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Steven Vanduffel Actuarial Science, KUL

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.

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Steven Vanduffel Actuarial Science, KUL

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  1. Risk measures and optimal portfolio selectionJ.Dhaene, M. Goovaerts, R. Kaas, Q. Tang, S. Vanduffel and D. Vyncke Steven Vanduffel Actuarial Science, KUL

  2. 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-0,5k2, k2) • kl=Covar(Yk,i Yl,i ) , k,l=1,2,…,n • = (1, 2,…,n)

  3. 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( -0,52,2) •  = .t • 2 = . .t (See also Emmer & Klüppelberg (2001) or Bjork (1998)

  4. 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:

  5. 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) ?

  6. 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

  7. 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

  8. Distortion risk measures • Positive homogeneous (PH) • Translation invariance (TI) • Monotonicity (M) • Additivity for commonotonic risks (a>0)

  9. 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)

  10. 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

  11. 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

  12. Optimal strategies • Stochastically discounted value: Following Kaas,Dhaene & Goovaerts (2000) we define the following approximations for S()

  13. 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())

  14. 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())

  15. 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

  16. 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

  17. Numerical illustration (2)

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