INFORMS Salt Lake City                                              May 2000

 

 

 

 

 

 

 

 

 

 

Balance Sheet Optimization:

Disjunctive Programs to Satisfy

Accounting Requirements

 

 

 

 

Katherine Wyatt

 

 

 

Logic Based Systems Lab

Brooklyn College

City University of New York

 

 

 

 

 

 

wyatt@sci.brooklyn.cuny.edu

http://www.sci.brooklyn.cuny.edu/~wyatt

                                                              

 

Research partially supported by Grant ONR-N00014-96-1-1057

 

 

 

 

 

 

 

The Statement of Financial Accounting Standards

 No. 133, Accounting for Derivative Instruments and Hedging Activities,

 

 

·      sets out guidelines for the reporting and accounting of derivative instruments on a company’s income statement and balance sheet

 

·      in particular, describes requirements for “hedge accounting”

 

 

·      goes into effect for financial reports issued after

       June 15, 2000

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

What is “hedge accounting?”

 

®      In hedge accounting,

 

 

§      the gain or loss from a derivative is

             netted with the offsetting loss or gain

             from the item being hedged.

 

§      only the amount of gain or loss on a

        derivative that is not offset is reported.

        

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

New reporting requirements under SFAS 133:

 

 

1.   All financial instruments will be reported

         at fair value on the balance sheet

 

    2.  Four risks, market risk, interest rate risk,

        foreign exchange risk, and credit risk,

         can be hedged against

 

    3.  Hedged item / derivative pairs must be

         designated at the beginning of a financial

         period.

 

4.  Companies must demonstrate at the

     beginning of the period, and on an ongoing   

     basis, that they expect the hedge to be  

     effective.

 

 

 

 

 

 

 

 

 

 

 

Selecting Optimal Hedge Assignments

 

 

objective:        minimize gain or loss from

                         derivatives that is not offset

 

 

 

 

“Allow” hedges when

 

Item and derivative have demonstrated offsetting gains and losses

 

                                              AND

 

 

Item and derivative share sensitivity to same

risk indicators

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Complying with SFAS 133 can lead to four

different programs:

 

 

§      hedging against market risk only

 

 

§      hedging against all four risks

 

 

§      hedging complete items

 

 

§      hedging with basis swaps and written options

 

 

 

 

 

 

 

 

 

 

 

 

 

Shared sensitivity to risk indicators and offsetting gains and losses can demonstrate an effective hedge

 

 

 

 

risk

risk indicator
 
Market risk
 
S&P 500 Index

Natural gas 6M forward price

Coffee 6M futures price

 
 
Market interest rate risk
 
USD LIBOR spot

USD LIBOR 6M

USD LIBOR 12M

DM LIBOR spot

DM LIBOR 6M

 
 
FX risk
 
USD/Brazilian Real exchange rate

USD/DM exchange rate

 
 
Credit risk
 
Industry sector index

 

 

 

 

 

 

 

 

 

 

 

 

 

Hedging against all four risks:

 

§      If market risk is hedged for any item, then only market risk can be hedged for that item.

 

 

§      Interest rate risk, foreign exchange risk, and credit

        risk can be jointly hedged against for an item, as

        long as market risk is not hedged against.

 

 

§      A portion of an item can be hedged against.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hedging against all 4 risks

 

Variables:

 

vj Î R+            continuous variable for absolute value of

                       difference between gain or loss on

                       derivative and loss or gain on item

 

 

Yjk,a                 continuous variable for amount of item k

                       hedged by derivative j for each risk a   

 

 

Z jk,a                0-1 variable signifies whether hedge is    

                       allowed

 

 

Wj                   0-1 variable signifies whether dervative j

                       satisfies lower bound requirement

 

Data Input:

 

qjk,a                 risk sensitivity for item k/derivative j hedge

                       against risk a

 

Igk,a                 item gains relative to risk a

Ilk,a                  item losses relative to risk a

 

Dgj                  derivative gains

Dlj                   derivative losses

 

 

 

 

Optimal hedge program

 

maximize         -åj vj

 

subject to:

 

                " j  " k  " a    qjk,a * Yjk,a, ³ 0

 

                " j " a      åk=1...M  Igk,a * Yjk,a  £ 1.25Dl j

 

                 " j " a      åk=1...M  Ilk,a * Yjk,a  £ 1.25Dg j

 

                         " k  åa=1...4 åj=1...N   Yjk,a  £ 1

 

                " j  " k  " a   Yjk,a - Zjk,a  ³ 0

              

                " j  " k  Zjk,1 +  Zjk,2   £ 1

 

                " j  " k  Zjk,1 +  Zjk,3   £ 1

 

                " j  " k  Zjk,1 +  Zjk,4   £ 1

 

                " j   åa=1...4 åk=1...M  Iglk,a * Yjk,a - lb* Wj ³ 0

 

                " j   åa=1...4 åk=1...M  Iglk,a * Yjk,a  - v j £ Dgl j

 

                        " j   -åa=1...4 åk=1...M  Iglk,a * Yjk,a  - v j £ - Dgl j

 

 

 

 

 

 

Program Characteristics

 

 

§      Use CPLEX 5.0

 

 

§      Large sparse coefficient matrix; most entries 0, ± 1

 

 

§      Large diagonal submatrices

 

 

§      Most of the constraints on 0-1 variables are clique constraints

 

 

§      Many rows and columns eliminated during presolve

 

 

§      Could do preprocessing as separate phase

 

 

 

 

 

 

 

 

 

Experimental Trials

 

 

Ø  Ran program in batches of 100 or 225 trials on UltraSparc

 

 

Ø  Number of items and derivatives ranged between 15 and 35

 

 

Ø  Used random number generator and benchmark to set risk sensitivities for items and derivatives

 

 

Ø  Varied benchmark among 3 values: 0.6, 0.75, 0.85

 

 

Ø  Recorded number of trials in each batch where

§     MIP optimum equaled relaxed optimum

§     both equaled 0

 

 

 

Ø   Recorded number of rows, columns, and nonzeros eliminated during presolve

 

 

 

 

 

 

 

 

Experimental Results

 

§      Number of rows and columns eliminated tied very closely to benchmark

 

 

§      Number of trials where MIP optimum reached relaxed optimum also linked to benchmark value

 

 

§      Time to complete usually under 5 seconds, but most batches had outliers; number of outliers seemed to increase when benchmark higher