Need help conceptualizing the constant \"e\"
 

I've used the constant e many times in my finance classes and know how to use it, but I don't understand what e is. To me it's just a number and I use it for discounting.

I looked it up in wikipedia and that page confuses me.

Can anyone explain e to me in kindergarten language?

Re: Need help conceptualizing the constant \"e\"
 

By definition e is the limit of (1+1/n)^n as n approaches infinity .

Try this out yourself .

Plug a large number for n in your calculator and you'll ~ match the number e you get on your calculator .

If we expand this using the binomial theorem we can represent as a limit of infinite sums .

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! +....
e ~ 2.718281828....

e is an irrational number much like pi so it cannot be represented as a fraction a/b . There are many interesting properties about e but it's probably worthwhile to start with the definition before you get into other topics .

Re: Need help conceptualizing the constant \"e\"
 

ok makes sense.

now, and you might not know this, but why do i use it in finance so much? we use it for continuously compounding interest rates. for example $100 at 5% interest continuously compounded for 2 years is $100e^.05*2. so, why e?

Re: Need help conceptualizing the constant \"e\"
 

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ok makes sense.

now, and you might not know this, but why do i use it in finance so much? we use it for continuously compounding interest rates. for example $100 at 5% interest continuously compounded for 2 years is $100e^.05*2. so, why e?

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Because the definition of e...
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the limit of (1+1/n)^n as n approaches infinity

[/ QUOTE ] ...is the formula for compounding interest with the number times you compound limiting to infinity.

Say you compound interest 1 time over a period at rate r.

You get (1+r)^1 = 1+r

Say you compound interest 4 times over a period.

You get (1+r/4)^4

Say you compound monthly.

You get (1+r/12)^12.

Say you compound continuously.

You get lim(n to infinity) (1+r/n)^n = e^r

I forget the proof of the very last equality, but I imagine it's not hard to come by. It should be pretty easy to see where the definition of e comes into play though based on its similarity to the formula for compound interest.

Re: Need help conceptualizing the constant \"e\"
 

Pretty much what Omaha said but I'll expand some more .

When we talk about compound interest , we have a familiar formula A = P(1+r/n)^(nt)

P = Principal amount
r= annual interest rate
n = the number of compounded periods per annum .
t= t years


The above formula may be re-written as

A= P*[(1+r/n)^(n/r)]^(rt)
Substitute n/r = x

A= P*[1+1/x)^x]^(rt)

So as x becomes large , the quantity (1+1/x)^x approaches e.

A=p*e^(rt)

Example : Find the amount after 3 years if $1000 is invested at an interested rate of 12% per annum compounded continuously .

Solution : Using A=P*e^(rt) , r=0.12 and t=3,
A=1000*e^(0.12*3)
A= $1433.33

Re: Need help conceptualizing the constant \"e\"
 

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By definition e is the limit of (1+1/n)^n as n approaches infinity .

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I think this definition is arbitrary. e shows up in all sorts of places and you could use any one of them as the starting point to define it. My first introduction to e was by way of the area under the curve 1/x. e is that number such that the area from 1 to e under the curve 1/x is 1. The natural log function, ln, is defined as the area under the curve 1/x so from this definition e is that number such that ln(e)=1. From this definition properties like the one above can be derived.

PairTheBoard

Re: Need help conceptualizing the constant \"e\"
 

Mathematicians were looking for a function that is equal to its own derivative. They narrowed down the search to functions of the form f(x) = a^x, where a is real.

For a fixed x,

f'(x) = lim (1/h)( f(x+h) - f(x) ) where h--> infinity

a^x = lim (1/h)( a^(x+h) - a^x ) where h---> infinity

factoring out a^x from the right hand side

a^x = a^x * lim (1/h)( a^x - 1) where h--> infinity

1 = lim (1/h) (a^x - 1 ) where h--->infinity

e is defined to be the unique value of a such that the equation above is true. You can massage the equation above and substitute h = 1/n to get the definition provided by previous posters.

Re: Need help conceptualizing the constant \"e\"
 

e dollars is the amount of money you would have at the end of the year if you put one dollar into a bank that offered 100% interest compounded continuously.

For those who don't know what this means lets change it to a bank that compounded your million dollars every 3.65 days. At a 100% annual interest rate. So every 3.65 days they gave you one percent. After 3.65 days you would have 1,010,000. After 7.30 days you would have 1,020,100. After 10.95 days you would have 1,030,201. At the end of the year you would have just short of "e" million dollars (as opposed to two million with no compounding or 2.25 million if interest was compounded every six months.) The thing is that even though shrinking the time period for compunding makes you more and more money, you run into one of those limit thingies that jason and boris love and you can't get past e.

The more important thing about e concerns making prop bets when poker tournaments redraw. If there are more than a few players left from 20 to a googol, the chances that everybody will draw a new seat is almost exactly one out of e.

Re: Need help conceptualizing the constant \"e\"
 

oops, in the last 2 lines of my post, it should be a^h not a^x inside the limit =/

Re: Need help conceptualizing the constant \"e\"
 

[ QUOTE ]
[ QUOTE ]
By definition e is the limit of (1+1/n)^n as n approaches infinity .

[/ QUOTE ]

I think this definition is arbitrary. e shows up in all sorts of places and you could use any one of them as the starting point to define it. My first introduction to e was by way of the area under the curve 1/x. e is that number such that the area from 1 to e under the curve 1/x is 1. The natural log function, ln, is defined as the area under the curve 1/x so from this definition e is that number such that ln(e)=1. From this definition properties like the one above can be derived.

PairTheBoard

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Somebody else must have written this post for you.