Time Value of Money Part IX
Present Value of Annuities – Shortcuts on the Road Trip to Annuities
In the last post, we talked about when I went on a road trip with my buddy Steve to San Diego over spring break. When asked “What are annuities?” we can now say that annuities are payments of a set amount of money that occur on a routine basis such as every year. In this lesson, we’ll use a shortcut method to calculate the present value of annuities.
To refresh on the previous example, instead of paying me $400 right away for the road trip gas money, we said that my friend would pay me $150 starting in one year for a total of three years. To figure out how much this deal was worth today (the present value of annuities), we calculated the present value of each of the three $150 payments. Remember, the first $150 payment would come 1 year from today, another payment 2 years from today and the final $150 would be three years from now. Since the first payment comes one year from today, this is referred to as an “ordinary annuity.”
After adding up how much each of the payment is worth to us today, we got a total present value of the annuity of $416.25. If you’d like to review how we did that, it’s all in the last annuities lesson. While this way is perfectly fine to calculate the present value of an annuity, there are a couple of shortcuts that we can use which we’ll go through below.
Shortcuts to Calculate the Present Value of Annuities
One way that we can quickly calculate the how much an annuity is worth to us today is to use a table of annuity factors. I created one which you can take a look at below:
Let’s work through this problem again using the table instead of having to do calculations for each of the payments. Using this method, we need to know how much money we’re going to get for each payment, the number of times we get the payment and the interest rate that we could get in our savings account. Using the annuity example above, we said that we were going to get $150 for each payment and we’d get this 3 times starting in one year. We also said that the interest rate in our savings account was 4%.
Since we know the annuity has a total of 3 periods and the interest rate we could save at is 4%, we can go into the Annuity Factors Present Value Table and find the annuity factor. For this case, we find the period of 3 and then go to the right until we’re in the 4% column which gives us the annuity factor of 2.7751. To find how much the annuity is worth to us today, all we have to do is multiply the value of each payment ($150) by the annuity factor (2.7751). We’ll do this below:
Present Value of Annuity = Value of Each Payment x Present Value Annuity Factor
Present Value of Annuity = $150 x 2.7751 = $416.26
Notice that this is the same value that we got in the last lesson (off by one cent due to rounding error)? As long as we know how many years we’re going to get the payment, what the interest rate is in our savings account and the amount of each payment, we can use the table to calculate the present value of the annuity. Let’s try one more example.
Finding Value in the Present Value Annuities Table
Let’s say that you just sold your car to your uncle and he has promised to pay you $1200 per year for 7 years starting in one year. We’ll also say that you could currently save in your savings account at 3%. Using the Annuity Factors Present Value Table, we can find that the annuity factor is 6.0021 by looking up 7 periods with 3% interest. To find the present value of the ordinary annuity, we multiply the annuity factor by the amount we get each period:
Present Value of Annuity = $1200 x 6.2303 = $7476.36
With this arrangement, your uncle paying you $1200 for 7 years (starting in one year) is the same as him giving you $7476.36 today.
Key Takeaway: We can find the present value of annuities by multiplying the annuity factor by how much money we’ll get each period.
In the next small business financial management lesson, we’ll talk about how to find the future value from an annuity.