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# Annuities – Present Value and Future Value

An annuity is defined as a stream of payments made over time. An annuity is typically an investment in which one party puts money in with the promise of the other paying it back. There are several categories of annuities:

• The time between payments doesn't vary, the interest rate stays the same, and the amount of the payments is always the same. You know what you are getting with a fixed annuity.

• If any of the time, interest rate, or payment amounts are not fixed, it becomes a variable annuity. Variable annuities sold as investments are subject to securities regulations.

• A specialized variable annuity where interest/investment returns are indexed to equities (stock market).

Each of these categories of annuities can come in two flavors – ordinary, and annuity due:

• Ordinarily, annuity payments are due at the end of each period, so we call those an ordinary annuity.
• Sometimes payments are due at the start of each period and we call those an annuity due. Lease payments usually work like an annuity due.

We said an annuity is usually an investment where one party puts money in with the promise of the other paying it back. The time when money is going into the annuity is the accumulation phase. The money comes back out during the distribution phase.

Either phase could be a single payment, and there may or may not be much time between the last payment in and the first payment out. If either phase is more than a single payment, an annuity may exist. (If each is a single payment, there is no annuity, and you can calculate present value or future value of a lump sum.)

Figuring the present value or future value of a series of payments (annuity) can be done just like figuring PV or FV of a single amount, but doing it again and again for each payment and adding them together. That works, but it is cumbersome. Some math genius figured out a formula for doing it all at once for fixed annuities.

To calculate the present value for an ordinary fixed annuity (payment and interest rate don't change during life of annuity), there are four variables. With any three we can solve for the fourth:

• PV(OA), or Present Value of Ordinary Annuity: the value of the annuity at time t=0
• PMT: Payment amount (value) of the individual payments in each period
• i: interest rate compounded for each period of time
• n: number of payment periods

PV(OA) = (PMT/i) · [1 – (1 / (1 + i)n)]

The difference between an ordinary annuity (above) and an annuity due, is the annuity due had the payment at the beginning of each period, so it should get one more period of compounding than an ordinary annuity. All you have to do to get the PV of an annuity due is multiply the above equation by (1 +i ) to calculate the value for one period sooner.

PV(AD) = PV(OA)  ·  (1 + i)

As alternatives to these formulas, tables in the back of finance textbooks provide factors for calculating present and future values of annuities and single amounts. Also, financial calculators and electronic spreadsheets include financial functions and allow for entering the three variables you know and solving for the fourth.

To calculate the future value for an ordinary fixed annuity (payment and interest rate don't change during life of annuity), there are four variables. With any three we can solve for the fourth:

• FV(OA), or Future Value of Ordinary Annuity: the value of the annuity at

time t=n

• PMT: Payment amount (value) of the individual payments in each period
• i: interest rate compounded for each period of time
• n: number of payment periods

FV(OA) = PMT · [((1 + i)n – 1) / i ]

The difference between an ordinary annuity (above) and an annuity due, is the annuity due had the payment at the beginning of each period, so it should get one more period of compounding than an ordinary annuity. All you have to do to get the FV of an annuity due is multiply the above equation by (1 +i ) to calculate the value for one period sooner.

FV(AD) = FV(OA)  ·  (1 + i)

As alternatives to these formulas, tables in the back of finance textbooks provide factors for calculating present and future values of annuities and single amounts. Also, financial calculators and electronic spreadsheets include financial functions and allow for entering the three variables you know and solving for the fourth.

On Marion's 35th birthday, her insurance company told her she is expected to live until age 85. She wants to retire at age 60. Many of her expenses will be eliminated by then, so she estimates she will only need 15,000 per year to live comfortably.

Marion has a family history of disease, so she plans to have home health care starting at age 70 which will cost 45,000 per year.

She doesn't want to outlive her income, so she allows another 3 years of life beyond the actuary's estimate.

She wants 40,000 to be left to cover her final expenses, including cremation.

Long term interest rates suggest that her opportunity cost of cash approximates the 20-year treasury bond rate of 8% per annum.

She has not started saving yet, but wants to start right away.

1. How much money does she need to have when she retires to achieve her goals?
2. How much money does she need to save each year from now until the time she retires in order to have enough money when she retires to achieve her goals?

This problem tests your understanding of present and future values of sums and annuities, not your ability to do financial planning, since it ignores things like inflation.

With present and future value problems we need to understand the stages of accumulation and distribution. (You may want to draw a timeline to make the problem easier to visualize.) This one starts with an accumulation phase starting now and continuing for 25 years until age 60. Then the distribution phase kicks in, with distributions continuing for 28 years, with an increase along the way, and then a final distribution.

Marion has given us several goals to include in the solution. It is possible to solve for each part separately and have annual savings goals for each. Instead, we are going to first determine how much Marion will need to have accumulated at retirement (question 1), and then calculate the accumulation phase only once, to meet that (question 2).

To calculate the amounts each part will require be available at Marion's retirement, I set up a separate section for:

• ordinary retirement income
• home health care
• final expenses

Then we come back together, combine these requirements, and figure the savings requirement to solve the two problems.

Marion says she needs 15,000 annually from age 60 through age 88. That will be 28 years of 15,000 payments and we will use her 8% interest factor.

The Excel formula for present value of an annuity looks like this:=PV(0.08,28,-15000)

=165,766.18 required at Age 60

Category: Annuity