motivated by email from Leon G.
When one speaks of a portfolio "return", one often means something like this:
If my portfolio went from $P_{0} to $P_{1} in one year, my annual return would be P_{1}/P_{0}  1.
>Huh?
Example 0:
I start with $P_{0} = $10,000 and, after a year, it's worth $P_{1} = 11,230.
Then P_{1}/P_{0}  1 = 0.123 and that's a 12.3% return.
On the other hand, if my portfolio had returns R_{1}, R_{2}, ... R_{N} over N successive years,
my average annual return, or arithmetical mean return (or simply my Mean return), would be:
[1]
For N successive annual returns of R_{1}, R_{2}, ... R_{N},
Mean Return = (1/N) (R_{1} + R_{2} + ... + R_{N})

When people talk about "Expected" Return, they're (usually) talking about this Mean Return.
Example 1:
If the returns are 1%, 6% and 10% over 3 years, then the Mean Return is (1/3)(0.01 + 0.06 + 0.10) = 0.05 or 5%.
>Why 0.06? Why not say 6%?
Uh ... yeah, we could do that, but later we'll have to use the decimal equivalent. Just wait, okay?
Example 2:
For the above example, the total gain factor or cumulative gain, over 3 years, would be (10.01)(1+0.06)(1+0.10) = 1.154.
... meaning that an initial portfolio of $10,000 would be worth $10000 (1.154) = $11,540, a gain of $1,540.
Sometimes, by "Total Return", one means that cumulative gain of $1,540.
Sometimes, by "Total Return", one means that gain expressed as a percentage of the original amout, namely 1,540/10,000 = 0.1540 or 15.4%
We'll use the latter, so we might then say:
[2]
For N successive annual returns of R_{1}, R_{2}, ... R_{N},
Total Return = (1+R_{1})(1+R_{2}) ... (1+R_{N})  1

Note that we now use the decimal equivalent for the returns (as in Example 2), eh?
Example 3:
Okay, suppose we have that Total Return of 15.4% over 3 years.
What "constant" annual return would generate the same final portfolio value?
In other words, starting with $10,000, what constant return would give us $11,540 in 3 years?
That'd require 10,000(1+R)^{3} = 11,540 or (1+R)^{3} = 11,540/10,000 = 1.154
... and that'd require (1+R) = (1.154)^{1/3} = 1.0489 so R = 1.0489  1 = 0.0489 or 4.89% each year.
That's called the Annualized Return or Compound Annual Growth Rate (CAGR) of maybe the Geometric Mean Return or maybe ...
>Please pick just one, okay?
[3]
For N successive annual returns of R_{1}, R_{2}, ... R_{N} (which changes $P_{0} into $P_{0}):
Annualized Return = { (1+R_{1})(1+R_{2}) ... (1+R_{N})}^{1/N}  1
= {1+Total Return}^{1/N}  1
= {P_{N}/P_{0}}^{1/N}  1
= mean of the logarithmic

Now some math types like to talk about "continuous compounding" which goes like this:
 Suppose we have a return of 12.34% in one year ... that's R = 0.1234.
 We then consider the consequences of having a weekly return of R/52 for 52 weeks.
That'd turn $1.00 into (1+R/52)^{52}, in 52 weeks.
 We then consider the consequences of having a daily return of R/365 for 365 days.
That'd turn $1.00 into (1+R/365)^{365}, in 365 days.
 We then consider the consequences of having a return each second of R/31,536,000.
That'd turn $1.00 into (1+R/31536000)^{31536000}, in 31,536,000 seconds.
 We then consider the consequences ...
>Could you just skip to the end?
Okay, for "continuous" compounding we'd look at (1+R/m)^{m} as m got larger and larger.
Example 4:
Indeed, if we look at the limit as m we'd see something like Figure 1
which has R = 0.10 (or 10%).
The red line is the limiting value of (1+0.1/m)^{m}. It's e^{0.1} = 1.105170918
That means $1.00 turns into e^{0.1} = $1.105170918.
That'd would imply a "continuous compounding rate" of
e^{0.1}  1 = 0.105170918 or about 10.52%
In general:
[4]
For an annual return of R
the corresponding Continous Compounding Return = e^{R}  1

A portfolio that goes from $P_{0} to $P_{1} in 1 year, using a "continuous Return" of R, has
P_{1} = P_{0} e^{R}.
If the change is over n years, P_{1} = P_{0} e^{nR}.
[4a]
For an Total Return of R over n years
the corresponding Continous Compounding Return = e^{R}  1

I might point out that the magic BlackScholes option pricing formula
has a term with e^{rt}.
That generates a Present Value ... using "continuous compounding".
Of course, some define Return somewhat differently.
For example, if your portfolio goes from $P_{0} to $P_{1}, we can define the Return as log(P_{1} / P_{0}).
Example 5:
Starting with $P_{0} = $10,000, we end up with $P_{1} = $11,230.
The Return might also be calculated as: log(11,230/10,000) = 0.116003676 or about 11.60%.
 Figure 1 
[5]
If an investment goes from $P_{0} to $P_{1},
another Return = log(P_{1} / P_{0})

