Pension Mathematics: Defined Benefit Plan Valuation
Value defined benefit pension plans using actuarial cost methods for Exam LTAM.
Benefit Formulas
Defined benefit (DB) pension plans promise retirement income based on a formula involving salary and years of service. A typical formula might be 1.5% times years of service times final average salary (FAS). For example, an employee who works 30 years and has a final average salary of 80,000 would receive an annual pension of 0.015 * 30 * 80,000 = 36,000. Variations include career average formulas (which use average salary over the entire career rather than the final years) and flat benefit formulas (a fixed dollar amount per year of service).
Valuing these benefits requires several key inputs. The salary scale s_x represents the ratio of salary at age x to salary at a reference age, allowing projection of future salary growth. The multiple decrement model captures the probability that an active employee survives to retirement age without exiting due to death, disability, or voluntary withdrawal. The valuation interest rate discounts projected future benefits to the present. Each of these assumptions significantly affects the calculated liability.
Salary Projections
The salary scale s_x is used to project an employee's current salary to their expected salary at retirement. If an employee age x currently earns S_x, the projected salary at age r (retirement) is S_x * s_r / s_x. The final average salary is typically the average over the last 3 or 5 years before retirement, which requires projecting salary to ages r-4 through r (for a 5-year average) and averaging.
For example, suppose an employee is age 40 with current salary 60,000 and the salary scale gives s_40 = 2.0 and s_60 = 4.5 (with retirement at 60). The projected salary at age 60 is 60,000 * (4.5/2.0) = 135,000. If the final average salary uses the last 3 years, you would compute projected salaries at ages 58, 59, and 60 using the appropriate salary scale values and average them.
Actuarial Liability
The actuarial liability (AL) represents the present value of benefits attributed to past service as of the valuation date. The calculation depends critically on the actuarial cost method. Under the projected unit credit (PUC) method, the benefit attributed to past service is based on projected final salary times years of past service. This means the AL reflects the full projected salary, not just current salary, making it larger than under the traditional unit credit method.
Under the entry age normal (EAN) method, the AL equals the present value of future benefits minus the present value of future normal costs. The normal cost under EAN is a level amount (or level percentage of salary) determined at the employee's entry age such that, if contributed each year from entry to retirement, the accumulated contributions would exactly fund the projected benefit. The EAN method produces the most stable year-to-year normal cost, which is why it is widely used in practice.
Normal Cost
The normal cost (NC) is the portion of the present value of benefits attributed to the current year of service. Under PUC, the normal cost for an employee age x is the present value of the incremental benefit earned in the current year: NC = (B/n) * v^(r-x) * (r-x)_p_x^(tau) * a-ddot_r, where B/n is one year's worth of benefit accrual, v^(r-x) discounts from retirement to the current age, (r-x)_p_x^(tau) is the probability of surviving in the multiple decrement environment to retirement, and a-ddot_r is the annuity value at retirement.
Under EAN, the normal cost rate P satisfies P * a-ddot_{e:r-e}^(tau) = PVFB_e, where PVFB_e is the present value of future benefits calculated at entry age e, and a-ddot_{e:r-e}^(tau) is the annuity of salary (or unit payments) from entry to retirement weighted by multiple decrement survival probabilities. The normal cost at age x is then P (or P times current salary if expressed as a percentage).
Unfunded Actuarial Liability
The unfunded actuarial liability (UAL) equals the actuarial liability minus the actuarial value of plan assets: UAL = AL minus F, where F represents the fund assets. A positive UAL means the plan is underfunded and requires additional contributions beyond the normal cost to bring funding up to the target level. The total contribution required each year is the normal cost plus an amortization payment on the UAL.
The actuarial value of plan assets may differ from the market value. Many pension plans use smoothing methods that spread investment gains and losses over 3 to 5 years. This reduces the volatility of contribution requirements but means the actuarial asset value can lag behind or ahead of the true market value.
Common Pitfalls
A frequent error is confusing the traditional unit credit and projected unit credit methods. TUC uses current salary to compute the accrued benefit, while PUC uses projected final salary. This difference means PUC produces a larger AL and a normal cost that grows with salary, whereas TUC produces a normal cost that grows even faster because each year's benefit increment reflects both the salary increase and the additional service year.
Another common mistake is forgetting to apply the multiple decrement survival probability when discounting benefits from retirement to the valuation date. The probability (r-x)_p_x^(tau) accounts for the chance the employee does not leave before reaching retirement. Omitting it overstates the liability. Also watch for problems that specify different retirement ages or early retirement provisions, as these change the discounting period and survival probabilities.
Study Tips
Exam LTAM tests pension mathematics by asking you to compute actuarial liabilities and normal costs under PUC and EAN methods, often requiring salary projections using a salary scale and multiple decrement survival probabilities. Practice setting up the full computation from raw data: start with the salary scale, project final average salary, compute the projected benefit, discount to the valuation date using survival probabilities and interest, and separate the result into past service (AL) and current year (NC) components. Work through problems comparing PUC and EAN to understand how each method allocates costs differently over an employee's career.