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 20year treasury bond rate of 8% per annum.
She has not started saving yet, but wants to start right away.
 How much money does she need to have when she retires to achieve her goals?
 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.
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.
Comments
33 Responses to “Annuities – Present Value and Future Value”
Leave a Reply
Get More Help!
Click one of these links to get more help from another Cramlet in this same chapter:
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!
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.
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
1.1591=i
0.159=i
0.159*100=i
15.90 or 16%=i
Everyone loves what you guys tend to be up too. This kind
of clever work and reporting! Keep up the very good works guys I’ve added you guys
to blogroll.
Hello there I am so happy I found your web site,
I really found you by accident, while I was browsing on
Yahoo for something else, Regardless I am here now and
would just like to say thank you for a fantastic post and a all round enjoyable
blog (I also love the theme/design), I don’t have time to browse it all at the moment but
I have bookmarked it and also included your RSS feeds,
so when I have time I will be back to read a great deal
more, Please do keep up the fantastic b.
Thanks for the auspicious writeup. It actually was a amusement
account it. Look complicated to far delivered agreeable
from you! By the way, how can we communicate?
My brother recommended I might like this blog.
He was entirely right. This post truly made my day.
You cann’t imagine just how much time I had spent for this info!
Thanks!
Cancer cells deprive normal body cells of most their nutrition and causes anemia.
 If you happen to be using an organization you will require to review their contract to see exactly the kinds of things they are going to cover.
This identical principle could make a massive difference in longterm care.
Some could be large and unintrusive, being that they
can only give you a studio apartment and little supervision, although some could be small , devoted to providing
assistance in most day to day activities.
Nice weblog here! Additionally your website loads up very fast!
What host are you using? Can I get your associate hyperlink to your host?
I wish my site loaded up as quickly as yours lol
Thanks for finally writing about > Annuities – Present Value and Future Value 
Finance: Time Value of Money  CollegeCram.com < Loved it!
Body scrubs are one of the easiest homemade beauty products to make.
The US National Library of Medicine warns against ingesting
this petroleum product. We don’t want to end up old and wrinkled because we
spent every other day in a tanning booth.
Look into my webpage: Top skin care products for men
Yesterday, while I was at work, my sister stole my iphone and
tested to see if it can survive a 25 foot drop, just so
she can be a youtube sensation. My apple ipad is now broken and she has 83 views.
I know this is entirely off topic but I had to share it with someone!
I am in fact grateful to the holder of this web page who has shared this impressive post at here.
But there are also those who are not amenable to this kind of treatment such as Dr.
For example, you might be seeking rhinoplasty as way to achieve a functional goal.
If nothing else, it is more “beach and tan” culture than the West LA So many people seeking cosmetic surgery and there are
many thousands of them who can afford it.
Here is my homepage – facial plastic surgery videos
Finally, mini face lift provide the sufferers an all natural experience
by themselves as well as at the same time improve their own self esteem and confidence amounts.
Just like other facilities, medical facilities at Salt Lake City are quite advance and it has been serving different needs of the patients for the
past many years. This will give an assurance that the treatment which has offered is effective and suitable for you.
My blog post; nose job vs rhinoplasty [Jose]
Now when it comes to the actually tweezing part, you want to make sure that you’re using
a good pair of tweezers. This is because targeting the hair
shaft without damaging the pigment of the skin can be quite a tricky process.
Simply put, the heat destroys the whole hair follicle
permanently.
Here is my webpage skin care products celebrities use
They include a children’s clothing size chart for ease of shopping online and carry other accessories such as hair bows, backpacks, hats and shoes
and boots. While most dresses are basic silk, satin, or tulle, some are nestled underneath an overlay of
cotton lace to create a more textured appearance. Full body lace is finest worn with a smooth tank top.
my weblog; Celebrity dress up Games For girls and boys
However, sometimes we need to do something
a little more adventurous, or perhaps a little more pointedly
selfserving. Sometimes, our decisions could be clouded by pressure
from our surroundings. Vanity is when you have
something taken care of because you want everyone to
notice and look at you.
Visit my web page :: facial plastic surgery for acne scars
If you are looking for the best way to file for extension electronically, then you should use
IRS efile. Users can contact support service via e mail, live chat, toll free phone number or any other user friendly method.
If you obtain CP 503 in your mailbox, you almost certainly previously obtained an IRS Notice.
My weblog earl cohen cpa boca raton fl
Customers handle their own antifreeze, flsh bulbs, and
lighter fluid. And anyone reading these installments are now
aware of how laid bare my life is becoming.
• Must know how to assess injury risks that comes with hazardous manual handling.
Here is my homepage … food storage companies
As we said, there are changes that your mate could also make.
Overcrowded schools are also using mobile restroom facilities to address the needs of the growing student populations.
With conscientious use, a programmable thermostat can save about $150 year in energy costs.
Here is my website; install tub Pompano beach real Estate
Running the compressor of the above room air conditioners involves high electricity consumptions and bills.
It”s tough to keep things dry on a center console boat. As they are huge, RVs also have a huge checklist of parts and components that you need to inspect after long storage to ensure smooth and safe driving.
Feel free to surf to my page: flatbed dunnage box
2 ANALYSIS ON THE CONCENTRATION BY AREA IN TERMS OF
THE MARKET SHARE 26. Velocity boaters like finding around at a quickly pace even though most fishermen want a boat to get
to the fish so they usually shift at a slower pace.
The articles and segments written on its pages are written by fishing aficionados and specialists themselves.
Here is my webpage :: deck box with seating
Hence, considering the basic factors in mind, one can certainly be rest assured about the comfortable
and smoother relocation. Have all the materials needed to pack ready and
available to everyone at the beginning of the project.
Many of the refugees are moving towards the capitol to receive whatever assistance that the government can provide which is not much.
Feel free to visit my website :: self storage companies in nj
Now that you know a little more about the CPA exam scoring process and how
it works, you can stop checking your email every five minutes and stop accosting your mail carrier when they deliver your mail.
The AICPA course is eleven hours long, obtained via correspondence
or selfguided classes over the Internet. Learning how to get your gambling winnings tax back can be quite beneficial to you,
especially if you find a refund management company that can assist you in getting this rebate without a lot of hassle and fuss.
My web page boca raton accounting jobs
Hi there, I desire to subscribe for this web site to get most
uptodate updates, so where can i do it please help.
I am surprised by the information in this article I found it to be not only very interesting however it also made me think. It is hard now a days to get appropriate information to ones search, so I am pleased that I found this article post
Pretty section of content. I just stumbled upon your website and in accession capital to claim that I
acquire actually enjoyed account your blog posts. Any way I’ll be subscribing in your feeds or even I success you get right of entry to consistently
quickly.
Wonderful goods from you, man. I’ve understand
your stuff previous to and you are just extremely magnificent.
I really like what you’ve acquired here, really
like what you are saying and the way in which you say it.
You make it enjoyable and you still take care of to keep it smart.
I can’t wait to read much more from you. This is actually a wonderful
website.
I think I am getting my head around this – thanks for all the detailed info. You know your onions!
An English language version of one’s book also can lead to foreign translation rights.
Give away books like you give away business cards as well as your business is certain to grow.
Article Source: abo ut design competition plus read about digital stream
and norwegian language.
Hello, this weekend is nice designed for me, because this occasion i am reading this great informative paragraph here at my home.
Please let me know if you’re looking for a author for your blog. You have some really great articles and I believe I would be a good asset. If you ever want to take some of the load off, I’d absolutely love to write some content for your blog in exchange for a link back to mine. Please shoot me an email if interested. Regards!