Stochastic Programming (SP) with EMP - Gams

9
634 Stochastic Programming (SP) with EMP
10 Equations
11 defr return of portfolio
12 budget budget constraint
13 obj_eq objective eqn;
14
15 defr.. r =e= sum(j, v(j)*w(j));
16 budget.. sum(j, w(j)) =e= 1;
17 obj_eq.. objective =e= EV_r;
18 model portfolio / all /;
19
20 file emp / ’%emp.info%’ /;
21 emp.nd=4;
22 put emp ’* problem %gams.i%’
23 / ’ExpectedValue r EV_r’
24 / ’stage 2 v defr r’
25 / ’stage 1 objective obj_eq EV_r’
26 / "jrandvar v(’att’) v(’gmc’) v(’usx’)"
27 loop(s,
28 put / p(s) vs(s,"att") vs(s,"gmc") vs(s,"usx"));
29 putclose emp;
30
31 Parameter
32 s_v(s,j) return from assets by scenario /s1.att 1/
33 s_r(s) return from portfolio by scenario;
34
35 Set dict / s .scenario.’’
36 v .randvar. s_v
37 r .level. s_r /;
38
39 solve portfolio using emp max objective scenario dict;
40 display s_r, w.l;
In this model the objective to be maximized is EV_r, the expected return. In the emp file EV_r is declared as the
ExpectedValue of the random variable r. Note that the variables and the equation of stage 2 remain unchanged, but
the new variables EV_r and obj belong to stage 1. Since the expected value of r is not scenario dependent, its value is
known in the preceding stage to the resolution of r, namely stage 1. Note that in a 3-stage-problem with r in the third stage,
the expected value of r will be known with certainty in the second stage. Both models have the same solution. We prefer
the second model since the syntax is more explicit and clearer.
4.2 Value at Risk (VaR)
Value at Risk is an attempt to provide a single number summarizing the total risk in a portfolio of financial assets. It gives
a threshold value for the worst expected loss over a specified time horizon at a given confidence level. Another way of
expressing this is that VaR is the lowest quantile of the potential losses that can occur within a given portfolio during a
predetermined time period. The time period and the confidence level (the quantile) are the two major parameters. For
example, if a portfolio of assets has a one-day 5% VaR of 1 million US Dollars, there is a 0.05 probability that the portfolio
will fall in value by more than 1 million US Dollars over a one day period under normal market conditions (compare the
figure below).
Mathematically,
VaRθ (L) = inf{l ∈ R|P(L > l) ≤ 1 − θ}, (4.36)
where L is the loss of the portfolio and θ ∈ (0,1) is the confidence level. VaR can be modeled using chance constraints as
introduced in section 3 above. Another risk measure derived from VaR is Conditional Value at Risk, it is the topic of the
next section.