Pension Systems and Long-Term Liabilities

Pension funds manage the world’s most long-term financial obligations—benefit payouts are promised for 30, 50, sometimes 70 years into the future. Actuarial analysis of pension systems covers three components: liability valuation (PVO), asset and reserve formation, and management of interest rate and longevity risk. Errors in these calculations have led to the collapse of such pension systems as UK Equitable Life (2000), Detroit (2013), Puerto Rico (2017). Proper actuarial modeling is critically important for billions of future retirees.

Types of Pension Plans

Defined Benefit (DB) — pensions with defined payouts. The employer promises to pay a pension according to a formula, usually: pension = b × years of service × final salary, where b ≈ 1.5–2.5%. Example: 25 years of service × 2% × 100,000 rub. = 50,000 rub./month for life.

All actuarial risks (longevity, investment returns, inflation) are borne by the employer/sponsor. This is the classic model of the 20th century (US Social Security, UK State Pension, Russian insurance pension).

Defined Contribution (DC) — pensions with defined contributions. The employer and/or employee contribute a fixed % of salary (usually 5–15%) to an individual account. Pension = accumulated account ÷ $e_{65}$ (or converted into an annuity).

All risks are borne by the employee. The dominant model of the 21st century (US 401(k), Australia Superannuation, Chile AFP).

Hybrid (CDC, cash balance). A hybrid of DB and DC: the employer guarantees a basic return, surpluses go to retirees. Used in the Netherlands, Denmark.

Longevity risk. Growing life expectancy increases DB liabilities. An error in $e_{65}$ by 1 year ≈ 4–5% additional liabilities. Over the last 30 years, $e_{65}$ increased by ~5 years → ~25% "unplanned" liabilities. This is one of the reasons for the massive shift from DB to DC.

Actuarial Valuation of Pension Liabilities

Present Value of Obligations (PVO): $ PVO = \sum_i \sum_t B_i(t) \cdot v^t \cdot,t p{x_i}, $ where $B_i(t)$ is the payment to employee $i$ in year $t$, $v^t$ is the discount, $t p{x_i}$ is the probability that employee $i$ is alive in year $t$.

Projected Unit Credit (PUC) method — standard in IAS 19, US GAAP, IFRS: For every worker, the present value of the pension "earned" to date is calculated, projecting salary up to retirement.

$ PVO_i = (\text{service}i / \text{service at retirement}) \cdot b \cdot \text{final salary} \cdot \ddot{a}{\text{retire age} | \text{curtate}} \cdot v^{\text{years to retire}} \cdot {}{\text{years to retire}}p{x_i}. $

Service Cost (current year accrual): increase in PVO over the year due to an extra year of service.

Interest Cost: $PVO_{begin} \cdot i$ — "increase" due to the approach of payment time.

Discount rate. A critical assumption that catastrophically affects PVO.

• IAS 19 (Europe, IFRS countries): Yield on high-quality corporate bonds (AA-rated). If the rate decreases by 100 bps, PVO grows by 10–15%.
• ASC 715 (US GAAP): similar.
• GASB (US, public plans): historical portfolio return (often inflated, 7–8%). Creates an illusion of stability.

In 2010–2020, falling global rates "sank" many DB plans—for example, General Electric Pension, Boeing.

Asset-Liability Management (ALM)

Goal: to build a portfolio of assets such that changes in its value offset changes in PVO with rate movements.

Immunization (Redington 1952). Conditions:

1. PV(assets) = PV(liabilities).
2. Duration of assets = duration of liabilities: $D_A = D_L$.
3. Convexity of assets ≥ convexity of liabilities: $C_A \geq C_L$.

This ensures: for small parallel shifts in the yield curve, the change in net position ($A - L$) ≈ 0, and for large shifts—even a benefit.

Duration matching: basic level of immunization. In practice, Macaulay duration (for timing) or modified duration (for sensitivity) is used.

LDI (Liability-Driven Investing). Modern strategy:

• Hedging portfolio: 60–80% of assets in long-term bonds, swaps, for duration matching.
• Return portfolio: 20–40% in equities, alternatives, to generate additional yield.

UK pension funds massively transitioned to LDI after the Pensions Act 2004 and regulation by The Pensions Regulator. In September 2022, the UK faced an LDI crisis: a >100 bps spike in rates in a week triggered margin calls → panic gilts selloff → Bank of England was forced to intervene. Lesson: leverage in LDI requires caution.

Numerical Example: Plan Liability Valuation

Plan: 1,000 participants, average age 45. Each promised 100,000 rub./year for life from age 65.

Step 1. Discount factor to 65: $v^{20} = 1.04^{-20} \approx 0.456$. Step 2. Probability of surviving from 45 to 65: ${20}p{45} \approx 0.85$ (for a mixed population). Step 3. $\ddot{a}{65}$ (life annuity at age 65, $i=4%$) $\approx 14.8$. Step 4. PVO per participant: $100,000 \cdot v^{20} \cdot {}{20}p_{45} \cdot \ddot{a}_{65} = 100,000 \cdot 0.456 \cdot 0.85 \cdot 14.8 \approx 573,600$ rub. Step 5. PVO for the whole plan: $1,000 \times 573,600 = 573.6$ million rub.

Sensitivity.

• At $i = 3%$ (instead of 4%): $v^{20} = 0.554$, $\ddot{a}_{65} \approx 16.3$. PVO ≈ 776 million (+35%).
• At $e_{65} + 1$ year ($\ddot{a}_{65} \to 15.7$): PVO ≈ 608 million (+6%).

Liability duration: $D_L \approx 25$ years (highly dependent on age distribution).

Real-World Applications

• Gazprom, Gazfond. The largest NPF (Non-State Pension Fund) in Russia, liabilities > 500 billion rub. Actuarial calculations using IPS system.
• CalPERS (California Public Employees). $440 billion in assets, 2 million participants. Discount rate reduced from 7.5% to 6.8% (2021), triggered liability revision +$100 billion.
• UK USS (Universities Superannuation Scheme). £75 billion, 460,000 participants. Funding crisis 2018–2023—a dispute over discount rate, faculty strikes.
• GPIF (Government Pension Investment Fund of Japan). The world’s largest pension fund ($1.6 trillion). Hybrid LDI with a large equity allocation to combat longevity of Japanese ( $e_{65} = 21$ years).

Exercise. Pension plan: 1,000 participants of average age 45, promised pension 100,000 rub./year for life from 65. Discount rate $i = 4%$. Use Gompertz–Makeham with $A = 0.0005$, $B = 5 \cdot 10^{-5}$, $c = 1.10$. (a) Implement a function for $\ddot{a}_x$ (life annuity) and $_t p_x$. (b) Calculate PVO. (c) Sensitivity: calculate PVO at $i = 3%$ and $i = 5%$. What is the duration $D_L = -d,\ln(PVO)/di$? (d) If the plan’s assets are 600 million rub. in bonds with duration 8 years—estimate the deficit for a -100 bps shift in rates. (e) Propose an ALM strategy: what fraction of assets should be moved into long-term bonds ($D = 25$) so that $D_A = D_L$?