1) Determine how much youwill have to save each month at 3, 6, 9, and 12 percent compoundedmonthly for you to accumulate a nest egg for retirement. Thevariables are current age, age of retirement,...

Question 1: Determine how much you will have to save each month at 3, 6, 9, and
12 percent compounded monthly for you to accumulate a nest egg for retirement. The
variables are current age, age of retirement, nest egg size, and interest rate. Show
all work and cite as appropriate.
Variable definitions:
ca = current age
ra = retirement age
nest egg size = N
interest rate = r
Total Amount after compounding interest = A
A = P
(
1 + r
n
)nt
where
A = Final Amount = N (nest egg size)
P = Principal Amount or Initial Investment
r = Annual Interest Rate
n = no. of times interest is compounded per year = 12
t = no. of yrs amount is deposited = ra-ca
Let k = nt
Calculations:
Final Amount from 1st month’s deposit = P ×
(
1 + r
12
)k
Final Amount from 2nd month’s deposit = P ×
(
1 + r
12
)k−1
...
Total Amount at the end = P
(
1 + r
12
)k
+ P
(
1 + r
12
)k−1
+ . . . P
(
1 + r
12
)1
= [P
(
1 + r
12
)k
+ P
(
1 + r
12
)k−1
+ . . . P
(
1 + r
12
)1
+ P ] − P
Let
(
1 + r
12
)
= λ
Total Amount at the end = [Pλk + Pλk−1 + · · · + Pλ+ P ] − P
= P
(
λk+1−1
λ−1
)
− P
= Pλ
(
λk−1
λ−1
)
= N
Hence,
P =
(
N(λ−1)
λ(λk−1)
)
For r = 3 %,
λ = 1.0025
P =
(
N(2.49×10−3)
1.0025k−1
)
For r = 6 %,
1
λ = 1.005
P =
(
N(4.975×10−3)
λk−1
)
For r = 9 %,
λ = 1.0075
P =
(
N(7.4×10−3)
λk−1
)
For r = 12 %,
λ = 1.01
P...