Financial Applications of Linear
Programming
Prof.Dr.Aydın Ulucan
Multiperiod Cash Balancing
 A company forecasts monthly cash inflow and outflows as
shown below for 2013.
Month
Cash Flow
(000 TL)
Month
Cash Flow
(000 TL)
Jan.
-120
Jul.
-70
Feb.
-100
Aug.
-20
Mar.
-80
Sep.
150
Apr.
-100
Oct.
120
May.
-40
Nov.
-70
Jun.
50
Dec.
450
Multiperiod Cash Balancing
 In order to proceed the payments company should get credit.
 There are two types of credit optins: Long and short term.
 Long term credit might be taken in the beginning of 2013 January.
Every month 2.5% interest is paid and the credit is paid back in
2014 January.
 Short term credits might be taken every month. Monthly credit
interest rate is 3.75%.
 Every month, excess cash earns 1% interest.
Multiperiod Cash Balancing
 Company would like to maximize its cash balance in 2014
January using linear programming approach.
 They also would like to develop the model that minimize the
total payment to credit interest.
 There is 65000 TL cash in hand at the beginning of 2013 an at
the and of each month company would like to have at least
50000 TL cash in hand.
Multiperiod Cash Balancing
 ND(t)= ND(t-1)+ FaizGeliri(t-1) +AlınanBorç(t) +
NakitGirişi(t) – UDönemBorçFaiziÖdeme(t) KDönemBorçFaiziÖdeme(t-1) –
UDönemKrediGeriÖdemesi (Ocak2011) –
KDönemKrediGeriÖdemesi(t-1)
Multiperiod Cash Balancing
Bond Portfolio
 You are holding hold a bond portfolio of four bonds, 100 units of
each of them.You are considering rebalancing it. The bid and ask
prices of the bonds are given in the table below.
Bond 1
Bond 2
Bond 3
Bond 4
Bid Price
$980
$970
$960
$940
Ask Price
$990
$985
$972
$954
 The cash payments of the bonds in the next three years are as follows.
 Cash on hand earns 5% interest.You want to rebalance the portfolio in such a
way that at any time in the future your cash position will be at least as good as
the position that would result from your current portfolio. How much money
you can take out today under this condition?
Year 1
Year 2
Year 3
Bond 1
$100
$110
$1100
Bond 2
$80
$90
$1120
Bond 3
$70
$80
$1090
Bond 4
$60
$50
$1110
Excel Modeli
Pension Funds
 The pension fund manager of the Association of Business Professors identified
three reliable mutual funds, which have a long time successful record of
operation: the Growth Fund, the Global Fund and the Income Fund. All these
funds invest in four asset categories: U.S. large capitalization stocks, U.S. small
capitalization stocks, foreign stocks and U.S. bonds, in proportions given in the
following table.
Asset Categories
U.S. Large Capitalization Stocks
U.S. Small Capitalization Stocks
Foreign Stocks
U.S. Bonds
Average Annual Return
Growth
45%
40%
10%
5%
17%
Funds
Global
20%
10%
60%
10%
14%
Income
30%
5%
5%
60%
10%
 The table also shows an average annual return of these funds observed over a long time
period.
 The pension fund offers to its participants a Balanced Package. The minimum and maximum
investments of the package in each asset category package is given in the table below.
 The pension fund has 5 million dollars. How much should the manager invest in the Growth
Fund, Global Fund and Income Fund to construct the package and to maximize the annual
return?
Asset Categories
Minimum Maximum
U.S. Large Capitalization Stocks
25%
35%
U.S. Small Capitalization Stocks
15%
25%
Foreign Stocks
20%
30%
U.S. Bonds
20%
30%
Pension Fund-Matching
 Bir şirket emeklilik fonunu yönetiyorsunuz. Fon her yıl başında aşağıda
görülen ödemeleri yapmak zorunda.
Ödeme
Ödeme
Ödeme
2010
11
2015
18
2020
25
2011
12
2016
20
2021
30
2012
14
2017
21
2022
31
2013
15
2018
22
2023
31
2014
16
2019
24
2024
31


Bu ödemeleri bono satın alarak finanse etmek istiyorsunuz. Üç bono arasından
portföy oluşturacaksınız.
 Bono 1. Maliyet $980, 2011-2014 kupon ödemesi $60, 2015 ödeme
$1060
 Bono 2. Maliyet $970, 2011-2020 kupon ödemesi $65, 2021 ödeme
$1065
 Bono 3. Maliyet $1050, 2011-2023 kupon ödemesi $75, 2024 ödeme
$1075
1 Ocak 2010’da bonoları satın alacak ve 11000 dolarlık ödemeyi yapacaksınız.
Her yıl elinizdeki nakit %4 faiz kazanıyor. 2010-2024 arası nakit gereksinimini
karşılamak için 2010’da almanız gereken bonoları minimize eden modeli
kurunuz
Arbitraj Modeli