Future Loss Random Variable¶

Definition¶

For a policy still in force at duration $t$ , let $Z_t$ , $H_t$ and $Y_t$ be present values at time $t$ of future benefits, expenses and premium income. The gross future loss is $L_t^g=Z_t+H_t-Y_t$ $Y_{t}$ . The net loss omits expenses. All quantities refer to the policy conditional on its being in force at $t$ .

Interpretation¶

The insurer faces random future obligations and random future premium income. Positive loss means future outgo exceeds income; negative loss means a future surplus.

Two bars for one realized scenario: outgo made of benefits Z_t = 100 plus expenses H_t = 5, against premium income Y_t = 80, with the excess of 25 bracketed as the gross loss.

The figure takes the realized scenario from the Examples: benefits $Z_t=100$ and expenses $H_t=5$ make up the outgo bar, premium income $Y_t=80$ the income bar, and the gap between their tops is $L_t^g=105-80=25$ $25$ . A different realized lifetime would change all three bars, which is why $L_t^g$ is a random variable rather than a number.

Actuarial present value averages a random discounted cash flow. In contrast, $Z_t$ , $H_t$ and $Y_t$ retain the realized uncertainty; their conditional expectations, rather than the realized values themselves, enter an expected policy liability.

Key Results¶

The time- $t$ discount origin differs from the issue-date discount origin.
A policy ended by death before $t$ does not have an in-force future-loss distribution at $t$ .
The course normally measures the loss after any benefit at $t$ but before the premium at $t$ . Annuity timing must be specified separately.
An equivalence premium gives $\mathbb{E}[L_0^g]=0$ only under the same valuation and expense basis.

Examples¶

If the future benefit present value (PV) is 100, expense PV 5 and premium-income PV 80 in a realized scenario, the gross future loss is 25. These realized quantities need not equal their expectations.

Prerequisites¶

Prospective Policy Value

Sources¶

Life Insurance · Lecture 7

Aliases: gross future loss, net future loss, future loss, loss random variable, prospective loss random variable Keywords: benefits plus expenses minus premiums, gross loss L t g, loss at issue L 0, expected loss zero under the equivalence premium

Last updated: 16 October 2026 22:28