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Team Maverick Bond Portfolio Managment Projekt Predrag Pesic, Bhavneesh Shukla, Sandesh Gn, Eleftherios Ninos, Nermeen Kishk Part 1Step 1: Adjust the bond price with accrued interestThe formula was applied to calculate the accured interest (A I)of ich Bond AI=days since last cupon/days in current cupon period*F*C%/mDays since last coupon date:57 days (11,10,2012-15,02,2013)Days in current coupon period:182Face value (F): $100 Coupon rate: C % Annual coupon periods (m): 2TodaysDate 11.10.2012Days inYear 365 Bond Quotes Price with Coupon Maturity Accrued Rate Ask Price Interest Date (%) x/32 Decimal 15.2.2013 4,625% 101 22 0,6875 $ 101,72 15.2.2013 0,875% 100 4 0,1250 $ 100,14 15.8.2013 4,375% 103 20 0,6250 $ 103,69 15.8.2013 1,750% 101 8 0,2500 $ 101,27 15.2.2014 3,875% 105 18 0,5625 $ 105,61 15.2.2014 1,375% 101 18 0,5625 $ 101,22 15.8.2014 4,250% 107 16 0,5000 $ 107,67 15.8.2014 0,750% 100 17 0,5313 $ 100,12 15.2.2015 4,000% 108 18 0,5625 $ 108,63 15.2.2015 1,875% 103 12 0,3750 $ 103,29 15.8.2015 4,250% 110 21 0,6563 $ 110,67 15.8.2015 0,500% 100 1 0,0313 $ 100,08 15.2.2016 4,000% 111 15 0,4688 $ 111,63 15.2.2016 2,375% 106 9 0,2813 $ 106,37 15.8.2016 4,250% 113 23 0,7188 $ 113,20 15.8.2016 1,250% 102 19 0,5938 $ 102,20 15.2.2017 4,500% 115 13 0,4063 $ 115,70 15.2.2017 2,125% 105 9 0,2813 $ 105,33 Table 1: Adjusted Bond Price with Accrued InterestStep 2: Term Structure with Zero-bondZero-bonds at each maturity dates are calculated first using the term structure. Since the firsttwo bonds maturing on 15,02 2013 have no coupon payments, the spot rate for 15,02,2013 wasobtained by direct averaging the first two bonds’ spot rates. The rest of the bonds were pairedtogether according to the maturity date. Each pair was used to construct zero-coupon bond at theaccording maturity date with the following equations: Price:xP1+yP2=P0 Cuopon:xC1+yC2=0 Face:xF1+yF2=F0P1 and P2 are adjusted price of pair bonds with accured interest P0 is the price ofcorresponding zero-coupon bond at maturity date.
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The spot rate St at each maturity date can be calculated with following equations: P0=D(t) F0=exp(-Si*t) St=in(F0/P0)/t Zero Coupon Bonds with Face Value $100 Poly Derived Short Rates Maturity Spot Rates Price of Zero Time to Maturity Spot Rate Date 15.2.2013 $99,96 0,347945205 0,112% 0,093% 0,169% 15.8.2013 $99,67 0,843835616 0,396% 0,176% 0,282% 15.2.2014 $99,36 1,347945205 0,474% 0,227% 0,345% 15.8.2014 $99,04 1,843835616 0,524% 0,270% 0,442% 15.2.2015 $98,80 2,347945205 0,515% 0,326% 0,640% 15.8.2015 $98,61 2,843835616 0,491% 0,407% 0,965% 15.2.2016 $98,70 3,347945205 0,391% 0,525% 1,438% 15.8.2016 $97,96 3,846575342 0,536% 0,681% 2,030% 15.2.2017 $96,22 4,350684932 0,885% 0,876% 2,710%Step 3: Term Structure with Polynomial The spot rate at each maturity date can also be approximated by a 4th order polynomial St= D(t)=exp( *t)=exp(-( ))The price from the polynomial approximation Qj(t) can be obtained by summing the discountedcoupon payments and face value of bond j using the corresponding D(t). In order to find the termstructure coefficients, we setup the following least squares optimization: min With constraint ao≥0 the term structure coefficients that minimizes the sum of squarewe show in the Tabel 4 Term Structure Coefficients Sum of Squared Errora0 0,0000026518253 1,06a1 0,0031998339503a2 -0,0017122129494a3 0,0004826096945a4 -0,0000348844366 Tabel 4
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Step5aCash matchingwith reinvestmentat zero ratePortfolio 0 99,80413576 0 49,80850219 0 399,8128604 0 69,84034757 0 799,8429666 0 119,9179519 0 499,9209498 0 59,98031542 0 149,9840642 233032,2 CF from Portfolio 10000 5000 40000 7000 80000 12000 50000 6000 15000Step 5: Cash Matching of Liabilities A) Simple Cash Matching: excess periodic cash flows are held at zero interest. Main objective of cash matching is to minimize the portfolio costWe contains ( +
