Question

Your child was just born and you are planning for his/her college education. Based on your wonderful experience in Managerial Economics you decide to send your child to Binghamton University as well. You anticipate the annual tuition to be $60,000 per year for the four years of college. You plan on making equal deposits on your child’s birthday every year starting today, the day of your child’s birth. No deposits will be made after starting college. The first tuition payment is due in exactly 18 years from today (the day your child turns 18 – no deposit required, i.e. last deposit is on 17^th birthday). Assume the annual expected return on your investments is 10% over this period.

Calculate the annual deposit.

Calculate the amount needed if only equal annual deposits are made on birthday’s 5-10 inclusive.

Calculate the amount needed if two equal annual deposits are made on birthday’s 5 and 13.

Answer part (i), now assume tuition rises 10% per year.

Answer part (i) assuming first deposit will be made on your child’s 1^st birthday. All other information is the same. What is the annual tuition payment? How does it compare to part (i)? Is your answer surprising?

show work

Answer 1

annual tuition fee , T = $60,000

expected return on investment , r = 10% = 0.10

no.of years for which the fee will be paid , n = 4

value of the tuition fee at the end of 17 years = T*PVIFA

PVIFA = present value interest rate factor of annuity= ((1+r)ⁿ-1)/((1+r)ⁿ*r) = ((1.10)⁴-1)/((1.10)⁴*0.10) = 3.1698654

value of the tuition fee at the end of 17 years, V = T*PVIFA = 60,000*3.1698654 = $190191.926781

present value of tuition fee,(PV) = V/(1+r)¹⁷ = 190191.926781/(1.10)¹⁷ = 190191.926781/5.054470285 = 37628.45878

(i) Let A be the Annual deposit

PVIFA1 = ((1+r)ⁿ-1)/((1+r)ⁿ*r) = ((1.10)¹⁷-1)/((1.10)¹⁷*0.10) = 8.0215533

PV = A+ (A*PVIFA1) = A*(1+PVIFA1) = A*(1+8.0215533) = A*9.0215533

A = 37628.45878/9.0215533 = 4170.951219 or $4170.95 ( rounding off to 2 decimal places) or $4171 ( rounding off to nearest dollar value)

(ii) present value of tuition fee at the end of 4 years, P4 = V/(1+r)¹³ = 190191.926781/(1.1)¹³ = 190191.926781/3.452271214 = $55091.826502

no. of years of equal annual deposit = n = 6

let the annual deposit then = A

PVIFA =  ((1+r)ⁿ-1)/((1+r)ⁿ*r) = ((1.10)⁶-1)/((1.10)⁶*0.10) = 4.3552607

A = P4/PVIFA = 55091.826502/4.3552607 = 12649.489963 or $12,649.49 ( rounding off to 2 decimal places) or $12,649

(iii)

if only 2 equal installments are made at the end of 5th and 13th year

let A = equal deposit

P4 =[ A/(1+r) ] + [A/(1+r)⁹] = [ A/(1.10) ] + [A/(1.10)⁹] = 0.909091A + 0.424098A = 1.33318853A

55091.826502 = 1.33318853A

A = 55091.826502/1.33318853 = $41323.35778 or $41,323.36 ( rounding off to 2 decimal places) or $41,323 ( rounding off to nearest dollar value)

(iv) if tuition fees rises 10% every year, growth rate , g = 10% = 0.10

tuition fee in at the end of 18 years, c1 = 60,000

tuition fee in at the end of 19 years , c2= 60,000*(1+g) = 60,000*1.10 = 66000

tuition fee in at the end of 20 years, c3 = 66,000*(1+g) = 66,000*1.10 = 72,600

tuition fee in at the end of 21 years, c4 = 72,600*(1+g) = 72,600*1.10 = 79,860

value of the tuition fee at the end of 17 years, V = [c1/(1+r)] + [c2/(1+r)²] + [c3/(1+r)³] + [c4/(1+r)⁴]

V =  [60,000/(1.10)] + [66000/(1.10)²] + [72,600/(1.10)³] + [79,860/(1.10)⁴] = $218181.818182

present value of tuition fee,(PV) = V/(1+r)¹⁷ = 218181.818182/(1.10)¹⁷ = 218181.818182/5.054470285 = 43166.10958

Let A be the Annual deposit

PVIFA1 = ((1+r)ⁿ-1)/((1+r)ⁿ*r) = ((1.10)¹⁷-1)/((1.10)¹⁷*0.10) = 8.0215533

PV = A+ (A*PVIFA1) = A*(1+PVIFA1) = A*(1+8.0215533) = A*9.0215533

A = 43166.10958/9.0215533 = $4784.776 or $4784.78 ( rounding off to 2 decimal places) or $4785 ( rounding off to nearest dollar value)