Section 11.6 Ordinary Annuities, Sinking Funds, and Retirement Investments

What You Will Learn Ordinary Annuities Sinking Funds Variable Annuities Immediate Annuities IRA’s 401ks and 403bs

Annuity An annuity is an account into which, or out of which, a sequence of scheduled payments is made.

Ordinary Annuity An annuity into which equal payments are made at regular intervals, with the interest compounded at the end of each interval and with a fixed interest rate for each compounding period, is called an ordinary annuity or fixed annuity.

Ordinary Annuity The amount of money that is present in an ordinary annuity after t years is known as the accumulated amount or the future value of an annuity. To determine the accumulated amount of an ordinary annuity, we could use a computer to generate a spreadsheet, as on the next slide.

Ordinary Annuity 1st half of table

Ordinary Annuity 2nd half of table

Ordinary Annuity Another way, and perhaps a more efficient way, to calculate the accumulated amount in an ordinary annuity is to use the ordinary annuity formula on the next slide.

Ordinary Annuity Formula The accumulated amount, A, of an ordinary annuity with payments of p dollars made n times per year, for t years, at interest rate, r, compounded at the end of each payment period is given by the formula

Example 1: Using the Ordinary Annuity Formula Bill and Megan Lutes are depositing $250 each quarter in an ordinary annuity that pays 4% interest compounded quarterly. Determine the accumulated amount in this annuity after 35 years.

Example 1: Using the Ordinary Annuity Formula Solution p = $250, r = 0.04, n = 4, t = 35.

Example 1: Using the Ordinary Annuity Formula There will be about $75,677.48 in Bill and Megan’s annuity after 35 years.

Timely Tip An ordinary annuity is used when you wish to determine the accumulated amount obtained over n years when you contribute a fixed amount each period. A sinking fund is used when you wish to determine how much money an investor must invest each period to reach an accumulated amount at a specific time.

Sinking Fund A sinking fund is a type of annuity in which the goal is to save a specific amount of money in a specific amount of time.

Sinking Fund Payment Formula In the formula, p is the payment needed to reach the accumulated amount, A. Payments are made n times per year, for t years, into a sinking fund with interest rate r, compounded n times per year.

Example 2: Using Sinking Fund Payment Formula Abigail and John Clayton would like to have $55,000 in 10 years to remodel their home. The Claytons decide to invest monthly in a sinking fund that pays 3.3% interest compounded monthly. How much should the Claytons invest in the sinking fund each month to accumulate $55,000 in 10 years?

Example 2: Using Sinking Fund Payment Formula Solution A = $55,000, r = 0.033, n = 12, t = 10

Example 2: Using Sinking Fund Payment Formula Solution We often round to the nearest cent, or in this case, to $387.48. However, because $387.48 is slightly less than the answer we obtained, we will round our answer up to $387.49 to ensure that the Claytons reach their goal of $55,000.

Variable Annuities A variable annuity is an annuity that is invested in stocks, bonds, mutual funds, or other investments that do not provide a guaranteed interest rate. The term variable is used because the value of the annuity will vary depending on the performance of the investment options chosen by the investor.

Variable Annuities Like an ordinary annuity, an investor usually invests in a variable annuity by making regular periodic payments. Unlike an ordinary annuity, however, a variable annuity does not have a fixed rate of interest.

Immediate Annuities An immediate annuity is an annuity that is established with a lump sum of money for the purpose of providing the investor with regular, usually monthly, payments for the rest of the investor’s life. In exchange for giving the investment company a lump sum of money, the investor is guaranteed to receive a monthly income for the duration of the investor’s life.

Individual Retirement Accounts Individual retirement accounts, also known as IRAs, are savings accounts that allow individuals to invest up to a certain amount of money each year for the purpose of saving for retirement. IRAs have distinct tax advantages over ordinary savings accounts. There are several different kinds of IRAs, but most can be classified as either a traditional IRA or a Roth IRA.

Individual Retirement Accounts A traditional IRA is an IRA into which pretax money is invested, meaning that any money invested into a traditional IRA is not subject to income taxes. However, when money is withdrawn from a traditional IRA, the money is subject to income taxes.

Individual Retirement Accounts A Roth IRA is an IRA into which post tax money is invested, meaning that the investor has already paid income taxes on the money invested. When money is withdrawn from a Roth IRA after the investor reaches retirement age, the money is not subject to income taxes.

401ks A 401k plan is a retirement savings plan that allows employees of private companies to make contributions of pre-income tax dollars that are then pooled with other employees’ money. These funds are then invested in a variety of stocks, bonds, money markets, and mutual funds.

401ks Although 401k plans share many of the same tax advantages as IRAs, they also have several distinct advantages.

401ks First, employees are generally allowed to invest more money annually in a 401k plan than in an IRA. Often, employees are eligible to invest up to 15% of their annual income into a 401k plan. A second advantage is that some employers will match employee contributions up to a certain limit.

401ks A Roth 401k plan is a retirement savings plan that differs from a 401k plan in that an investor first pays income tax on the money used to invest in a Roth 401k plan. Like a Roth IRA, money withdrawn from a Roth 401k plan after the individual reaches retirement age is not subject to income tax.

403bs A 403b plan is a retirement plan similar to a 401k plan, but it is available only to employees of schools, civil governments, hospitals, charities, and other not-for-profit organizations. A 403b plan shares many of the same advantages as 401k plans. One major difference, though, is that a 403b plan is not necessarily established by the employer as is a 401k plan.

403bs Rather, a 403b plan is frequently established independently by the employee with a representative of an investment or insurance company.

Summary As you can see from this brief discussion of annuities, IRAs, 401k plans, and 403b plans, many important details can affect your decisions as you begin to invest for retirement. For more information, and for other types of retirement accounts not mentioned here, consult a financial advisor or visit a library or Internet Web site.

Summary Regardless of the source of your information, it is very important to do your own research and ask many questions prior to investing.