Understand the non-stochastic interest rate models used to calculate present values and accumulated values of cash flows and calculate present values and accumulated values of cash flows.
For fixed interest rates (level or varying over time); yield curves (spot and forward interest rates); and interest rate scenarios models:
• Describe the models.
Chapter 7
Emerging Costs
7.1 Cash Values
Policy values quantify the obligations, or liabilities, of a life insurance company.
• They are used by managers and investors to determine the financial position of the company.
• They are also used by managers and regulators to determine the likelihood that a company will continue as an ongoing operation to fulfill promises made to policyholders.
Policy values are also used to determine the allocation of investment assets needed to fulfill company obliga- tions from issuing policies - this is the topic of Section
S:AssetShares
. This section is devoted to the application of policy values to determine a policy’s value to the insured upon cancelation of the policy, itscash value.
An insurance policy provides a combination of protection against unforeseen risks (the insurance element) as well as a savings plan (the investment component). When a policyholder decides to terminate the contract prematurely (before called for in the plan design), in many cases the policyholder will receive an amount known as a cash value, withdrawal benefit, ornonforfeiture benefit. You can think of the cash value as the policy value minus a surrender charge, that is,
kCV = kV − kSC.
Because of the difficulty of collecting additional funds from a withdrawing policyholder,kCV ≥0.
Determining the cash value (or, equivalently the surrender charge) is not straightforward. One needs to think about fairness to the continuing policyholders as well as fairness to those that terminate the contract. One viewpoint is that those that surrender their policy have not fulfilled their contract and therefore are not entitled to any benefits. Another viewpoint is that those that surrender their policy should receive all of their premiums minus a charge for the insurance protection received.
It is also important to recognize the timing and incidence of expenses because most insurance contracts are costly to underwrite and incur substantial expenses in the early years. In Section
S :DAExpenses
, we discuss an alternative reserving basis that acknowledges high early expenses.
7.2 Asset Shares
The “asset share” of a policy in force is the share of the insurer’s assets attributable to that policy. Here, expenses, interest, mortality and so forth are calculated based on the insurer’s experience for similar policies over the period.
Asset share calculations are similar to those used for policy values although the purposes of the two concepts are quite different. The policy value represents an amount the insurerneeds to have, the asset share represents the amount that the insureractually does have.
Asset shares are typically calculated recursively, such as using relationship similar to those for policy values, e.g., (kAS+Gk−ek)(1 +ik) = q (d) [x]+k(bk+1+Ek+1) +q (w) [x]+k k+1CV +p (τ) [x]+k k+1AS where
• at the beginning of the year, kAS is the asset share, Gk is the contract premium, ek are annual expenses per contract
• at death,q([xd]+) kis the probability of death,bk+1is the insurance benefit, andEkare settlement expenses per contract
• at withdrawal,q([xw]+)k is the probability of withdrawal, k+1CV is the cash value,
• at survival to the end of the year, p([xτ]+) k = 1−(q[(xd]+) k+q[(xw]+)k)is the probability of survivorship and k+1AS is the asset share.
Asset shares define the notion of how quickly a policy builds policy values (assets) and hence profits. The main advantage of this recursive approach is that we can examine the incidence (timing) of profits. The recursive relation can be expanded to include other ancillary benefits, taxes, more complex expense structures, and so forth.
Asset shares are primarily used to set premiums by defining a profit goal. For example, • PV(profit) = x% PV (premium)
• A return on investment =x%
• The product breaks evennyears or sooner.
Asset shares can easily be used for flexible (variable) plans.
They can also be used to determine reserves, dividend schedules, withdrawal benefits or so-called “model offices” (combining asset shares over several plans, age at issue, new versus old business, and so on).
Example (SoA # 336).
For a fully discrete insurance of 1000 on (x), you are given: • 4AS= 396.63is the asset share at the end of year 4.
• 5AS= 694.50is the asset share at the end of year 5.
• G= 281.77is the gross premium.
• 5CV = 572.12is the cash value at the end of year 5.
• c4= 0.05is the fraction of the gross premium paid at time 4 for expenses.
• e4= 7.0is the amount of per policy expenses paid at time 4.
7.3. EMERGING COSTS 85
• qx(2)+4= 0.26is the probability of decrement by withdrawal. Calculatei.
