ALM and interest rates

Asset liability management (ALM) is the discipline that coordinates investment decisions with the nature of the obligations. It does not seek the highest-yielding portfolio, but the one that makes it possible to pay what was promised with the lowest probability of surprises, given a risk appetite and a set of regulatory constraints.

The central risk is interest rate risk. If an insurer sells 30-year life annuities and backs them with 5-year instruments, a fall in rates forces it to reinvest maturing assets at yields below those promised: this is reinvestment risk. If it does the opposite and makes its assets longer than its liabilities, a rise in rates reduces the market value of its portfolio by more than the value of its obligations. In both cases the economic surplus, the difference between assets and liabilities valued on the same curve, moves.

For the actuary, ALM is where technical reserves, the capital requirement, the mortality tables that define the cash flows and the valuation of employee benefits all meet. The tools are the same in every case: project cash flows, discount them with an appropriate curve and measure how value changes when the curve moves.

Although the principle is the same, each institution faces a different liability profile, and that changes what it means to be well matched.

In all three cases the analysis starts the same way: a projection of liability cash flows by year (or month), a discount curve and a description of the asset cash flows. What changes is the metric being protected: economic surplus in an insurer, funded status in a pension plan or the expected replacement rate in an individual account (see pensions and the pension simulator).

Macaulay duration is the average time of the cash flows weighted by their present value. For a zero-coupon bond it equals the maturity; for a coupon bond it is shorter than the maturity, because part of the value is received earlier. Modified duration measures the percentage sensitivity of price to a small change in yield: if a bond has a modified duration of 6, a 50 basis point rise reduces its price by about 3%.

Duration is a linear approximation. The price-yield relationship is convex, so the tangent line underestimates price whether rates rise or fall. Convexity captures that curvature and is larger the more spread out in time the cash flows are. Two portfolios with the same duration can have very different convexities: a maturity ladder concentrated near the duration (bullet strategy) has less convexity than one combining very short and very long maturities (barbell strategy).

When cash flows depend on interest rates, as in a callable bond, a mortgage with prepayment or an insurance policy with surrenders, the analytical formula no longer applies. Effective duration is used instead: the instrument is revalued with the curve shifted up and down, letting the cash flows respond to each scenario, and the numerical slope is computed. It is the appropriate measure for life liabilities with policyholder options.

Economic surplus is S = A − L, with assets and liabilities valued on the same curve. Its sensitivity depends on the peso duration of each side, that is, value multiplied by modified duration. If the peso duration of assets exceeds that of liabilities, surplus falls when rates rise; if it is lower, surplus falls when rates fall. The second situation is typical of life annuity portfolios and mature pension plans, because the liabilities are longer than the bonds available.

The duration gap summarizes that exposure in years. When there is a surplus, equal durations are not enough: a portfolio with D_A = D_L and A > L still has more peso duration on the asset side. To neutralize the surplus against parallel shocks requires D_A = (L/A)·D_L, whereas protecting the funding ratio A/L requires D_A = D_L. Choosing which of the two to protect is a risk appetite decision that should be documented.

An operational measure is the surplus DV01: how many pesos it changes per basis point. Investment committees use it to set limits, for example, that a 100 basis point move may not consume more than a certain percentage of the solvency margin.

Frank Redington, a British actuary, formulated the theory of immunization in 1952: choose assets so that surplus does not decrease under a small parallel change in interest rates. His three conditions are that the present value of assets equal that of liabilities, that their durations coincide and that asset cash flows be more spread out around that duration than liability cash flows, which is equivalent to greater convexity.

The proof is short. With force of interest δ, let S(δ) = A(δ) − L(δ). Expanding in a Taylor series around the current rate δ₀, the first condition gives S(δ₀) = 0, the second makes the first derivative vanish and the third makes the second derivative positive. Then δ₀ is a local minimum of S and any small move, up or down, leaves a non-negative surplus.

Immunization has well-known limitations. It erodes with the passage of time and with every rate move, because asset and liability durations do not age at the same pace, so it requires rebalancing. It assumes parallel shifts in a flat curve, while the real curve changes in level, slope and curvature. And it offers no protection against risks unrelated to interest rates: issuer default, changes in mortality or mass surrenders.

Cash flow matching is the most demanding alternative: buying assets whose cash flows replicate the expected liabilities year by year. It eliminates reinvestment and interest rate risk for certain cash flows, but it is expensive, requires instruments at every maturity and in the right currency, and does not resolve the uncertainty in the liabilities themselves. In practice the two are combined: cash flow matching for the early years and duration control for the long tail.

To capture changes in slope, partial durations or key rate durations (KRD) are used. Key maturities on the curve are chosen (for example 1, 3, 5, 10, 20 and 30 years), a single key point is shifted, interpolating its effect linearly to the neighboring points, and the relative change in value is measured. The sum of the KRDs is approximately the effective duration. Matching asset and liability KRDs point by point also protects against twists in the curve.

