Time Value of Money Part XI
Using the Annuities Due Formula
In the previous Future Value of Annuities lesson, we calculated the how much I would have in four years if I saved $1000 at the end of every year. We were able to find that I’d have $4246.50 saved up for the Ford Ranger truck I wanted to buy using the future value annuities formula. Since the first payment started at the end of the year, we called this an ordinary annuity. In this small business financial management lesson, let’s say that I was able to save for four years and could put the first $1000 in the savings account today. Since I can now deposit the money at the beginning of the period instead of the end, this is called an annuity due. With a couple of tweaks of the annuities formula, we can calculate how much we’ll have by saving $1000 for four years using the annuities due formula. As in the last example, we’ll say that I could save at 4% interest in my savings account.
For an ordinary annuity, we used the formula below:
Future Value of Ordinary Annuity = Value of Each Payment x Future Value Annuity Factor
Since this example is for an annuity due (meaning we are able to put the first payment in today instead of at the end of the year), we need to change it up a bit as shown below:
Future Value of Annuity Due = (1 + Interest Rate) x Value of Each Payment x Future Value Annuity Factor
So, to solve the annuities due formula how much we’ll have in 4 years (the future value of the annuity due), we just need the interest rate, how many years we’re planning on saving and how much we’ll save each time. For the future value annuity factor, we’ll use the Annuity Factors Future Value Table like we did in the last lesson. Since we’ll save for four years (meaning four periods) and can save at 4% interest rate, we can look in the table and find that our annuity factor is 4.2465 (same as the last example). Now all we have to do is plug what we know into the annuities due formula to calculate how much we’ll have in our savings account in four years:
Future Value of Annuity Due = (1 + 4%) x $1000 x 4.2465 = (1.04) x $1000 x 4.2465 = $4416.36
So if we start saving today and deposit $1000 into our savings account every year for four years, we’ll have $4416.36 in four years to spend on the Ford Ranger.
Annuities Due Formula to Calculate the Present Value
In the present value of annuities lesson, we covered an example where my friend couldn’t pay for all the gas for our spring break road trip. Instead of paying the $400 up front, he offered to pay $150 starting in one year for a total of three years. Since he gave each payment at the end of the year, this was an example of an ordinary annuity. Let’s pretend that instead he would pay at a beginning of each year and his first $150 payment would be today. In this case, we want to figure out how much these three payments are worth to us today if we can save at 4% in a savings account. Since he makes the first payment starting today instead of at the end of the year, this is another example of an annuities due. From our previous example, we showed that the present value of an ordinary annuity formula was:
Present Value of Ordinary Annuity = Value of Each Payment x Present Value Annuity Factor
Like in the future value example above, we need to make a little change to make it into the annuities due formula:
Present Value of Ordinary Annuity = (1 + Interest Rate) x Value of Each Payment x Present Value Annuity Factor
For this example, we know that he will give us three payments at the beginning of each period and the interest rate is 4%. Using the Annuity Factors Present Value Table, we can find that the annuity factor we need is 2.7751. Since we know that each payment will be $150 and the interest rate is 4%, we can plug in the rest of the values into the annuities due formula to solve:
Present Value of Ordinary Annuity = (1 + 4%) x $150 x 2.7751 = (1.04) x $150 x 2.7751 = $432.92
Key Takeaway: While an ordinary annuity pays the first payment at the end of the year, an annuity due pays at the beginning of the year. To calculate the present or future value of an annuity due, we can use the annuities due formulas listed in this lesson.
In the final lesson for the time value of money series, we’ll finish talking about getting paid money every year for the rest of your life. Sounds pretty good right?