Question

Antonia receives a $10,000 benefit payment at the end of each year. She can invest these payments in an account yielding 4% interest, compounded annually. Assuming she just received this year’s payment, what is the present value of her next five payments?

Answer 1

$44, 520.

Explanation:

Because the equal payments occur at the end of each year, we know we have an ordinary annuity.

The equation for calculating the present value of an ordinary annuity is:

PVOA = FV {[1 - (1 / (1 + i)ⁿ)] / i}

PVOA = $10,000 {[1 - (1 / (1 + 0.04)⁵)] / 0.04}

PVOA = $10,000 {4.452}

Here PVOA Factor is 4.452 for n = 5 and i = 4%

PVOA = $44,520

This PVOA calculation tells you that receiving $44,520 today is equivalent to receiving $10,000 at the end of each of the next five years, if the time value of money is 4% per year. If the 4% rate is Antonia's required rate of return, this tells you that Antonia could pay up to $44,520 for the five-year annuity.

1st Year:

PV = $10,000 [1 / (1 + 0.04)]

PV = $10,000 [1 / 1.04]

PV = $10,000 [0.962]

Here PV Factor is 0.962 for n = 1 and i = 4%

PV = $9,620

$10,000 at the end of First year has a present value of $9,620

2nd Year:

PV = $10,000 [1 / (1 + 0.04)²]

PV = $10,000 [1 / (1.0816)]

PV = $10,000 [0.925]

Here PV Factor is 0.925 for n = 2 and i = 4%

PV = $9,250

$10,000 at the end of Second year has a present value of $9,250

3rd Year:

PV = $10,000 [1 / (1 + 0.04)³]

PV = $10,000 [1 / 1.1249]

PV = $10,000 [0.889]

Here PV Factor is 0.889 for n = 3 and i = 4%

PV = $8,890

$10,000 at the end of Third year has a present value of $8,890

4th Year:

PV = $10,000 [1 / (1 + 0.04)⁴]

PV = $10,000 [1 / 1.1699]

PV = $10,000 [0.855]

Here PV Factor is 0.855 for n = 4 and i = 4%

PV = $8,550

$10,000 at the end of Fourth year has a present value of $8550

5th Year:

PV = $10,000 [1 / (1 + 0.04)⁵]

PV = $10,000 [1 / 1.2167]

PV = $10,000 [0.822]

Here PV Factor is 0.822 for n = 5 and i = 4%

PV = $8,220

$10,000 at the end of Fifth year has a present value of $8,220

The total of those five present values:

$9,620 + $9,250 + $8,890 + $8,550 + $8,220

= $44,520

The difference between the $50,000 of total future payments and the present value of $44,520 is the interest Antonia money earns while she wait to receive the payments. This $5,480 difference is referred to as Antonia's return on its investment.

Answer 2

Answer: $44,518

Explanation:

Using the formula:

PV = P(
`(1-(1+r)^(-n) )/(r)` )

PV = present value

P = Principal

r = rate

n = number of periods

From the question :

PV = ?

P = $10,000

r = 0.04

n = 5

Substituting into the formula , we have

PV = 10,000 (
`(1-(1+0.04)^(-5) )/(0.04)` )

PV = 10,000 (
`(1-0.8219271068)/(0.04)` )

PV = 10,000 (
`(0.1780728932)/(0.04)`

PV = 10,000 ( 4.451822331 )

PV = 44518.22331

Therefore :

PV≈ $ 44,518