Immunising a Pension Fund with Two PIB Tenors

The fund has PKR 6.7bn and must hold 5% cash, or PKR 0.335bn. The invested portfolio is therefore PKR 6.365bn. Because cash has zero duration, the invested portfolio must have a duration of 8.6/0.95 = 9.0526 years. Let w be the weight of the 10-year PIB in the invested portfolio and (1-w) the weight of the 20-year PIB. Then 6.9w + 11.8(1-w) = 9.0526. This gives 11.8 - 4.9w = 9.0526, so 4.9w = 2.7474 and w = 0.5607. The 10-year PIB therefore receives 56.1% of the invested portfolio and the 20-year PIB 43.9%. In rupee terms, the 10-year holding is 0.561 × 6.365 = PKR 3.57bn, the 20-year holding is 0.439 × 6.365 = PKR 2.79bn, and cash is PKR 0.335bn. Both tenor buckets are below the 60% limit, so the constraints are satisfied. The resulting portfolio is a barbell: it combines a 10-year and a 20-year bond to match an 8.6-year liability. A barbell usually has higher convexity than a bullet portfolio of the same duration, which is favourable if large yield moves occur, but it also has more exposure to curve steepening and flattening. The fund should record that the duration match is exact only at the current yields and that convexity does not protect against non-parallel shifts.

For a 100 bp parallel decline, the first-order gain is approximately modified duration × 1.0% = 8.6% of PKR 6.7bn, or about PKR 576m, before convexity. The liabilities' present value rises by roughly the same 8.6%, so the fund remains hedged on a first-order basis. Convexity adds a small positive amount to the assets, which is favourable. For the twist, the 10-year yield is assumed to fall 150 bp and the 20-year yield 50 bp. The 10-year holding gains about 6.9 × 1.5% = 10.35%, or PKR 369m on PKR 3.57bn. The 20-year holding gains about 11.8 × 0.5% = 5.9%, or PKR 165m on PKR 2.79bn. Cash is unchanged. Total asset gain is about PKR 534m, or 8.0% of the fund. The liabilities are long-dated, with payments at 8 and 12 years; if the relevant discount rates fall closer to the long-rate move of 50 bp, the liability gain is about 8.6 × 0.5% = 4.3%, or PKR 288m. The fund would show a surplus gain of about PKR 246m. If the twist is reversed-short rates rise and long rates fall-the 10-year holding loses while the 20-year holding gains, and the liability may gain; the outcome is less predictable. The lesson is that duration matching protects against parallel shifts but not against curve twists, so the fund should monitor key-rate durations and consider adding a 12-year or longer cash-flow match if available.

A quarterly full rebalance is too expensive at 25 bp per transaction, especially if both legs are traded. A trigger-based policy is better. Rebalance when the portfolio's modified duration deviates from the liability duration by more than 0.25 years, or when the duration gap implies a surplus sensitivity greater than 2% of assets for a 100 bp move. Review monthly, but trade only when a trigger is hit. The expected 100 bp rate cut argues for reducing the frequency of sales: the fund should not sell 20-year paper after a rally merely to fine-tune duration if the liability is still well hedged. Use the 8-year and 12-year cash-flow dates to build a maturity ladder rather than relying only on duration. Because the 10-year PIB matures before the 12-year liability, part of the 12-year payment should be matched with 20-year paper or a reinvestment reserve; the 8-year payment can be matched with 10-year paper plus a cash reserve. Hold the 5% cash buffer for collateral and transaction costs. When the fund receives coupon income, direct it first to the nearest liability gap rather than reinvesting pro rata. Rebalance the barbell if the curve steepens by more than 75 bp, because the 20-year leg will extend duration and increase surplus volatility. Document the triggers and obtain trustee approval for any change in the 60% bucket limit before trading.