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Portfolio generated cash flow-ceash leaved for the next period≥liabilityFor the intermediate period (From 15,02,2013-15,08,2016) ( for j=2,…,8Portfolio generated cash flow + previous excess – cash leaved for the next period ≥ liability(From15,02,2016-15,02,2017) ( Portfolio generated cash flow + previous excess ≥ liability ≥ 0 for j=1,2,…,8 Cash leaved for the next period ≥ 0Step6 A. Present Value and Derivative FormulasLet Ck (k=1,…,9) be the cash flows occurring on the dates of the liabilities, the present valueof this cash flow is: PV= *t )The spot rate( S )ist the first replace in the 4th other polynomial equation. Then, we tookderivative of PV with respect to each of the coefficients Duration-MatchingThere are two requirements for matching the durations:12-Since ,
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The objective of Duration Matching optimization is to minimize the number of bonds:With constraints: exp =0 PV of portfolio cash flow = PV of liability cash flow exp =0 i=0,1,2,3,4 The sensitivity of the present value of the portfolio cash flow to the small change in theCash Matching Cash matching Cash matching with Cash Matching reinvestment at with reinvestment (poly spot rates) zero rateMaturity Coupon Dirty Price Portfolio Portfolio Inputs Minimize15.2.2013 0,04625 $ 102,41 0 Outputs Constraints15.2.2013 0,00875 $ 100,26 99,80413576 Decision Variables15.8.2013 0,04375 $ 104,31 0 Intermediate Results15.8.2013 0,01750 $ 101,52 49,8085021915.2.2014 0,03875 $ 106,17 015.2.2014 0,01375 $ 101,78 399,812860415.8.2014 0,04250 $ 108,17 015.8.2014 0,00750 $ 100,65 69,8403475715.2.2015 0,04000 $ 109,19 0 <===== Decision Variables15.2.2015 0,01875 $ 103,67 799,842966615.8.2015 0,04250 $ 111,32 015.8.2015 0,00500 $ 100,11 119,917951915.2.2016 0,04000 $ 112,10 015.2.2016 0,02375 $ 106,65 499,920949815.8.2016 0,04250 $ 114,38 015.8.2016 0,01250 $ 102,79 59,9803154215.2.2017 0,04500 $ 116,11 015.2.2017 0,02125 $ 105,61 149,9840642 Total Cost 233032,2 0 <===== Objective Function (Minimize) CF from CF from
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Date Obligation Portfolio Portfolio15.2.2013 10000 < 1000015.8.2013 5000 < 500015.2.2014 40000 < 4000015.8.2014 7000 < 700015.2.2015 80000 < 80000 <===== Cash Flow Constraints15.8.2015 12000 < 1200015.2.2016 50000 < 5000015.8.2016 6000 < 600015.2.2017 15000 < 15000Step 7: Comparison Advantage of using the simple cash flow matching method is the portfolio produces sufficient capitalat the exactly times of the liability regardless on whether the spot rate changes. Simple cash flowmethod has the highest portfolio cost among all three methods. The portfolio also does not take in to account the reinvestment opportunity of excessive cash flowgenerated at each period, which makes this method a conservative one. On the contrast, theportfolio constructed by complex cash matching method accounts for the reinvestment of excessivecash flow, which could generate a lower portfolio cost. Conversely, since this reinvestment strategyis dependent on the forward rates, any chance in the forward rate can drastically affect the portfoliocost since the complex cash matching portfolio obtained at time zero is no longer optimal.The immunization portfolio method produces the lowest cost among the three methods. It is alsoless sensitive to small changes in the term structure by combining the portfolio cash flows andliabilities. However, the disadvantage of immunization portfolio method is that it may not producesufficient capital at each time of liability.Since each method has its advantages and disadvantages. It is based on the objective of theinvestor to decide which method is best suited for him or her. When the goal of the investor is topay off the liabilities with minimum risk, simple cash flow matching should be preferred. If theinvestor prefers a lower cost at the expense of higher risk, he or she then can choose complex cashflow matching. It will likely still generate enough capital for each liability. Lastly, Immunizationportfolio should be used when the investor is indifferent about receiving enough capital at eachtime of liability and is more concerned with the overall yield of the portfolio given the cost.
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