The 8th Wonder of the World

Dr. Carmen Bruni

Centre for Education in Mathematics and ComputingUniversity of Waterloo

http://cemc.uwaterloo.ca

October 19th, 2016

Question

What did Albert Einstein call the 8th Wonder of the World?

Answer

Compound Interest

Quote (Paraphrased)

“Those who don’t understand it pay it and those that dounderstand it earn it.” (Einstein)

How Does Compound Interest Work?

• Idea: We give money to certain companies so that they cangenerate more money.

• However, when that money is returned to us, we want moremoney than we originally gave to you.

• This is the idea of interest.

How Does Compound Interest Work?

• Idea: We give money to certain companies so that they cangenerate more money.

• However, when that money is returned to us, we want moremoney than we originally gave to you.

• This is the idea of interest.

How Does Compound Interest Work?

• Idea: We give money to certain companies so that they cangenerate more money.

• However, when that money is returned to us, we want moremoney than we originally gave to you.

• This is the idea of interest.

Compound Interest

• With compound interest, you repeat the above computationwith not only your original amount (the principal) but also onthe interest received.

• This way you can make money faster by continually addingthe previous value to this one.

Compound Interest

• With compound interest, you repeat the above computationwith not only your original amount (the principal) but also onthe interest received.

• This way you can make money faster by continually addingthe previous value to this one.

Simple example

Suppose you go to a bank that offers you a 1% interest rate peryear. You invest $1000 in the bank. After three years if you nevertouch the money, how much money do you have?

Solution

Year Amount

0 $10001 $10102 $1020.13 $1030.3

Therefore, after 3 years, your money would be worth $1030.3dollars.

Compounding Period

• In the previous example, we only compounded the interestonce per year.

• What if I wanted to compound it more frequently?

• You wouldn’t get paid 1% each time, rather, you would make1%/n where n is the number of times you are calculatinginterest per year.

Compounding Biannually

Year Amount

0 $10000.5 $10051 $1010.03

1.5 $1015.082 $1020.15

2.5 $ 1025.253 $1030.38

So after three years, you would make an additional 8 cents.

Compounding Quarterly

Year Amount

0 $10000.25 $1002.50.5 $1005.01

0.75 $1007.521 $1010.04

1.25 $1012.561.5 $1015.09

1.75 $1017.632 $1020.18

2.25 $1022.732.5 $1025.28

2.75 $1027.853 $1030.42

An extra 12 cents versus just an annual interest rate.

Question

Can we play this game forever?

More Frequently CompoundingSuppose I gave you $1000000000 dollars and we started tocompound it at 100%. What would be the result if wecompounded it...

• Annually?

• Biannually? (Twice a year)

• Quarterly? (Four times a year)

• Monthly? (Twelve times a year)

• Daily? (365 times a year)

• Hourly? (8760 times a year)

• Minutely? (525600 times a year)

• Secondly? (31536000 times a year)

• Faster? (????)

Can you Compound?

In this case, money after one year is

109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

Can you Compound?

In this case, money after one year is 109 · (1 + 1/n)n where n is thecompounding interval. We’ll round to the nearest dollar below.

• Annually: $2000000000

• Biannually: $2250000000

• Quarterly: $2441406250

• Monthly: $2613035290

• Daily: $2714567482

• Hourly: $2718126692

• Minutely: $2718279243

• Secondly: $2718281785

• Faster: $2718281828

A Limit

• It turns out that no matter how quickly you want tocompound, you’re limited by this value called e.

• Think of e as this speed limit for interest.

A Limit

• It turns out that no matter how quickly you want tocompound, you’re limited by this value called e.

• Think of e as this speed limit for interest.

A Limit

• It turns out that no matter how quickly you want tocompound, you’re limited by this value called e.

• Think of e as this speed limit for interest.

What is e?

• If instead of starting with 109 we began with 1, The numberwe would get is 2.7182818284590... and we denote thisnumber using the letter e.

• The line corresponding to “faster” above is 109e.

• We can describe it mathematically via the following notationfrom Calculus:

e = limn→∞

(1 +

1

n

)n

What is e?

• If instead of starting with 109 we began with 1, The numberwe would get is 2.7182818284590... and we denote thisnumber using the letter e.

