VaR Breaches, Duration Mismatch, and a 60/40 Pension
A defined-benefit pension has $1.8 billion in assets, liabilities with a 14-year duration, and a funded ratio of 92%. The current allocation is 60% global equity and 40% aggregate bonds; the bond portfolio has a duration of 6.5, and equity duration is assumed to be zero. A one-day 95% VaR model reports $18 million. Over the past 250 trading days there were 15 breaches, including four in the same week. A stress test shows that a 100 bp rise in the 10-year yield reduces liabilities by 11% and assets by 4%. The plan has 15% in private equity with a 10-year lockup and pays $12 million in monthly benefits. The board wants to “de-risk” but is divided between raising cash, buying long duration, and moving to a risk-parity allocation.
1. Evaluate the VaR model. Compute the Kupiec proportion-of-failures test and explain why the breach count alone does not prove the model is wrong.
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Expected breaches at 95% over 250 days = 5% × 250 = 12.5. Observed = 15, or 6.0%. Kupiec LR = −2 ln{[(1−p)^(n−x) p^x] / [(1−x/n)^(n−x)(x/n)^x]}, with p = 0.05, n = 250, x = 15. The restricted log-likelihood is 235 ln(0.95) + 15 ln(0.05) = −12.05 − 44.94 = −56.99. The unrestricted is 235 ln(0.94) + 15 ln(0.06) = −14.54 − 42.20 = −56.74. LR = −2(−56.99 + 56.74) = 0.50, below the 3.84 critical value at 5%. So the breach frequency is not statistically excessive.
But the model still deserves scrutiny: four breaches in one week shows clustering, which a one-day 95% VaR and Kupiec test do not capture. The model may be calibrated to a low-volatility regime and may not capture nonlinearity, liquidity, or private-equity smoothing. The stress test is the more important diagnostic: a 100 bp rise cuts assets 4% ($72m) but liabilities 11% (about $215m), so the surplus actually improves in a rate rise-meaning the plan is hurt by falling rates. The current 60/40 mix has only about 2.6 years of asset duration (0.40 × 6.5), far short of the 14-year liability duration. The VaR number is not measuring the dominant risk.
2. Quantify the duration gap and propose a hedging allocation. Address the board's three options: raising cash, buying long duration, and risk parity.
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Liability value = $1.8bn / 0.92 = $1.957bn. Dollar duration of liabilities ≈ 14 × $1.957bn = $27.4bn per 1% rate move, or about $274m per 100 bp. Asset dollar duration is only 0.40 × 6.5 × $1.8bn = $46.8m per 100 bp. The plan is running an enormous negative duration gap of roughly $227m per 100 bp. Raising cash reduces both risk and return and makes the duration gap worse; it is not a hedge. Risk parity may diversify equity risk but, unless it explicitly holds long-duration bonds or swaps, it will not match the liability.
The rational de-risking is a liability-driven investment (LDI) overlay: extend bond duration using long Treasuries, stripped bonds, or receive-fixed interest-rate swaps. A rough hedge would add about $227m of dollar duration-for example, long bonds with 15-year duration would require roughly $227m/(15 × 1%) ≈ $1.5bn notional, or a swap overlay with a similar DV01. The plan can keep growth assets but should measure the surplus duration gap, not just asset volatility. It should also set a funded-ratio trigger: at 100%, shift more into long duration; at 92%, a rate fall of 100 bp could cut the funded ratio materially, so hedging is urgent. The hedge should be sized to the liability, not to a 60/40 benchmark.
3. Assess the liquidity position and the private-equity lockup. What governance and risk-reporting changes would you recommend?
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Monthly benefits are $12m, or $144m per year-about 8% of assets. The plan holds 15% private equity ($270m) with a 10-year lockup, so it cannot rely on that sleeve for benefits. It needs a liquidity buffer of at least 12–24 months of benefit payments plus collateral calls from any swap overlay. If the plan uses derivatives to extend duration, a rate rise could trigger variation margin; the buffer must cover that. Holding 40% aggregate bonds at 6.5 duration gives some liquidity, but the long end may be less liquid in stress.
Recommendations: (1) build a cash/short-Treasury buffer of $150–250m; (2) separate the growth portfolio from the LDI overlay and measure surplus VaR and surplus duration, not standalone asset VaR; (3) stress test jointly-rates ±100/200 bp, equity −30%, credit spreads +200 bp, and private-equity write-downs-and report the funded-ratio impact; (4) replace the one-day 95% VaR with a multi-horizon, scenario-based risk report and track breach clustering; (5) set a private-equity pacing plan so commitments do not force asset sales; (6) document a governance policy with clear triggers for de-risking and re-risking. Risk parity alone is not a liability hedge; it can be a growth sleeve only if the surplus duration gap is managed separately.