Time Value of Money Part VIII
So What Are Annuities? Small Business Financial Management Lesson
To answer the question of “What are annuities?” we’ll start off with a quick example. For my junior year of undergrad, my friend Steve and I didn’t have any plans for spring break. While a ton of our friends were flying to tropical locations, we decided that we were going to take a road trip in my little Ford Ranger. Instead of planning on where we were going, we decided to just start driving and see where we ended up. While starting in Minneapolis, within five days we ended up about 2000 miles away in San Diego and were able to put our feet in the Pacific Ocean.
After everything was said and done, we ended up putting on about 4500 miles on my truck on the trip which was about $800 in gas. Let’s say that my travel buddy didn’t plan that we’d end up all the way in San Diego and didn’t have enough money to cover his portion of the gas bill. Instead of giving me the $400 today, we’ll say that he offered to give $150 starting in one year and do that for the next three years. So breaking this up, he’d give me $150 in one year, $150 in two years and $150 in three years. In financial terms, what he offered me was an annuity over the next three years. This means that he’ll give me equal payments each year for a certain period of time.
We’ll define an annuity as a fixed amount of money that is distributed on a routine basis such as every year.
Splitting Up the Value of an Annuity
As we know from the time value of money post, getting money in the future is worth less to us than getting it now. So, to calculate what this annuity is worth to us now, we could break the three payments up and calculate the present value for each. For this example, let’s say that I could save at 4% in a savings account. Let’s go through that exercise below. To calculate how much the payment in a year is worth to us, we’ll use the formula that we’ve covered in the last lesson:
Present Value = Future Value / ( 1 + Interest Rate ) ^ Number of Years
So for the first payment, the future value is $150, the interest rate is 4% and the number of years is 1:
Present Value in 1 Year= $150 / (1 + 4%) ^ 1 = $150 / (1.04) ^ 1 = $144.23
So from this example, the value of first $150 in a year is worth $144.23 to us today. Now let’s calculate the value of the payments of $150 in years 2 and 3:
Present Value in 2 Years = $150 / (1.04) ^ 2 = $138.68
Present Value in 3 Years = $150 / (1.04) ^ 3 = $133.34
Notice how the present value goes down as the time until we’ll get the payment increases? This makes sense since money is worth more to use today than getting it tomorrow, and worth more to us getting it tomorrow rather than 2 days from now. So to calculate the full value of this annuity (in this case my friend paying me $150 for three years), all we have to do is add up the three present values that we calculated above.
Present Value of Annuity = $144.23 + $138.68 + $133.34 = $416.25
So by my friend offering to give me $150 starting in a year for the next three years, it’s actually worth $416.25 to me today.
Key Takeaway: When asked “What are annuities?” we can now reply that they are equal payments at set intervals for a certain period of time. To calculate the present value of an annuity, all we have to do is calculate the present value of each payment using methods we’ve already learned and then add them all together.
In the next lesson, we’ll cover some shortcuts on how to calculate the present value of an annuity.