2 Sums and Approximations or just buy something fun. However, if you don't get the $20,000 for another 50 years then someone else is earning all the interest or investment profit. Furthermore, prices are likely to gradually rise over the next 50 years, so you when you finally get the money, you won't be able to buy as much. Finally, people only live so long, if you were 60 years old, a payout 50 years in the future would be worth next to nothing! But what if your choice were between $40,000 a year for 50 years and a million dollars today? Now which is better? What is an annuity is actually worth? 1.1 The Future value of Money In order to address such questions, we have to make an assumption about the future value of money. Let's put most of the complications aside and think about this from a simple-minded perspective. The average rate of inflation in the United States from 1980 to 2004 was about p= 3.5% per year. This means that the price of a selection of basic goods increases by about 3.5% each year. If this trend continues, then goods costing $100 today will cost 81001+p)=810350in1year 1001+p)2=81072in2year S100(1+p) in k years Looked at another way, k years from now, $100 will have the buying power of just 100/(1+ p) dollars today. Now we can work out the value of an annuity that pays m dollars at the start of each ear for the next n years payments current value Sm today y Sm in 1 vear t p Sm in 2 years Sm in n- l years (1+p)n- Total current value: V (1+p)2 Sums and Approximations or just buy something fun. However, if you don’t get the $20,000 for another 50 years, then someone else is earning all the interest or investment profit. Furthermore, prices are likely to gradually rise over the next 50 years, so you when you finally get the money, you won’t be able to buy as much. Finally, people only live so long; if you were 60 years old, a payout 50 years in the future would be worth next to nothing! But what if your choice were between $40,000 a year for 50 years and a million dollars today? Now which is better? What is an annuity is actually worth? 1.1 The Future Value of Money In order to address such questions, we have to make an assumption about the future value of money. Let’s put most of the complications aside and think about this from a simple­minded perspective. The average rate of inflation in the United States from 1980 to 2004 was about p = 3.5% per year. This means that the price of a selection of basic goods increases by about 3.5% each year. If this trend continues, then goods costing $100 today will cost: $100(1 + p) = $103.50 in 1 year $100(1 + p) 2 = $107.12 in 2 year . . . $100(1 + p) k in k years Looked at another way, k years from now, $100 will have the buying power of just 100/(1+ p)k dollars today. Now we can work out the value of an annuity that pays m dollars at the start of each year for the next n years: payments current value $m today m $m in 1 year m 1 + p $m in 2 years m (1 + p)2 · · · · · · $m in n − 1 years m (1 + p)n−1 �n−1 m Total current value: V = (1 + p)k k=0