CASE STUDY: Cash Matching Bond Portfolio (Linear, Linearmulti, Max_risk )

Background

This case study demonstrates several optimization setups for a simple cash matching problem described in

Luenberger (1998), p.108. The model matches cash obligations over some periods with payments from a

portfolio of bonds. Bonds of various maturities pay coupons as well as face values at different time

periods. We design a portfolio providing cash flows to cover liabilities at all periods and minimizing the

initial portfolio cost. Four optimization problems are formulated. The first and the second problems

disregard surplus cash at every time period, i.e., surplus is not reinvested. The first problem is a Linear

Programming problem. The second problem, which is equivalent to the first problem, is formulated with

PSG nonlinear function max_risk. The third and the fourth problems carry forward extra cash with some

interest. This carry-over is done with additional carry-over variables which can be interpreted as artificial

bonds. The third problem is a Linear Programming problem. The fourth problem, which is equivalent to the

third problem, is formulated with the PSG nonlinear function max_risk.

References

[1] Luenberger, D.G. (1998): Investment Science, Oxford University Press, 494 p.

Notations

J = number of scenarios (time periods), Jj ,,1 index of scenarios (corresponding years);

jb = liability at the end of year j (on scenario j );

I = number of bonds in the replication portfolio; Ii ,,1 index of bond in the portfolio;

ix = number of shares of bond i in the portfolio; ),,( 21 Ixxxx is a decision vector;

ia = price of bond i at initial time;

ijc = amount of receipts at the end of year j (on scenario j ) resulting from one share of bond i ;

r = yearly risk free rate (constant over time Jj ,,1 ;

j

j

jr

b

10 = liability jb discounted to the present time, Jj ,,1 ;

j

ij

jir

c

1 = bond payment ijc , discounted to present time, Ii ,,1 , Jj ,,1 ;

0 = random value having equally probable realizations 010 ,, J ;

i = random value having equally probable realizations Jii ,,1 , Ii ,,1 ;

I ,,, 10 = random vector having equally probable scenarios jIjj ,,1

, Jj ,,1 ;

I

iijijj xxL

10),(

= underperformance of replicating portfolio compared to liability at the end of

year j (on scenario j ) discounted to initial time:

I

iiixxL

10),(

= loss function having scenarios ),(,),,( 1 JxLxL

;

js = cash surplus at year j (can be interpreted as the number of shares of an artificial bond j absorbing

extra cash in the end of year j and paying it at year j +1 );

111 ),(),( sxLxLs

= underperformance of replicating portfolio compared to the liability at the end of

year one with additional carry-over variable 1s ;

jjjjs ssrxLxL 1)1(),(),(

= underperformance of replicating portfolio compared to the liability

at the end of year j in case with carry over variables, Jj 1 ;

),(

xLs= loss function having scenarios, ),(,),,( 1 J

ss xLxL

;

),(max)),((1

jJj

xLxLmax_risk

.

Problem 1. (Linear Programming) Surplus (extra cash) is not reinvested.

minimizing portfolio cost

I

iii

xxa

1

min (CS.1)

subject to

constraints assuring zero underperformance of replicating portfolio

JjxLj ,,1,0),(

(CS.2)

lower bounds on positions

Iixi ,,1,0 (CS.3)

Problem 2. (Using Maximum Risk (max_risk) PSG function) Surplus (extra cash) is not reinvested. This

problem is equivalent to Problem 1.

Minimizing portfolio cost

I

iii

xxa

1

min (CS.4)

subject to

constraint assuring zero underperformance of replicating portfolio

0)),((

xLmax_risk (CS.5)

lower bounds on positions

Iixi ,,1,0 (CS.6)

Problem 3. (Linear Programming) Extra cash is carried forward at interest r .

minimizing portfolio cost

I

iii

xxa

1

min (CS.7)

subject to

cash balance constraints

JjxL js ,,1,0),(

(CS.8)

lower bounds on variables

Iixi ,,1,0 (CS.9)

Jjs j ,,1,0 (CS.10)

Problem 4. (Using Maximum Risk (max_risk) PSG function ) Extra cash is carried forward at interest r .

minimizing portfolio cost

I

iii

xxa

1

min (CS.11)

subject to

constraint assuring zero underperformance of replicating portfolio

0)),((

xLmax_risk s (CS.12)

lower bounds on variables

Iixi ,,1,0 (CS.13)

Jjs j ,,1,0 (CS.14)