Natural logs are frequently used in finance.
ROW/COL	B	C	D	E	F	G	H	I
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6
7	The compound value of a $ is:
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9		V = P *(1+ i/n)^(n*t)
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12
13
14	where:
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16	V is the ending value
17	P is the present value you start with
18	I is the rate of interest
19	n is the frequency of compounding
20	t is the length of time
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22	Notice that part of the formula looks a lot like the expansion to get the value of "e"
23	If we let P=1 and t=1 it is exactly the same formula.
24
25		V = 1 *(1+ i/n)^(n*1)
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29
30	If n =1 then we get:  (which is frequently the form of the compound amount formula)
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32		V = P *(1+ i/1)^(1*t)
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36	Impact of compounding
37	Using an interest rate of:			5.0000%
38	Assume the time period is 1 year and the amount is $1
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40	Frequency of compounding		n	V	Annual Rate of return			ln(Annual Rate of Return)
41	Annual		1	$1.05000	5.0000%			4.8790%
42	Semi-Annual		2	$1.05063	5.0625%			4.9385%
43	Quarterly		4	$1.05095	5.0945%			4.9690%
44	Monthly		12	$1.05116	5.1162%			4.9896%
45	Daily		365	$1.05127	5.1267%			4.9997%
46	Hourly		8760	$1.05127	5.1271%			5.0000%
47	Minutely		525600	$1.05127	5.1271%			5.0000%
48	Secondly		31536000	$1.05127	5.1271%			5.0000%
49	Infinite		=EXP(i)	$1.05127	5.1271%			5.0000%
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51	Notice that as the frequency of compounding increase our rate of return increase, but at a decreasing rate.
52	Until the difference between daily, secondly and infinite is nil.
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54
55	So if you allow infinite compounding the future value of a dollar formula becomes:
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57		V = P*exp(i*t)
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61
62	Remembering that ln and e^x are opposite transactions we can determine an equivalence between rates of interest and compounding
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65	You would indifferent between the following rates compounded once a year, or the rates compounded continuously.
66	Compounded once a year	Compounded continuously	Compound amount of $1 at the end of a year
67	Annual Rate of return	ln(1+Annual Rate of Return)	Once a year	Continuously
68	5.0000%	4.8790%	$1.0500	$1.0500
69	5.0625%	4.9385%	$1.0506	$1.0506
70	5.0945%	4.9690%	$1.0509	$1.0509
71	5.1162%	4.9896%	$1.0512	$1.0512
72	5.1267%	4.9997%	$1.0513	$1.0513
73	5.1271%	5.0000%	$1.0513	$1.0513
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75	You can always find a rate that will cause you to be indifferent between an annual rate and continuously compounded one by taking the ln(1+i)
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77	Annual Rate	Continuously Compounded
78	5.0000%	4.8790%	<=LN(1+B78)
79
80	Or you find a continuously compounded rate that is equal to an annual rate by using exp(i)-1
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82	Continuously Compounded	Annual Rate
83	5.5000%	5.6541%	<=EXP(B83)-1
84
85	Try it by changing the green numbers
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88	In many cases theoretical consideration require that we use continuously compounded rates in finance.
89	In general when using the CAPM and Black/Scholes you should use continuously compounded rates.
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