Company Value with Ruin Constraint in a Discrete Model
Abstract
:1. Introduction
2. Methods
2.1. A Modified Bellman Equation
2.2. Iteration Method
2.3. Running Allowed Ruin Probabilities
2.4. The Barrier Method
2.5. The Lagrange Multiplier Approach
3. Numerical Example
DeFinettiModel ds=1; S0=300; S=0:ds:S0; KS=length(S); W=zeros(1,KS); V1=W; V2=W; V0=W; p=0.7; q=1-p; r=1/1.03; a1=1.0714285; a2=0.4; b2=-.5957446812; b1=1-b2; for k=1:KS W(k)=b1*a1^(k-1)+b2*a2^(k-1); end kk=6; C=1/(W(kk)-W(kk-1)); for k=1:kk V0(k)=W(k)*C; end g=(1-p)/p; Psi=g.^(1:1:KS); for i=(kk+1):KS V0(i)=V0(i-1)+1; end Policy improvement with Bellman, with new formulas for beta1 and beta2 and interpolation DeFinettiModel; da=1/100000; Alpha=da:da:1; KA=length(Alpha); V1=zeros(KS,KA); V2=V1; for L=1:400 M=zeros(1,KA); al0=ceil(Psi(1)/da); for al=al0:KA alpha=al*da; y1=(alpha-q)/p/da; u1=floor(y1); z1=(y1-u1); if u1==0 v1=z1*V1(2,u1+1); else v1=V1(2,u1)+z1*(V1(2,u1+1)-V1(2,u1)); end V2(1,al)=r*p*v1; end for s=2:KS-1 for al=1:KA alpha=al*da; if Psi(s)>=alpha V2(s,al)=0; else ga=(1-alpha)/(1-Psi(s)); y1=(1-ga+ga*Psi(s+1))/da; y2=(1-ga+ga*Psi(s-1))/da; u1=floor(y1); u2=floor(y2); z1=(y1-u1); z2=(y2-u2); if u1==0 v1=z1*V1(s+1,u1+1); else v1=V1(s+1,u1)+z1*(V1(s+1,u1+1)-V1(s+1,u1)); end if u2==0 v2=z2*V1(s-1,u2+1); else v2=V1(s-1,u2)+z2*(V1(s-1,u2+1)-V1(s-1,u2)); end x=r*p*v1+r*q*v2; if Psi(s-1)<alpha & x<V2(s-1,al)+1 if M(al)==0 M(al)=s; end x=V2(s-1,al)+1; end V2(s,al)=max(V1(s,al),x); end end end V1=V2; V1(:,KA)=V0; [L V1(5,round(0.2/da)) V0(5)]’ end Policy improvement without dynamic equation clear; DeFinettiModel; da=1/100000; Alpha=da:da:1; KA=length(Alpha); V1=zeros(KS,KA); V2=V1; V1(:,KA)=V0; M0=round(0.2/da); for L=1:150 M=zeros(1,KA); for s=1:KS for al=max(round(Psi(s)/da),1):KA-1 Feld=zeros(1,KS); alpha=al*da; if M(al)>0 && s>M(al) && Psi(s-1)<al V1(s,al)=V1(s-1,al)+1; end for B=s+1:KS x1=(Psi(s)-Psi(B))/(1-Psi(B)); x2=1-x1; y=(alpha-x1)/x2*KA; aB=floor(y); z=y-aB; if aB==0 VF=z*V1(B,aB+1); end if aB>0 VF=V1(B,aB)+z*(V1(B,aB+1)-V1(aB)); end Feld(B-s)=W(s)/W(B)*VF; end x=max(Feld); if s>1 y=V2(s-1,al)+ds; if (Psi(s-1)<al*da) & (x<y) V2(s,al)=max(V1(s,al),y); if M(al)==0 M(al)=s; end else V2(s,al)=max(V1(s,al),x); end; end end V2(s,KA)=V0(s); end V1=V2; end Lagrange method DeFinettiModel; T0=2000; T=0:T0; KT=length(T); V=zeros(KS,KT); W=V; M=zeros(1,KT); L=2.93; s0=5; a0=0.2; p=0.7; r=1/1.03; % computation of value function V(:,T0)=-L*Psi; for k=1:T0-1 t=T0-k; rt=r^(t-1); V(1,t)=p*V(2,t+1)-q*L; for i=2:KS-1 V(i,t)=max(p*V(i+1,t+1)+q*V(i-1,t+1),V(i-1,t)+rt); if p*V(i+1,t+1)+q*V(i-1,t+1)<V(i-1,t)+rt if M(t+1)==0 M(t+1)=i-1; end end end end % computation of corresponding ruin probability W(:,T0)=Psi; for k=1:T0-1 t=T0-k; W(1,t)=p*W(2,t+1)+q; for i=2:KS-1 W(i,t)=p*W(i+1,t+1)+q*W(i-1,t+1); if i>M(t+1) W(i,t)=W(M(t+1),t); end end end [V(5,1) W(5,1) V(5,1)+L*W(5,1)]’ And finally the MAPLE code for the barrier method: Barrier.mw restart; Digits := 25; p := .7; q := 1-p; r := 1/1.03; Ps := s->(q/p)^(s+1); z := solve(r*(p*x^2+q) = x, x); B0 := solve((1-B)*z[2]+B*z[1] = 0, B); W := s->(1-B0)*z[1]^s+B0*z[2]^s; s0 := 4; a0 := .2; for i from 0 to 6 do B[i] := 4 end do; for i from 7 to 14 do B[i] := 5 end do; for i from 15 to 19 do B[i] := 8 end do; for i from 20 to 30 do B[i] := 12 end do; for i from 31 to 40 do B[i] := 15 end do; for i from 41 to 50 do B[i] := 18 end do; for i from 51 to 101 do B[i] := 24 end do; g[0] := (1-a0)/(1-Ps(s0)); a[0] := 1-g[0]+g[0]*Ps(B[0]); for i from 0 to 100 do a[i] := 1-g[i]+g[i]*Ps(B[i]); g[i+1] := (1-a[i])/(1-Ps(B[i]-1)) end do; g[100]; A1 := (103/33)*W(s0)/W(B[0]+1); C := 10/11; U[1] := 1; for i from 2 to 100 do U[i] := U[i-1]*C*W(B[i-1]-1)/W(B[i]+1) end do; F := evalf(A1*(sum(U[k], k = 1 .. 100)));
Conflicts of Interest
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Hipp, C. Company Value with Ruin Constraint in a Discrete Model. Risks 2018, 6, 1. https://doi.org/10.3390/risks6010001
Hipp C. Company Value with Ruin Constraint in a Discrete Model. Risks. 2018; 6(1):1. https://doi.org/10.3390/risks6010001
Chicago/Turabian StyleHipp, Christian. 2018. "Company Value with Ruin Constraint in a Discrete Model" Risks 6, no. 1: 1. https://doi.org/10.3390/risks6010001