Solution.
At the beginning of the year, the asset share plus net income available is
4AS+G(1−e4)−e4= 396.63 + 281.77(1−0.05)−7 = 657.3115.
At the end of the year, funds must be sufficient to pay those who survive, die and withdraw:
p(xτ+4 5) AS+ 1000qx(1)+4+ 5CV q (2)
x+4
= (1−0.09−0.26)(694.50) + 1000(0.09) + (572.12)(0.26) = 690.1762. Using the relation657.3115(1 +i) = 690.1762, we geti= 4.9999%≈5%.
Example (SoA # 242).
For a fully discrete whole life insurance of 10,000 on (x), you are given: • 10AS= 1600is the asset share at the end of year 10.
• G= 200is the gross premium.
• 11CV = 1700is the cash value at the end of year 11.
• c10= 0.04is the fraction of gross premium paid at time 10 for expenses.
• e10= 70is the amount of per policy expense paid at time 10.
• Death and withdrawal are the only decrements. • qx(d+10) = 0.02
• qx(w+10) = 0.18 • i= 0.05
Calculate 11AS, the asset share at the end of year 11.
Solution.
At the beginning of the year, the asset share plus net income available is
10AS+G(1−c10)−e10= 1600 + 200(1−0.04)−70 = 1722.
At the end of the year, funds must be sufficient to pay those who survive, die and withdraw:
p(xτ+4 11) AS+ 10000qx(d+10) + 11CV q (w)
x+10
= (1−0.02−0.18)11AS+ 10000(0.02) + (1700)(0.18) = 0.8 11AS+ 506.
Thus, with i= 0.05, we have(1.05)1722 = 0.8 11AS+ 506, or 11AS = 1627.625.
7.3
Emerging Costs
Under “cash accounting,” a person or firm counts income when it is received and claims deductions when money is paid. Annual cash flows may consist of income items: premium income, interest and dividends on assets, maturity proceeds of assets, and reinsurance recoveries in respect of claims. The outflows may consist of: claims settle or amounts paid on account, reinsurance premiums, expenses, tax, and dividends. In cash accounting, the profit, or net income, is simply the excess of money received over money paid. Following
cash makes life easy - one simply has to look at the checkbook to see when money is coming in and when it goes out. Projecting annual cash flows as they “emerge,” or emerging cash flows, is an important task. However, firms cannot simply follow cash to determine profitability. Consider the case of a savings bank that receives deposits from their customers without making any payments. This money received is not a “profit” to the bank - rather the bank receives the deposit as money coming in but also has an obligation to return the money when demanded by the customer. That is, the bank’s obligations have increased with the customer deposit.
Under “accrual accounting,” revenues and costs are recognized when they are incurred, not necessarily when related payments are received or made. In many businesses, the timing of incurral is easy to identify. Typically, a company records revenue as incurred when a product or service is shipped or billed to a customer, not when payments are actually made (they could be much later if you are buying on a payment plan). To assess profitability, a company’s costs need to be matched to the revenue.
Matching costs to revenue is difficult in life insurance because of the potential time lag between premiums (one type of revenue) and benefits. To understand the cost of a life insurance product, let us begin by
considering a net basis, ignoring expenses from doing business as well as any profit motivation.
Suppose that we have 1,000 identical policyholders age xwho purchase a policy with a generic premium payment schedule {Ph} and benefit schedule {bh}. At some future time point, sayhyears later, we expect there to be 1000 hpx policyholders alive with each policy having value hV. Thus, the company’s total obligation is1000hpx hV. One year later, the company’s obligation will be1000v h+1px h+1V. The increase
in this obligation is a cost to the company. Using the recursive reserve formula, this cost may be expressed as
1000vh+1px h+1V− 1000hpx hV
= 1000hpx(vpx+h h+1V − hV)
= 1000hpx(vpx+h h+1V − {vqx+hbh+1−Ph+vpx+h h+1V})
= 1000hpx(Ph−vqx+hbh+1),
with the relation h+1px= hpxpx+h. Thus, the cost is simply the expected cash inflow, premiums in excess of benefits. This is anticipated as we are working on a net basis which implies zero profits.
