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Math 472 Spring 2011 Midterm 1 Formulas

Math 472 Spring 2011 Midterm 1 Formulas - h 1 • b h 1-h 1...

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MATH 472/MATH 567 Actuarial Theory II/Topics in Actuarial Theory I Midterm 1: Formula Summary This review sheet contains the key concepts from Chapter 8 (8.1-8.5) of the text. BENEFIT RESERVES FOR GENERAL INSURANCE: Fully Discrete Life Insurance on (x): Prospective Loss @ time h: h L = b K ( x )+1 v K ( x )+1 - h - K ( x ) j = h π j v j - h for K ( x ) = h , h + 1, ... Prospective Benefit Reserve @ time h: h V = j =0 b h + j +1 v j +1 j | q x + h - j =0 π h + j v j j p x + h Retrospective Benefit Reserve @ time h: h V = 1 h E x [ h - 1 j =0 π j v j j p x - h - 1 j =0 b j +1 v j +1 j | q x ] Equivalence Principle: 0 V = 0 Fully Continuous Life Insurance on (x): Prospective Loss @ time t: t L = b T ( x ) v T ( x ) - t - R T ( x ) t π u v u - t du for T ( x ) > t Prospective Benefit Reserve @ time t: t ¯ V = R 0 b t + u v u u p x + t μ x + t + u du - R 0 π t + u v u u p x + t du Retrospective Benefit Reserve @ time t: t ¯ V = 1 t E x [ R t 0 π u v u u p x du - R t 0 b u v u u p x μ x + u du ] Equivalence Principle: 0 ¯ V = 0 1
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Recursions for Fully Discrete Benefit Reserves: ( h V + π h )(1 + i ) = q x + h × b h +1 + p x + h × h +1 V OR ( h V + π h )(1 + i ) = h +1 V + q x + h [ b h +1 - h +1 V ] h V + π h = initial benefit reserve for year ( h + 1) h V = terminal benefit reserve for year h
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Unformatted text preview: h + 1) • b h +1-h +1 V = net amount at risk for year ( h + 1) Interim Benefit Reserves : Exact formula: h + s V = v 1-s ( b h +1 ) 1-s q x + h + s + v 1-s ( h +1 V ) 1-s p x + h + s where h = 0, 1, . .. and 0 < s < 1 Approximate formula, UDD: h + s V ≈ [Linear interpolation of successive terminal benefit reserves at h , h + 1] + [Unearned benefit premium during the valuation period]. Note: You may use the approximate formula when given UDD to calculate the interim benefit reserve. Hattendorf Theorem : var [ h L | K ( x ) ≥ h ] = var [Λ h | K ( x ) ≥ h ] + v 2 p x + h var [ h +1 L | K ( x ) ≥ h + 1] where var [Λ h | K ( x ) ≥ h ] = v 2 ( b h +1-h +1 V ) 2 p x + h q x + h 2...
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