Question

1. The buyer of a certain machine is given 2 options to pay it: 1^st option is paying it P2,000 downpayment and P2,000 annually for the next 6 years; 2^nd option is to pay it P3,500 downpayment and P2,000 annually for the next 5 years. If the money is worth 12% cpd. annually, which option is better and by how much?

2. Today, a man invested P20,000 for the college education of his 4-year old daughter. If the fund earns 12% cpd. annually, how much will the daughter get each year starting from her 16^th to 20^th birthday?

Answer 1

1) Rate of interest = 12% = 0.12

Present value is calculated as: [Annual Payment / (1 + Rate of Interest)^^Year]

Total cost of machine = Downpayment + Present value of annual payment

Machine 1:

2,000 downpayment and 2,000 for 6 years

Present value of each downpayment is:

Year	Annual Payment	Present Value
1	2000	1785.71
2	2000	1594.39
3	2000	1423.56
4	2000	1271.04
5	2000	1134.85
6	2000	1013.26
		8222.81

Total cost of this machine = 2,000 + 8,222.81 = 10,222.81

Machine 2:

3,500 downpayment and 2,000 for next 5 years

Year	Annual Payment	Present Value
1	2000	1785.71
2	2000	1594.39
3	2000	1423.56
4	2000	1271.04
5	2000	1134.85
		7209.55

Total cost of this machine = 3,500 + 7,209.55 = 10,709.55

Machine 1 is better option as cost is lower in that case.

2) Man Invested = 20,000

Rate of Interest = 12% = 0.12

His daughter will get let say equal payment of X from 16 - 20 birthday

Amount is deposited for 12 years which will make is 20,000 * (1 + 0.12)^¹² = 77,919.51

Let say her daughter withdraws X from this amount on 16th birthday which will leave (77,919.51 - X). This amount will earn rate of interest till 17th birthday which will make is (1.12 * (77,919.51 - X))

Her daughter withdraws X from this amount on 17th birthday which will leave (1.12 * (77,919.51 - X) - X). This amount will earn rate of interest till 18th birthday which will make is 1.12 * (1.12 * (77,919.51 - X) - X)

Her daughter withdraws X from this amount on 18th birthday which will leave (1.12 * (1.12 * (77,919.51 - X) - X) - X). This amount will earn rate of interest till 19th birthday which will make is 1.12 * (1.12 * (1.12 * (77,919.51 - X) - X) - X)

Her daughter withdraws X from this amount on 19th birthday which will leave (1.12 * (1.12 * (1.12 * (77,919.51 - X) - X) - X) - X). This amount will earn rate of interest till 20th birthday which will make is 1.12 * (1.12 * (1.12 * (1.12 * (77,919.51 - X) - X) - X) - X)

Her daughter withdraws X from this amount on 20th birthday which will leave zero in the account. Thus, 1.12 * (1.12 * (1.12 * (1.12 * (77,919.51 - X) - X) - X) - X) - X = 0

4.037X = 77,919.51

X = 19,301.34

Her daughter will get 19,301.51 each year from 16-20th birthday.