a roadmap for maintaining your lifestyle post-retirement. It involves understanding your future financial needs and taking necessary steps to meet those needs.
When planning for retirement, consider whether you'll have additional sources of income, such as pensions or Social Security, and ensure your savings can cover any gaps. Keep in mind factors like longevity, health care costs, and desired retirement lifestyle. These factors influence how much you need to save and how those savings should be invested.
The key is to start early, develop a savings strategy, and consistently monitor your financial plan. Adjust your savings goals as your financial situation and retirement aspirations evolve. A well-thought-out retirement plan ensures that you'll have enough funds to support yourself throughout your retirement years.
Annuities
Annuities play a significant role in retirement planning, acting as a financial product that provides a steady income stream in retirement. They are particularly useful for ensuring you do not outlive your resources.
Annuities come in various forms, including fixed and variable, each with its unique features. A fixed annuity offers a predetermined payout, providing certainty and stability. Variable annuities, on the other hand, fluctuate based on the performance of the invested assets and carry higher risk and potential reward.
Choosing the right annuity depends on your financial needs and risk tolerance. A well-structured annuity can help bridge any retirement income shortfalls and provide financial peace of mind. For instance, in the problem, the father aims to create an annuity that maintains purchasing power for 25 years, ensuring consistent income throughout his retirement.
Inflation Adjustment
Inflation adjustment is crucial in retirement planning and involves estimating how the value of money will diminish over time. Inflation erodes purchasing power, meaning that what costs $40,000 today will cost more in the future.
To ensure that your savings align with potential increases in costs, it’s important to calculate how much your retirement fund will need to grow to maintain its power. This is done by adjusting your estimates using the expected inflation rate.
In our exercise, the annual inflation rate is projected at 5%. This means retirement income needs to grow each year to allow the retiree to purchase the same goods and services as they do now. Factoring in inflation helps in setting realistic savings goals and ensuring that your retirement plan remains robust over time.
Future Value Calculation
Future value calculation is a financial principle used to determine how much present money will be worth in the future. It's crucial for understanding how investments grow over time and for planning how much you need to save today to meet future financial goals.
To calculate the future value, you take the current amount and apply an expected rate of return over a chosen period. The formula: \[FV = PV \times (1 + r)^n\] allows you to find the amount you will have in the future, accounting for compound interest or growth rates.
In the context of our problem, this formula is used multiple times to figure out how much current savings will grow and how much needs to be saved annually to meet the father’s retirement goal. The calculations help to make informed decisions about saving and investing sufficient amounts each year until retirement is reached.

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Most popular questions from this chapter

Nonannual compounding a. You plan to make 5 deposits of \(\$ 1,000\) each, one every 6 months, with the first payment being made in 6 months. You will then make no more deposits. If the bank pays 4 percent nominal interest, compounded semiannually, how much would be in your account after 3 years? b. One year from today you must make a payment of \(\$ 10,000\). To prepare for this payment, you plan to make 2 equal quarterly deposits, in 3 and 6 months, in a bank that pays 4 percent nominal interest, compounded quarterly. How large must each of the 2 payments be?

Effective rate of interest Find the interest rates earned on each of the following: a. You borrore \(\$ 700\) and promise to pay back \(\$ 749\) at the end of 1 year. b. You lend \(\$ 700\) and the borrower promises to pay you \(\$ 749\) at the end of 1 year. c. You borrow \(\$ 85,000\) and promise to pay back \(\$ 201,229\) at the end of 10 years. d. You borrow \(\$ 9,000\) and promise to make payments of \(\$ 2,684.80\) at the end of each year for 5 years.

Time for a lump sum to double How long will it take \(\$ 200\) to double if it earns the following rates? Compounding occurs once a year. a. \(\quad\) 7 percent. b. 10 percent. c. 18 percent. d. 100 percent.

Future value: annuity versus annuity due What's the future value of a 7 percent, 5 -year ordinary annuity that pays \(\$ 300\) each year? If this were an annuity due, what would its future value be?

Time value of money analysis You have applied for a job with a local bank. As part of its evaluation process, you must take an examination on time value of money analysis covering the following questions. a. Draw time lines for (1) a \(\$ 100\) lump sum cash flow at the end of Year 2,(2) an ordinary annuity of \(\$ 100\) per year for 3 years, and (3) an uneven cash flow stream of \(-\$ 50, \$ 100, \$ 75,\) and \(\$ 50\) at the end of Years 0 through 3. b. (1) What's the future value of \(\$ 100\) after 3 years if it earns 10 percent, annual compounding? (2) What's the present value of \(\$ 100\) to be received in 3 years if the interest rate is 10 percent, annual compounding? c. What annual interest rate would cause \(\$ 100\) to grow to \(\$ 125.97\) in 3 years? d. If a company's sales are growing at a rate of 20 percent annually, how long will it take sales to double? e. What's the difference between an ordinary annuity and an annuity due? What type of annuity is shown here? How would you change it to the other type of annuity? $$\begin{array}{cccc} 0 & 1 & 2 & 3 \\ \hline 1 & 100 & \$ 100 & \$ 100 \end{array}$$ f. (1) What is the future value of a 3 -year, \(\$ 100\) ordinary annuity if the annual interest rate is 10 percent? (2) What is its present value? (3) What would the future and present values be if it were an annuity due? 8\. A 5-year \$100 ordinary annuity has an annual interest rate of 10 percent. (1) What is its present value? (2) What would the present value be if it was a 10 -year annuity? (3) What would the present value be if it was a 25 -year annuity? (4) What would the present value be if this was a perpetuity? h. \(\quad\) A 20 -year-old student wants to save \(\$ 3\) a day for her retirement. Every day she places \(\$ 3\) in a drawer. At the end of each year, she invests the accumulated savings \((\$ 1,095)\) in a brokerage account with an expected annual return of 12 percent. (1) If she keeps saving in this manner, how much will she have accumulated at age \(65 ?\) (2) If a 40 -year-old investor began saving in this manner, how much would he have at age \(65 ?\) (3) How much would the 40 -year-old investor have to save each year to accumulate the same amount at 65 as the 20 -year-old investor? i. What is the present value of the following uneven cash flow stream? The annual interest rate is 10 percent. $$\begin{array}{ccccc} 0 & 1 & 2 & 3 & 4 \\ \hline 1 & \$ 100 & \$ 300 & \$ 300 & -\$ 50 \end{array}$$ j. (1) Will the future value be langer or smaller if we compound an initial a mount more offen than annually for example, semianmually, holding the stated (nominal) rate constant? Why (2) Define (a) the stated, or quoted, or nominal, rate, \((b)\) the periodic rate, and (c) the effective annual rate EAR or FFF (3) What is the EAR corresponding to a nominal rate of 10 percent compounded semiannually? Commn pounded quarterly? Compounded daily? (4) What is the future value of \(\$ 100\) after 3 years under 10 percent semiannual compounding? Quartelly compounding? k. When will the FAR equal the nominal ( quoted) rate?

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