[MUSIC] Last time, we saw how to calculate the present and future values of a single lump sum cash flow. What if you have a stream of cash flows, especially the same cash flow occurs every period. In this video, we will look at what are called annuities. We'll also distinguish between ordinary annuity and annuity due. Further we will see how to calculate the present and future values of an ordinary annuity. Let's start off by defining what an annuity is. An annuity is a stream of identical cash flows made or receive separated by equal intervals of time. A $10 a month payment for a year is an example of annuity. However, $10 monthly payments that occasionally skip a month on a stream of payments that alternates between $10 and $20 payments is not an annuity. Both the amount and the time interval between the 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. 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 full year, $1,000 a year annuity due. Let us compute the future value after four years of the four year $1,000 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 one to year four. So it's future value is 1000(1 + 0.10) the whole cubed which equals $1,331. The payment at the end of the second year has to be moved from year two to year four. It's future value is 1,000 times (1 + 0.10) the whole squared. Which equals $1,210. The payment at the end of the third year has to be moved from year three to year four. So, its future value is 1,000 times 1 + 0.10. Which equals $1,100. The payment at the end of the fourth year has to be moved from year 4 to year 4. So its future value is 1,000 (1 + 0,1), the whole raised to the power of 0 which simply equals $1,000. So the future value of the 4 year ordinary annuity is 1,331 + 1,210 + 1,100 + 1,000, = $4,461. To get a more general form let's denote the future value of a ordinary annuity as FVA. The periodic payment by PMT, and the interest rate as r. Then we have FVA4 = PMT times (1 + r) raised to the power of 3 + PMT times (1 + r) raised to the power of 2 + PMT times ( 1 + r) raised to the the power of 1 + PMT times (1 + r) raised to the power of 0. For a n-payment ordinary annuity we have FVA sub n = PMT times (1 + r) raised to the power of n- 1 + PMT times (1 + r) raised to the power n- 2 + so on. + PMT times (1+r) raised to the power of 0. This can be simplified to FVA sub n = PMT times ((1+r) whole raised to the power of n- 1) / r. What if you want to calculate the present value of an ordinary annuity. We have already calculated the future value of an ordinary annuity. Essentially we can merger a stream of cash flows into a single lump sum number. We can now move the single lump sum from the future back to the present by discarding it back by any others. Denoting the present value today of an ordinary annuity as PVA sub 0. We simply have PVA sub 0 = FVA sub n / (1 + r) the whole raised to the power of n. But we know that FVA sub n = PMT times 1 + r to the power of n- 1 / r. So, we now have PVA sub 0 = PMT / r times [1-1 / (1+r) the whole raised to the power of n]. Let's look at an example of how you would use these formulas. Say you have your heart set on buying this beautiful condo. Its current price is $500,000 all of which you plan to 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 of, 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 = $500,000. It is a 30-year loan with annual payments. So, there will be 30 payments in this ordinary annuity, little, and easy called 30. r is given to be 10%. We're interested in calculating the annual payment that is PMT. We will use the formula PVA sub 0 = PMT / r times [1- 1 divided by (1 + r) raised to the power of n]. Plugging all known values, we have 500,000 = PMT / 0.10. Times 1- 1 / (1 + 0.10) raised to the power of 30. We need to solve for PMT which comes out to be $53,039.62. You will pay $53,039.62 each year for the next 30 years to pay off the loan. Until now, we will assume that interest is paid only once a year. In reality, this is usually not the case. Banks typically pay or charge interest once a month. In the next video, we will talk about the concept of compounding frequency and effective annual rates. And how it impacts present value and future value calculations. [MUSIC]
