Learning outcomes. After watching this video you will be able to explain the difference between ordinary annuity and annuity due. Calculate the present value of an ordinary annuity. Ordinary annuities. In this video we show how to calculate the present and future values of identical periodic cash flows which are called annuities. An annuity is a stream of identical cash flows Made or received, separated by equal intervals of time. A $10 a month payment for a year is an example of an annuity. However, a $10 monthly payment that occasionally skips a month or a stream of payments that alternates between 10 and $20 is not an annuity. Both the amount and the time interval between cash flows must be the same for it to be called an annuity. There are two types of annuities. Ordinary annuity and annuity due. An ordinary annuity is one where the cash flows occur at the end of each time period. And annuity due is one where the cash flows occur at the start of each time period. The figure that you see compares a four year $1,000 a year ordinary annuity to a four year $1,000 a year annuity due. As you can see in the ordinary annuity, the payments occur at years one, two, three and four. Whereas with an annuity due, the payments occur at years zero, one, two, and three. Let us compute the future value after four years of the four year $1000 a year ordinary annuity. Let's say that the interest rate is 10% a year. The payment at the end of the first year has to be moved from year 1 to year 4. So its future value is 1,000(1 + 0.1) the whole thing raised to the power of 3, which equals $1,331. The payment at the end of the second year has to moved forward from year 2 to year 4. So its future value is 1000 times 1 plus 0.1. The whole thing ratio to the power of 2, which equals $1,210 dollars. The payment at the end of the third year has to be moved from year 3 to year 4. So it's future value is 1000 times 1 plus 0.1 which equals $1,100 dollars. The payment at the end of the fourth year is already at the end of the fourth year, so it's future value is simply $1,000. So, the future value of the fourth year ordinary annuity is 1,331 plus 1,210, plus 1,100, plus 1,000 which equals $4,641. To get a more general form, let us denote the Future Value of an Ordinary Annuity as FVA, the Periodic Payment by PMT, and the Interest rate as r. Then we have FVA sub 4 = PMT x (1 + r) raised to the power of 3 plus PMT times 1 plus r raised to the power of 2. Plus PMT times 1 plus r raised to the power of 1 plus PMT. This can be extended to an n-payment ordinary annuity, in which case, we have FVA sub n, which equals PMT times 1 plus R raised to the power of n minus 1. Plus PMT times 1 plus r raised to the power of n minus 2, so on plus PMT. This can then further be simplified to FVA sub n is equal to PMT times within square brackets, one plus r, the whole thing raised to the power n minus 1. The whole thing divided by r. What if we want to calculate the present value of an ordinary annuity? We have calculated the future value of the ordinary annuity. Essentially, we can order the stream of cash flows into a single lump sum number. We can now move the single lump sum from the future to the present by discounting it back n years. Denoting the present value today of an ordinary annuity as PVA sub 0. We have PVA sub zero equals FVA sub n divided by one plus r raised to the power of n. What we already have a formula of FVA sub n, which we can now plugin for. So now PVA sub zero is equal to PMT divided by r times within square brackets 1 minus 1 over (1 + r ) raise to the power of n. Let's look at an example of how you would use the formulas. Say you have your heart set on buying this beautiful condo. Its current price is $500,000, all of which you will borrow from the bank. The bank will charge you an interest rate of 10% per year and the loan will be for 30 years. Assuming that you make annual payments, at the end of each year, how much will you repay the bank each year? To start off, I hope you recognize that this is an ordinary annuity. As the payments are made at the end of each year. Since you borrow $500,000 today, PVA sub 0 is equal to 500,000. It is a 30-year loan with annual payments, so there'll be 30 payments in this ordinary annuity. So n is 30, r is given to be 10%. Now we're interested in calculating the annual payment that is PMT. We will use the formula PVA sub 0 is equal to PMT divided by r times 1 minus 1 over (1 plus r) the whole thing raised to the power of n. Plugging in all the non values. We have 500,000 equals PMT divided by 0.1 times within square brackets, 1 minus 1 over 1 plus 0.1 raised to the power of 30. We need to solve for PMT which comes out to be $53,039.62. That is you will pay $53,039.62 every year for 30 years to pay off the loan. In the next video, we will talk about the concept of compounding frequency and effective annual rates and how it impacts present and future value calculations.