Asset Shares and Surrender Values for LTAM

Asset Share Calculation

The asset share represents the accumulated fund per surviving policyholder using actual (or expected realistic) experience. Unlike reserves, which use prescribed assumptions, asset shares use realistic or actual assumptions for mortality, interest, expenses, and lapse rates. The asset share recursion is: AS_t = (AS_{t-1} + G_t - e_t) * (1 + i) - q_x^d * b - q_x^w * CV_t, all divided by (1 - q_x^d - q_x^w). Here G is gross premium, e is expenses, b is death benefit, CV is cash surrender value, and the decrements are death and withdrawal.

Asset shares typically start at negative values due to heavy first-year acquisition expenses (commissions, underwriting costs, policy issuance). Over time, as renewal premiums are collected and expenses are lower, the asset share grows and eventually becomes positive. The point where the asset share crosses zero is important because it indicates when the policy begins contributing positively to the company's surplus.

Understanding the Recursion Components

The asset share recursion can be broken down into clear steps. At the start of year t, the fund per survivor is AS_{t-1}. The gross premium G_t is added and expenses e_t are deducted. This net amount earns interest at rate i for the year, producing (AS_{t-1} + G_t - e_t)(1 + i). From this accumulated fund, the expected cost of death claims (q_x^d * b) and the expected cost of surrenders (q_x^w * CV_t) are subtracted. The remainder is shared among the survivors, so we divide by the survival probability (1 - q_x^d - q_x^w).

For example, suppose AS_4 = 2,000, the gross premium is 500, renewal expenses are 50, the interest rate is 5%, q_65^d = 0.015, q_65^w = 0.05, the death benefit is 10,000, and the cash surrender value is 1,800. Then the accumulated fund is (2,000 + 500 - 50)(1.05) = 2,572.50. The expected death cost is 0.015 * 10,000 = 150. The expected surrender cost is 0.05 * 1,800 = 90. So AS_5 = (2,572.50 - 150 - 90) / (1 - 0.015 - 0.05) = 2,332.50 / 0.935 = 2,494.12.

Asset Shares vs. Reserves

The asset share and the reserve serve different purposes and use different assumptions. The reserve is a regulatory or accounting measure calculated using prescribed mortality, interest, and expense assumptions. It represents the insurer's liability for future benefits. The asset share reflects the actual accumulated value of the policy based on realistic experience. In the early policy years, the asset share is typically less than the reserve because of high acquisition costs. Over time, the asset share may exceed the reserve, creating surplus.

The difference between the asset share and the reserve at any duration has practical implications. If AS_t < tV, the policy has not yet recovered its acquisition costs, and a surrender at that point would result in a loss for the insurer. If AS_t > tV, the insurer has accumulated more than needed to fund future liabilities, and the excess contributes to profit or surplus.

Surrender Values and Cash Values

The cash surrender value (CSV) is the amount paid to a policyholder who voluntarily terminates the policy. It typically equals the reserve minus a surrender charge: CSV_t = max(tV - SC_t, 0), where SC_t is the surrender charge at duration t. Surrender charges decrease over time and eventually reach zero, often by duration 10 or 20. Their purpose is to recover unamortized acquisition costs and protect the insurer from anti-selection (healthy lives surrendering while unhealthy lives retain coverage).

Nonforfeiture laws require that policies provide minimum cash values. The Standard Nonforfeiture Law specifies that minimum values be computed using the adjusted premium method. The adjusted premium equals the net level premium plus a first-year expense allowance spread over the premium-paying period. The minimum cash value at duration t is the reserve computed using this adjusted premium, which is lower than the net level premium reserve.

Determining Appropriate Surrender Values

Actuaries use asset shares as a guide for setting surrender values. A common approach sets the cash surrender value equal to the asset share (or a percentage of it) once the asset share becomes positive. This ensures that surrendering policyholders receive a fair share of the accumulated fund while the insurer recovers acquisition costs. The surrender value scale must also satisfy nonforfeiture minimums and be competitive in the marketplace.

Setting surrender values too high creates anti-selection risk: policyholders who are healthy will surrender for the generous cash value, while those in poor health will maintain coverage. Setting them too low may make the product uncompetitive or violate nonforfeiture requirements. The balance between these considerations is a key product design decision.

Common Pitfalls

A frequent error in asset share calculations is mishandling the timing of cash flows. Premiums are received and expenses are paid at the start of the year, while death benefits and surrender values are paid at the end. Placing cash flows at the wrong time produces incorrect interest calculations. Another common mistake is forgetting to divide by the survival probability when computing the asset share per survivor.

Students also sometimes confuse the adjusted premium with the net level premium or the gross premium. The adjusted premium is specifically used for nonforfeiture calculations and includes a prescribed expense loading. Make sure you understand which premium concept is being used in each context.

Study Tips

Exam LTAM tests asset share calculations with multiple decrements, requiring you to build a year-by-year accumulation table. Practice the recursion formula until it becomes automatic. Be prepared to compare asset shares to reserves at various durations and explain the financial implications of the difference. Understand the nonforfeiture framework, including how adjusted premiums and minimum cash values are determined. Work through problems that combine asset shares with surrender value determination, as these integrate several important concepts.