• The line corresponding to “faster” above is 109e.

• We can describe it mathematically via the following notationfrom Calculus:

e = limn→∞

(1 +

1

n

)n

What is e?

• If instead of starting with 109 we began with 1, The numberwe would get is 2.7182818284590... and we denote thisnumber using the letter e.

• The line corresponding to “faster” above is 109e.

• We can describe it mathematically via the following notationfrom Calculus:

e = limn→∞

(1 +

1

n

)n

e: A Special Number

• In 1683, Jacob Bernoulli posed the above questions andconcluded that no matter how quickly you do thecompounding, there was some upper limit.

• The constant e first appeared in the 1700s when LeonardEuler began studying it and he was the first to use the letter eto describe it.

• Euler showed this constant was equivalent to

e =1

0!+

1

1!+

1

2!+

1

3!+

1

4! + ...

where n! is the factorial notation (see the worksheet).

• He showed that e is irrational, found a continued fractionexpansion for it as well as derived many other properties ofthis constant.

e: A Special Number

• In 1683, Jacob Bernoulli posed the above questions andconcluded that no matter how quickly you do thecompounding, there was some upper limit.

• The constant e first appeared in the 1700s when LeonardEuler began studying it and he was the first to use the letter eto describe it.

• Euler showed this constant was equivalent to

e =1

0!+

1

1!+

1

2!+

1

3!+

1

4! + ...

where n! is the factorial notation (see the worksheet).

• He showed that e is irrational, found a continued fractionexpansion for it as well as derived many other properties ofthis constant.

e: A Special Number

• In 1683, Jacob Bernoulli posed the above questions andconcluded that no matter how quickly you do thecompounding, there was some upper limit.

• The constant e first appeared in the 1700s when LeonardEuler began studying it and he was the first to use the letter eto describe it.

• Euler showed this constant was equivalent to

e =1

0!+

1

1!+

1

2!+

1

3!+

1

4! + ...

where n! is the factorial notation (see the worksheet).

• He showed that e is irrational, found a continued fractionexpansion for it as well as derived many other properties ofthis constant.

e: A Special Number

• In 1683, Jacob Bernoulli posed the above questions andconcluded that no matter how quickly you do thecompounding, there was some upper limit.

• The constant e first appeared in the 1700s when LeonardEuler began studying it and he was the first to use the letter eto describe it.

• Euler showed this constant was equivalent to

e =1

0!+

1

1!+

1

2!+

1

3!+

1

4! + ...

where n! is the factorial notation (see the worksheet).

• He showed that e is irrational, found a continued fractionexpansion for it as well as derived many other properties ofthis constant.

Importance• This number has many important properties in Calculus, for

example, the growth rate at any point of the function y = ex

is proportional to itself.• Many places compound compound interest rates using this e

number and we call this continuous probability

Summary of Compound Interest

Discrete Interest:

A = P0

(1 +

r

n

)nt

where P0 is the principle amount, n is the number of interestperiods per year, r is the interest rate (as a decimal), t is the timein years and A is the final amount.

Continuous Interest:A = P0e

rt

Rule of 72

How long would it take to double your money using compoundinterest?

• Let’s go back to the $1000000000 example with 100% annualinterest.

• If compounding once a year, this takes a year.

• If compounding twice a year, then after 6 months, we wouldhave:

$1000000000(1 + 1/2) = $1500000000

after 6 months and $2250000000 after a year. So it takes us afull year to [at least] double our investment.

Rule of 72

How long would it take to double your money using compoundinterest?

• Let’s go back to the $1000000000 example with 100% annualinterest.

• If compounding once a year, this takes a year.

• If compounding twice a year, then after 6 months, we wouldhave:

$1000000000(1 + 1/2) = $1500000000

after 6 months and $2250000000 after a year. So it takes us afull year to [at least] double our investment.

Rule of 72

How long would it take to double your money using compoundinterest?

• Let’s go back to the $1000000000 example with 100% annualinterest.

• If compounding once a year, this takes a year.

• If compounding twice a year, then after 6 months, we wouldhave:

$1000000000(1 + 1/2) = $1500000000

after 6 months and $2250000000 after a year. So it takes us afull year to [at least] double our investment.

