Exam (elaborations)
APS502 Financial Engineering Individual Assignment
- Course
- Institution
This assignment only includes the solution and the question is on another PDF file.
[Show more]
Seller
Follow
xxxxxx1
Content preview
PROBLEM-IBOND PORTFOLIO OPTIMIZATION:
Problem Statement:
To Formulate a linear programming model to find the lowest-cost dedicated bond portfolio that
covers the stream of liabilities given.
Let,
Xn = Number of bonds
Pn = Price of nth bond
Zk = Excess amount at the end of kth date
Si = Spot rate
fi,j = Forward rate
Portfolio Details:
Spot rates
s1 = 1%; s2 = 1.5%; s3 = 2.0%; s4 = 2.5%; s5 =3.0%; s6 = 3.5%
Date 1 2 3 4 5 6
Obligations
Required 500 200 800 400 700 900
Bond 1 2 3 4 5 6 Price Rating
1 10 10 10 10 10 110 108 B
2 7 7 7 7 7 107 94 B
3 8 8 8 8 8 108 99 B
4 6 6 6 6 106 92.7 B
5 7 7 7 7 107 96.6 B
6 6 6 6 106 95.9 B
7 5 5 5 105 92.9 A
8 10 10 110 110 A
9 8 8 108 104 A
10 6 6 106 101 A
11 10 110 107 A
12 7 107 102 A
13 100 95.2 A
Table 1: Bonds & Obligation Details
,Cost function:
F(X) = P1X1 + P2X2 + P3X3 + P4X4 + P5X5 + P6X6 + P7X7 + P8X8 + P9X9 + P10X10 + P11X11 +
P12X12 + P13X13
Substituting the price of bonds, we get,
F(X) = 108 X1 + 94 X2 + 99 X3 + 92.7 X4 + 96.6 X5 + 95.9 X6 + 92.9 X7 + 110 X8 + 104 X9 +
101 X10 + 107 X11 + 102 X12 + 95.2 X13
Constraints:
Objective: To Minimize the Cost Function based on the following constraints.
1. 1st year liability is 500, equating 1st year cash inflow to match the liability and Z1 extra
amount is carried over to the next year.
10 X1 + 7 X2 + 8 X3 + 6 X4 + 7 X5 + 6 X6 + 5 X7 + 10 X8 + 8 X9 + 6 X10 + 10 X11 + 7 X12
+ 100 X13 – Z1 ≥ 500
2. 2nd year liability is 200, equating 2nd year cash inflow and Z1 amount along with forward
rate to match the liability and Z2 extra amount is carried over to the next year.
10 X1 + 7 X2 + 8 X3 + 6 X4 + 7 X5 + 6 X6 + 5 X7 + 10 X8 + 8 X9 + 6 X10 + 110 X11 + 107
X12 + Z1(1+f1,2) - Z2 ≥ 200
3. 3rd year liability is 800, equating 3rd year cash inflow along with Z2 amount with forward
rate to match liability and Z3 extra amount is carried over to next year.
10 X1 + 7 X2 + 8 X3 + 6 X4 + 7 X5 + 6 X6 + 5 X7 + 110 X8 + 108 X9 + 106 X10 + Z2
(1+f2,3) – Z3 ≥ 800
4. 4th year liability is 400, equating 4th year cash inflow along with Z3 amount with forward
rate to match liability and Z4 extra amount is carried over to next date.
10 X1 + 7 X2 + 8 X3 + 6 X4 + 7 X5 + 106 X6 + 105 X7 + Z3(1+ f3,4) – Z4 ≥ 400
5. 5th year liability is 700, equating 5th year cash inflow along with Z4 amount with forward
rate to match liability and Z5 extra amount is carried over to next date.
10 X1 + 7 X2 + 8 X3 + 106 X4 + 107 X5 + Z4 (1+ f4,5) – Z5 ≥ 700
6. 6th year liability is 900, equating 6th year cash inflow along with Z5 amount with forward
rate to match the liability in that year.
110 X1 + 107 X2 + 108 X3 + Z5 (1+ f5,6) ≥ 900
, Forward rate calculation:
(𝑗−𝑖) 𝑗
(1 + 𝑠𝑖 )(𝑖) (1 + 𝑓(𝑖,𝑗) ) = (1 + 𝑠𝑗 )
Spot rates for the dates are,
S1 = 1%, S2 = 1.5%, S3 = 2%, S4 = 2.5%, S5 = 3%, S6 = 3.5%
(1+f 1, 2) = (1+s2)2 / (1+s1)1 = (1+0.015)2 / (1+0.01)1 = 1.02
(1+f 2,3) = (1+s3)3 / (1+s2)2 = (1+0. 02)3 / (1+0.015)2 = 1. 03
(1+f 3,4) = (1+s4)4 / (1+s3)3 = (1+0. 025)4 / (1+0.02)3 = 1.04
(1+f 4,5) = (1+s5)5 / (1+s4)4 = (1+0. 03)5 / (1+0.025)4 = 1.05
(1+f 5,6) = (1+s6)6 / (1+s5)5 = (1+0. 035)6 / (1+0.03)5 = 1.06
Substituting the forward rates on the above constraints. We get,
1. 10 X1 + 7 X2 + 8 X3 + 6 X4 + 7 X5 + 6 X6 + 5 X7 + 10 X8 + 8 X9 + 6 X10 + 10 X11 +
7 X12 + 100 X13 – Z1 ≥ 500
2. 10 X1 + 7 X2 + 8 X3 + 6 X4 + 7 X5 + 6 X6 +5 X7 +10 X8 +8 X9 +6 X10 +110 X11 +107 X12
+ 1.02 Z1 - Z2 ≥ 200
3. 10 X1 + 7 X2 + 8 X3 + 6 X4 + 7 X5 + 6 X6 + 5 X7 + 110 X8 + 108 X9 + 106 X10 +1.03Z2
– Z3 ≥ 800
4. 10 X1 + 7 X2 + 8 X3 + 6 X4 + 7 X5 + 106 X6 + 105 X7 + 1.04 Z3– Z4 ≥ 400
5. 10 X1 + 7 X2 + 8 X3 + 106 X4 + 107 X5 + 1.05 Z4 – Z5 ≥ 700
6. 110 X1 + 107 X2 + 108 X3 + 1.06 Z5 ≥ 900
PART-I:
Objective:
Minimize Cost function:
i.e., Min. 108 X1 + 94 X2 + 99 X3 + 92.7 X4 + 96.6 X5 + 95.9 X6 + 92.9 X7 + 110 X8 +
104 X9 + 101 X10 + 107 X11 + 102 X12 + 95.2 X13
Constraints: (No restrictions based on asset rating)
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller xxxxxx1. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
77254 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now