Back to the Story of Lanna Loaner

• Lanna Loaner has just graduated from College with a debt of $51,596

• 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 5

years in monthly installments– Lets also say the interest rate changes to 8% 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

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5 6 7 8 9 10

This time I’m going to sweep all the money into a potat year #5. (Partially because I’ve already done half theproblem and I’m lazy).

What I already Know

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5 6 7 8 9 10

If I sweep all that money the bank loaned forwardto year 5, it is equal to the bank having $51,596 dollarsout on loans.

New Picture

55y 1m ---------------------------------------------------------------- 10y

-$51,956

I have to get my banker paid back over a period of 60 equalpayments with 8% 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 60 numbers to be swept back - if I have

to do 60 P/F magic numbers I’m going to puke• I don’t know how big these 60 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 $51,596.• 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– $51,596 * 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 8% per year after her

graduation BUT• We ALSO know that after she graduates the

banker is going to ream her one - its compounding monthly– 8%/12 months/year = .667%/month– i is equal to 0.00667

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 5 years or– 60 payments

• n = 60• Plug and Crank

– A/P i, n = {( 0.00667 * [ 1 + 0.00667 ] 60)/( [ 1 + 0.00667 ] 60 - 1) } = 0.0202783

Turning on our Sweeper

55y 1m ---------------------------------------------------------------- 10y

-$51,956

$51,956 * 0.0202783 = $1046.28/month

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