Rule of 72

How long would it take to double your money using compoundinterest?

• Let’s go back to the $1000000000 example with 100% annualinterest.

• If compounding once a year, this takes a year.

• If compounding twice a year, then after 6 months, we wouldhave:

$1000000000(1 + 1/2) = $1500000000

after 6 months and $2250000000 after a year. So it takes us afull year to [at least] double our investment.

Quarterly Comopunding

• If compounding quarterly, the amount of money we have aftereach quarter is given in this table:

Period Amount

3 months $12500000006 months $15625000009 months $1953125000

12 months $2441406250

• However for 12 months...

Quarterly Comopunding

• If compounding quarterly, the amount of money we have aftereach quarter is given in this table:

Period Amount

3 months $12500000006 months $15625000009 months $1953125000

12 months $2441406250

• However for 12 months...

Monthly Compounding

Period Amount

1 month $10833333332 months $11736111113 months $12714120374 months $13773630405 months $14921432936 months $16164885687 months $17511959488 months $18971289449 months $2055223023

Thus, after only 9 months if interest is compounded monthly, youwill have doubled your initial investment.

How do we quickly solvethis in general?

Monthly Compounding

Period Amount

1 month $10833333332 months $11736111113 months $12714120374 months $13773630405 months $14921432936 months $16164885687 months $17511959488 months $18971289449 months $2055223023

Thus, after only 9 months if interest is compounded monthly, youwill have doubled your initial investment. How do we quickly solvethis in general?

Question

We want to know the value of nt, the number of time periods,where

P0

(1 +

r

n

)nt= 2P0

or more simply, (1 +

r

n

)nt= 2

where r and n are fixed. Can we solve this?

Simplify the problem

Put in a simpler format, we are looking for a number x satisfyingcx = 2 where c is some known positive constant. For example, wewant to solve 3x = 2. How can we do this?

Inverse Function

• Ideally, we would like to find a new function, let’s call it f (x)where f (cx) = x .

• We don’t need to know exactly what this function is but wedo need to be able to compute it (or at least get a computerto do so).

• This magical function we give a name and we call it logc(x).This is the function that satisfies:

logc(cx) = x

• In the exercises, you will also justify to yourself thatlogc(x) = logd (x)

logd (c)where d is any other positive number.

Inverse Function

• Ideally, we would like to find a new function, let’s call it f (x)where f (cx) = x .

• We don’t need to know exactly what this function is but wedo need to be able to compute it (or at least get a computerto do so).

• This magical function we give a name and we call it logc(x).This is the function that satisfies:

logc(cx) = x

• In the exercises, you will also justify to yourself thatlogc(x) = logd (x)

logd (c)where d is any other positive number.

Inverse Function

• Ideally, we would like to find a new function, let’s call it f (x)where f (cx) = x .

• We don’t need to know exactly what this function is but wedo need to be able to compute it (or at least get a computerto do so).

• This magical function we give a name and we call it logc(x).This is the function that satisfies:

logc(cx) = x

• In the exercises, you will also justify to yourself thatlogc(x) = logd (x)

logd (c)where d is any other positive number.

Inverse Function

• Ideally, we would like to find a new function, let’s call it f (x)where f (cx) = x .

• We don’t need to know exactly what this function is but wedo need to be able to compute it (or at least get a computerto do so).

• This magical function we give a name and we call it logc(x).This is the function that satisfies:

logc(cx) = x

• In the exercises, you will also justify to yourself thatlogc(x) = logd (x)

logd (c)where d is any other positive number.

Special Number

As e is so important in this setting, the function loge(x) gets itsown special name, we denote this function by ln(x).

Graphs (courtesy Desmos.com)

Calculate

Using this new function we can compute the values that we need!In general, (

1 +r

n

)nt= 2

ln

((1 +

r

n

)nt)

= ln(2)

nt ln(

1 +r

n

)= ln(2)

nt =ln(2)

ln(1 + r

n

)

Approximations of Taylor and MacLaurin

This last value, while exact, is difficult for humans to do on thespot so we sometimes use the rule of 72 (or the rule of 70 or therule of 69). This is based on the fact that ln(1 + x) ≈ x followingfrom theorems of calculus due to Taylor and MacLaurin.

nt =ln(2)

ln(1 + r

n

) ≈ 0.693147rn

Rule of 72

As we deal primarily with percentages, we change rn to R%

100%n . Thisgives

nt ≈ 0.693147rn

=0.693147

R%100%n

=69.3147%n

R%

or more simply t ≈ 69.3147R . So to compute the doubling time t, we

simply take 69 and divide it by R. Since 69 doesn’t have a lot ofgreat factors, people will often use the number 70 or 72 to helpmake the computations easier depending on what R is.

