Annuities – Present Value and Future Value

Posted by Professor Cram in Time Value of Money

Annuities – a stream of payments

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:

  • Fixed Annuity

    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.

  • Variable 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.

  • Equity Indexed Annuity

    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.

Annuity phases: Accumulation and Distribution

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.

Present Value of Annuity with Fixed Payments for n periods (Ordinary or Annuity Due)

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.

Future Value of Annuity with Fixed Payments for n periods (Ordinary or Annuity Due)

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.

Annuities – An example homework problem

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?

Annuities – laying out the example homework problem

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.

Timeline for annuity problem

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.

Present Value of an Annuity: solving for retirement income

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

We will add this sum to her other requirements later to determine her total required savings.

Present Value of an Annuity: solving for home health care

Marion needs 45,000 additional annually from age 70 through age 88. We will first solve for how much Marion will need when that distribution starts.

This is a "Present Value of an Annuity" problem. There will be 18 periods, 8% interest, and 45,000 annual payments.

The Excel formula looks like this:
=PV(0.08,18,-45000)   and it resolves to
=421,734.92 required at Age 70 (when this distribution begins)

To find the PV of that amount at Age 60, we discount it back 10 years by taking the Present Value of a sum. Again, I use Excel, and the formula looks like this:
=PV(0.08,10,0,-421734.92)
=195,344.87 required at Age 60, which we will combine with the other values later.

Present Value of an Annuity: solving for final expenses

Marion has a fixed amount (40,000 for final expenses) she needs at age 88.

This is a straightforward "present value of a sum" problem. I am using the =PV financial function in an Excel spreadsheet to calculate the the value needed at retirement (age 60, which is 28 years before she expects to die.)

I use her 8% interest rate, 0 payment amounts, start with a present value of 0, since she hasn't started saving for this yet, and the future value of 40000 since that is the goal.

The format in Excel looks like this: 

=PV(i,n,Payment,FV)   so my input is:

=PV(0.08,28,0,-40,000)  which results in:

=4,646.55 required at Age 60, which we will combine with the other values in the next section.

Present Value of an Annuity: solving for annuity value

If we combine all of the required amounts at age 60, we get:

   4,646.55 for final expenses
 165,766.18 for ordinary retirement income
 195,344.87 for home health care
+__________
 365,747.60 accumulation required at age 60

Present Value of an Annuity: solving for annuity payment

Now we need to know the annual saving required to accumulate 365,747.60. This time we know the future value and are solving for the Payment.

There are 25 years to save, and we still use the 8% interest rate. The Excel formula for solving for a payment is: =PMT(int,n,pv,fv).

We will use 0 as the present value since she hasn't started saving yet. Here is my entry:

=PMT(0.08,25,0,365747.60) which equals:
=-5,002.98

She needs to save 5,002.98 each year until she retires.

Annuity Example – recap of annuity payments and value

We just figured out that Marion needs to save $5,002.98 per year for 25 years. Thus, she is paying $125,074.50 for her retirement plan.

Since she will receive 8% interest all along the way (good luck with that, Marion) she will really have $365,747.60 set aside when she retires (at least that is the goal we calculated). That means she will make $240,673.10 in interest by the time she retires. That is more than she will put in herself!

Let's see how much she is going to receive from the $365,747.60 she is saving. She expects to receive retirement income of $15,000 per year plus another $45,000 per year for home health care for 18 years, and still have $40,000 left for final expenses.

  • $15,000 per year x 28 years =  $420,000
  • $45,000 per year x 18 years =  $810,000
  • And she willl provide $40,000 for final expenses.

She will get $1,270,000 for investing $125,000.

If she had more years to invest, it would work even better. I love compounding. Are you ready to start your retirement fund now?

Annuities resources

  • Future Value of Annuity Due – Calculate the Future Value of an Annuity Due with a step by step example using your values for the periodic interest rate, number of periods, and periodic payment amount.
  • Future Value of an Ordinary Annuity – Calculate the Future Value of an Ordinary Annuity with a step by step example using your values for the periodic interest rate, number of periods, and periodic payment amount.
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Comments

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    Tremendous Blog here on the finance sector. I love reading blogs that have to do with business and finance, so thank you for keeping us up to date with your blog! Ill be coming back!

  2. name removed says:

    Can you help me solve this problem?

    If I save 2,820 per year and earn 1,150 interest year in 25 year (that mean 1st year I save RM2,820 and earn 1,150 interest, 2nd year i also save 2,820 and earn 1,150 interest and 3rd year until 25 years also same). At the end of 25 years, i can take back 113,007. What is the interest factor? Please show the formula to me. Thank a lot.

  3. Ali Rehman says:

    fv/pv=(1+i)^n
    113007/2820=(1+i)^25
    40.07=(1+i)^25
    40.07^(1/25)=1+i
    40.07^0.04=1+i
    1.159=1+i
    1.159-1=i
    0.159=i
    0.159*100=i
    15.90 or 16%=i

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