Liability-driven investing

The funded ratio is assets over the present value of liabilities and the surplus is the difference, and that arithmetic is already yours. What matters is what makes the ratio move, and the answer is rarely what a trustee expects.

Worked example: a pension trust's funded ratio and hedge ratio

Take the trust from the opening. Its benefit schedule pays ₹38.00 crore at the end of the first year and escalates 5.00% a year for twenty-five years, ending at ₹122.55 crore, with an undiscounted total of ₹1,813.63 crore. Discounted at the trust's stated actuarial rate of 7.10%, the present value is ₹706.56 crore and the modified duration is 11.18 years. Against ₹640.00 crore of assets that is a funded ratio of 90.58% and a deficit of ₹66.56 crore.

The asset book sits at 52% government securities with a sleeve duration of 9.20, 30% corporate and public-sector debt at 4.60, 3% short-dated debt at 0.40, 12% equity index units and 3% units of real-estate and infrastructure trusts. Treat the last two as carrying no direct rate sensitivity, which is the standard simplifying convention and is stated here as a convention rather than a fact, and the asset book's modified duration is 6.18 years.

The duration gap is 5.00 years, and a trustee who stops there has the wrong number. Duration is a percentage sensitivity, and percentages of different bases are not comparable. What moves the surplus is currency duration: assets times asset duration against liabilities times liability duration. Here that is ₹3,952.64 crore against ₹7,899.82 crore, so the trust is hedged to 50.03% of its liability's rate exposure and not to the 55% the ratio of durations suggests.

That gap is the reason the hedge ratio exists as a separate concept, and it widens as funding worsens. A plan at 90% funded that matches asset duration to liability duration is hedged to 90%, because it has only ninety paise of assets moving for every rupee of liability. Underfunding is itself an unhedged rate exposure, and no amount of duration matching fixes it.

Defined-benefit pensions, part two: the hedge-ratio decision and the glide path

The hedge ratio is the decision variable. Write it as assets times asset duration, divided by liabilities times liability duration, and then solve it backwards for the asset duration a target hedge requires:

required asset duration = target hedge ratio x (PV of liabilities / assets) x liability duration

For the trust, a 70% hedge requires an asset duration of 8.640428 years. A full hedge requires 12.343468 years, which is longer than the liability's own 11.18 because the assets are the smaller pool and have to work harder.

Now test that against the mandate. The trust's investment pattern permits 45% to 65% in government securities, 20% to 45% in other debt, up to 5% short-dated, 5% to 15% in equity and equity-linked units, and up to 5% in units of real-estate and infrastructure trusts. (Those limits are a synthetic mandate stated for the example. The genuine article for an Indian provident or superannuation trust is a notified pattern of investment that changes by circular, so check the pattern in force before using any of it on a real mandate.)

What hedge ratio can the mandate deliver, and what does it cost?

The longest liquid government bond available is a forty-year issue priced at par to yield 7.10%, carrying a Macaulay duration of 14.114104 years and a modified duration of 13.178435. Push the government sleeve to its 65% ceiling at that duration, run the debt sleeve at 30% with its duration extended to 8.00, and hold equity at its 5% floor. The maximum asset duration the mandate can produce is 10.965983 years, a hedge ratio of 88.84%.

A 70% hedge is therefore available and a full hedge is not, which is the most useful output of the exercise. The trust cannot immunise inside its cash bond mandate at any weighting. Closing the last eleven points requires either a change to the pattern of investment, which is a sponsor and regulator conversation, or a swap overlay, which is a governance conversation about whether the committee can supervise a derivative. Both are answers, and neither is a portfolio construction.

Build the 70% hedge and the policy portfolio falls out. Hold equity at its 15% cap and the trust units at their 5% cap, keep short-term debt at 3%, and solve the remaining 77% between the long government bond and the 4.60-duration corporate sleeve:

w_gsec x 13.178435 + w_corp x 4.60 + 0.03 x 0.40 = 8.640428
w_gsec + w_corp = 0.77          ->   w_gsec = 59.29%, w_corp = 17.71%

That portfolio has a modified duration of 8.640428 years by construction and an expected return of 7.950094% against the current portfolio's 7.859000%. The hedge did not cost expected return here. It was paid for out of the corporate sleeve while equity moved up to its cap, leaving the trust nine basis points better off than where it started.

What is a glide path in liability-driven investing?

A glide path is that decision run repeatedly as funded status changes, and its rule is asymmetric. A plan at 130% funded has a surplus worth protecting and little to gain from another ten points of equity, while a plan at 80% has a hole deep enough that pure de-risking guarantees it stays. The target hedge ratio therefore rises as the funded ratio rises, in steps written into the policy document before the moment arrives, because the moment always arrives during a market event when nobody wants to be the trustee who sold equities at the bottom. The rebalancing discipline from the allocation work is the machinery, and the glide path is that machinery with the funded ratio as its trigger instead of a calendar.

{
  "skill": "Building and stress-testing a liability schedule with its duration and hedge ratio",
  "use": "Have an assistant turn a stated benefit or claims schedule into a discounted cash-flow table with present value, Macaulay duration, modified duration and convexity, then solve for the asset duration a target hedge ratio requires and re-run the whole thing under a parallel rate shift.",
  "tools": ["Chat assistants with code execution - Claude, ChatGPT, Gemini", "Python: NumPy and pandas for the cash-flow table", "A spreadsheet for the trustee-facing version, because the committee will want to see the rows"],
  "prompt": "Here is a benefit schedule as (year, amount) pairs and a discount rate. Compute the present value, Macaulay duration, modified duration and convexity from first principles, showing the per-period present values and weights. Then, given asset value A and target hedge ratio h, solve for the required asset modified duration using the currency-duration definition h = (A x D_A) / (PV_L x D_L). Finally, revalue the liability EXACTLY at a discount rate 100 basis points lower and report the exact percentage change alongside the duration-only and duration-plus-convexity approximations, with the error of each. Do not round intermediate values.",
  "verify": "Recompute the present value by an independent route before trusting anything downstream: a level schedule must tie to the annuity formula and an escalating one to the growing-annuity closed form, to at least six decimal places. Confirm the duration-only approximation understates the gain on a rate fall and that adding convexity closes most of the gap, because a model that gets that sign wrong has the convexity term inverted. Trace the discount rate to its primary source, which for an Indian trust is the notified or actuarially stated basis and for a US plan is the disclosed discount-rate assumption in the sponsor's filing, and date-check it. A hedge ratio that comes back above 100% on an underfunded plan is arithmetically impossible with matched durations and means the model has divided by the wrong base.",
  "diy": "You must be able to write the currency-duration hedge ratio from memory, say why matching durations under-hedges an underfunded plan, and test a required asset duration against a mandate's limits with a calculator. The assistant will produce a duration table in seconds and will not tell you that the number it produced is unreachable inside the pattern of investment the trust actually operates under. That test is the judgement, and it is yours.",
  "market": "IN"
}