LECTURE NOTES Mathematical Modeling and its Application in Finance

Transcription

1 ECTURE OTES Mathematical Modeling and its Application in Finance This file is copyright protected. You can make hardcopies for your own use during the semester and in the contet of the course. If you would like to use the material for other purposes or make softcopies of this Powerpoint presentation you are hereby granted permission to do so, provided d that each and every slide you copy caries the copyright information on the lower left corner. Stavros A. Zenios, 5/8/0

2 ecture 9: Fied-Income Portfolio Dedication and Immunization Stavros A. Zenios Operations and Information Management Department The Wharton School University of Pennsylvania Stavros A. Zenios, 5/8/0

3 OUTIE Portfolio Dedication Principles Portfolio Immunization Principles Mathematical Programming Models Reading: S. Zenios, Financial Optimization, pp Fabozzi, Handbook of Fied Income Securities, ch , fifth edition, 997. Stavros A. Zenios, 5/8/0 3

4 Portfolio Dedication Fied income Asset/iability management strategy: Eample : Guaranteed Investment Contracts in the 70 s: Upward sloping yield curve 3- to 7-year 7 maturity of GICs with low interest payments Proceeds reinvested in 0- to 30-year mortgages, public bonds What happened as interest rates rose in the late 970s? Eample : Pension fund management may assume very conservative values for reinvestment opportunities. Result? Eample 3: Court settlements. How to manage such risks? CASH-FOW MATCHIG Stavros A. Zenios, 5/8/0 4

5 Cashflow Matching Assets iabilities Time o interest rate risk. Can we achieve cashflow matching? Stavros A. Zenios, 5/8/0 5

6 Portfolio Dedication Assets Reinvestment iabilities Time Maturing assets + Proceeds from reinvestment iabilities, at each e point in time Stavros A. Zenios, 5/8/0 6

7 Portfolio Dedication Determining the iabilities for a Pension Fund Closed block of retirees: Very accurate estimate benefits are known (+) Only a small part of the fund s liabilities (-)( Declining stream of liabilities due to mortality Retired-lives plus terminated vested participants Broader universe of participants (+) Vested benefits are known (+) Anticipated retiree obligations Even broader universe (+) Benefits are not known with certainty (-)( iabilities increase the first years and decline thereafter Stavros A. Zenios, 5/8/0 7

8 Portfolio Dedication Setting constraints: Restrict investments to high-quality bonds Restrictions by agency, sector, issuer (diversification of industry specific risk) Eample: Stavros A. Zenios, 5/8/0 Quality Treasury 0 to 00% Agency 0 to 00% Sector AAA 0 to 00% AA 0 to 00% A 0 to 50% BBB 0 Treasury 0 to 00% Agency 0 to 00% Industrial 0 to 30% Utility 0 to 30% Concentration: o more than 0% in one issue of issuer 8

9 Portfolio Dedication Choosing the reinvestment rollover---rate: Assuming high rollover rate will prefund liabilities if high-yield securities are available with maturity before the liability dates Assuming low reinvestment rates leads to tighter cashflow matching, at the potential loss of portfolio yield Eample: Current actuarial practices allow 5 to 8 percent Obective function: Minimize cost of portfolio Stavros A. Zenios, 5/8/0 9

10 Portfolio Dedication: mathematical program p : Cost (price) of security : Amount invested in security : iability obligation at period t t t :Cashflow generated from security ρ : Reinvestment rate at period t Constraint: at each period t, the cashflow received from assets + income from reinvestment the liability obligation Stavros A. Zenios, 5/8/0 0

11 Stavros A. Zenios, 5/8/0 Portfolio Dedication: mathematical program 3 3 ) ( (3) ) ( () () i.e., Reinvestmentsfrom period () () Subect to: Minimize p ρ ρ ρ ρ ρ ρ ρ Dedication condition: Present value of Assets Present Value of Dedication condition: Present value of Assets Present Value of iabilities iabilities

12 Eample: Portfolio Dedication for a Pension Fund Total liabilities: $83,758,000 over 35 years (retired lives only) Reinvestment rate: 5% Present value of liabilities: $59,88,000 Portfolio cost (market value): $3,60,000 Portfolio yield: 7.83% Savings (portfolio cost over present value of liabilities: 3% Portfolio reoptimization. Bond swaps. The importance of accurate price databases. Stavros A. Zenios, 5/8/0

13 Portfolio Immunization Broad fied-income income asset/liability management strategy Actuary F.M.Reddington (95) the investment of the assets in such a way that the eisting business is immune to a general change in the rate of interest How to measure interest rate sensitivity of assets and liabilities? Stavros A. Zenios, 5/8/0 3

14 P r :Price (or present va lue) of bond t : Cashflow from bond at period :Yield of bond t P T t t ( + r Taking derivative with respect to ) t yield we obtain the dollar duration d T t t t ( + r ) ( t + ) ote: Dollar duration of a portfolio of bonds is additive Stavros A. Zenios, 5/8/0 d d p 4

15 Mathematical Program for Portfolio Immunization p r d D P : Cost (price) of security : Amount invested in security :Yield of security :Duration of liability :Present value of liability :Duration of security Stavros A. Zenios, 5/8/0 5

16 Mathematical Program for Portfolio Immunization Maimize Portfolio Yield Subect to : (Present value matching) (Duration matching ) 0 d P d r d P Stavros A. Zenios, 5/8/0 6

17 Understanding Portfolio Immunization: Interpretation of duration Effect of changing interest rates on asset value: Asset value Reinvestment gains greater than market loss Interest rates rise instantaneously to 5% 9% bond in a 9% interest rate env. 30 years bond 00 Duration Market loss greater than reinvestment gains Years Stavros A. Zenios, 5/8/0 7

18 Understanding Portfolio Immunization: Duration matched portfolios Assume a target (liability) duration of 6.79 years Barbell Portfolio Time Even ladder Bullet portfolio Time Stavros A. Zenios, 5/8/0 8

19 Yield of structured duration matched portfolios Yield Bullet structure Barbell portfolio 5 0 Years Stavros A. Zenios, 5/8/0 9

20 What is the risk eposure of barbell and even ladder portfolios? Shape risk. Add conveity constraints The price equation is give by : P Q T t t( t + ) t t T t ( + r ( + r Taking second derivative with respect to yield we dollar conveity ) ) ( t + ). t obtain the The portfolio conveity is give by Q P Q Stavros A. Zenios, 5/8/0 0