148 Optimizing Optimization
De Rossi and Harvey (2009) assume that a time series y t , a risk tolerance
parameter 0 ω 1, and a signal to noise ratio q are given. They then decom-
pose each observation into its unobservable ω -expectile, μ (^) t ( ω ), and an error
term having ω -expectile equal to zero:
ytt t
tt t
μω εω
μω μ ω ηω
() ()
() 1 () ()
The ω -expectile, μ (^) t ( ω ), is assumed to change over time following a (slowly
evolving) random walk that is driven by a normally distributed error term η (^) t
having zero mean. 4 In the special case, ωμ 05 ., (0.)t 5 is just the time-varying
mean and therefore y t is a random walk plus noise. The signal μt(.)05 can be
estimated via the Kalman filter and smoother (KFS).
4 The dynamics could be specified alternatively as following an autoregressive process or an inte-
grated random walk.
2
1
0
–1
2
1
0
–1
2
1
0
–1
2
1
0
–1
5101520
Return, pct
q = 0.1
5 101520
Return, pct
q = 1
5 101520
Return, pct
q = 10
5 101520
Return, pct
q = 1000
Figure 6.2 Time-varying expectiles for alternative values of q. The data is a simulated
time series. The dotted line represents the sample 5% expectile, which corresponds to
the case q 0.