Question

You decided to quit smoking today. Now that you are going to save 600 €/month, you decided to save those 300€ per month in a bank account that offers a 6% interest rate compounded monthly, till the day you retire (that is to say, in 30 years). Please answer the following questions: If you do your deposits at the end of every month (so your first deposit will be in one month from today) in a bank account that offers a 6% interest rate compounded monthly, and you continue doing so till the day you retire (that is to say, in 30 years): Draw the timeline (at least the first five periods) with its corresponding numeration of periods and cashflows in their corresponding points How many cashflows will be there? how much money will you have at the end? Show the workout If you decide to do your deposits at the beginning of every month (so your first deposit will be done today) in a bank account that offers a 5% interest rate compounded monthly, and you continue doing so till the day you retire (that is to say, in 30 years): Draw the timeline (at least the first five periods) with its corresponding numeration of periods and cashflows in their corresponding points (10 points) What is the difference between this case and the previous one? (10 points) According to your opinion, what is it better? To bring the amounts to the bank at the beginning of every month (as described in this ex.) or at the end of every month (as described in the previous ex.)? Explain the reason (10 points) How much money will you have at the end? Show the workout (10 points) If you decide to do your deposits at the end of every month (similarly to ex.1, so your first deposit will be in one month from today) in a bank account that offers a 6% interest rate compounded monthly, and the deposits will increase in a 0.2% month after month, and you keep on doing so for the next 30 years: Draw the timeline (at least the first five periods) with its corresponding numeration of periods and cashflows in their corresponding points (10 points) how much money will you have at the end? Show the workout (10 points) What would it happen if the growth rate and the interest rate is the same? (10 points)

Answer 1

1. Deposit at the end of month, 6% compounded monthly

a. The table of cash flows looks like this for first 10 periods

Deposits at the end of the month
Interest Rate	6%
Year	Opening Balance (A)	Interest Earned (6% on Monthly compounding) B = A x 6%/12	Deposits C	Closing Balance D = A+B-C
1	-	-	300.00	300.00
2	300.00	1.50	300.00	601.50
3	601.50	3.01	300.00	904.51
4	904.51	4.52	300.00	1,209.03
5	1,209.03	6.05	300.00	1,515.08
6	1,515.08	7.58	300.00	1,822.65
7	1,822.65	9.11	300.00	2,131.76
8	2,131.76	10.66	300.00	2,442.42
9	2,442.42	12.21	300.00	2,754.63
10	2,754.63	13.77	300.00	3,068.41

Since the deposits are made at the end of month, interest is not earned on the deposit during the month.

The closing balance is sum of Opening Balance + Interest earned + Deposits during the month.

This closing balance becomes next months opening balance. and this continues for 360 periods.

1. b

The value of deposits at the end of 30 years or 12 months X 30 years = 360 months can be calculated using the FV value function in excel

Here,

Rate = 6% compounded monthly, which means = 6%/12 (12 months in a year)

NPER = Number of periods = 30 years = 30 x 12 = 360 months (Since we are doing monthly compounding)

PMT = Deposits every month = -300 (Negative sign indicates, it is a cash outflow)

PV = Present Value = 0 (Begining value of investments)

Type = 0 (Deposits are made at the end of the period/ month)

Using the above values in FV function excel (One can use the same logic in financial calculators)

Thus Future value = FV(6%/12,360,-300,0,0) = $301,354.51

1. C.  If the interest rate = 5% compounded monthly, and deposits are done at the start of month, the table would look like

Deposits made at the start of the month
Interest Rate	5%
Year	Opening Balance (A)	Deposits B	Interest Earned (5% on Monthly compounding) C = (A+B) x 5%/12	Closing Balance D = A+B+C
1	-	300.00	1.25	301.25
2	301.25	300.00	2.51	603.76
3	603.76	300.00	3.77	907.52
4	907.52	300.00	5.03	1,212.55
5	1,212.55	300.00	6.30	1,518.85
6	1,518.85	300.00	7.58	1,826.43
7	1,826.43	300.00	8.86	2,135.29
8	2,135.29	300.00	10.15	2,445.44
9	2,445.44	300.00	11.44	2,756.88
10	2,756.88	300.00	12.74	3,069.62

In this case, since the deposits are made at the start of month, interest on the deposit as well as opening balance for the month is earned. Remember, in the previous case, deposits were done at the end of month, and therefore interest was not earned on deposits for the month, but was earned only on opening balance of the month.

D.

The value of deposits at the end of 360 months can be calculated using FV function.

Here  

Rate = 5% compounded monthly, which means = 5%/12 (12 months in a year)

NPER = Number of periods = 30 years = 30 x 12 = 360 months (Since we are doing monthly compounding)

PMT = Deposits every month = -300 (Negative sign indicates, it is a cash outflow)

PV = Present Value = 0 (Begining value of investments)

Type = 1 (Deposits are made at the begining of the period/ month)

Using the above values in FV function excel (One can use the same logic in financial calculators)

Thus Future value = FV(5%/12,360,-300,0,1) = $250,717.91

E. Deposits done at the start of month are better, you earn interest on the deposit as well as opening balance during the month.