I recently learned something about Regret and ...

>Yeah! Did I tell you the time I didn't ask Sally for a date and ...
Pay attention. I don't mean that regret. I'm talking about the regret if a financial decision falls below your expectations.
It's a notion due (at least in part) to Ron Dembo. Suppose we have two choices:

1. Invest in a risk-free asset with guaranteed return k ... where k = 0.03 means 3%.
2. Invest in another asset with a random (hence, perhaps, riskier) return.

>Huh?
If we pay $1.00 for a lottery ticket which would pays $1 million to the winner - and we lose - we wouldn't cry. It's only $1.00, eh?
However, if we pay $1K for a lottery tocket that pays $1 billion, we would cry if we lost!
This implies that differences in scale matter! See Utility Theory.
Hence there's a difference in "regret" ... whatever "regret" means.

Consider deciding whether to sell your stock or keep it a bit longer.
If we sell and the price goes up, we're unhappy.
If we keep it and the price goes down, we're unhappy.
What to do, to minimize our "regret" ... whatever that means.

When asked how he invested his retirement funds, Markowitz (a father of Modern Portfolio Theory) answered:
"I should have computed the historic co-variances of the asset classes and drawn an efficient frontier.
Instead ... I split my contributions 50/50 between bonds and equities."
He wanted, he added, to "minimize my future regret."

>So you want to measure ... regret?
Yes. We'd like to compare the two choices on the basis of how we'd feel, having made one or the other.

Let's consider a specific problem:
Should I invest 80% in stocks and 20% in bonds ... or 100% in some risk-free financial instrument that gives a guaranteed return?

Suppose that, for the 80/20 portfolio, the returns r have a Mean M and Standard Deviation SD. Suppose, too, that the risk-free return is k. Let's also assume that, for the 80/20 portfolio, we can generate a (density) distribution function something like Figure 1A and a cumulative distribution function like Figure 1B. If we choose to invest in the 80/20 portfolio, we'd expect a return of M ... but it isn't guaranteed. Let's consider the difference between the returns, namely r - k. Our first task would be to calculate the mean and standard deviation of this new random variable: To this end we consider the average value of the squared difference:   (r - k)2. That's     (r - k)2 f(r) dr = (the mean of the squares). However, here's something that's well known:   The (mean of the squares) = (the square of the mean) + (the standard deviation)2. >Beg pardon? That's shown here ... and it's true for any random variable r. Note that   (the mean of r - k) = (the mean of r) - (the mean of k) = M - k. See Stat stuff #1. We also note that the variable r - k has the same standard deviation as r itself, namely SD. Subtracting a constant doesn't change the SD.

Figure 1A Figure 1B

Conclusion?

[1a]       A = (r - k)² f(r) dr = (M - k)² + SD²     that's (the square of the mean) + (the standard deviation)²
A can also be written:

[1b]       A = r² f(r+k) dr

Observations: A is some "average" measure of deviations between our portfolio return and the constant, risk-free return. The deviations are squared so it doesn't matter whether they're positive or negative. If k = M, the square root of this quantity is just the standard deviation of our portfolio returns ... which some regard as "risk" ! Staring intently at [1b] we note that it's the variance or (standard deviation)2 of the excess of our portfolio returns over and above k ... and the distribution for this excess if just f(r), but shifted left by the amount k, as indicated in Figure 1C.

Figure 1C

We'd also worry about our whether our portfolio returns will be less than k, so we write down the probability of that happening.

[2]       Probability that r < k = f(r) dr = F(k)

Where, exactly are you heading?
I want to make sense of this formula which is sometimes used to measure regret ... I understand  

or

We note the following: The first piece is just A = (r - k)2 f(r) dr = (M - k)2 + SD2 ... a measure of how far returns are from k. If M = k, then "regret" is just SD2 (Remember: SD = Standard Deviation). F(k) is the probability that our return is less than k. Regret will decrease if F(k) decreases ... so there's a smaller probability of getting less than k. The ... >Are you gonna derive that formula ... or not? Look at the behaviour of "regret", as given by that magic formula We take regret as a percentage and note that higher volatility SD means more regret. Also, increasing our Mean portfolio returns naturally decreases regret. Also, increasing the benchmark return, k, increases regret. Also ... >Are you gonna derive that formula ... or not? Uh ... I doubt it.

Regret, eh?

Happiness and Regret

We have a portfolio with returns characterized by the random variable r.
There's a return which we'd like to exceed some benchmark: k (which may be random, like the S&P 500 or some risk-free rate).
We look at the following items:

• The probability that our returns are above k. Call it Pr[r > k]. ... this is good; the UPSIDE.
• The probability that our returns are are less than k. Call it Pr[r < k] ... this is bad; the DOWNSIDE ... we're concerned.
• Define H = Pr[r > k] - λ Pr[r< k] = {1 - F(k)} - λ F(k) = 1 - (1+λ)F(k) ... where increasing λ increases our concern about returns less than k.

>How can you be happy with returns less than k?
We're still getting a positive return, in Figure 2.

>So, are you gonna derive a "regret" formula?
Uh ... I doubt it.