by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 [email protected]

Except for this comment explaining that it is blank for some deliberate reason, this page is intentionally blank!

/ The Number of the Natural Logarithm Base The mathematical number / ¸ 2.71828182846 can be understood to be a special number which occurs in mathematics, somewhat like 1 ¸ 3.14159265359. Although 1 occurs naturally in the geometry of the circle, it is surprising that / occurs in nature as part of compound growth. Both 1 and / are irrational transcendental numbers. Being irrational means they cannot be written as exact fractions, or another way of saying the same thing, they cannot be represented by repeating decimals. Being transcendental means they are also not roots of any polynomial equation which has rational coefficients. In the following we give the financial formula for the future value associated with the compound interest earnings of a fixed amount and also the formula for the future value of a series of periodic payments. The following five variables are the standard variables used in the math of finance. JZ TZ 3 8 T QX

œ œ œ œ œ

the future value amount the present value amount the periodic interest rate applied over one time period the number of time periods the amount of the periodic payment

There are only two financial formulas which are required to solve most problems involving compound interest of either a single fixed payment or a series of periodic payments. The future value J Z is the same in both formulas and can be used to equate a loan amount with the amount of each periodic payment for that loan. Only the first formula is needed to introduce the famous number /.

JZ œ TZ † ( " 3 Ñ8

and

JZ œ TQX † Ò

Ð " 3 Ñ8 " Ó 3

A TYPICAL CAR LOAN For example, suppose you come to me to get a 3-year $8,000 car loan. Assuming an annual interest rate of 12% per year, which is equivalent to 1% per month, I can use the first formula to decide the future value of my $8,000 thirty-six months from now. I figure this amount before I decide to loan you the money! In fact, I need to know this amount to be able to determine what your monthly car payments to me will be. Substituting, T Z =8000, 3 œ 0.01 and 8 œ 36 we use the first formula to calculate the future value J Z . JZ

œ TZ † Ð 1 3 Ñ 8 œ $8000 † Ð 1 0.01 Ñ 36 œ $8000 † Ð 1.01 Ñ 36 œ $8000 † Ò 1.43076878359 Ó œ $11,446.15

The Number / page 1

Ö rounded to the nearest penny ×

So I expect my money “multiply" by the factor in the square brackets which is about 1.43 which is not quite 1 12 times. The quantity Ð" 3Ñ8 is significant because it always acts as the multiplier of your money (or mine!). We will next show how this quantity also determines the number /! Although car loan problems are usually figured on a monthly basis, it will be interesting to consider the value of money when it is compounded more often that once every month. Note that the values of 8 and 3 must constantly be adjusted to be consistent with the stated interest rate. In fact, there is always a time period associated with an interest rate. If your bank tells you your money is earning 8% interest they are not telling you the whole truth. The bank's quoted 8% interest rate is probably a nominal yearly rate which would correspond to 8% ƒ 12 œ 23 % or 0.666% per month. Similarly, when your credit card company tells you they are only charging you 1 12 % interest, that is a monthly rate which translates to 1.5% ‚ 12 œ 18% per year. To keep things simple, let us just consider the future value of $1 which is compounded for the time periods listed in the table below. You can choose any interest rate per year, but to keep things simple we will use 100% per year. We don't mean to be greedy, but please note that this is only 8.3% per month, or 0.27397% per day, or 0.01141% per hour, or 0.000190% per minute!

Time Period Length

Time Periods Per Year œ 8

Year

"

Month

"#

Day

$'&

Hour

$'& † #% œ )('!

Minute

)('! † '! œ &#&'!!

Second

&#&'!! † '! œ $"&$'!!!

To compound every instant, take

Interest Rate Per Time Period œ 3

Compounding Factor œ Ð" 3Ñ8

"Þ! " œ !Þ!)$$$$$$$$ "# " œ !Þ!!#($*(#'!#( $'& " œ !Þ!!!""%"&&#&""%# )('! " !Þ!!!!!"*!#&)(&"*!$ &#&'!! " œ !Þ!!!!!!!!$"(!*(*"*)$) $"&$'!!!

#Þ! #Þ'"$!$&#*!"$ #Þ("%&'(%)#!$ #Þ(")"#''*"'$ #Þ(")#(*#%#&) #Þ(")#)"()&$'

637 Ð" 8" Ñ8 œ / ¸ 2.71828182846 8Ä∞

WARNING! If you attempt the above calculations on an ordinary calculator using a CB key you may get very inaccurate results. Most calculators are not designed to compute Ð" 3Ñ8 when 3 is very small and 8 is very large. Special precautions must be used to insure round-off errors don't interfere with the calculation.

