Twice a Year Scientific Journal

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46 Economic Analysis (2010, Vol. 43, No. 1-2, 44-49) To calculate the 1 year zero coupon we can use the price of a 1 year coupon bond and the theoretical spot rate of 0,5 year zero coupon bond, i.e. where is the face value, is the 0,5 year coupon paid at the end of 0,5 year and 1 year, and is the 1 year zero coupon yield. After simplifying we get (5) (6) Using 0,5 zero coupon yield form equation (4) in the equation (6) we compute the theoretical spot rate of 1 year zero coupon bond. Following the same approach the theoretical spot rates (or zero coupon yields) fall of the N maturities are calculated using the zero coupon yields of the previous maturities. In general about 15 maturities are sufficient to produce the whole yield curve of zero coupon bonds. Using the matrix theory in the bootstrapping method To obtain a direct solution we can apply matrix theory there. Let we consider bonds maturing at dates , and let be the total cash flow payments of ith bond (for ) on the date (for ). Then the prices of bonds equal to (7) Generally described matrix equals to matrix product of matrix and matrix , then After simplifying to the matrix , where the matrix presents the square matrix to the matrix . To determine the inverse, we calculate using determinant of , , where (9) and where represents square matrix of type from the matrix . As you could see in (7), the upper triangle of the cash flow matrix on the right side has zero values. By multiplying both sides of equation (7) by the inverse matrix of the cash flow, we get the discount functions corresponding to maturities, then (8) (10)
Glova, J., Matrix Theory Application in the Bootstrapping, EA (2010, Vol. 43, No, 1-2, 44-49) 47 Note that this solution requires that the number of bonds equal the number of cash flow maturity dates. Matrix theory application in the bootstrapping method To demonstrate the bootstrapping method using the matrix approach we assume 10 coupon bonds and their parameters given in the table below. For simplicity we assume all bonds make annual coupon payments. The face value of bonds is 100 €. Bond Price (in €) Maturity(in years) Coupon rate (in per cent p.a.) 1 96,60 1 2 2 93,71 2 2,5 3 91,56 3 3 4 90,24 4 3,5 5 89,74 5 4 6 90,04 6 4,5 7 91,09 7 5 8 92,82 8 5,5 9 95,19 9 6 10 98,14 10 6,5 Using the matrix theory in the bootstrapping method given by equation (10) we get the discount functions corresponding to maturities, then Multiplication of these two matrices gives the solution The theoretical spot rates are obtained from the corresponding discount functions derived from equation (1) or
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