Intuitive explanation for expectiles

I am looking for an intuitive explanation for expectiles.

Bellini and Di Bernardino: Risk Management with Expectiles, European Journal of Finance, May 2015

Definition of expectiles by paper above

Since ES lacks elicitability it seems that some researchers are moving on to expectiles.

Can anybody give an intuitve explanation for expectiles and what they represent.

This isn't 100% true but maybe one could argue that $ES_{97.5}$ is the equivalent to $VaR_{99}$. What would be the equivalent expectile?

• Notice that $(X-x)_-=(x-X)_+$. So the expectile can be interpreted as the strike of a put and call option such that their prices are in proportion $q/(1-q)$ to one another, by condition (2). If the expectation is taken w.r.t. risk-neutral pricing measure of course. Jan 25, 2018 at 14:03
• I don't think this has much to do with risk neutral pricing. But I am not sure. As far as I understand q stands for quantile not the risk neutral probability. Jan 25, 2018 at 14:06
• I never said it did. I just gave a possible interpretation within the context of $\mathbb{E}$ being taken w.r.t. risk-neutral pricing measure. P.S. what definition of ES do you use? Jan 25, 2018 at 14:08
• True, thank you. Jan 25, 2018 at 14:11
• This is my definition of ES mean above corresponding VaR $ES_\alpha = \frac{1}{1-\alpha} \int_\alpha^1 VaR_u(L)du$ the empirical version is mean of all observations above VaR Jan 25, 2018 at 14:20

No reply has been given so I wanted to at least give a visualisation of the expectiles.

Suppose the curvy dashed line in my picture represents a cumulative distribution function of some random variable X. Then blue part corresponds exactly to $\mathbb{E}[(X-x)_+]$, while the orange surface corresponds to $\mathbb{E}[(X-x)_-]$. In the picture $x=1$. Now if the proportion of the blue and orange surface is equal to $(1-q)/q$, then we can say that $x$ is the $q$-expectile for this distribution.

How does this connect to the expected shortfall? Well, the ES as you defined it is exactly the blue surface divided by $1-\alpha$ for the value of $\alpha$ such that $\text{VaR}_{\alpha}[X]=x$, i.e.

$$\mathbb{E}[(X-x)_+]=(1-\alpha)\text{ES}_{\alpha}[X] \; .$$

• Thanks. That is the intuition I was looking for. The concept seems similar to Omega, a measure of return. Feb 7, 2018 at 11:18
• Indeed, Omega is the ratio of the blue and orange surfaces. So, setting the ratio to a fixed number determines a value $x$ and that's your expectile. So, expectile is the inverse function of Omega. Feb 7, 2018 at 11:58

That picture in the other answer is pretty slick (+1), so I will just add a note on why one can interpret the colors of those areas like that:

1. Blue:

Define $$Y = (X-x)_+$$. This is nonnegative r.v., so you can take advantage of the formula

$$\mathbb{E}[Y] = \int_0^\infty [1-F_Y(y)] dy \tag{1}.$$

where $$F_Y$$ is the cdf of $$Y$$. The image plots $$F_X(x)$$, the cdf of $$X$$, though. For any $$y \ge 0$$

\begin{align*} F_Y(y) &= \mathbb{P}[Y \le y] \\ &= \mathbb{P}[(X-x)_+ \le y] \\ &= \mathbb{P}[(X-x)_+ \le y, \{X > x \}] + \mathbb{P}[(X-x)_+ \le y, \{X \le x \}] \\ &= F_X(y+x) \end{align*} Using a change of variables $$t = y + x$$, we get $$\mathbb{E}[(X-x)_+] = \int_0^\infty [1-F_X(y+x)] dy = \int_x^\infty [1-F_X(t)] dt.$$ The last expression $$\int_x^\infty [1-F_X(t)] dt$$ is exactly the area of the blue region.

1. Orange:

Define $$W = (X-x)_-$$. This is also nonnegative, and we can take advantage of (1) again. The only additional bit is the identity mentioned by @raskolnikov: $$(X-x)_-=(x-X)_+$$.

Let $$w \ge 0$$; then

\begin{align*} F_W(w) &= \mathbb{P}[W \le w] \\ &= \mathbb{P}[(X-x)_- \le w] \\ &= \mathbb{P}[(x-X)_+ \le w, \{X > x \}] + \mathbb{P}[(x-X)_+ \le w, \{X \le x \}] \\ &= \mathbb{P}[X > x ] + \mathbb{P}[x - w \le X \le x] \\ &= 1- F_X(x-w) \end{align*}

After a change of variables you get the formula for the orange region:

$$\mathbb{E}[(X-x)_-] = \int_{-\infty}^x F_X(t)dt.$$