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Unformatted text preview: Primbs/Investment Science 1 Topic 6: Bond Portfolios Reading: Luenberger Chapter 3, Section 57 Chapter 4, Section 9 Chapter 5, Section 2 Primbs/Investment Science 2 Bond Portfolios Bond Portfolios Duration of a Portfolio Immunization General Principles and Convexity Cash Matching Primbs/Investment Science 3 Duration of a Portfolio Given fixed income securities with prices P i and duration D i , i=1...m . The portfolio consisting of the aggregate of these has price P and duration D given by m P P P P + + + = ... 2 1 m m D w D w D w D + + + = ... 2 2 1 1 where m 1,..., i , = = P P w i i Formula works for both Macaulay and all forms of Modified duration. Note : This looks just like the Macaulay duration formula except with time replaced by duration! Primbs/Investment Science 4 Duration of a Portfolio Assume we have two cash flow streams. n ∑ = = n k P PV t A A A k k D A n ∑ = = n k P PV t B B B k k D B Primbs/Investment Science 5 Duration of a Portfolio What is the duration of A+B. ∑ = + + + = n k P P PV PV t B A B A B k A k k D ) ( ∑ ∑ = + = + + = n k P P PV t n k P P PV t B A B k k B A A k k ( 29 ( 29 ∑ ∑ = + + = + = n k P P P P PV t P P P n k P PV t B A B B B k k B A A A A k k ( 29 ( 29 B P P P A P P P D D B A B B A A + + + = Primbs/Investment Science 6 Bond Portfolios Bond Portfolios Duration of a Portfolio Immunization General Principles and Convexity Cash Matching Primbs/Investment Science 7 Immunization Construct a portfolio of bonds which is protected (or immunized) against changes in interest rates. Assume: You have an obligation to pay $1,000 two years from now. But, you can only purchase bonds of maturity 1 and 5 years. 1 year: reinvestment risk 5 year: too sensitive to interest rates Solution: Construct a portfolio of 1 and 5 year securities with the same present value and duration as your obligation. Primbs/Investment Science 8 Immunization Construct a portfolio with present value P and duration D equal to those of your obligation: Two Equations: PV: P yP xP = + 2 1 Duration: D D P yP D P xP = + 2 2 1 1 Solve for x and y to determine your portfolio. Primbs/Investment Science 9 Example You have an obligation of $1000 in 2 years, and you wish to invest now to meet that obligation. You can invest in a 1 year and 5 year zero coupon bonds. (use discrete yearly compounding.) Obligation $1000 in 2 years $841.68 2years Coupon Maturity Yield Price Duration (Mac) Bond 1 0% 1 years 9% $91.74 1year Bond 2 0% 5 years 9% $64.99 5years Primbs/Investment Science 10 Example Obligation $1000 in 2 years $841.68 2years Coupon Maturity Yield Price Duration (Mac) Bond 1 0% 1 years 9% $91.74 1year Bond 2 0% 5 years 9% $64.99 5years We need to form a portfolio with the same price and duration as the obligation:...
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 Fall '06
 PRIMBS
 Bond duration, Macaulay, Zerocoupon bond, Primbs/Investment Science

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