Question

1. When you were born, your parents decided to pay for four years of your education it looks like tuition will be $4000, your books would be $1000, and they are estimating $5000 per year for you to live on, you will receive equal monthly payments. They could not afford to save for the first two years of your life, but they started when you were 2 and saved for 8 years until you started playing competitive hockey at the age of 10. At that time the money continued to earn interest at 5% compounded monthly, how much did your parents save monthly to meet their goal of paying for four years of study when you turn 18, giving you equal payment over four years.

Answer 1

I am making following three assumptions that are customary to such kind of question:

• Savings are deposited at the end of every month.
• Interest rate is 5% per annum compounded monthly
• At the time of education, you need the money at the beginning of every month. So you get first payment for the first month of your education on the first day of the month.

Let's shift the time frame to the day you turned 18. Let's assume you have turned 18 today i.e. t = 0. At this time, you need a total amount = 4000 (for tuition) + 1000 (for books) + 5,000 (per year for living) x 4 years (of education) = 25,000

Period, N = 48 months

Interest rate = 5% per annum

Interest rate per period of 1 month, r = 5% / 12 = 0.417%

Amount required, per month = 25,000 / 48 = 520.83

I have made an assumption that you will need the monthly amount at the beginning of the month. So as as soon as you turned 18, you need your first monthly amount of 520.83.

Second installment is required at the beginning of second month ie. at the end of first month i.e. at t = 1 and so on. You need the last installment at the beginning of 48th month i.e. at the end of 47th month i.e. at t = 47.

So, present value of all your monthly requirement = Monthly amount x Sum of PV factor (over t=0 to t=47)

Sum of PV factor (over t= 0 to t=47) will be same as (1 + r) x sum of PV factor (over t=1 to t=48)

$=\frac{(1+r)}{r}\times [1 - (1+r)^{-N}] = \frac{(1+0.00417)}{0.00417}\times [1 - (1+0.00417)^{-48}]$

= 43.6039

PV of all my future requirement at t=0 i.e. as soon as you turn 18 years old = Monthly amount x sum of PV factor = 520.83 x 43.6039 = 22,710

So, this is the size of kitty required when you turn 18.

Let's turn the clock back now and come to a point in time when you were 2 years old. So, now at t = 0, you are two years old. Let's say the monthly saving is M.

r = 0.417%

Period, N = nos. of month in 8 (= 10 - 2) years = 12 x 8 = 96

So when you turn 8, future value of all your monthly savings = monthly saving x sum of FV factor

$=M\times \frac{1}{r}\times [(1+r)^{N} - 1] = M\times \frac{1}{0.00417}\times [(1+0.00417)^{96} - 1]$

= 117.7405M

From the time you were 10 years old till the time you turned 18, no investments were made but the amount above continued to earn interest at the rate of 5% compounded monthly.

So, from t = 10 till t=18, period, N = nos. of months in 8 (=18 - 10) years = 12 x 8 = 96

So, amount above will grow to an amount = 117.7405M x FV factor = 117.7405M x (1 + r)^N = 117.7405M x (1 + 0.00417)⁹⁶ = 175.5023M

The kitty size available when you will turn 18 = 175.5023M

The kitty size required when you turn 18 = 22,710 (calculated earlier in the solution)

Hence, 175.5023M = 22,710

Hence, M = 129.40

Hence, the monthly saving of your parents should be M = 129.40