$TITLE Dedication models * Dedication.gms: Dedication models. * Consiglio, Nielsen and Zenios. * PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 4.3 * Last modified: Apr 2008. SET Time Time periods /2001 * 2011/; ALIAS (Time, t, t1, t2); SCALARS Now Current year Horizon End of the Horizon; Now = 2001; Horizon = CARD(t)-1; PARAMETER tau(t) Time in years; * Note: time starts from 0 tau(t) = ORD(t)-1; SET Bonds Bonds universe /DS-8-06, DS-8-03, DS-7-07, DS-7-04, DS-6-11, DS-6-09, DS-6-02, DS-5-05, DS-5-03, DS-4-02 /; ALIAS(Bonds, i); SCALAR spread Borrowing spread over the reinvestment rate; PARAMETERS Price(i) Bond prices Coupon(i) Coupons Maturity(i) Maturities Liability(t) Stream of liabilities rf(t) Reinvestment rates F(t, i) Cashflows; * Bond data. Prices, coupons and maturities from the Danish market $INCLUDE "BondData.inc" * Copy/transform data. Note division by 100 to get unit data, and * subtraction of "Now" from Maturity date (so consistent with tau): Price(i) = BondData(i,"Price")/100; Coupon(i) = BondData(i,"Coupon")/100; Maturity(i) = BondData(i,"Maturity") - Now; * Calculate the ex-coupon cashflow of Bond i in year t: F(t,i) = 1$(tau(t) = Maturity(i)) + coupon(i) $ (tau(t) <= Maturity(i) AND tau(t) > 0); * For simplicity, we set the short term rate to be 0.03 in each period rf(t) = 0.04; spread = 0.02; DISPLAY F; PARAMETER Liability(t) Liabilities /2002 = 80000, 2003 = 100000, 2004 = 110000, 2005 = 120000, 2006 = 140000, 2007 = 120000, 2008 = 90000, 2009 = 50000, 2010 = 75000, 2011 = 150000/; POSITIVE VARIABLES x(i) Face value purchased surplus(t) Amount of money reinvested borrow(t) Amount of money borrowed; VARIABLE v0 Upfront investment; EQUATION CashFlowCon(t) Equations defining the cashflow balance; CashFlowCon(t).. SUM(i, F(t,i) * x(i) ) + ( v0 - SUM(i, Price(i) * x(i)) ) $(tau(t) = 0) + borrow(t) $(tau(t) < Horizon) + ( ( 1 + rf(t-1) ) * surplus(t-1) ) $(tau(t) > 0) =E= surplus(t) + Liability(t) $(tau(t) > 0) + ( 1 + rf(t-1) + spread ) * borrow(t-1) $(tau(t) > 0); MODEL Dedication 'PFO Model 4.2.3' /CashFlowCon/; SOLVE Dedication MINIMIZING v0 USING LP; PARAMETER PortYield(t) Portfolio yield at liability dates; PortYield(t)$(tau(t) > 0) = - LOG(CashFlowCon.M(t)) / tau(t) ; DISPLAY PortYield, v0.l, borrow.l, surplus.l, x.l; * Simulation for different values of the reinvestment rate rf SCALAR ReinvRate; FILE DedicationHandle /"DedicationPortfolios.csv"/; DedicationHandle.pc = 5; PUT DedicationHandle; FOR ( ReinvRate = 0.005 to 0.05 by 0.005, rf(t) = ReinvRate; SOLVE Dedication MINIMIZING v0 using LP; LOOP ( i, PUT ReinvRate:10:4,v0.l:10:4,i.tl,BondData(i,"Maturity"),Coupon(i),(x.l(i)*Price(i)):10:4/; ); LOOP ( t, PUT t.tl,(-borrow.l(t)):10:3,surplus.l(t):10:3/; ); );