>Is that anything like P_{1}/P_{0}  1?
Have you forgotten Example 0?
In fact, if we write R = P_{1}/P_{0}  1 (as in Example 0), then
this new calculation for Return would be: Return = log(1+R).
Stare intently at Figure 2 and you'll see that, at least for smallish Rvalues, there ain't much difference.
Indeed, log(1+R) ≅ R when R is small.
>When R is small? Yeah, like the returns in my portfolio.
We could then modify the above definition like so:
[5a]
If an investment goes from P_{0} to P_{1} = P_{0} (1+R),
the logarithmic Return = log(P_{1} / P_{0}) = log(1+R)

 Figure 2 
I assume you're talking about "natural" logs, to the base e?
Yes. Is there any other kind?
Indeed, using [5a], P_{1} = P_{0} (1+R) = P_{0} e^{Return}
>Is that it?
Heavens, no!
The evolution from P_{0} to P_{1} may be over N years, so we'd want to define some "Annualized" or
"Mean" using this other Return and we might also ...
>zzzZZZ
Note that if we consider a sequence of stock prices, P_{0}, P_{1}, P_{2} ...
then their logarithms would be: log(P_{0}), log(P_{1}), log(P_{2}) ...
The change in these logs would be:
log(P_{1})  log(P_{0}) = log(P_{1}/P_{0})
log(P_{2})  log(P_{1}) = log(P_{2}/P_{1})
etc. etc.
and we recognize these as the logarithmic Returns.
Of course, as you might expect, there are lots of relationships between the myriad of "returns".
Consider them logarithmic Returns, calculated from the logarithm of the prices ratios:
The Mean is (1/N) {log(P_{1}/P_{0})
+ log(P_{2}/P_{1}) + ... + log(P_{N}/(P_{N1})}
= (1/N) {log(P_{N}/(P_{0})}
= log({P_{N}/P_{0}}^{1/N})
= log(1+Annualized Return)
Note:
log(A/B) = log(A)  log(B), so lots of stuff cancels out and m log(X) = log(X^{m})
... and we're using the "Annualized Return" from [3].
[6]
the Mean of Logarithmic Returns = log(1+Annualized Return)

Then, too, if you have a bunch of "returns" (of any flavour), you can calculate a plethora of Standard Deviation (or Volatility) values.
The Standard Deviation (or Volatility) of a sequence of annual Returns is defined like so:
[7]
For N annual returns: R_{1}, R_{2}, ... R_{N},
Variance = (Standard Deviation)^{2} = (1/N){(R_{1}M)^{2} + (R_{2}M)^{2} + ... + (R_{N}M)^{2}}
where M = Mean Return = (1/N) (R_{1} + R_{2} + ... + R_{N})

That is, (Standard Deviation)^{2} (also called the Variance) is the average deviationfromthe Mean^{2}.
Alas, some people use 1/(N1), in the definition of (Standard Deviation)^{2}, instead of 1/N,
(regarding the returns as a "sample" from some larger universe of returns).
We'll use 1/N.
It turns out that:
[8]
(Standard Deviation)^{2} =
(1/N){R_{1}^{2} + R_{2}^{2} + ... + R_{N}^{2}}  M^{2}

See SD stuff.That is, (Standard Deviation)^{2} is: (average of the squares of the returns)  (the square of the average return).
If we used 1/(N1) we wouldn't get this neat formula!
Now let's consider, say, Mean and Annualized Returns.
We show here that, very nearly:
[9]
Annualized Return = (Mean Return)  (1/2) (Standard Deviation)^{2} ... very nearly

