The variance of the loss variable L, where L = benefits + expenses - premiums, can be expressed as which of the following forms (in terms of B, F, V, P, d, Ax, and V)?

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Multiple Choice

The variance of the loss variable L, where L = benefits + expenses - premiums, can be expressed as which of the following forms (in terms of B, F, V, P, d, Ax, and V)?

The main idea is how variance behaves for a life-contingent cash flow with a fixed net amount at risk. The randomness comes from when payments occur, governed by the lifetime distribution encoded in Ax. For a unit stream of payments, the second-moment factor that appears in this context is (2Ax − Ax^2). If the total net amount at risk is the fixed quantity B + F + (1 − V)P/d, then the variance of the loss is this net amount multiplied by the unit second-moment factor: (B + F + (1 − V)P/d) × (2Ax − Ax^2). That matches the given correct form. The other options would misstate the moment structure: introducing a square on the net amount would overstate the variance, and using Ax or Ax^2 alone does not reflect the correct second-moment factor for a unit life-contingent payment stream.

The main idea is how variance behaves for a life-contingent cash flow with a fixed net amount at risk. The randomness comes from when payments occur, governed by the lifetime distribution encoded in Ax. For a unit stream of payments, the second-moment factor that appears in this context is (2Ax − Ax^2). If the total net amount at risk is the fixed quantity B + F + (1 − V)P/d, then the variance of the loss is this net amount multiplied by the unit second-moment factor: (B + F + (1 − V)P/d) × (2Ax − Ax^2).

That matches the given correct form. The other options would misstate the moment structure: introducing a square on the net amount would overstate the variance, and using Ax or Ax^2 alone does not reflect the correct second-moment factor for a unit life-contingent payment stream.

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