Math 55(31) - Warren D. Smith - Homework #2 - due Tues 4 Feb 2003
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Read sec 4B and start 4C of book.

SOLVE exercises 1, 3, 5, 9, 11, 13, 21 page 240 sec 4B.

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ANSWERS

1. evaluate

 3
4 = 64

 3  5   8
2  2 = 2 = 256

 6     2    4
3  /  3  = 3  = 81

 -2     2
4  = 1/4  = 1/16

 5  -2   3
4  4  = 4  = 64

 3    -4   7
5  / 5  = 5 = 78125


3. solve these for x:

 2
x = 100    ans  x=10 or x=-10

 3
x = 27   ans  x=3  (but not x = -3)

 1/3                 3
x   = 2   ans   x = 2 = 8

 1/5                  5
x    = 2    ans  x = 2  = 32


5. At age 25 you set up an IRA; deposit $50 every month, it pays APR=8%
   interest, how much in the IRA when you reach age 65?  Compare this
   to the total amount of money you deposited.

Ans: use saving plan formula (book page 221 top)
                 NY
        (1 + i/N)   - 1
A = PMT ----------------
               i/N

to compute final amount A with N=12, PMT=50, i=0.08, Y=65-25=40.
So
                12 x 40
   1 + (0.08/12)        - 1
50 ------------------------ = $174550.39.
        0.08/12

9. Yolanda deposits $100 per month into an account with APR=5%,
Zach deposits $1200 at end of each year in account with APR=5%.
Who comes out ahead? (Assume payment and compunding periods are the
same so can use the Savings Plan Formula.) Compare balances in
their plans after 10 years.
                          120
             (1 + 0.05/12)   -1
Yolanda: 100 ------------------  = 15528.23.
                    0.05/12
                          10
              (1 + 0.05/1)   - 1
Zach:    1200 ------------------ = 15093.47.
                     0.05/1
Yolanda comes out ahead. More frequent compounding (monthly) better.

11. Juan deposits $200/month in account APR=6%,
Maria deposits $2500 per year in an account with APR=6.5%.
Who comes out ahead? (Assume payment and compunding periods are the
same so can use the Savings Plan Formula.) Compare balances in
their plans after 10 years.
                         120
            (1 + 0.06/12)    - 1
Juan:   200 -------------------- = 32775.87.
                   0.06/12

                          10
             (1 + 0.065/1)    - 1
Maria:  2500 -------------------- = 33736.06.
                   0.065/1
Maria ahead. This because of her higher APR; fact Juan getting
monthly compunding (vs annual, which is worse) not enough to make up
for that.

13.  You want $150,000 in 18 years. How much should you deposit
     monthly in an APR=7.5% account?

Ans: savings plan formula (version that is solved for PMT) page 225
top
               i/N
   PMT = A -------------
                  NY
           (1+i/N)   - 1

gives (for us N=12, Y=18, A=150000, i=0.075, NY=12x18=216)
                 0.075/12
   PMT = 150000 ------------------- = 329.96.
                            216
                (1+0.075/12)     -1

21. You are 30. You want to retire at 60 with a fund from which you
    can draw $50,000/year forever. How can you do it,
    assuming an APR of 8%?

You want the total amount A you have in the account at age 60
to obey   0.08 A = 50000.
That way you just take the 8% interest every year from 60 on
out of your account, which stays at A forever.
Solving: A = 50000/0.08 =  625000.

Acquiring this $625000 nest egg can be done by depositing PMT
each year in your 8% APR account, where (using formula above with N=1,
Y=60-30=30, i=0.08, A=625000)
                 0.08
PMT = 625000 ------------ =   5517.15.
                   30
             (1.08)   - 1
Note your total payments here are   5517.15 x 30 =  $ 165514.50
which is a lot smaller than your nest egg $ 625000.00, and
also a lot smaller than the $50000 retirement "salary" you
will be paying yourself each year forever. Such are the
wondrous effects of compound interest at this (rather high)
8% interest rate.  (Also you could have deposited
$419.36 each month if it were monthly compounding, etc.; so this was
not the only possible way to do it.)