Example

Suppose that we put our money in a bank that offers us 9%interest annually. Approximately how many years would it take forus to double our money?

Solution: Approximately 72/9 = 8 years. This is remarkably closeto the actual answer of

t =ln(2)

ln(1 + r

n

) =ln(2)

ln(1 + 0.09/1)≈ 8.043231727...

(Note from the previous formula, n = 1 since we compoundannually).

Example

Suppose that we put our money in a bank that offers us 9%interest annually. Approximately how many years would it take forus to double our money?

Solution: Approximately 72/9 = 8 years. This is remarkably closeto the actual answer of

t =ln(2)

ln(1 + r

n

) =ln(2)

ln(1 + 0.09/1)≈ 8.043231727...

(Note from the previous formula, n = 1 since we compoundannually).

Baltimore Stock Broker Problem

A story of a stock broker’s newsletter.

Weekly Newsletter

Week 1 headline: Stock ABC will go up this weekStock ABC price at the beginning of the week: 20Stock ABC price at the end of the week: 21

Weekly Newsletter

Week 2 headline: Stock ABD will go up this weekStock ABD price at the beginning of the week: 33Stock ABD price at the end of the week: 37

Weekly Newsletter

Week 3 headline: Stock ABE will go down this weekStock ABE price at the beginning of the week: 40Stock ABE price at the end of the week: 32

Weekly Newsletter

Week 4 headline: Stock ABF will go up this weekStock ABF price at the beginning of the week: 10Stock ABF price at the end of the week: 12

Weekly Newsletter

Week 5 headline: Stock ABG will go up this weekStock ABG price at the beginning of the week: 22Stock ABG price at the end of the week: 25

Weekly Newsletter

Week 6 headline: Stock ABH will go down this weekStock ABH price at the beginning of the week: 20Stock ABH price at the end of the week: 15

Weekly Newsletter

Week 7 headline: Stock ABI will go up this weekStock ABI price at the beginning of the week: 100Stock ABI price at the end of the week: 137

Weekly Newsletter

Week 8 headline: Stock ABJ will go down this weekStock ABJ price at the beginning of the week: 46Stock ABJ price at the end of the week: 39

Weekly Newsletter

Week 9 headline: Stock ABK will go up this weekStock ABK price at the beginning of the week: 16Stock ABK price at the end of the week: 20

Weekly Newsletter

Week 10 headline: Stock ABL will go up this weekStock ABL price at the beginning of the week: 22Stock ABL price at the end of the week: 25

Question

In week 11, the newsletter comes with an offer to take you on as anew client charging a hefty commission fee. Should you trust thebroker?

Our Viewpoint

Surely he must be pretty good. After all, you are a goodmathematician and know the odds of being right 10 times in a rowon a 50-50 proposition are (1/2)10 = 1/1024 < 1%.

The Stock Broker’s Viewpoint

• The stock broker is shrewd and sends in week 1, half of thenewsletters saying the stock will go down and half saying itwill go up.

• Then to the half that was correct (the up ones) he againsends in week 2 newsletters to half saying the stock will godown and half saying it will go up.

• He repeats with all the correct ones until he ends up with one(un?)lucky person (you!)

• Even though events are unlikely, they can still occur. Peoplewin lotteries everyday despite the fact that lottery chances areextremely remote (worse than continuing the stock brokerexample for another 10 days!)

The Stock Broker’s Viewpoint

• The stock broker is shrewd and sends in week 1, half of thenewsletters saying the stock will go down and half saying itwill go up.

• Then to the half that was correct (the up ones) he againsends in week 2 newsletters to half saying the stock will godown and half saying it will go up.

• He repeats with all the correct ones until he ends up with one(un?)lucky person (you!)

