Actuarial Notes

Valuation

Study guides: Exam ALTAM, Exam CP-321, Exam ILA-201U, Exam INV-201, Exam RET-201, Exam RET-301

Valuation is the actuarial process of placing a value, at a valuation date, on a set of future cash flows: an insurer's policy obligations, a pension plan's promised benefits, a block of assets, or a derivative. The cash flows are projected under stated assumptions and methods, then discounted. The result depends on the purpose of the valuation as much as on the cash flows. A funding valuation, an accounting valuation and a wind-up or solvency valuation of the same plan give different numbers.

V_0 = \sum_{t} v^t\, E\left[CF_t\right]

\text{Surplus} = \text{Assets} - \text{Liabilities}

V_0^{\text{derivative}} = e^{-rT}\,E^{\mathbb{Q}}\left[\text{Payoff}_T\right]

  • Liabilities. For an insurer, V_0 is the expected present value of benefits and expenses less premiums, the policy value or reserve. For a DB plan, a funding method splits the value of benefits into the actuarial accrued liability for past service and a normal cost for the coming year. Projected unit credit (PUC) projects salary to retirement. Traditional unit credit (TUC) uses the current salary.
  • The basis follows the purpose. Going-concern funding uses best-estimate assumptions, sometimes with margins. Accounting valuations follow a standard (GAAP, IFRS 17, pension accounting standards). Solvency and wind-up valuations assume the plan or company ends at the valuation date and use settlement-type discount rates. A valuation principle is the rulebook for one such purpose.
  • Assets are valued at market value or a smoothed actuarial value. Derivatives are valued by risk-neutral valuation or replication, as in the Black-Scholes model (INV-201, ALTAM embedded options).
  • Assumptions cover interest, mortality, salary scale, lapse and expense. Their selection is governed by ASOPs, such as ASOP 4 for pension obligations, and in Canada by CIA standards.

Accrued Liability and Normal Cost, PUC vs TUC A DB plan pays 1.5\% of final salary per year of service, as a pension from age 65. A member aged 55 has 20 years of service and a current salary of $100,000. Final salary is projected as 100{,}000(1.03)^{10}. The probability of reaching 65 in service is 0.94, no benefit is paid on earlier exit, i = 5\%, and the pension annuity factor at 65 is \ddot{a}^{(12)}_{65} = 12.0. Find the accrued liability under TUC and PUC, and the PUC normal cost.

Answer The EPV factor at 55 for each $1 of annual pension at 65 is 0.94\,v^{10}(12.0).

\begin{align*}
0.94\,v^{10}(12.0) &= 0.577078(12.0) \\
&= 6.92494 \\
\text{AAL}^{\text{TUC}} &= 0.015(20)(100{,}000)(6.92494) \\
&= 207{,}748 \\
\text{AAL}^{\text{PUC}} &= 0.015(20)(134{,}392)(6.92494) \\
&= 279{,}196 \\
\text{NC}^{\text{PUC}} &= 0.015(1)(134{,}392)(6.92494) \\
&= 13{,}960
\end{align*}

PUC reserves for the projected salary now, so its accrued liability is 34\% higher. TUC recognises the salary growth later, through higher normal costs.

One Plan, Two Valuation Purposes A closed pension plan's benefit outgo is approximated by $120,000 a year, paid at the end of each year for 15 years. Assets are $1,400,000. Value the liability and the funded ratio at a going-concern discount rate of 5.5\% and at a wind-up rate of 3.5\%.

Answer

\begin{align*}
L_{5.5\%} &= 120{,}000\,a_{\overline{15}|5.5\%} \\
&= 120{,}000(10.03758) \\
&= 1{,}204{,}510 \\
L_{3.5\%} &= 120{,}000(11.51741) \\
&= 1{,}382{,}089
\end{align*}

The funded ratio is 1{,}400{,}000/1{,}204{,}510 = 116.2\% on the going-concern basis, but only 101.3\% on wind-up. The cash flows are identical. Only the purpose, and therefore the discount rate, changed.

How it appears on the exam

  • ALTAM asks for the actuarial accrued liability and normal cost under PUC and TUC. It gives a service table, a salary scale and an annuity factor, sometimes with early-exit benefits that need a sum over decrements.
  • RET-201/RET-301 ask written and calculation questions on valuations for a stated purpose: funding, accounting disclosure, plan termination, wind-up and solvency, mergers, and actuarial equivalence. The question asks which assumptions and methods fit the purpose, and how plan investments interact with them.
  • ILA-101/ILA-201U compare reserve valuations under the U.S. statutory, GAAP and IFRS frameworks. CP-321 covers valuing retiree group benefits.
  • INV-201 values derivatives by replication and risk-neutral pricing.

Exam traps

  • PUC vs TUC salary. PUC uses the projected final salary. TUC uses the current salary. Swapping them gives the other method's answer, which will be among the choices.
  • Normal cost under TUC is not one year's accrual on current pay. It also includes the effect of this year's salary increase on all past service.
  • Basis for the wrong purpose. Quoting a going-concern liability where a wind-up or solvency figure is asked for overstates the funded position. A wind-up rate below the funding rate raises the liability.
  • Risk-neutral vs real-world. A derivative's payoff is averaged under \mathbb{Q} and discounted at r. Averaging under real-world probabilities and then discounting at r misprices it.