Duration answers what happens if the rate changes; stochastic models answer how likely it is to change. The most widely used in actuarial education are mean-reverting short rate models, in which the instantaneous rate r_t tends toward a long-term level b at speed a. They make it possible to simulate economic scenarios for dynamic solvency testing, value guarantees and obtain closed-form curves for zero-coupon bonds.

In the Vasicek model (1977) volatility is constant, which gives simple formulas and a normal distribution for r_t, but allows negative rates. In the Cox, Ingersoll and Ross model (CIR, 1985) volatility is proportional to the square root of the rate, which rules out negative values if the Feller condition holds. Both are affine: the zero-coupon bond price is exponential in the short rate, so the entire curve is determined by r_t and the parameters.

Neither reproduces the observed curve exactly with constant parameters. For derivative valuation, extensions that fit it by construction, such as Hull and White, are preferred; for long-term ALM scenarios, two- or three-factor models or economic scenario generators calibrated to history and to the current curve are used.

The risk-free peso curve is built from Federal Government securities: CETES (zero coupon up to one year, plus two-year issues), Bonos M (fixed rate with semiannual coupons, 3 to 30 years) and Udibonos (real rate, denominated in UDIs, which adjust with inflation). Banxico, as the government's financial agent, publishes the results of the weekly auction, and authorized pricing vendors such as Valmer and PiP build complete daily curves from those and other market transactions.

The table brings together the most recent primary yields available as of October 16, 2026. Not every maturity is auctioned every week, so the dates differ; for formal work it is better to use a pricing vendor's daily curve with a single cutoff date.

Three takeaways for ALM. First, the curve is steeply upward sloping: from 6.25% at 28 days to 10.15% at 30 years, about 390 basis points. Anyone funding long liabilities with short assets forgoes that term premium and is exposed to the short rate continuing to fall: Banxico cut its target by 25 basis points on May 7, 2026 and has held it at 6.50% since then, most recently on September 24. Second, the spread between the 10-year Bono M and Udibono suggests implied inflation of about 4.1% a year, by the Fisher relation (1.0935 / 1.0500 − 1), although with different dates and without netting out inflation risk and liquidity premiums. Third, the long end of the real curve is scarce and illiquid, which makes matching UDI-denominated life annuities more expensive.

Discount rate for reserves. Under the LISF, the best estimate of the unearned premium reserve and the reserve for outstanding obligations is discounted with the market risk-free interest rate curves for each currency or monetary unit supplied by the pricing vendor the institution has contracted, in accordance with the criteria in Annex 5.1.3-a of the CUSF (Single Insurance and Surety Circular) (Provision 5.1.3, section II). In other words, there is no fixed technical interest rate: the reserve moves with the market and ALM must reflect that. Pension insurance arising from the social security laws is the relevant exception, with a real technical interest rate (see the In Mexico section).

Discount rate for NIF D-3. For employee benefit obligations, NIF D-3 allows discounting with the government bond curve or with the curve of high-quality corporate bonds that have a deep market, with a term consistent with the duration of the obligation; IAS 19 starts from high-quality corporate bonds and falls back on government bonds when there is no deep market for the former. In Mexico the Bonos M curve is customary for nominal assumptions. The details are in the employee benefits guide; for ALM what matters is that this rate defines the liability against which the fund is measured.

The Mexican framework does not prescribe a single ALM metric, but it requires ALM in several ways. Article 247 of the LISF requires the investment policy, approved by the board of directors, to invest assets consistently with the nature, duration and currency of the obligations, with adequate maturity and interest rate matching and an appropriate liquidity ratio. The CUSF develops this in Title 8: Provision 8.1.3 requires a description of the mechanisms for mismatch risk, considering at least duration, currency, interest rate, exchange rate, price indices and valuation criteria, and Provision 8.17.2 defines matching as a structural correspondence in which liability positions are covered by equivalent offsetting asset positions. Asset selection is the responsibility of an investment committee (article 248), and coverage of the Investment Base is verified under the regime described in the solvency guide, where mismatch is also among the risks the RCS must capture.

Pension insurance arising from the social security laws is the most intensive ALM case in the country. Life annuities are paid in UDIs, and the pension policy reserve is calculated with a real technical interest rate: for policies under the Nuevo Esquema Operativo (New Operating Scheme) it is the discount rate implied in the offer if it is less than or equal to 3.5% real, and 3.5% if the offer used a higher rate (Provision 5.8.4); earlier policies are valued at 3.5% real. Provision 5.8.13 allows, with prior CNSF authorization, the offer rate to be used in the time buckets where asset cash flows cover liability cash flows, which explicitly rewards matching. And the investment fluctuation reserve (Chapter 5.11) is funded with 25% of the excess of real returns over the minimum creditable returns, as a cushion against falling yields.

Reinvestment risk in these portfolios is structural: the liabilities extend beyond the available Udibono maturities, so part of the cash flows in the fourth and fifth decades will be reinvested at unknown rates. If the real rate falls below the technical rate, the reserve becomes inadequate; if the insurer offered annuities at a high rate to win the case, its margin depends on sustaining that yield for decades.