annuities

28. Annuity Present Values – what is the present value of an annuity of $5,000 per year, with the first cash flow received three years from today and the last one received 25 years from today? Use a discount rate of 8 percent.

Answer:

T = 25 year – 2 year = 23 year

PVA = C (1-1/(1+r)^t / r )

PVA = $5,000 (1 – 1/(1,08)²³/0.08)

PVA = $5,000 (1 – 1/5.87146 / 0.08)

PVA = $5,000 ( 1 – 0.17032 /0.08)

PVA = $5,000 (0.82968 / 0.08)

PVA = $5,000 (10.371)

PVA = $51855.29

This is the value of the annuity one period before the first payment, or year 2. So, the value of the cash flows today is:

PV = FV/(1+r)^t

PV = $51855.29 / (1+0.08)²

PV = $44457.56

Jihan & 17 years from today

Zafran & 15 years from today

52. Annuities – You are saving for the college education of your two children. They are two years apart in age; one will begin college 15 years from today and the other will begin 17 years from today. You estimate your children’s college expenses to be $ 35,000 per year per child, payable at the beginning of each school year.  The annual interest rate is 8.5 percent. How much money must you deposit in an account each year to fund your children’s education? Your deposit begin one year from today. You will make your last deposit when your oldest child enters college. Assume four years of college.

Answer:

First, we will calculate the present value of the college expenses for each child.

Total expenses: $35,000 x 2 = $70,000

T = 4 years of college

PVA = C (1-1/(1+r)^t / r )

PVA = $70,000 ( 1- 1/ (1+0.085)⁴ / 0.085)

PVA = $ 70,000 (1 – 1/1.3858 / 0.085)

PVA =$70,000 (1 – 0.72160 /0.085)

PVA = $70,000 (0.2784 / 0.085)

PVA = $70,000 (3.2753)

PVA = $229270.59 total expenses for two Childs

@ = $229270.59 /2

@=$114635.29 expenses for each child.

This is the cost of each child college expenses one year before they enter college. So, the cost of the oldest child’s college expenses today will be:

PV = FV/(1+r)¹⁴

PV = $114635.29 / 1.085¹⁴

PV = $114635.29 / 3.1334

PV = $36584.95

And the cost of the youngest child college expenses today will be:

PV = FV/ (1+r)¹⁶

PV = $114635.29 / 1.085¹⁶

PV = $114635.95 / 3.6887

PV = $31077.43

Therefore, the total cost today of your children’s college expenses is:

Cost today = $36584.95 + $31077.43

Cost today = $67662.38

This is the present value of your annual savings, which are an annuity. So, the amount you must save each year will be:

PVA = C (1-1/(1+r)^t / r )

$67662.38 = C (1 – 1/(1.085)¹⁵ /0.085)

$67622.38 = C (1 – 1/ 3.3997 / 0.085)

$67622.38 = C (0.7059 / 0.085)

$67622.38 = 8.3047 C

C =$ 8142.66

Note:

“An annuity is a level stream of regulars payments that last for a fixed number of periods. Not surprisingly, annuities are among the most common kinds of financial instruments. The pensions that people receive when they retire are often in the form of an annuity. Leases and mortgages are also often annuities…”

Reference: Corporate Finance Book, Stephen A.Ross, Randolph W.Westerfield and Jeffrey Jaffe, Ninth Edition. Chapter 4, questions number 28 & 52, page 126 & 129.