Var and Expected Shortfall I am struggling to find an example which has 2 random variables (say L1 and L2) with same VaR but different Expected Shortfall.
 A: There's really nothing unusual about that. A simple example would be to just take a standard normal distribution and a distribution with fatter tails, say a Student-t distribution, and just shift it so their VaR's line up.
More explicitly, fix the level $\alpha$ and take:
$$L_1 \sim \mathcal{N}(0,1)$$
And:
$$L_2' \sim \mathcal{T}_\nu$$
Then define:
$$L_2: = L'_2 + (q^{(1)}_{\alpha}-q^{(2)}_{\alpha})$$
where $q^{(i)}_{\alpha}$ is the $\alpha$-quantile for $L_i$. Then:
$$P[L_2 \leq q^{(1)}_{\alpha}] = P[L'_2 + (q^{(1)}_{\alpha}-q^{(2)}_{\alpha})\leq q^{(1)}_{\alpha}] = P[L_2'\leq q^{(2)}_{\alpha}] = \alpha$$
So, the VaR of $L_1$ and $L_2$ is the same for this value of $\alpha$.
alpha <- 0.01
nu <- 4
adj <- (qnorm(alpha)-qt(alpha,nu))

xs <- seq(-5,0,length.out=1000)

plot(xs,pnorm(xs),log="y",type="l",col='blue', main = "CDF",ylab="Probability",xlab="")
lines(xs,pt(xs-adj,nu), col = 'red')
abline(h=alpha)
legend("bottomright", legend = c("L1 ~ N(0,1)","L2 ~ T_nu + adj"),col=c("blue","red"),lty=1)


However, the expected shortfall is not the same because $L_2$'s left tail is fatter, so more of the mass is concentrated further to the left of their common $\alpha$ quantile, so ES will be larger. Referencing the formulas here, in this specific example you can compute that:
es1 <- dnorm(qnorm(alpha))/alpha
es2 <- -adj+(nu+qt(1-alpha,nu)^2)/(nu-1) * dt(qt(1-alpha,nu),nu)/(alpha)

c(es1, es2)
# [1] 2.665214 3.799985

The VaR is legitimately fundamentally not informative about the mass to the left of that quantile, at all. This is a more contrived example, but consider $L_1$ with any distribution whatsoever, then define $L_2$ as having the same distribution to the right of the VaR, and then concentrate all the mass on the left at an arbitrarily distant point. This doesn't change the VaR, but you can get an arbitrarily large difference in expected shortfall.
The VaR cannot distinguish between the normal "it could get a little bit worse than this" scenario and the "if it gets any worse than this, the world will end" scenario.