7.4
Annual Profits
It is important to monitor the actual experience of a block of business. To this end, we use carats, or “hats,” to denote actual values. We use
• ˆik is the actual rate of return from invested assets, • eˆk is the actual annual expenses per contract, • at death,qˆ([xd]+) k is the realized fraction of deaths,
• at withdrawal,qˆ([xw]+)k is the realized fraction of withdrawals, and
• at survival to the end of the year,pˆ([xτ]+) k = 1−(ˆq([xd]+) k+ ˆq([xw]+)k)is realized fraction that survived. With these quantities, the profit during the year (at timek+ 1) is
P rof itk+1= (kAS+Gk−eˆk) (1 + ˆik)−qˆ (d) [x]+k(bk+1+Ek+1)−qˆ (w) [x]+k k+1CV −pˆ (τ) [x]+k k+1AS. For this illustration, we have that assumed premiumsGk and settlement expensesEk are the same as actual values. We also assume that $ _k AS$ represents the terminal (year-end) company obligation (that does not depend on experience). Recall the recursive asset share calculation
(kAS+Gk−ek)(1 +ik) =q
(d)
[x]+k(bk+1+Ek+1− k+1AS) +q
(w)
7.5. PROFIT TESTING 87
Substituting fork+1AS, we may write the profit during the year is
P rof itk+1= (kAS+Gk−ˆek) (1 + ˆik)−qˆ (d) [x]+k(bk+1+Ek+1− k+1AS) − qˆ[(xw]+)k(k+1CV − k+1AS)− k+1AS = (kAS+Gk) (ˆik−ik) + ek(1 + ˆik)−ˆek(1 +ik) + (bk+1+Ek+1− k+1AS) (q (d) [x]+k−qˆ (d) [x]+k) + (k+1CV − k+1AS) (q (w) [x]+k−qˆ (w) [x]+k).
This is one way to decompose the profit into identifiable components, known as theanalysis of surplus. Here, we interpret these portions of the profit:
• (kAS+Gk) (ˆik−ik)- due to favorable investment experience, • ek(1 + ˆik)−ˆek(1 +ik)- due to favorable expense experience, • (bk+1+Ek+1− k+1AS) (q
(d) [x]+k−qˆ
(d)
[x]+k)- due to favorable mortality experience, and • (k+1CV − k+1AS) (q
(w) [x]+k−qˆ
(w)
[x]+k)- due to favorable withdrawal experience.
7.5
Profit Testing
In the prior section, we defined profits based on actual experience. It is also helpful to define a profit at the plan design stage, before experience is realized. Instead of using actual experience, we now remove the carats and employ aprofit test basis which is the set of assumptions used to examine the incidence and timing of profits.
The profit during the year (at time k+ 1) is P rk+1= (kAS+Gk−ek) (1 +ik)−q (d) [x]+k(bk+1+Ek+1)−q (w) [x]+k k+1CV −p (τ) [x]+k k+1AS. When summarizing profits, it can be helpful to remind ourselves that profits are only available for those policies in force at the beginning of the year. Thus, we might define a term such as Πk+1 = kp
(τ) [x]P rk+1
for profits discounted for survivorship. The termprofit signature is used for the vector of discounted profits
Π= (Π0,Π1. . .)′.
Profits depend on all the assumptions, including assumed interest. To summarize profits, one measure used is theinternal rate of return(IRR), defined to be the solution of the equation
∑ k ( 1 1 +j )k Πk= ∑ k ( 1 1 +j )k k−1p (τ) [x]P rk = 0,
where profits are calculated using interest rateik ≡j. TheIRRis the solution of a nonlinear equation and so may not exist or may have multiple solutions. For the multiple solution problem, we can use the hurdle rate, defined to be the smallestIRRso that the contract is deemed adequately profitable.
We can summarize profits using an assumed discount rate, r. Define thenet present value, also known as theexpected present value of future profits, to be
N P V =∑ k ( 1 1 +r )k Πk= ∑ k ( 1 1 +r )k k−1p (τ) [x]P rk,
Another profit measure is the discounted payback period, the first time that the sum of discounted profits is non-negative. Here, the discounting is for (1) survival and (2) for interest, using risk discount rate r. That is, the discounted payback period is the smallest value ofmsuch that
m ∑ k=0 ( 1 1 +r )k Πk≥0.