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Utility theory involves the utility function (e.g., $U(x)$) that models the satisfaction or happiness derived from wealth $x$, and is used in actuarial science to assess the risk aversion and pricing strategies based on expected utility rather than expected value.

Commutation functions are a suite of actuarial functions (e.g., $D_x, N_x, M_x, R_x$) based on mortality tables that simplify the calculation of premiums and reserves in life insurance and pensions. Used to create a more efficient computation of present values and probabilities.

$p_{x:t} = \prod_{k=0}^{t-1} (1-q_{x+k})$, showing the probability of surviving each year consecutively. Used in actuarial models to predict life expectancy and other longevity-related measures.

$\frac{d}{dt} V_t = V_t i(t) - \delta_t + \mu_t (1 - V_t)$, where $V_t$ is the reserve at time $t$, $i(t)$ is the interest rate, $\delta_t$ is the premium rate per unit, and $\mu_t$ is the force of mortality at time $t$. Governs the accumulation of funds in a life insurance reserve.

Ruin theory involves models such as the probability of ruin $\psi(u)$, which calculates the chance that a company's reserve will be depleted to zero or negative. Helpful in assessing the sustainability and risk of an insurance company's business model.

$a_x = \sum_{t=1}^{\omega-x} v^t p_{x:t}$ for a discrete life annuity, where $a_x$ is the present value of an annuity for a person at age $x$, $v$ is the discount factor, $p_{x:t}$ is the probability of survival from age $x$ to age $x+t$, and $\omega$ is the maximum age possible. It's used to price an annuity product based on mortality tables.

A tabular representation of the probability that a person of a given age will die before their next birthday. Often represented as $q_x$ for the probability of dying between age $x$ and $x+1$. Mortality tables form the basis for life insurance and annuities pricing.

$q_{x:t} = 1 - p_{x:t}$, where $q_{x:t}$ is the probability of death between the ages of $x$ and $x+t$, and $p_{x:t}$ is the probability of survival from age $x$ to age $x+t$. Used in life insurance to calculate risk of death over a certain period.

The principle stating that the expected present value of premiums should be equal to the expected present value of benefits, used to determine fair and balanced premiums in life insurance and annuity contracts. Ensures that there is no expected gain or loss for the insurer at the policy's inception.

$FV = PV(1 + i)^n$, where $FV$ is future value, $PV$ is present value, $i$ is the interest rate, and $n$ is the number of periods. Used to calculate the future worth of a present sum of money after accruing interest over time.

$\sigma_a = \sqrt{Var(a)}$, where $Var(a)$ is the variance of the annuity present value. Measures the dispersion of possible outcomes around the expected present value of the annuity.

$PV = \frac{FV}{(1 + i)^n}$, where $PV$ is present value, $FV$ is future value, $i$ is the interest rate, and $n$ is the number of periods. Used to determine the value today of a sum of money to be received in the future.

$V_x(T) = \frac{A_x(T) - NP \times a_x(T)}{1 + i}^n$, where $V_x(T)$ is the reserve at time $n$ for a term insurance on (x), $A_x(T)$ is the net single premium of the insurance, $NP$ is the net premium per period, and $a_x(T)$ is the present value of a term annuity. Calculated to understand the amount that needs to be held at any point in time to ensure future liabilities can be met.

$APV = \sum (Benefits \times v^t \times p_{x:t}) - \sum (Premiums \times v^t \times p_{x:t})$, where $v^t$ is the discount factor and $p_{x:t}$ is the survival probability. Used to determine the current value of future uncertain payments, considering time value of money and the insured's probability of living or dying.

$PV = Pmt \times \frac{1 - (1 + i)^{-n}}{i}$, where $PV$ is present value, $Pmt$ is the payment per period, $i$ is the interest rate, and $n$ is the number of periods. Used to determine the current value of a series of future payments.

$\mathbf{e}_x^{\prime} = \sum_{t=0}^\infty t p_{x:t} q_{x+t}$, where $\mathbf{e}_x^{\prime}$ is the curtate expected lifetime for a person aged $x$, $p_{x:t}$ is the probability of surviving $t$ years, and $q_{x+t}$ is the probability of dying within the next year. Gives the expected number of complete years remaining for a person aged $x$.

$FV = Pmt \times \frac{(1 + i)^n - 1}{i}$, where $FV$ is future value, $Pmt$ is the payment per period, $i$ is the interest rate, and $n$ is the number of periods. Calculates how much a series of future payments will be worth at a given point in the future.

$NP = \frac{A_x}{a_x}$, where $NP$ is the net premium for life insurance, $A_x$ is the present value of the death benefit, and $a_x$ is the present value of an annuity. Used to calculate the premium required to cover the expected costs of insurance without profit or expenses.

$SLP = E[L] + \alpha \sqrt{Var[L]}$, where $SLP$ is the stop-loss premium, $E[L]$ is the expected value of loss, $Var[L]$ is the variance of loss, and $\alpha$ is the safety loading factor. Used in reinsurance to set a threshold for loss coverage.