Question

Find the future values of these ordinary annuities. Compounding occurs once a year. Do not round intermediate calculations. Round your answers to the nearest cent.

Find the future values of these ordinary annuities. Compounding occurs once a year. Do not round intermediate calculations. Round your answers to the nearest cent.
a $500 per year for 6 years at 8%.
b $250 per year for 3 years at 4%.
c $1,000 per year for 2 years at 0%.
d Rework parts a, b, and c assuming they are annuities due.
Future value of $500 per year for 6 years at 8%: $
Future value of $250 per year for 3 years at 4%: $
Future value of $1,000 per year for 2 years at 0%: $

Answer 1

Alright, let's take this step by step.

First, let's understand what an ordinary annuity is. An ordinary annuity is a series of equal payments made at the end of consecutive periods over a fixed length of time. For example, if you save $100 every year for 5 years, that’s an ordinary annuity.

Now, let’s understand the formula to calculate the future value (FV) of an ordinary annuity:

FV = P x ((1 + r)^n - 1) / r

Where:

- FV is the future value of the annuity.

- P is the payment per period (how much you save each time).

- r is the interest rate per period (in decimal form).

- n is the number of periods (how many times you save).

Let’s solve each part:

a) $500 per year for 6 years at 8%.

P = 500, r = 8% = 0.08, n = 6

FV = 500 x ((1 + 0.08)^6 - 1) / 0.08

≈ 500 x (1.59385 - 1) / 0.08

≈ 500 x (0.59385) / 0.08

≈ 500 x 7.4231

≈ 3701.55

So, the future value of $500 per year for 6 years at 8% is about $3,701.55.

b) $250 per year for 3 years at 4%.

P = 250, r = 4% = 0.04, n = 3

FV = 250 x ((1 + 0.04)^3 - 1) / 0.04

≈ 250 x (1.12486 - 1) / 0.04

≈ 250 x (0.12486) / 0.04

≈ 250 x 3.1215

≈ 780.38

So, the future value of $250 per year for 3 years at 4% is about $780.38.

c) $1,000 per year for 2 years at 0%.

P = 1000, r = 0% = 0.00, n = 2

FV = 1000 x ((1 + 0.00)^2 - 1) / 0.00

= 1000 x (1 - 1) / 0.00

= 1000 x 0

= 0

Wait, something went wrong, because we know that if we save $1000 for 2 years with no interest, we should have $2000. This is a special case, where we just sum the contributions because there's no interest:

FV = 1000 x 2

= 2000

So, the future value of $1,000 per year for 2 years at 0% is $2,000.

Now, for annuities due:

An annuity due is similar to an ordinary annuity, but the payments are made at the beginning of each period instead of the end. To convert the future value of an ordinary annuity to an annuity due, you can use the following formula:

FV of Annuity Due = FV of Ordinary Annuity x (1 + r)

a) Reworked

FV of Annuity Due = 3701.55 x (1 + 0.08)

≈ 3701

.55 x 1.08

≈ 3997.67

b) Reworked

FV of Annuity Due = 780.38 x (1 + 0.04)

≈ 780.38 x 1.04

≈ 810.80

c) Reworked

FV of Annuity Due = 2000 x (1 + 0.00)

= 2000 x 1

= 2000 (This doesn't change because there's no interest).

And there you have it! The future values for both ordinary annuities and annuities due!