Actuarial Exam 2/FM Prep: Continuous Modified Duration as a Function of n

Exercise *7.1.9 (modified): Consider a bond with continuous coupons at rate r per period and redemption amount 1 in n periods, valued at (continuous) yield rate \[Delta]. (a) Express (d L)/(d \[Delta]) (where L=L(\[Delta]) is the present value of the income stream) in terms of continuous annuity symbols and the factors r, n, and \[Delta]. (b) The modified duration is defined to be Subscript[D, mod]=-(1/L)\[CenterDot](d L)/(d \[Delta]). Find an expression for (d Subscript[D, mod])/(d n). Show that it has the same sign as \[Delta]+(r-\[Delta])[r(n-Subscript[Overscript[a, _], Overscript[n, _]|])+\[Delta] Subscript[Overscript[a, _], Overscript[n, _]|]], so that if r greater than or equal to \[Delta], then Subscript[D, mod] increases with n, but if r less than \[Delta], then Subscript[D, mod] is increasing for n=0 to n=Subscript[n, 0], and then decreasing for n greater than Subscript[n, 0]. (This accounts for the smaller durations for the 60-coupon bonds than the 30-coupon bonds in line 1 of Table 7.1b and lines 1 and 2 of Table 7.1c...on page 398). (Also see Video #155 for a discrete Macaulay duration version of this.) (c) Give a financial interpretation to the results of part (b). (Also see Video #155 for a discrete Macaulay duration version of this.)

Financial Math for Actuarial Exam 2 (FM), Video #165. Exercise *7.1.9 (modified) from "The Mathematics of Investment and Credit", 7th Edition, by Samuel A. Broverman.

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[samario_torres]

Do you plan on making any videos for more in depth time series? Thanks!