Asymptotic behavior of sequences UIP exercise lesson Angelo Lucia - PowerPoint PPT Presentation

Asymptotic behavior of sequences UIP exercise lesson Angelo Lucia September 29th, 2017

A document from 1648 Figure 1: Yale’s Beinecke Rare Book & Manuscript Library This is a bond from the Dutch water board Stichtse Rijnlanden, issued on May 15th 1648. In exchange of 1000 “Carolus guilder” paid at the time, the bearer of the document receives 20 guilder every year of interest. 1

A document from 1648 • The bond is still valid today and Yale in 2010-2015 received ∼ 10E/year from it. • It is hard to estimate the purchasing power of 1000 guilders in the 17th century, but as a reference: a pastor earned 500 guilder per year, roughly 20 guilders every two weeks. • Interestingly, the bond never expires, but is perpetually valid. 2

An investment opportunity? Perpetual bonds Perpetual bonds do no expire, they cannot be redeemed, but they only pay the owner an annual interest forever. You are offered to buy such a perpetual bond. You pay 1,000,000kr today, and you will receive 20,000kr every year for the rest of eternity. Questions: 1. Is is a good deal? Would you accept it? 2. Will the bank go bankrupt? Will you be infinitely rich? 3

Inflation Things get nominally more expensive as time passes. Example Let us assume for simplicity that there is an annual inflation rate of 2%. This means that what we can buy today for 20,000kr, it will cost 20,400kr next year, 20,808kr in two year, 24,380kr in 10 years. The interest that we will receive from the bond will be always the same number (20,000kr), but it will be worth less and less relative to the cost of life. 4

Inflation So in today money, our bond interest will be worth 1. (1 − 2%) × 20,000kr = 19,600kr the first year; 2. (1 − 2%) × 19,600kr = 19,208kr the second year; 3. (1 − 2%) × 19,208kr = 18,824kr the third year... and so on. So the total gain we obtain from our potential bond is 1. M 1 = 19 , 600kr the first year; 2. M 2 = 19 , 600 + 19 , 208 = 38 , 808kr the second year; 3. M 3 = 38 , 808 + 18 , 824 = 57 , 632kr the third year... 5

Handling infinities We can calculate “easily” M Y for any given year Y . But to to properly estimate the predicted income from the bond (and thus, how much we would be available to pay), we need to know the total amount gained over “infinity” years. Problem How can we describe (mathematically) this “value after infinite time” of our investment? 6

A sequence of numbers M 6 M 5 M 4 M 3 M 2 M 1 M 0 n 7

Exercise Divide these cases into two groups which share some common property. c 0 a 6 a 5 a 4 c 1 a 3 b 0 b 2 b 4 b 6 a 2 c 2 c 3 c 4 c 5 c 6 a 1 a 0 n b 1 b 3 b 5 n n d 6 d 4 d 2 f 0 f 1 d 0 f 2 f 3 f 4 f 5 f 6 e 0 n e 2 e 4 e 6 d 1 d 3 n e 5 e 3 d 5 e 1 n 8

Exercise a 6 a 5 a 4 a 3 b 0 b 2 b 4 b 6 a 2 a 1 n a 0 n b 1 b 3 b 5 9

Exercise a 6 a 5 a 4 a 3 b 0 b 2 b 4 b 6 a 2 a 1 n a 0 n b 1 b 3 b 5 (a) Monotone (b) Non monotone [Back] 9

Exercise d 6 d 4 d 2 d 0 b 0 b 2 b 4 b 6 n d 1 n d 3 b 1 b 3 b 5 d 5 10

Exercise d 6 d 4 d 2 d 0 b 0 b 2 b 4 b 6 n d 1 n d 3 b 1 b 3 b 5 d 5 (a) Bounded (b) Unbounded [Back] 10

Back to investments 1. If the gains from our investment M Y are monotone , we will get more money every year. 2. Otherwise, we risk loosing money (for example with company shares). 11

Back to investments 1. If the gains from our investment M Y are monotone , we will get more money every year. 2. Otherwise, we risk loosing money (for example with company shares). 3. If the gains M Y are monotone and unbounded , the bank will bankrupt and we will be infinitely rich (not really realistic). 11

Monotone and bounded What if M Y is monotone and bounded ? 1. Can it oscillate between two values? n 12

Monotone and bounded What if M Y is monotone and bounded ? 1. Can it oscillate between two values? 2. Will it approach a maximum? n 12

Monotone and bounded What if M Y is monotone and bounded ? 1. Can it oscillate between two values? 2. Will it approach a maximum? M Y will get closer and closer to some maximum n value which is called the limit. The expected gains are finite! 12

Perpetual bonds In the case of perpetual bond, it is possible to actually calculate the total gain possible (the limit): annual interest M ∞ = annual inflation rate . 13

Perpetual bonds In the case of perpetual bond, it is possible to actually calculate the total gain possible (the limit): annual interest M ∞ = annual inflation rate . So that in our case M ∞ = 20 , 000kr / 0 . 02 =1,000,000kr (we break even)! 13

ILOs After the lecture, the students should be able to: 1. classify sequences according to the properties of monotonicity and boundedness ; 2. argue that monotone bounded sequences have a well-defined limit ; 3. explain the value of perpetual bonds via asymptotic analysis of predicted gains. 14