Sharpe ratio
... and related stuff.

In an interesting paper
(in .PDF format), Victor Norton (Bowling Green State University) considers variations on the Sharpe Ratio in order to decide when to switch allocations from one set of
assets to another and ...
>Sharpe Ratio?
Yes. If your average portfolio return is R and a riskfree return is R_{o} (for example, cash or money market),
then the excess is r = R  R_{o}.
That excess is what you get for accepting the risk associated with the risky assets in your portfolio.
It's your Reward and you'd expect R
to be greater than R_{o} (unless your portfolio is entirely in riskfree assets).
>That r is the Sharpe Ratio?
Not yet.
If the Volatility or Standard Deviation (SD) of your portfolio is small, then you should be happy. Chances are you'll get the Reward r.
On the other hand, if your portfolio assets are volatile (so SD is large) then there's a good chance you may get a return less than the riskfree return, R_{o}.
The Sharpe Ratio measures Reward/Volatility, that is
Sharpe Ratio = r / SD = (R  R_{o}) / SD 
>So the red graph is better, eh?
It'll have a larger Sharpe Ratio, if that's what you mean. That's because the Volatility = SD is small.
>But both the red and blue guys have the same Reward, eh?
Yes, because they have the same Mean (or Average) Return R.
 Figure 1

>So what does Norton do? Maximize the Sharpe Ratio?
Not exactly. The question is: What to use as the Portfolio Return R?
>You said it was the Average Return, so why not just ...?
But there are averages and averages. For example, if the returns over the last N months were
R_{1}, R_{2}, ... R_{N}, where R_{1} is the most recent and R_{N}
the most remote (N months ago), then the average ...
>It's (1/N)(R_{1}+R_{2}+ ...+R_{N}), right?
That's one kind of average; the garden variety Arithmetic Mean. However, the are lots of Weighted Averages.
For example, you may want to place more weight on the more recent returns. In general, if you want a Weighted Average of N returns, then:
 Pick N weights w_{1}, w_{2}, ... w_{N} which add to "1" (that is, w_{1}+w_{2}+ ...+w_{N}=1)
 Calculate a Weighted Average = w_{1}R_{1} + w_{2}R_{2} + ... + w_{N}R_{N}.
 Note that, if all returns are equal to, say, R, the Weighted Average = w_{1}R+w_{2}R+...+w_{N}R = R
which explains why the weights must add to "1".
Now the problem is to select the "best" weights.
>What's "best"?
Aah, that's the question, but ...
>And what if your portfolio has several assets? What's the "best" allocation then?
Okay, but first consider Norton's weighted average ritual with a single stock (or mutual fund):
 If r_{2}, ... r_{N} are the monthly rewards
... meaning: (monthly stock return)  (monthly riskfree return), then
 r_{av} = W_{1}r_{1}+W_{2}r_{2}+...+W_{N}r_{N}
is the weighted average reward (for that stock).
 If we let w_{1} = SQRT[W_{1}], w_{2} = SQRT[W_{2}], etc. be the Square Root of the weights,
then r_{av}, which is the sum of products, can be written as
r_{av} = w_{1}(w_{1}r_{1})+w_{2}(w_{2}r_{2})+...+w_{N}(w_{N}r_{N})
 Following Norton, we introduce a vector E with N components w_{1}, w_{2}, ... w_{N}:
E = [w_{1}, w_{2}, ... ,w_{N}]
 The length of this vector is E
= SQRT(w_{1}^{2}+w_{2}^{2}+...+w_{N}^{2})
= SQRT(W_{1}+W_{2}+...+W_{N}) = SQRT(1) = 1
(since the weights add to "1"). Hence E is a unit vector.
 In addition, we introduce another vector:
r = [w_{1}r_{1},w_{2}r_{2},...,w_{N}r_{N}].
 With these two vectors we can write the (weighted) average stock reward as a dot product (or scalar product):
r_{av} = rE
(meaning we add the product of components of E and r)
However, (the Average Square)  (the Square of the Average) is just the Variance.
>Huh?
Remember, the Variance is the average squared deviation of the monthly rewards from the average reward.
That is:
Variance = ΣW_{k}[r_{k}  r_{av}]^{2}
= ΣW_{k}[r_{k}^{2}  2r_{k}r_{av}
+ r_{av}^{2}]
= ΣW_{k}r_{k}^{2}  2r_{av}ΣW_{k}r_{k}
+ r_{av}^{2}ΣW_{k}
= ΣW_{k}r_{k}^{2}  2r_{av}r_{av}
+ r_{av}^{2}(1)
That is:
Variance = ΣW_{k}r_{k}^{2}  r_{av}^{2}
= (the Average Square)  (the Square of the Average).
>You're talking "weighted" averages, eh?
Yes. Didn't I say that? Pay attention.
>Okay, but that's just for 2dimensional vectors. That means just two months of rewards and ...
Actually, I lied. It's for any number of months. In fact, the 2D part comes from the fact that we just looked at
the plane defined by E and
r ... as in Figures 2 and 3 (below).
>That's pretty slick, eh?
Isn't it! Notice that, if we refer to the Standard Deviation as "Risk" (as Sharpe does), then "Risk" and "Reward" are orthogonal.
Indeed, we can now write:
r
= f
+ (rE)E
= f + r_{av}E
where
E = 1
f = Standard Deviation of Monthly Rewards
r_{av} = the (weighted) Average of Monthly Rewards
Sharpe Ratio = (rE) / f
= r_{av} / f

 Figure 3

>But we're talking a single asset, eh?
So far, yes. But our single, weighted reward vector r could be the reward associated
with a single portfolio which just happens to have umpteen components.
>Huh?
Here we consider a portfolio with M assets:
 Our portfolio has fractions x_{1}, x_{2}, ... x_{M} devoted to each.
 With assets "1" to "M" we associate (weighted) reward vectors
r_{1}, r_{2}, ...
r_{M}
where r_{j} =
f_{j}
+ (r_{j}E)E
= f_{j} + r_{avj}E
(j = 1, 2, ... M)
 Our portfolio reward vector is then:
r
= x_{1}r_{1}
+ x_{2}r_{2}+ ...
+ x_{M}r_{M}
= Σx_{j}r_{j}
= Σx_{j}f_{j}
+ (Σ(x_{j} r_{j})E)
E
= f
+ (rE)E
= f + r_{av}E
where r_{av} = Σx_{j} r_{avj}
is the average portfolio reward.
>The "weighted" average, eh?
Yes. Do I have to keep saying that? If you'd just pay ...
>It's like the single investment case. I guess the Sharpe Ratio for your portfolio is r_{av} / f, eh?
Very good. In fact, f is the Standard Deviation for our portfolio and
f^{2}
= ff
= Σx_{j}f_{j}Σx_{k}f_{k}
is a sum of products of our fractions x_{j}, containing stuff involving x_{1}^{2} and
x_{1}x_{3} and x_{7}x_{5} and ...
>Okay. I get the idea.
