Universal Life Account Value Projection and Asset-Liability Duration Gap

Background

Beacon Life issued a universal life (UL) policy with an initial account value of $10,000**, a level planned annual premium of **$3,000, a face amount of $100,000** (level death benefit option), and standard UL mechanics: each year, a **5$= \text{face} - AV$) at an annual COI rate of **0.4$1,000 of NAAR times 12 (a simplified annualized approximation of a monthly COI structure), then a flat $60 policy expense charge, with interest credited at the end of the year on the resulting balance at a scenario-dependent credited rate.

Five-year projection under the base credited-rate scenario

Beacon's ALM team projects the following credited rates for years 1–5, reflecting a gently declining-then-recovering new-money rate environment: $3.5\%, 3.0\%, 2.8\%, 3.2\%, 3.0\%$.

The COI charge shrinks slightly each year (from $4.32 to$3.68) simply because the account value is growing while the face amount is fixed, so NAAR $=100{,}000-AV$ shrinks and with it the dollar COI, even though the rate per $1,000 of NAAR is held flat in this projection — a reminder that UL charges depend on the interaction of a contractual rate and a policy-specific, dynamically-changing base (NAAR), not the rate alone.

Asset-liability duration mismatch

Beacon's asset segment backing this UL block has market value $1,250,000** (across the pooled cohort of similar policies), effective (modified) duration$D_A = 6.2$and convexity$C_A = 60$. The associated UL reserve liabilities have market value **$1,150,000, duration $D_L = 4.5$, convexity $C_L = 35$. Under a hypothetical parallel downward shock of 100 basis points ($\Delta y = -0.01$), the standard duration-convexity approximation for the change in value of an asset or liability with value $V$, duration $D$ and convexity $C$ is

$\Delta V \approx -D\cdot \Delta y \cdot V + \tfrac12\, C \cdot (\Delta y)^2 \cdot V.$

Applying this:

$\Delta A \approx -6.2\times(-0.01)\times 1{,}250{,}000 + 0.5\times 60\times(0.0001)\times 1{,}250{,}000 = \$81{,}250,$ $\Delta L \approx -4.5\times(-0.01)\times 1{,}150{,}000 + 0.5\times 35\times(0.0001)\times 1{,}150{,}000 = \$53{,}762.50.$

Starting surplus (assets minus liabilities) is 1{,}250{,}000-1{,}150{,}000=\100{,}000$. After the shock, surplus becomes$100{,}000+81{,}250-53{,}762.50 = $127{,}487.50$, a **gain of$27,487.50** — because Beacon's assets have longer duration than its liabilities (a positive duration gap, $D_A>D_L$), a rate decrease raises asset values by more than liability values, helping surplus. The leverage-adjusted duration of surplus itself, defined as

$D_{\text{surplus}} = \frac{A\cdot D_A - L\cdot D_L}{A-L} = \frac{1{,}250{,}000(6.2) - 1{,}150{,}000(4.5)}{100{,}000} = 25.75,$

is far larger than either $D_A$ or $D_L$ individually — a direct consequence of leverage (small surplus base relative to assets/liabilities), which is the reason insurers with even modest duration mismatches can see outsized percentage swings in surplus from parallel rate moves.

The risk this creates

Although a rate decrease helped surplus in this scenario, the same duration gap means a rate increase would hurt surplus by a similar order of magnitude — Beacon is exposed to interest-rate risk in both directions depending on the sign of the shock, and management must decide whether to accept this asymmetric-cost/asymmetric-benefit profile, hedge it (e.g., pay-fixed swaps or duration-extending bond purchases to shrink the gap), or hold additional capital against it. UL blocks are particularly exposed to this because policyholders can react to rate moves via lapses or additional premium ("disintermediation risk"), which was not modeled in the simple projection above but is a standard qualitative concern examiners expect candidates to raise.

Your task

Using the figures above as given, answer the following. Where asked to extend the analysis (e.g., a different shock size or COI/credited-rate scenario), apply the same formulas shown here.