• Even though events are unlikely, they can still occur. Peoplewin lotteries everyday despite the fact that lottery chances areextremely remote (worse than continuing the stock brokerexample for another 10 days!)

The Stock Broker’s Viewpoint

• The stock broker is shrewd and sends in week 1, half of thenewsletters saying the stock will go down and half saying itwill go up.

• Then to the half that was correct (the up ones) he againsends in week 2 newsletters to half saying the stock will godown and half saying it will go up.

• He repeats with all the correct ones until he ends up with one(un?)lucky person (you!)

• Even though events are unlikely, they can still occur. Peoplewin lotteries everyday despite the fact that lottery chances areextremely remote (worse than continuing the stock brokerexample for another 10 days!)

The Stock Broker’s Viewpoint

• The stock broker is shrewd and sends in week 1, half of thenewsletters saying the stock will go down and half saying itwill go up.

• Then to the half that was correct (the up ones) he againsends in week 2 newsletters to half saying the stock will godown and half saying it will go up.

• He repeats with all the correct ones until he ends up with one(un?)lucky person (you!)

• Even though events are unlikely, they can still occur. Peoplewin lotteries everyday despite the fact that lottery chances areextremely remote (worse than continuing the stock brokerexample for another 10 days!)

Cash Money

Sometimes you see an advertisement like:

$300 for only $20

What does this mean?

Idea

No, they’re not giving away money. This actually means you canborrow 300 dollars and it costs you 20 dollars. How much interestis the money lending company charging you?

Payday Loans

• Take a cash advance on your paycheque.

• Maximum allowable amount under Ontario law for a paydayloan is $21 dollars per $100 dollars per two week period.

• Assuming you did this every two weeks (and always paid it offin full on your payday), determine the amount of money youpaid back in interest. How much is this in annual interest?

Payday Loans

• Take a cash advance on your paycheque.

• Maximum allowable amount under Ontario law for a paydayloan is $21 dollars per $100 dollars per two week period.

• Assuming you did this every two weeks (and always paid it offin full on your payday), determine the amount of money youpaid back in interest. How much is this in annual interest?

Law

• In Canada, the above example is the largest amount one canpay for a two week loan.

• This amounts to the 546% that we computed from before.

• What some companies now do to find a way around this issomething called “Rent-to-Own”.

• The idea is that you rent say your couch and you keep payingrent until you own it.

• This can result in annual interest rates of values even higherthan in the previous example.

Law

• In Canada, the above example is the largest amount one canpay for a two week loan.

• This amounts to the 546% that we computed from before.

• What some companies now do to find a way around this issomething called “Rent-to-Own”.

• The idea is that you rent say your couch and you keep payingrent until you own it.

• This can result in annual interest rates of values even higherthan in the previous example.

Law

• In Canada, the above example is the largest amount one canpay for a two week loan.

• This amounts to the 546% that we computed from before.

• What some companies now do to find a way around this issomething called “Rent-to-Own”.

• The idea is that you rent say your couch and you keep payingrent until you own it.

• This can result in annual interest rates of values even higherthan in the previous example.

Law

• In Canada, the above example is the largest amount one canpay for a two week loan.

• This amounts to the 546% that we computed from before.

• What some companies now do to find a way around this issomething called “Rent-to-Own”.

• The idea is that you rent say your couch and you keep payingrent until you own it.

• This can result in annual interest rates of values even higherthan in the previous example.

Law

• In Canada, the above example is the largest amount one canpay for a two week loan.

• This amounts to the 546% that we computed from before.

• What some companies now do to find a way around this issomething called “Rent-to-Own”.

• The idea is that you rent say your couch and you keep payingrent until you own it.

• This can result in annual interest rates of values even higherthan in the previous example.

References

• http://earthobservatory.nasa.gov/Features/

WxForecasting/wx2.php

• http://timemapper.okfnlabs.org/manunicast/

history-of-weather-forecasting#0

• “The Signal and the Noise” - Nate Silver

• http://www.hoyes.com/blog/

what-is-the-maximum-amount-of-interest

-i-can-be-charged-in-ontario/

• “How Not to Be Wrong, The Power of MathematicalThinking” - Jordan Ellenberg

• Slides from Alain Gamache