Value Insurance with a Term Rider and Base Whole-Life Cover

Introduction

A term rider adds a temporary death benefit to a base whole-life policy. It does not replace the base benefit: while the rider is in force, a death claim includes both amounts. Valuing the contract therefore requires attention to two different timelines: how long each benefit is payable and when premiums for each component are due.

This tutorial calculates component premiums and prospective reserves for a hypothetical policy with a whole-life base benefit and a term rider. The examples use annual premiums paid at the beginning of a year while the insured is alive, and death benefits paid at the end of the year of death. The methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Valuing the Two Benefit Components

Let \(B\) be the base whole-life sum insured and \(R\) the rider sum insured. Suppose the rider covers the first \(n\) policy years. Under an annual discrete model, if death occurs in one of those years the total benefit is \(B+R\); if death occurs later, it is \(B\). This addition assumes the rider is explicitly additional cover rather than a provision that changes or replaces the base amount.

For the calculations below, assume an effective annual interest rate \(i=10\%\), so \(v=1/1.10=10/11\). Assume the probability of death during each policy year, conditional on survival to its start, is \(q=0.10\), and the probability of survival for that year is \(p=0.90\). These probabilities are the same at every duration. Death benefits are paid at the end of the year of death.

The formula for the term benefit follows by separating the possible years of death. For death in year \(k+1\), the insured must first survive \(k\) years, with probability \(p^k\), then die in the next year, with probability \(q\); the payment is discounted \(k+1\) years. Summing those expected discounted payments gives:

For the assumed \(i\) and \(q\), \(vp=(10/11)(9/10)=9/11\) and \(vq=1/11\). Thus \(A_x=(1/11)/(1-9/11)=1/2\), while \(\ddot{a}_x=1/(1-9/11)=11/2=5.5\). The term value is obtained by using only the first \(n\) terms of the same geometric series.

Pricing Components With Different Premium Periods

Under the equivalence principle, the expected present value of a component’s premiums equals the expected present value of that component’s benefits. If premiums for a component are due at the beginning of each of \(m\) years while the insured is alive, their value per unit of annual premium is \(\ddot{a}_{x:\overline{m}|}=\sum_{k=0}^{m-1}(vp)^k\). The premium is the benefit value divided by the appropriate premium-annuity value. The annuity must match the actual times premiums are due.

For a contract with component-specific premiums, the total premium due at a particular time is the sum of the premiums then payable for the base and the rider. After the rider premium period ends, the remaining premium may consist only of the base premium.

Prospective Reserves by Component

A prospective reserve is the expected present value of future benefits less the expected present value of future premiums, conditional on the insured being alive at the valuation time. To make premium timing explicit, define \({}_kV\) here as the reserve at time \(k\), immediately before any premium due at that time. Future benefits start with the policy year beginning at time \(k\). A premium due at time \(k\) is included in future premiums if it is scheduled and the insured is alive to pay it.

The contract reserve is the sum of the component reserves when both benefits and premiums have been separated consistently. At duration \(k\), a rider with \(n\)-year coverage has \(n-k\) coverage years remaining if \(k<n\), and none if \(k\ge n\). The remaining rider premium annuity, however, depends on the rider’s premium schedule, not just on the remaining coverage.

With the constant mortality and interest assumptions used here, a whole-life base benefit priced with a level premium has zero prospective reserve at each duration before a premium. Conditional on survival, the remaining whole-life benefit value is still \(B/2\), and the remaining lifetime premium-annuity value is still \(5.5\). Their difference is zero when the base premium is \(B/11\). This simplification depends on the constant assumptions; it should not be carried over to a model with changing mortality rates.

Shorter Premium Periods and the Expiry Boundary

A rider can remain covered after all its premiums have been paid. In that case, the rider reserve generally includes the value of its remaining benefit without a corresponding future rider-premium value. Conversely, a rider can expire while the base policy and its premiums continue. At and after that boundary, the rider contributes no future benefits or premiums to the reserve.

Common Mistakes and a Timing Check

The most important calculation check is to keep three durations separate: the base benefit’s duration, the rider’s coverage duration, and the rider’s premium-payment duration. A rider may have a four-year benefit term but only three premiums, as in the previous example. Each duration determines a different cash-flow stream.

• Counting the rider as a replacement benefit. If the contract says the rider is additional, the death benefit during the rider term is \(B+R\), not just \(R\).
• Using the coverage term as the premium term. In the three-year rider example, dividing its benefit value by the three-year annuity value would give \(40{,}000/11=\$3{,}636.36\), not the correct rider premium of $4,975.21. That calculation assumes three rider premiums, but only two are scheduled.
• Removing the rider when its premiums end. A benefit can remain in force after its last premium. At time 2 in the three-year example, the rider has one year of coverage remaining and a positive reserve.
• Including expired coverage in a later reserve. After the rider’s coverage term ends, it contributes neither future benefit nor future rider premium to the prospective reserve.
• Mixing reserve timing conventions. The calculations here value reserves immediately before any premium due at time \(k\). A reserve immediately after that premium would differ by the premium just paid; state the valuation point before applying a reserve formula.

The component approach is useful because it makes contract boundaries visible. A single combined benefit value can be calculated, but separating the base and rider helps detect whether a payment has been counted during the correct years. It also makes clear why the total premium can change when the rider premium period ends, even though the base policy continues.