Question

You want to start saving for retirement. Your goal is to retire in 25 years. Assume that you have $25,000 to invest now and that you will contribute $4,800 per year.

1. What will your account be worth when you retire if you can earn 6% a year?
2. What will your account be worth if the 6% annual return is compounded monthly and instead of contributing $4,800 per year, you contribute $400 monthly (you still start with $25,000)

Assume all payments and withdrawals are made at the end of the period.

3. Why are the two amounts different?

Answer 1

a) First, let's understand the cash flows:

At year 0, we have $25,000 to invest and we will contribute $4,800 per year for 25 years.

Account worth when we retire if we can earn 6% a year = Sum of Future value of cash flows earning 6% a year

FV = Cash flow*((1+Interest rate)^n)

Years	Cash Flows	n	FV
0	25000	25	107,296.77
1	4800	24	19,434.89
2	4800	23	18,334.80
3	4800	22	17,296.98
4	4800	21	16,317.91
5	4800	20	15,394.25
6	4800	19	14,522.88
7	4800	18	13,700.83
8	4800	17	12,925.31
9	4800	16	12,193.69
10	4800	15	11,503.48
11	4800	14	10,852.34
12	4800	13	10,238.06
13	4800	12	9,658.54
14	4800	11	9,111.83
15	4800	10	8,596.07
16	4800	9	8,109.50
17	4800	8	7,650.47
18	4800	7	7,217.43
19	4800	6	6,808.89
20	4800	5	6,423.48
21	4800	4	6,059.89
22	4800	3	5,716.88
23	4800	2	5,393.28
24	4800	1	5,088.00
25	4800	0	4,800.00
Interest rate	6%
Account's worth when we retire	370,646.43

b)

First, let's understand the cash flows:

At year 0, we have $25,000 to invest and we will contribute $400 per month for 300 months.

Now, let's find the EAR:

EAR = ((1+(Nominal rate/Number of compounding periods)^ Number of compounding periods)  - 1

EAR= [(1+(6%/12)) ^12] - 1

EAR = 6.1678%

To solve this we will use BA 2 plus financial to find the FV of the cash flows:

PMT(Montly contribution): $400

I/Y(6.1678%/12): 0.5140%

PV(AMount depositing in the starting): $25,000

N: 25*12 = 300

CMPT FV

Account's worth when we retire = $400,862.8167

c)

Two amounts are different because in one the interest rate is compounding yearly and we are depositing $4800 yearly.

And in the other month the interest rate is compounding monthly and we are depositing $400 montlhy.