ALTAM - Section 2 (Multiple Decrement Models) Questions & Answers Rated 100% Correct!!

Multiple Decrement Model - Answer-A special type of multiple sate model where multiple decrements branch out from a single starting point. tq^(j)x - Answer-The probability (x) fails within t years due to decrement j. tq^(τ)x = - Answer-Σ[j=1 to m] tq^(j)x = td^(τ)x/l^(τ)x tq^(j)x = - Answer-Σ[k=0 to t-1] kp^(τ)x q^(j)x+k = Σ[k=0 to t-1] k|q^(j)x = td^(j)x/l^(τ)x tp^(τ)x = - Answer-1 - tq^(τ)x = l^(τ)x+t/l^(τ)x t|uq^(j)x = - Answer-tp^(τ)x * uq^(j)x+t = ud^(j)x+t/l^(τ)x l^(τ)x - Answer-the total number of lives at age x td^(τ)x - Answer-the total number of failures due to all causes between age x and x+t td^(j)x - Answer-the number of failures to to decrement j within t periods td^(τ)x = - Answer-l^(τ)x - l^(τ)x+t = Σ[j=1 to m] td^(j)x EPV(future benefits) = - Answer-Σ[j=1 to m] Σ[k=0 to ∞] v^k+1 * b^(j) * kp^(τ)x * q^(j)x+k EPV(annuity) = - Answer-Σ[k=0 to ∞] v^k * kp^(τ)x µ^(j)x+t - Answer-the force of failure due to decrement j at age x+tTotal Force of Decrement, µ^(τ)x+t = - Answer-Σ[j=1 to m] µ^(j)x+t tp^(τ)x (continuous) = - Answer-e^-∫[0 to t] µ^(τ)x+s ds tq^(τ)x (continuous) = - Answer-∫[0 to t] sp^(τ)x * µ^(τ)x+s ds µ^(τ)x+t = - Answer-d/dt tq^(τ)x / tp^(τ)x µ^(j)x+t = - Answer-d/dt tq^(j)x / tp^(τ)x tq^(j)x = - Answer-∫[0 to t] sp^(τ)x * µ^(j)x+s ds d/dt tq^(j)x = - Answer-tp^(τ)x * µ^(j)x+t t|uq^(j)x (continuous) = - Answer-∫[t to t+u] sp^(τ)x * µ^(j)x+s ds l^(τ)x+s (UDD) = - Answer-(1 - s)*l^(τ)x + s*l^(τ)x+1 spx^(τ) (UDD) = - Answer-1 - sq^(τ)x sq^(τ)x (UDD) = - Answer-s*q^(τ)x, 0≤s<1 sq^(j)x (UDD) = - Answer-s*q^(j)x, 0≤s<1 sp^(τ)x (UDD0 = - Answer-1 - s*q^(τ)x, 0≤s<1 µ^(j)x+s (UDD) = - Answer-q^(j)x / (1 - s*q^(τ)x)µ^(τ)x+s (CF) = - Answer-µ^(τ)x µ^(j)x/µ^(τ)x (CF) = - Answer-= q^(j)x/q^(τ)x sq^(j)x (CF) = - Answer-q^(j)x/q^(τ)x [1 - (p^(τ)x)^s] EPV(benefits) (continuous) = - Answer-∫[0 to ∞] v^t*tp^(τ)x*µ^(j)x+t dt EPV(benefits) (continuous, CF) = - Answer-µ^(j)x/(δt*µ^(τ)) EPV(annuity) (continuous) = - Answer-∫[0 to ∞] v^t*tp^(τ)x dt Associated Single Decrement Model - Answer-A model where decrements are independent of each other. Independent Probability/ Absolute Rate of Decrement, tq'^(j)x - Answer-The probability of (x) failing due to decrement j within t years, in the absence of other decrements. Dependent Probability, tq^(j)x - Answer-The probability of (x) failing due to decrement j within t years, in the presence of other decrements. tq^(j)x ___ tq'^(j)x - Answer-≤ tq'^(j)x ___ tq^(j)x - Answer-≥ tq'^(j)x + tp'^(j)x = - Answer-1 tq'^(j)x = - Answer-∫[0 to t] sp'^(j)x*µ^(j)x+s ds µ^(j)x+t (independent) = - Answer-d/dt t'^(q)x / tp'^(j)xtp'^(j)x = - Answer-e^-∫[0 to t] µ^(j)x+s ds tp^(τ)x (independent) = - Answer-∏[j=1 to m] tp'^(j)x tq^(τ)x (independent) = - Answer-∑[j=1 to m] tq^(j)x sp^(τ)x (CF) = - Answer-e^-s*µ^(τ)x sp'^(j)x (CF) = - Answer-e^-s*µ^(j)x = (sp^(τ)x)^(µ^(j)x/µ^(τ)x) = (sp^(τ)x)^(q^(j)x/q^(τ)x) ln(sp'^(j)x) / ln(sp'(τ)x) (UDDMDT) = - Answer-q^(j)x / q^(τ)x q'^(j)x (UDDASDT) = - Answer-sp'^(j)x*µ^(j)x+s sp'^(j)x (UDDASDT) = - Answer-1 - s*q'^(j)x sq^(1)x (UDDASDT, two decrements) = - Answer-q'^(1)x(s - s^2*q'^(2)/2), 0≤s≤1 sq^(2)x (UDDASDT, two decrements) = - Answer-q'^(2)x(s - s^2*q'^(1)/2), 0≤s≤1 sq^(1)x (UDDASDT, three decrements) = - Answer-q'^(1)x[s - s^2/2(q'^(2)x + q'^(3)x) + s^3/3(q^'(2)x*q'^(3)x)], 0≤s≤1 sq^(j)x (CF) = - Answer-q^(j)x/q^(τ)x * (1 - (p^(τ)x)^s) q^(2)x (Case 1) = - Answer-q^(τ)x - q^(1)x = (1 - q'^(1)x)*q'^(2)x q^(2)x (Case 3) = - Answer-(1 - s*q^(1)x)*q'^(2)xp^(τ)x (Case 3) = - Answer-(1 - q'^(1)x)(1 - q'^(2)x) q^(1)x (Case 3) = - Answer-s*q'^(1)x + p'^(2)x*(1 - s)*q'^(1)x q^(3)x (Case 4) = - Answer-q^(τ)x - q^(1)x - q^(2)x