where, we're using the definitions [1] Mean Return, [3] Annualized return and [7] Standard Deviation.
>So, using logarithmic Returns, what's the Standard Deviation?
We'd calculate, for N Returns equal to log(P_{1}/P_{0}), log(P_{2}/P_{1}), ...
log(P_{N}/P_{N1}), the value of:
(Standard Deviation)^{2} =
(1/N){log^{2}(P_{1}/P_{0})
+ log^{2}(P_{2}/P_{1}) + ...
+ log^{2}(P_{N}/P_{N1})}  log^{2}(1+Annualized Return)
... using [8] and [6]
>That's pretty messy. Is there a simpler formula?
Not that I know of.
Let's consider ten years worth of daily prices for GE stock, as in Figure 3.
The various returns are like so:
 Mean (daily) Return = 0.054% the average of all returns
 Total Return (over 10 years) = 153.8% (Final Price)/(Initial Price)  1
 Annualized Return = 9.76% (1+Total Return)^{1/10}  1
 Standard Deviation (of daily returns) = 1.850%
 Mean of daily logarithmic Returns = 0.037%
 Standard Deviation of daily logarithmic Returns = 1.848%
 Continuously Compounding Total Return = 365.7%
a Total Return of 153.8%, compounded continuously is e^{1.538}  1 or (1+1.538/m)^{m}  1 as m ∞
 Continuously Compounding Annual Return = 10.25%
an Annualized Return of 9.76%, compounded continuously is e^{0.0976}  1
 The ...
>zzzZZZ
Pay attention.
 Sometimes people take the Mean (daily) Return and "annualize" it by multiplying by the number of days in a year.
That'd give an "annualized Mean" of 365*(0.054%) = 17.7%.
 Sometimes they multiply by the number of "market days" in a year, let's say 250.
That'd give an "annualized Mean" of 250*(0.054%) = 13.5%.
 Sometimes people annualize the Standard Deviation of daily returns my multiplying by the square root of the number of days in a year.
That'd give an "annualized Standard Deviation" of SQRT(365)*(1.850%) = 35.3%.
 Sometimes people annualize the Standard Deviation of daily returns my multiplying by the square root of the number of market days in a year.
That'd give an "annualized Standard Deviation" of SQRT(250)*(1.850%) = 29.3%.
 Sometimes people take a Continuously Compounded Annual return and extend it to give a Continuously Compounded Return over a time t.
In general, we might use: e^{Rt}  1 where R is some "annual" return.
Using R = Annualized Return = 10.25%, that'd give e^{10(0.1025)}  1 = 175.7%
That explains the ubiquitous term e^{rt}, as in the BlackScholes formula:
C = S N(d_{1})  K e^{rt} N(d_{2})
 Sometimes
people mathematical investmenttypes use
Instantaneous Return r so that dP/P = r dt, where P(t) is the price at time t.
Here dP(t) is the change in Price over a microsecond (or less).
Note that this will change from day to day second to second, but if we adopt some constant "average" instantaneous return r,
then dP/P = r dt implies that P(t) = P(0) e^{rt}, an exponential growth in stock price.
 Sometimes people ...
>zzz... huh? Average instantaneous return?
Yes. It's a convenient mathematical construct which allows mathtypes to analyze stock price evolution and ...
>How do they calculate it?
Well, if r is some kind of constant average instantaneous return, so that
P(t) = P(0) e^{rt}, then after N years we'd have
P(N) = P(0) e^{rN}.
Solving for r, we'd get, using our earlier notation where P(N) is written P_{N}, etc.
[10]
The average instantaneous return, over N years, is:
r = log({P_{N}/P_{0}}^{1/N})
= log(1+Annualized Return)
= Mean of the logarithmic Returns

where we've used [6], above.
We wrote dP/P = r dt which is the same as d/dt log(P(t)) = r
and that allows us to solve for P(t) = P(0) e^{rt} ... when r is constant!
However, we should have written dP/P = r dt+ some random stuff.
Since we're talking "instantaneous", the return will vary randomly from second to second ... hence the "random stuff".
>And the solution of that differential equation is?
Well ... that's a long story, told by Kiyosi Ito.
>Yeah, but which "return" should one use in that magic formula?
You mean this magic formula?
where P is the price at time T and P_{o} is the current price.
Aah, that's a problem, eh? What to use for the return and standard deviation, r and s?
In our Ito tutorial, we suggested that they're the Mean and Standard Deviation of stock prices ... but there are other choices, eh?
I guess we should see which works best, right?
Alas, f(P) is a return distribution at some time T in the future, and how do we get such a distribution ... from real price data?
We can look at historical data, generate f(P) according to the above formula, then check the price at a time T in the future.
However, that'll give us a single price ... not a distribution of prices.
>But the formula can tell you the probability of getting a price less than, say $10.
Sure, and if that probability is 5% and the actual stock price turns out to be $9.00 we can say: "We're in that 5%".
If our stock is $11.00 then we can say: "We're in that 95%".
>Okay, so how do you get a distribution of prices ... a probability distribution?
Ay, there's the rub. The only way I can imagine doing that is like so:
 Generate a set of random "historical" data
 Extract some return and standard deviation
(staring intently at [1] to [6] and choosing Mean or Annualized or Continuous or logarithmic return and standard deviation...or whatever)
 Having made our choice, (r, s), we generate a distribution f(P) according to the above formula
 We then generate a jillion random prices P, at time T and compare to the distribution f(P)
 Then we repeat steps 2 to 4 a jillion times to see which choice (in step 2) gave the "best" match
>And how do you get the random historical data ... and random future price P?
We could select from ... uh, a lognormal distribution of returns.
>What! If you assume a lognormal distribution  to get your random data  aren't you already making some choice for r and s?
Uh ... we'll see.
click to continue

