Report Project I Zurich Lions

7/30/2019 Report Project I Zurich Lions

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Stanford University Finance Fall 2012 Bond Portfolio Management Project

Instrucor: Kay Giesecke

Team: Zurich Lions

Velican Usta & Yefreed Ditta

A.Part 1Step 1. Price Adjustment

Prices of the bonds were adjusted by calculating first the decimal part and adding that to the price,

and then the next step was to calculate the accrued interest:

Accrued Interest = Days elapsed since last maturity * Coupon / Days in a year

Results are shown in column F in the FrontEnd worksheet (Appendix A.1).

Step 2. Spot Rates using Zero Coupon Bonds

To construct the spot curve zero coupon bonds are constructed per maturity (except 15.02.2013

where the averages of two bonds were used). The amount of bonds per maturity was determined byselling 1 bond (the one with the higher coupon rate in the period) and buying n bonds of the ones

with the lower coupon rate, where n is the result of dividing the higher coupon rate by the lower.

Having this in mind is clear that the face value Fof the zero coupon bond is:

F = (n-1)*100 (1)

And todays price normalized to 100 is:

P = (n*price of 1st bondprice of 2ndbond)/ (n-1) (2)

Using continuous compounding,

D (t) = e r (t) t

(3)

Since the coupon payments cancel each other, the only cash flow will be the face value at the end of

the period, hence:

P = F * D (t) (4)

Replacing (3) in (4)

=


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For the first spot rate calculation (15.02.2013), the average of the two bonds are taken into account,

using the dirty price of the bond as P and the last payment at the maturity date as Fin (5):

P = dirty price including accrued interest

F = 100*(1+coupon rate/2)

The respective rates are averaged afterwards. Results are in the FrontEnd worksheet (Appendix

A.2).

Step 3. Spot Rates using a 4th Order Polynomial

The term structure of interest rates are approximated by a 4th

order polynomial of the form:

r(tj) = a0 + a1*tj + a2* tj2

+ a3* tj3

+ a4* tj4

The coefficients are estimated by minimizing the square of the price differences:

minimize 2

)( jj QP

where Pj: dirty price of the bond

Qj: NPV of the bond

The calculations of the NPV of the bonds (Appendix A.3) uses the discount factors which include the

rates approximated by the 4th

order polynomial. The results are in the FrondEnd wosksheet

(Appendix A.3).

Using these NPVs and the dirty prices, the sum of the squared errors are then minimized using the

Solver function of MS Excel (Appendix A.3). The resulting coefficients are:

a0=0,a1=0.001233,a2=0.000852, a3=-0.000479, a4= 0.000075

Step 4. Plots of Term Structure


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The first method is straightforward however gives increasing and decreasing trends in the spot rate

curve which is not realistic as the expected interest rate to invest money in longer periods should be

higher than the shorter periods that the spot rate curve is expected to be strictly increasing. The first

method is sensitive to the different bonds as the characteristics of the bonds for each maturity date

may differ from one another, therefore it is more advantageous to use an average method such as

the polynomial approximation as in the second method.

B.Part 2Step 5A. Simple Cash Matching

The total cost of the portfolio is minimized, with the number of bonds purchased as variables:

minimize ii Px , where xi is the number and Pi is the price of ith

bond.

The constraints are that the liabilities must be exceeded by the total cash flows at each maturity date

and the number of bonds purchased must be grater than or equal to 0.

The results (cash flows and the portfolio) are in the CashMatchFront worksheet (Appendix B.1).

Step 5B. Complex Cash Matching

In this step, the same approach is used as in step 5A, with an additional setting that the excess cash

flow at each payment date can be reinvested at the forward rates which are computed using a

continuous compounding and between two consecutive due dates:

12

112221

tt

tstsf tttt

=

The excess cash flow is reinvested using the forward rates between two consecutive dates using a

half-year compunding as it is assumed that the reinvestment instrument will be the purchase of

bonds which make payments in half-year periods.

The cash flow at time t2 resulting from the reinvestment at time t1 is:

CFt2 = (Excess Flow)t1* (1 + f12/2)

This cash flow is added to the cash flow resulting from the payments and coupons. The results

(forward rates, cash flows and the portfolio) are in the CashMatchFront worksheet (Appendix B.2).


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and u(ai) = iiiii ttatatataa ++++ )(4

4

3

3

2

210

Utilization of the chain rule gives:

iida

du

du

dy

da

dy=

The termsdu

dyfor each derivative is the same which is e

-u(the discount rate multiplied with -1).

Using this rule, the derivatives of the present value of the portfolio with respect to the polynomial

coefficients are:

)()(

0

44

33

2210

i

ttatatataa

tteCF

da

dPViiiii

i=

++++

)(

2)(

1

44

33

2210

i

ttatatataa

t teCFda

dPViiiii

i =++++

)(3)(

2

44

33

2210

i

ttatatataa

tteCF

da

dPViiiii

i=

++++

)(4)(

3

44

33

2210

i

ttatatataa

tteCF

da

dPViiiii

i=

++++

)(5)(

4

44

33

2210

i

ttatatataa

t teCFda

dPViiiii

i=

++++

The first constraint is that the PV of the cash flows are equal to that of liabilities. The immunization

formula (given the fact that the PVs are equal) results in the constraints that the derivatives are also

equal for cash flows and liabilities.

The next step is to set the constaints for 5 cases:

1) Equalize PVand dPV/da02) Equalize PV, dPV/da0and dPV/da13) Equalize PV, dPV/da0,dPV/da1 and dPV/da24) Equalize PV, dPV/da0,dPV/da1, dPV/da2 and dPV/da3


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advantage of the immunized portfolio over the portfolios with present value matching. Moreover as

a natural result of this method, there is not a need anymore to match the present values of each cashstream and the portfolio. Different cases immunize the portfolio against different kinds of changes in

interest changes. Case 1 immunizes the portfolio against parallel shifts in the spot rate curve in the

vertical axis (Figure 2.1) which was covered in the lecture.

Figure 2.1 Parallel shifts in spot rates: with respect to a0

To illustrate other shapes of the shifts in the spot rate curve, below is the change of spot rates for 3

representative a1 values (Figure 2.2).

Figure 2.2 Representative shift in the spot rate curve: with respect to a 1

In both graphs, the blue curves represent the spot rates with optimal a0 and a1.


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Appendix A

A.1 Results: Bond Prices with Accrued Interest

Figure A.1 Adjusted bond prices (FrontEnd Worksheet)


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A2. Results: Spot Rates using Zero Coupon Bonds

Figure A.2 Spot rates using the zero coupon bonds (FrondEnd worksheet)

A.3 Results: NPV of Bonds using 4thOrder Polynomial

Figure A.3 NPV of Bonds using 4thOrder Polynomial-after optimization (FrontEnd worksheet)


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Figure A.3 Detailed results coefficient estimation (FrontEnd worksheet)


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Appendix B

B.1 Results: Simple Cash Matching

Figure B.1 Cash flow of the resulting portfolio without reinvestment


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Figure B.2 Portfolio without reinvestment


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B.2 Results: Complex Cash Matching

Figure B.3 Forward rates of two consecutive due dates & cash flow of the resulting portfolio with reinvestment


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Figure B.4 Portfolio with reinvestment


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B.3 Results: Portfolio Immunization

Figure B.5 Immunization of the portfolio for 5 cases

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