Question

You are a financial planner. Your client’s primary saving objective is to provide for the college education of her two children. Her children are currently 4 and 6 years old. Assume that tuition will cost $30,000 per year for four years for each child. You will make the first tuition payment for the 6-year-old 12 years from today, and the first tuition payment for the 4-year-old 14 years from today. Regardless of the saving plan that you put into place, you believe it is reasonable to assume your client will earn an average annual return of 8% on her investments.

A. How much must your client have in the college account today so that she is able to pay for the tuition of both children?

B. Assume your answer to Part A is $75,000. If your client currently has no money saved for her children’s college, how much must she deposit in each year to fund their education? Assume the first deposit is made today and the last deposit is made 11 years from today (for a total of 12 annual deposits).

C. Assume your answer to Part A is $75,000. Your client wants to deposit $6,000 today and is willing to grow that deposit amount by 10% in each of the next 11 years for a total of 12 deposits. (For example, next year’s deposit will be $6,600.) If she follows this saving plan, will she have saved enough for her children’s education?

Answer 1

Part A
	PV of annuity
	P = PMT+PMT x (((1-(1 + r) ^- (n-1))) / r)
	Where:
	P = the present value of an annuity stream	P
	PMT = the dollar amount of each annuity payment	$       30,000
	r = the effective interest rate (also known as the discount rate)	8%
	n = the number of periods in which payments will be made	4
	PV of annuity at Year 12=	PMT+PMT x (((1-(1 + r) ^- (n-1))) / r)
	PV of annuity at Year 12=	30000+30000*(((1-(1 + 8%) ^- (4-1))) / 8%)
	PV of annuity at Year 12=	$107,312.91
	PV of annuity at Year 0=	107312.91/(1+8%)^12
	PV of annuity at Year 0=	$ 42,615.43
	PV of annuity at Year 14=	107312.9096
	PV of annuity at Year 0=	107312.91/(1+8%)^14
	PV of annuity at Year 0=	36535.86509
	Total PV required today=	42615.43+36535.87
	Total PV required today=	$ 79,151.30
Part B
	PV of annuity
	P = PMT+PMT x (((1-(1 + r) ^- (n-1))) / r)
	Where:
	P = the present value of an annuity stream	$ 75,000.00
	PMT = the dollar amount of each annuity payment	PMT
	r = the effective interest rate (also known as the discount rate)	8%
	n = the number of periods in which payments will be made	11
	PV of annuity=	PMT+PMT x (((1-(1 + r) ^- (n-1))) / r)
	75000=	PMT*(1+ (((1-(1 + 8%) ^- (11-1))) / 8%))
	Annual payment=	75000/(1+ (((1-(1 + 8%) ^- (11-1))) / 8%))
	Annual payment=	$    9,727.52
Part C
	PV of annuity for growing annuity due
	P = (PMT/(r-g)) x (1-((1+g)/(1 + r)) ^n) *(1+r)
	Where:
	P = the present value of an annuity stream	P
	PMT = the dollar amount of first payment	$    6,000.00
	r = the effective interest rate (also known as the discount rate)	8%
	n = the number of periods in which payments will be made	12
	g= Growth rate	10%
	PV of annuity=	(PMT/(r-g)) x (1-((1+g)/(1 + r)) ^n) *(1+r)
	PV of annuity=	(6000/(8%-10%)) x (1-((1+10%)/(1 + 8%)) ^12) *(1+8%)
	PV of annuity=	$ 79,805.44
	So PV of all annual payments comes to & 79,805 which is more than required 75,000.