170 likes | 393 Views
Risk measures and optimal portfolio selection J.Dhaene, M. Goovaerts, R. Kaas, Q. Tang, S. Vanduffel and D. Vyncke. Steven Vanduffel Actuarial Science, KUL. Investment strategies. We consider n assets.
E N D
Risk measures and optimal portfolio selectionJ.Dhaene, M. Goovaerts, R. Kaas, Q. Tang, S. Vanduffel and D. Vyncke Steven Vanduffel Actuarial Science, KUL
Investment strategies • We consider n assets. • Stochastic return process: 1 unit invested in asset « k » at time=0 grows to at time j. • For given k, the Yk,i are i.i.d. N(k-0,5k2, k2) • kl=Covar(Yk,i Yl,i ) , k,l=1,2,…,n • = (1, 2,…,n)
Investment strategies • Asset mix (t)=[1(t) 2(t),…, n(t)] • Continuously rebalancing: (t) = • Hence 1 unit invested in the asset mix at time=0 grows to at time j. • the Yi () are i.i.d.N( -0,52,2) • = .t • 2 = . .t (See also Emmer & Klüppelberg (2001) or Bjork (1998)
Provisons and discounting • Let a1, a2, …, an be deterministic cashflows due at time i=1,2,…,n • Let R0 be the initial provion in order to meet these future obligations. • Stochastic provision at time j: Rj (R0,) = R j-1 (R0, )eYj() -j; j=1,2…,n • Stochastically discounted value: • Relationship between stochastic provision and stoch. discounted value:
Optimal strategy • Probability of reaching the finish Pr(Rn (R0, )>0)=FS()(R0) • Minimisation problem : • Can we determine a strategy * that minimises the initial provision R0 for a given probability ‘p’ of reaching the finish • The optimal pair (R0* ,*) satisfies • R0* =Min(Qp(S()) • Look for the strategy that minimises the ‘value at risk’ of the stochastically discounted value of the future obligations • Does it make (more) sense to consider strategies that minimises another risk measure(s) ?
Intermezzo: distortion risk measures • Wang (1996), Wang,Young and Panjer (1997) f(x)=x g(x)=I(x>1-p) h(x)=Min(x/(1-p),1) 0<=x<=1 1 h f g 0 1-p 1 f is a distortion function f , f(0)=0 and f(1)=1
Distortion risk measures f(x)=x g(x)=I(x>1-p) h(x)=Min(x/(1-p),1) 0<=x<=1 1 h f • Distortion risk measure f • f(X) = Qq(X)df(q) • Take distorted expectations. g 0 1-p 1
Distortion risk measures • Positive homogeneous (PH) • Translation invariance (TI) • Monotonicity (M) • Additivity for commonotonic risks (a>0)
Concave distortion risk measures • In this case (recall that SL order represents the comment preferences of all risk averse decision takers) • This implies subadditivity (SA) • PH+TI+SA+M = coherent risk measure (Artzner,Delbaen,Eber & Heath 1999)
Coherent risk measures • VaR is not subadditive, hence not coherent • The TVaR a concave distortion risk measure hence coherent • TVaR is the smallest concave distortion risk measure above the VaR. • « In the class of concave distortion risk measures is TVaR the answer in case you want a coherent risk measure at a minimal extra cost compared with VaR » • Not all coherent risk measures are distortion risk measures, • e.g. • Coherency is trivial • Not comonotonic additive, hence not distortion risk measure • e.g. (X,Y) comonotonic, X and Y Bernoulli(q1),resp Bernoulli(q2) with 0<q1<q2<1, q1+q2>1 1 h f g 0 1-p 1
Optimal strategies • Minimisation problem 1 : • Can we determine a strategy * that minimises the initial provision R0 for a given probability ‘p’ of reaching the finish • The optimal pair (R0* ,*) satisfies • R0* =Min(Qp(S()) • Look for the strategy that minimises the ‘Value at risk’ of the stochastically discounted value of the future obligations • Minimisation problem 2: • R0 is now determined not as a VaR but as a TVaR • The optimal pair (R0* ,*) satisfies • R0* =Min(TVaRp(S()) • Look for the strategy that minimises the ‘TVaR’ of the stochastically discounted value of the future obligations
Optimal strategies • Stochastically discounted value: Following Kaas,Dhaene & Goovaerts (2000) we define the following approximations for S()
Optimal strategies • Observe that for p>1/2, the quantiles increase in for given • Two step procedure: • The first step is a mean variance optimisation « Among all portfolios with a given mean find the one with the minimal variance » • =>feasible portfolios= • 2. The second step is determining • R0,u* =Min(Qp(Su()) for all • and • R0,u* =Min(Qp(Su()) for all • Minimisation problem 1 : • R0* =Min(Qp(S()) Will be approximated by • R0,u* =Min(Qp(Su()) • R0,l* =Min(Qp(Sl())
Optimal strategies • The tailvars increase in for given • Two step procedure: • The first step is a mean variance optimisation « Among all portfolios with a given mean find the one with the minimal variance » • =>feasible portfolios= • 2. The second step is determining • R0,u* =Min(TVaRp(Su()) for all • and • R0,u* =Min(TVaRp(Su()) for all • Minimisation problem 2 : • R0* =Min(TVaRp(S()) Will be approximated by • R0,u* =Min(TVaRp(Su()) • R0,l* =Min(TVaRp(Sl())
Extensions • Look for portfolio’s that for a given provison maximise the probability of reaching the finish. • The terminal wealth problem (periodical savings) • For a given probability of obtaining a target capital after n periods, the optimal portfolio is the one that minimises the periodical savings • For a given amount of savings the optimal portfolio is the one that maximises the probability of obtaining a target capital
Numerical illustration • Let target capital = 1 • Let a be the periodical monthly savings • n=480 • 1 Riskless asset: • Market portfolio: • = .1 + (1-).2 • 2 = (1- ). 22 .(1-)0<=<=1 • Question: For a given probability p (=1-) of obtaining a target capital after n periods, compute the portfolio that minimises the periodical savings