Financial Mathematics: Asset-Liability Management & Duration - UConn, Math 3615 - Prof. Jo, Study notes of Mathematics

Download Financial Mathematics: Asset-Liability Management & Duration - UConn, Math 3615 - Prof. Jo and more Study notes Mathematics in PDF only on Docsity! University of Connecticut Math 3615: Financial Mathematics Problems Fall 2008 Summary – Module 7 - ASSET-LIABILITY MANAGEMENT, DURATION, AND IMMUNIZATION - Duration (or Macaulay Duration) – D, MacD, or d The “duration” of a stream of payments is the weighted average of the number of years until each future payment. The weight applied to the number of years for each payment is a fraction equal to the present value of that payment, divided by the total present value of all the payments. Note that duration depends on not only the amounts and dates of the payments, but also the interest rate at which the present values are calculated. (Alternate description: Duration is the first moment (about t=0) of the present values of a set of future payments, divided by the total present value of the payments.) Modified Duration – DM, ModD, or v The “modified duration” of a stream of payments is the duration (as defined above), divided by (1+i). The significance of modified duration is that it represents the proportionate (percentage) change in the payments’ present value as the result of a change in the interest rate used to value the payments (a duration of 1 implies that the percentage change in value equals the number of percentage points by which i changes). Note that the sign of the change in value is opposite to that of the change in i: '( ) 1 D P i DM i P − = = + where P is the price (or value) of the stream of payments (at interest rate i) Convexity Convexity (modified convexity, actually) measures the change in an investment’s duration due to a change in interest rate: Conv ( ) ( ) P i P i ′′ = For any single payment occurring in n periods (e.g., an n-year zero-coupon bond): D = n DM = n / (1+i) Conv = n (n+1) / (1+i) 2 For a series of level payments (an n-year level annuity-immediate): | | ( ) n n Ia D a = | | ( ) 1 n n IaD DM i a = = +

For a bond: |( ) Bond Price n n Fr Ia nCv D + = |( ) / (1 ) Bond Price n n Fr Ia nCv DM i + = + Note: If C (redemption value) = Face, and if r (coupon rate) = i, then |n D a=

and |n DM a= .