in this video I discuss perpetuities and how the valuation of a perpetuity can be used to derive the valuation of an annuity before watching this video you should already be familiar with the basic concepts of present value and future value let's start with a simple example let's say that you are able to invest at a risk-free annual rate of 5% forever if you invest $100 in this account you will receive a $5 return each year forever now let's put these $5 interest payments on a timeline note that these payments last forever assuming the 5% interest rate continues to Prevail into the future what is the present value of the $5 interest payments we could write this as a standard present value problem as I've done here on the left hand side of the equation I have the present value of the perpet it PVP for short on the right hand side I am discounting all future $5 payments at a 5% interest rate but the summation never ends so this is a calculation that I need to think about carefully on some reflection you might realize that the promise of these $5 payments forever when interest rates are 5% is worth $100 why because if I invest $100 today at a 5% interest rate I will generate an infinite stream of $5 payments each year this logic is sound but how can I prove this mathematically there's a clever trick that we can use to solve the problem mathematically let's start with the infinite sum I can multiply both sides of this equation by 1.05 or 1 plus the 5% interest rate notice how things cancel on the right hand side of the equation for each $5 payment the discounting is reduced by one year or in other words the exponent associated with 1.05 in the denominator of each term is reduced by one I end up with 1.05 * PVP on the left hand side of this equation on the right hand start with the $5 notice the remaining portion of the right hand side of the equation is the infinite summation of $5 discounted at 5% Forever This is where the clever trick comes into play notice that this summation is the same problem that we started with so we can merely substitute our abbreviation for the present value of a perpetuity or PVP for this infinite summation next subtract PVP from both sides after doing so we are left with 05 * PVP which equals $5 divide both sides by the interest rate or 5% this leaves you with the $5 annual payment divided by the interest rate of 5% which is $100 of course as we discussed before the valuation of this perpetuity makes sense because an investment of $100 today will generate a $5 payment forever now what about the more General case where we have a perpetuity that pays a constant cash flow C forever when you have an opportunity to invest at an interest rate R forever in general the present value of the perpetuity will be simply C the p periodic cash flow divided by R the prevailing interest rate as is shown on the board now I realize that the assumption that you are earning a constant interest rate on your Investments forever is not realistic anyone who pays passing attention to financial Market knows that the amount of Interest offered on a savings account charged for a home mortgage or a car loan changes over time nonetheless this simple example will be useful in thinking about the general problem of how to value a long lived or in this case infinite cash flow stream if you continue to study Finance this simplification will be quite useful for thinking about a number of valuation problems including how economists think about the valuation of stocks moreover we can use the perpetuity valuation formula to Value an annuity which has many real world applications let's start with the perpetuity we just discussed which has a constant cash flow C forever we now know the present value of this cash flow is C / R the prevailing interest rate recall that an annuity is a fixed number of cash flow payments over a fixed number of years as depicted on the timeline on the bottom the present value of an annuity is provided by the formula behind me while it may not be obvious yet we can derive the valuation of an annuity using the formula to value a perpetuity to see this let's engage in the following thought experiment let's start with a perpetuity where the First Cash payment occurs one year from today we will label this perpetuity with a red a on the board we are now going to subtract the cash flows associated with a second perpetuity B but the second perpetuity does not begin paying the promised cash flow C until your t+ one is depicted on this timeline you can think of this as entering one contract call it a where you buy the right to receive a constant cash payment forever beginning next year and a second contract call it B where you promise to pay a constant cash payment C forever beginning in year t+1 now note what happens when we calculate the difference between a and b or a minus B the resulting cash flow stream is that of an annuity with the First Cash Flow being being received in year one and the last cash flow being received in year t to see this let's look at the cash flows in a few years in year zero neither a nor B promises a cash flow so the difference is zero in year one you begin receiving your promised cash flow C from a but are not yet obligated to begin paying the cash flow C from B thus the difference of a minus B is merely C in year 1 this calculation is the same from year 2 through year t however in your t+1 things change though you continue to receive a payment from a you are now also obligated to begin paying a cash flow c as promised on the second perpetuity B thus the difference between the two perpetuities or a minus B is zero beginning in your t+ one and in every remaining year now let's pause and think about the last row here note that a minus B yields a cash flow payment C that begins in one year and ends in year T this is precisely how we Define a fixed annuity which makes a promise cash flow payment C for a fixed number of periods T so we should be able to Value this annuity by thinking about the valuations of the two perpetuities that generate the cash flow as a starting point let's value the two perpetuities A and B in this problem we know that the present value of perpetuity a is merely C over R the present value of perpetuity B requires two calculations first note that in your T the future value of the perpetuity B is C over R why because when I am in your T standing in your T I know that perpetuity B will begin paying in one year or your t+ one I also know that the value of a perpetuity that begins paying in one year is merely C over R it's important to note that the perpetuity formula values of perpetuity one year prior to the First Cash payment the second step in this calculation is to calculate the present value of the perpetuity by discounting the value of the perpetuity in Period T and additional T periods or dividing by 1 + r raised to the^ of T we now have the present value of our second perpetuity b as of period zero as is depicted on the board now we can pull this all together to calculate the value of an annuity if the difference in the cash flows between a and b in each year represent that of an annuity it follows that the difference in the present values of A and B represent the present value of an annuity thus we start with the present value of perpetuity a c over R we subtract the present value of perpetuity B which yields the present value of a minus B now this is not quite the annuity valuation formula that I presented earlier but if we simplify the resulting formula by factoring out C over R we end up with a simplified version of the annuity formula to sum up you should now be familiar with evaluation of a perpetuity and an annuity in financial economics the perpetuity formula has wide applications for developing theories about how we should value long lipped Investments like stocks the annuity formula has wide practical applications take a few minutes think about how you might be able to use these formula to analyze financial decisions that many people face in their lives