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 risk-free 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 risk-free 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 risk-free return, Ro. The Sharpe Ratio measures Reward/Volatility, that is Sharpe Ratio = r / SD = (R - Ro) / 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 R₁, R₂, ... R_N, where R₁ is the most recent and R_N the most remote (N months ago), then the average ...
>It's (1/N)(R₁+R₂+ ...+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 w₁, w₂, ... w_N which add to "1" (that is, w₁+w₂+ ...+w_N=1)
• Calculate a Weighted Average = w₁R₁ + w₂R₂ + ... + w_NR_N.
• Note that, if all returns are equal to, say, R, the Weighted Average = w₁R+w₂R+...+w_NR = R
  which explains why the weights must add to "1".

• Okay, let's consider a portfolio with two assets with monthly Rewards, over the past N months, of
    asset#1:   r_1,1, r_2,1, ... r_N,1   with Mean:   r_av,1 = (1/N)(r_1,1+r_2,1+ ...+r_N,1) = (1/N)Σr_k,1
    asset#2:   r_1,2, r_2,2, ... r_N,2   with Mean:   r_av,2 = (1/N)(r_1,2+r_2,2+ ...+r_N,2) = (1/N)Σr_k,2
  (where the sums are from k=1 to k=N and each "Reward" is the monthly return less the monthly risk-free return, as in: r = R - R_o).
• We devote fractions x and y to each asset, where x + y = 1.
  Hence, the Portfolio rewards are: (x r_1,1+y r_1,2), (x r_2,1+y r_2,2), ... (x r_N,1+y r_N,2).
  For month "k", the Portfolio reward is:   x r_k,1+y r_k,2.
• The Arithmetic Mean (or Average) Portfolio Reward (over the past N months) is then:
      r_av = (1/N)[(x r_1,1 + y r_1,2)+ (x r_2,1 + y r_2,2)+...+ (x r_N,1 + y r_N,2)] = (1/N) [x Σr_k,1 + y Σr_k,2] = x r_av,1 + y r_av,2
• For month "k", the Portfolio Reward deviates from the Average Portfolio Reward by
      x r_k,1+y r_k,2 - r_av = (x r_k,1+y r_k,2) - (x r_av,1 + y r_av,2) = x(r_k,1 - r_av,1) + y(r_k,2 - r_av,2).
• The Average Square of these deviations determines the Standard Deviation (or Volatility) according to:
      SD² = (1/N) Σ[x(r_k,1 - r_av,1) + y(r_k,2 - r_av,2)]².
• Performing the squaring and summing gives:
      SD² = (1/N) [x² Σ(r_k,1-r_av,1)² + y² Σ(r_k,2-r_av,2)² ] + a bunch of cross-product terms.
• Typical cross-product terms would look like xy(r_5,1-r_av,1)(r_7,2-r_av,2)
  ... or maybe   x²(r_3,1-r_av,1)(r_5,1-r_av,1) ... or maybe   y²(r_2,2-r_av,2)(r_7,7-r_av,7)
• If we assume that the rewards are independent/uncorrelated then the average of these cross-product terms is zero.
• Our Standard Deviation of our Portfolio Rewards is then:
      SD² = (1/N) [x² Σ(r_k,1-r_av,1)² + y² Σ(r_k,2-r_av,2)²] = x² SD²₁ + y² SD²₂
  where SD₁ and SD₂ are the Standard Deviations of the Rewards for assets 1 and 2.

>That's confusing. I mean ...
Okay, let's say it again:

1. R_k,1 and R_k,2 are the monthly returns for assets 1 and 2 (with k = 1, 2, ... N) and R_o is some Risk-free Return.
2. r_k,1 and r_k,2 are the monthly Rewards for assets 1 and 2 where r_k,1 = R_k,1-R_o and r_k,2 = R_k,2-R_o.
3. r_av,1 = (1/N)Σr_k,1 and r_av,2 = (1/N)Σr_k,2 are the Average (or Expected) Rewards for each asset.
4. If we devote fractions x and y to each asset (where x + y = 1) then the monthly Portfolio Rewards are
   r_k = x r_k,1 + y r_k,2   (k = 1, 2, ... N)
   and the Mean (or Expected) Portfolio Reward is
   r_av = x r_av,1 + y r_av,2
   and the monthly deviations from the average are
   r_k - r_av = x (r_k,1-r_av,1) + y (r_k,2-r_av,2)   (k = 1, 2, ... N)
5. The Standard Deviation of the Portfolio Rewards is then given by the average of the squares of these deviations, via::
   SD² = (1/N)Σ[r_k-r_av]² = (1/N)Σ[x (r_k,1-r_av,1) + y (r_k,2-r_av,2)]²
6. The Portfolio Sharpe Ratio associated with these Portfolio Rewards is then:
   Sharpe Ratio = r_av / SD
7. The Standard Deviations of the asset rewards are SD₁ and SD₂ where
   SD²₁ = (1/N)Σ[r_k,1-r_av,1]² and SD²₂ = (1/N)Σ[r_k,2-r_av,2]²
8. If the rewards are uncorrelated, from month to month and asset 1 vs asset 2, then the Standard Deviation of Portfolio Rewards is given by:
   SD² = x² SD²₁ + y² SD²₂
9. The normal Sharpe Ratio associated with these Portfolio Rewards is then:
       Sharpe Ratio = r_av / SD = [x r_av,1 + y r_av,2] / SQRT[x² SD²₁ + y² SD²₂]