Application of Exponential Growth to Finance Name: Period Date:

Introduction One of the most fundamental ideas in economics is that there is no reason to take something (like money) tomorrow if you can have it today. Between now and tomorrow a lot can happen—you might die, the object that you desire might be destroyed, or the entire world might blow up. With all of this in mind, answer the following questions—there are no right answers, but please try to have a debate with other people at your table to justify your answer.

(a) Would you rather receive $100 for sure, or be given a 90% chance of receiving $125?

(b) Would you rather have $100 now (for sure) or a promise from your friend that she will pay you $200 at the end of class?

(c) Would you rather receive $100 today or $105 when you graduate from West?

(d) Would you rather be given $100 today or $5,000 in the year 2060?

(e) Would you rather be given $100 now or be assured that your eldest great-grandchild will be given $1,000,000 in the year 2106?

Modeling Compound Interest with Mathematics Largely due to the ideas that you thought about above, it makes sense that money in a bank account grows over time. And as it turns out, it grows exponentially as one begins to earn interest on interest, as well as on any initial balance. When you deposit an amount of money into your bank account the principal, P, grows at an annual interest rate, r. To get the amount of money, A (principal + interest) in the account at the end of one year, multiply P times (1 + r). The 1 gives you the original principle P, while the r gives you the additional interest (what you gain).

A= P(1 + r )

Note that r is expressed as a decimal. For example, 4% interest means r = 0.04

(a) How much money would you have after one year if you started with $100 and earned 10% interest?

(b) How much money would you have after one year if you started with $100 and you earned 3% interest?

At the end of two years, you would obviously have even more money. But if you invested $100 and earned 10% interest over two years you would have more than just $120

(c) How much money would you have after two years?

In general, the amount of money after two years can be expressed by the following formula which multiplies what you have after one year by (1+ r).

2 A=(amount after one year )(1 + r ) = (P(1+ r ) )(1 + r ) = P (1 + r )

(d) Write a formula for the amount of money in the account after 7 years if you initially invest $100 and earn an interest rate r each year.

Extending this, we can develop the formula for the amount of money in the account at the end of "t" periods.

A= P(1 + r )t

D:\Docs\2018-04-23\0e7d07d99bc979b00e4e7d0b246b8d14.doc In words, you take a quantity of some item (e.g. dollars, bacteria, rabbits) and multiply it by a constant factor (1+r) each period t (years) during a given interval (e.g. seven years). You can thus determine the exponential growth of that quantity during the interval. The general exponential function that models the growth of money thus depends on P and r.

(e) If you initially invest P = $500 and earn r = 5% interest, how much money will you have in 15 years?

(f) If you initially invested $1 at 100% interest how much money would you have after one year?

Let us now consider what might happen if we were to only invest for part of a year.

(g) If you invested $100 at r = 12% each year, but pulled your money out of the bank after only six months, how much money should you receive? There is no completely right answer to this question since it is not immediately clear what it means to earn 12% annually for only half of a year.

(h) Suppose that you were able to convince your banker to give you $106 dollars after half of a year (as you may have argued in part (g)). How much would this be worth if you reinvested it for another half year, again at 12% annually?

(i) How much have you gained by withdrawing your money mid-year and then reinvesting it?

A modification needs to be made to our financial model if the compounding is done more than once a year. Suppose it is done "n" times a year. While you do not get the full rate r each time, you do get to the compound more frequently.

r A( t )= P (1 + )nt n

(j) Why is r divided by n in the above formula? And why is t multiplied by n?

Here n is the number of compounding periods per year. For example, if n = 1, we have annual compounding, if n = 12, we have monthly compounding, and so on.

(k) Again assuming 12% interest, how much money would you have after one year if you initially invest $100 and compound each month?

(l) How about every day? This will keep your banker very busy!

(m)If you initially invested $1 at 100% interest how much money would you have after one year if you compounded each month? Compare this to part (f).

(n) If you compounded each day how much would you have?

D:\Docs\2018-04-23\0e7d07d99bc979b00e4e7d0b246b8d14.doc (o) How about each second?

(p) What is the best you could do rounded to the nearest penny?

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