Give the consequence of a constante interest force µ(x):=µ
Given the provided conditions: Constant interest force: � ( � ) : = � μ ( x ) := μ Survival probability: � � � = � − � � > 0 tp x = e − μ t > 0 (where � � � tp x denotes the probability of survival from age � x to age � + � x + t ) Force of interest: � δ Net single premium: � − � = ∫ 0 ∞ � − � � ⋅ � − � � � � = 1 � + � a − x = ∫ 0 ∞ e − δ t ⋅ e − μ t d t = μ + δ 1 Expectation of future lifetime: � � ° = ∫ 0 ∞ � � � � � = ∫ 0 ∞ � − � � � � = 1 � e x ° = ∫ 0 ∞ tp x d t = ∫ 0 ∞ e − μ t d t = μ 1 Net premium reserve: � � = � − � + � − � − � − � + � = 0 t V = A − x + t − P − a − x + t = 0 Let's analyze the consequences of these conditions: Net Single Premium ( � − � A − x ) : � − � = ∫ 0 ∞ � ( � ) ⋅ � � � ⋅ � � ( � ) � � = ∫ 0 ∞ � − � � ⋅ � − � � ⋅ � � � = � � + � A − x = ∫ 0 ∞ v ( t ) ⋅ tp x ⋅ μ x ( t ) d t = ∫ 0 ∞ e − δ t ⋅ e − μ t ⋅ μ d t = μ + δ μ This represents the net present value of future benefits, discounted at the force of interest � δ ...