The Number / page 2

Given the above expression for /, we can prove: 637 Ð" 8B Ñ8 œ /B 8Ä∞ Proof: /

œ

637 Ð" 8" Ñ8 8Ä∞

/B œ 637 Ð" 8" Ñ8

ÐThis is a definition of /.Ñ B

ÐAssume B is a fixed real number. Take the Bth power on both sides of the above equation.Ñ

8Ä∞

œ

637 Ð" 8" Ñ8

B

ÐJustification for moving the Bth power inside the limit depends on continuity of the Bth power function. The Bth power of the limit is the limit of the Bth power.Ñ

8Ä∞

œ

637 Ð" 8" Ñ8B 8Ä∞

œ

637 Ð" 8Ä∞

œ

B 8B 637 Ð" 8B Ñ 8B Ä ∞

ÐAs 8p∞ so does 8B and vice versa!Ñ

œ

637 Ð" B5 Ñ5 5Ä∞

ÐReplace 8B with the single variable 5Ñ

œ

637 Ð" 8B Ñ8 8Ä∞

( 8 is just as good a variable as 5 !Ñ

B 8B 8B Ñ

ÐApply the multiplication rule for exponents since we are taking a power of a power.) ÐMultiply

" 8

The Number / page 3

by

B B

œ "Ñ

CONTINUOUS COMPOUND INTEREST The previous two standard financial formulas can also be used to determine formulas for continuous compounding. As is shown below, the same formulas hold with the factor Ð" 3Ñ8 replaced by /38 . A Lump Sum Compounded Continuously For 8 Time Periods JZ œ TZ † Ð 1 3 Ñ8 3 7

œ TZ † Ð 1

3 7

œ TZ † Ò Ð 1

Ö discrete compounding ×

Ñ

7

3 7

×

3 7 7Ñ

×

{ compounded 8 † 7 times at a rate œ

Ñ8†7 Ó8

To arrive at continuous compounding we simply take the limit as 7 approaches ∞. { /3 œ the limit as 7 Ä ∞ of Ð"

œ TZ † Ò /3 Ó8 œ TZ † /3†8

This example shows that when compounding continuously, Ð" 3Ñ8 can be replaced by / 3†8 , or, equivalently, 3 can be replaced by / 3 ". Ð" 3Ñ8 ¸ / 3†8

3 ¸ /3 "

Note that in the above and in what follows 3 is the periodic interest rate which can be described as the interest rate per time period, and 8 denotes the number of time periods. Series Of N Equal Payments Compounded Continuously Over 8 Time Periods J Z œ T Q X † Ð" 3Ñ8" T Q X † Ð" 3Ñ8# â T Q X † Ð" 3Ñ T Q X ¸ T Q X † Ò / 3 †Ð8"Ñ Ó T Q X † Ò / 3 † Ð8#Ñ Ó â T Q X † Ò / 3 † " Ó T Q X œ T Q X † Ò Ð / 3 Ñ 8" Ð / 3 Ñ 8# â Ð / 3 Ñ " Ó œ T Q X † Ò 1 Ð / 3 Ñ Ð / 3 Ñ # Ð / 3 Ñ $ â Ð / 3 Ñ 8" Ó œ T QX † Ò

1 Ð /3 Ñ8 1 Ð /3 Ñ

œ T QX † Ò

/3†8 " /3 "

œ T QX † Ò

/3†8 1 3

Ó Ó Ó

The Number / page 4

œ

TQX 3

† Ò /3†8 1 Ó

Ö this is the standard form ×

Ö for those who know calculus this standard form can be expanded as shown below to arrive at an equivalent integral form } œ

T QX †

" 3

† Ò /3†8 /! Ó

œ

T QX †

1 3

† Ò / 3 † > l !8 Ó

œ

T QX † Ò

œ T QX †

/3†> 3

08

l 80 Ó Ö this is the integral form ×

/ 3 † > .>

Unequal Payment And Compounding Periods For those interested in more mathematical details, below we provide formulas which can be used to calculate annuities where the payment and compounding frequencies are different. Sometimes a series of payments are made where the payment period is more frequent than the compounding period. Suppose a series of 8 † 7 equal payments will be made over 8 time periods in which there will be 7 payments per time period accumulating an interest rate 3 per time period. Then the discrete and continuous future value formulas are as follows: JZ

œ

7 3

† T QX † Ò Ð 1

JZ

œ

7 3

† T QX † Ò / 3 † 8 1 Ó

JZ

œ 7 † T QX

8 † 0

3 7

Ñ8†7 1 Ó

{ discrete case × { continuous case ×

/ 3 † > .> Ö this is the integral form of the continuous case ×

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Previously we applied a limit process to a discrete variable 8 which takes on counting values to establish relationships with the number /. An alternate proof of the limit approximation for / can be applied which employs L'Hospital's Rule. But this derivation requires a prior knowledge of some calculus. Let C œ Ð" B" ÑB Then,

68ÐCÑ œ B † 68Ð" B" Ñ 637 BÄ∞

68ÐCÑ œ

Take logs on both sides.

637 BÄ∞ B

œ

637 BÄ∞

œ 637

BÄ∞

œ 637

BÄ∞

† 68Ð" B" Ñ

68Ð" B" Ñ " B

" " B"

† HB B"

HB B"

" "

" B

lim 68ÐCÑ

637 BÄ∞

to both sides.

Dividing by

" B

is the

same as multiplying by B. Apply L'Hospital's Rule.

Cancel HB B" from both numerator and denominator.

" BÄ∞ "

= 1

/ BÄ∞

Apply

lim

" B

œ"

Exponentiate on both sides.

œ /"

lim / 68ÐCÑ œ /

Take / inside the limit.

lim / 68ÐCÑ œ /

Simplify.

lim Ò C Ó œ /

C œ / 68ÐCÑ

BÄ∞

BÄ∞

BÄ∞

lim Ð" B" ÑB œ /

Substitute for C.

BÄ∞

The Number / page 6