Sunrise Industries wishes to accumulate funds to provide a retirement
annuity for its vice president of research, Jill Moran.  Ms. Moran by
contract will retire at the end of exactly 12 years. Upon retirement,
she is entitled to receive an annual end-of-year payment of $42,000
for exactly 20 years.  If she dies prior to the end of the 20-year
period, the annual payments will pass to her heirs.  During the
12-year "accumulation period" Sunrise wishes to fund the annuity by
making equal annual end-of-year deposits into an account earning 9%
interest.  Once the 20-year "distribution period" begins, Sunrise
plancs to move the accumulated monies into an account earning a
guaranteed 12% per year.  At the end of the distribution period, the
account balance will equal zero.  Note that the first deposit will be
made at the end of year 1 and that the first distribution payment will
be received at the end of year 13.

answer B.
How large a sum must Sunrise accumulate by the end of year 12 to
provide the 20-year, $42,000. annuity?

Chicka --

This calculation requires 20 separate calculations to arrive at the
end-of-year 12 number.  Let's start with what we know, then work
backwards.
*  At the end of year 20, the last principal payment will zero the account
*  Interest will also be paid at 12%
*  Total payment = $42,000


YEAR 20:

So, we don't know the principal, but let's call it x:
.12*x + x = $42,000
solve for x: 1.12x = $37,500


YEAR 19:

Now we'll see the interest benefit of 12% on that $37,500 in
principal, plus another chunk of principal paid from the annuity. 
That additional chunk of principal is y:
.12 ($37,500 + y) + y = $42,000
$4,500 + .12y + y = $42,000
1.12y = $37,500
y = $33,482


YEAR 18:

Total principal on which interest is paid is now $37,500 + $33,482 +
the piece of principal paid this year (z), so we run the calculation
again:
.12 ($70,982 + z) + z = $42,000
$8518 + 1.12z = $42,000
1.12z = $29,895

We could keep going or perform the same calculation in a spreadsheet 17 more times:
.12 * (end-of-year principal + this year's principal) + this year's
principal = $42,000

The spreadsheet calculation is made easier by the fact that
.12*(end-of-year principal) is the next year's interest.  So, year 17
will have interest of .12 ($37,500 + $33,482 + $29,895) = $12,105;
principal paid during the year is now calculated as:
($42,000-$12,105)/1.12 = $26,692

The iterations are shown in the spreadsheet here, which is viewable in
your browser, even if you don't have Excel:
http://www.mooneyevents.com/annuity.xls

Thus, the end-of-year 12 number -- the one that STARTS the retirement
annuity -- is $313,716.74, which will provide Jill Moran with $37,646
in interest and $4,354 at the end of her first year of retirement.

We haven't done the hard part of this annuity calculation -- finding
the amount that needs be paid in each year.  Here is how that is done
(and thank goodness for spreadsheets):
http://answers.google.com/answers/threadview?id=122589

If any aspect of this answer is not clear, please post a clarification
request before rating this Google Answer.

Best regards,

Omnivorous-GA

---

Request for Answer Clarification by chicka-ga on 23 Nov 2003 07:14 PST

Hello,

Yes, .12*x + x = $42,000

solve for x 1.12x = 37,500

are you saying that...

.12*1.12 + x = 42,000
what is the visual equation to derive at 37,000?

---

Clarification of Answer by omnivorous-ga on 23 Nov 2003 09:24 PST

> .12*1.12 + x = 42,000
> what is the visual equation to derive at 37,000?

The above (marked with >) is incorrect.

For Year 20, it's the first line that you'd typed:
.12*x + x = $42,000

The first term is the interest; the second term the principal returned to Jill.

Best regards,

Omnivorous-GA

---

Request for Answer Clarification by chicka1-ga on 02 Dec 2003 08:14 PST

Hello,

where did you get 1.12x from to equal 37,500?

thank you.
chicka-ga

---

Clarification of Answer by omnivorous-ga on 02 Dec 2003 09:34 PST

Chicka --

In proper math terms it should be 1.12x = $42,500, as that's one year's interest;
x = $37,500

That's how you start working backwards on the interest + principal sides.

My apologies for the shorthand . . .

Best regards,

Omnivorous-GA

Problem answered thank you. Clarification is needed in order to
interpret equation answer to problem.
chicka

Hmmmmm

But where would you find 12% pa guaranteed over 20 years?

Or even 9% pa over 12 years?

Probono --

You remember 1980 -- you could get 15% returns!  Of course, inflation was 12%. . .

Yeah .... but 12% guaranteed over 20 years?

Never in my lifetime ...

And I've just turned 100.