Annuities ©Dr. B. C. Paul 2001 revisions 2008 Note – The subject covered in these slides is considered to be “common knowledge” to those familiar with the subject and books or articles covering the concepts are widespread. Information of repayment terms for student loans were based on such sources as Bankrate.com and Staffordloan.com or the Department of Education web site.

Back to the Story of Lanna Loaner  Lanna Loaner has just graduated from College with a debt of $102,325  Of course student loan programs don’t expect Lanna to pay off her loan on graduation day.  They’ll have her pay it off over the next say 10 years in monthly installments  Lets also say she consolidates at 5.5% with monthly compounding.

Step #1 in Problem Solving  Let pick the perspective for the story problem. (We have the bank that has money loaned out and is going to collect payments - or we have Lanna).  This time I’m going to pick the banks perspective (I could make it work either way)

Drawing Pretty Pictures This time I’m going to sweep all the money into a pot at year #5. (Partially because I’ve already done half the problem and I’m lazy).

What I already Know If I sweep all that money the bank loaned forward to year 5, it is equal to the bank having $102,325 dollars out on loans.

New Picture 5 5y 1m y -$102,325 I have to get my banker paid back over a period of 120 equal payments with 5.5% interest compounding monthly.

Magic Number Come Out and Play  I need magic number that will sweep these future payments of unknown size, back into my money pot.  Two Observations  I have 120 numbers to be swept back - if I have to do 120 P/F magic numbers I’m going to puke  I don’t know how big these 120 numbers are.

Equal Payments Have a Special Name  Annuity  An annuity is a series of equal payments  Common occurrences of this type of cash flow  Mortgage Payments  Payments out of Retirement Funds  Engineers projecting the same earnings from their project year after year.

Enter a New Super Hero  A/P  A/P stands for an Annuity  who's Present Value  A/P * Present Value =  An Annuity with the same  total value

What do I know  I know I have a banker who is out $102,325.  How much money do I have to sweep back into his pot before he is going to be happy?  Because I’m not paying him off on graduation day - I’ll have to sweep the money back with interest  I have a present value  $102,325 * A/P = size of those annuity payments

OK, Now I Have Everything but the Stupid Formula for A/P  A/P i, n = {( i * [ 1 + i ] n )/( [ 1 + i ] n - 1) }  This sounds like a formula to put in a spread sheet or to save in a calculator so that nimble fingers can’t punch it in wrong  I didn’t do a derivation of the formula  Thing I remember most about that derivation was that I never wanted to see it again  Look at the Formula and Say “I Believe”!

Ok - It’s a really cool formula but what does it all mean  i is the interest rate  Oh that’s not so bad  We know the interest rate will be 5.5% per year after her graduation BUT  We ALSO know that after she graduates the banker is going to ream her one - its compounding monthly  5.5%/12 months/year =.4583%/month  i is equal to

More Coolness with the Formula  n is the number of payments and  the number of compounding periods  In this case Lanna will make  monthly payments for 10 years or  120 payments  n = 120  Plug and Crank  A/P i, n = {( * [ ] 120 )/( [ ] ) } =

Turning on our Sweeper 5 5y 1m y -$102,325 $102,325 * = $ per month

Try That With Class Assistant Out comes A/P Apply it $102,325 * = $ % compounded 12 times a year 10years*12 months = 120 payments

$ ! – one way to reduce is to spread out over more time Could Make the Payments over 20 years 102,325 * = $704/month

I Could Also Spread it Over 30 years $102,325* = $581.2

A Nasty Fact of Life  Lanna borrowed $22,137+4*15,137 = $82,685  Over 10 year Lanna Pays Back  $1,108.69*120 = $133,048  $133,048 - $82,685 = $50,357 interest  Over 20 years Lanna would Pay  $704*240 = $168,959  $168,959 - $82,685 = $86,273  Over 30 years Lanna would Pay  $581.2*360 = $209,234  $209,234 - $82,685 = $126,548  I Wonder Why Everyone Wants to Tell You How Easy it is to borrow money, but no one wants to tell you what it will be like to pay it back?

Example Disclaimers  Interest rate I used was for a Stafford Student loan  But there are borrowing limits on Stafford loans – you could not borrow $82,000  The loan accumulated interest while Lanna was in school – Stafford loans don’t (Plus loans to parents do – and have a higher interest rate)  There were no up front fees for the loans (Stafford and Plus loans can have fees)  Student loans are distributed at the start of each semester – not the start of each year so the example borrowed some money sooner than would be allowed.  Most Private Student loans have variable interest rates that can explode upwards  Over-all the example is a little optimistic  I’m adding a spreadsheet under Resources that can help estimate Student loan options

Limitations and Point  My calculation did not consider that the value of that money will change over time  It Does Work in the My Money Equation Income = Necessities + Good Stuff + Taxes + Insurance + Savings + Investments + Interest - Debts Income is some number If interest is a big number then some of the other stuff such as Good Stuff Insurance, Savings, or Investments will have to be smaller (Interest plunders your future quality of life)

Can You Ever Win with Debt?  If income rises more rapidly than debt + interest  Example – College  With debt and interest paid off over 30 years Lanna spends $209,234  Using early 2000s data (its even more now)  HS lifetime earnings average $1.2 million  College Grad $2.1 million  $900,000 > $209,234

If We Consider That Lanna Was Wise and Chose Engineering the Outlook is Even Better

Other Potential Winning Moves  Maybe buying a house?  Most home loans have lower interest rate than the rate of return for rental businesses  Homes have lower property tax and interest is usually a tax deduction  Might not work if  You move a lot – there are fixed buying and selling costs  Buying a house causes you to “splurge” for the perfect residence

Good Loosing Moves  Taking vacations and buying consumables using debt  The goods are gone – the debt remains  Consumer Durables  Debt makes them cost more and they usually don’t add to income or reduce expenses

Observations About A/P  A/P is sometimes called a capital recovery factor  In many problems you will have an initial capital outlay.  If you multiply this initial outlay by the A/P factor it tells you how big the payments will have to be starting with the next compounding period to pay back the capital

A Sucker Mistake Warning  Sneaky people will put things that look like annuities into cash flows  A/P (and other annuity magic numbers you haven’t met yet) work only on true annuities  Characteristics of True Annuity  They consist of a series of equal payments  The payments occur at the end of each compounding time interval  The payments begin one compounding period into the future after present time (time 0)

Now Its Your Turn Do Assignment #4 You will be helping Harry Homebuyer figure out his house payments (another form of annuity).