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The nature of valuation When a life insurance product is created, its premiums are established by the pricing actuary. But a policy that is in force already still requires attention of an actuary in order to assure that the benefits or claims are paid properly, and that the insurance company s profit objectives are realized. While these two objectives may appear contradictory, they should not be. The company must remain solvent if it wants to earn profits at all, so actuaries make certain that all scheduled liabilities cash flows: benefits (for life insurance, annuities, etc.) and claims (for property/casualty and liability insurance, etc.) are met. In order to do this, an actuary effectively designates a portion of the company s assets as set aside to meet those liabilities cash flows. To put it simply, this actuary marks a part of the assets held by the insurance company as customers money. This work of marking customers money is called valuation, or calculation of reserves. Most of liabilities of insurance companies are reserves. It should be noted that insurance companies rarely, or ever, have liabilities typical in regular industrial companies, i.e., loans and bonds, because such liabilities would have to be, by law, junior to insurance liabilities (i.e., reserves) and this would make them very unattractive investments. The actuary who calculates reserves is called the valuation actuary in the United States, or appointed actuary in Canada. For a property/casualty or liability insurer, most reserves are for insured events that have already occurred. They are called claim reserves. Interestingly enough, the largest portion of claim reserves is typically those claims that the insurance company has not yet been notified about, incurred-but-not-reported reserves (IBNR). It is the job of the actuary who calculates the reserves to estimate IBNR. In health insurance, IBNR reserves are mostly for incurred but not fully developed claims, as reporting is faster. For life insurance, on the other hand, most reserves are for insured events that have not occurred yet. They are called benefit reserves. In the basic life contingencies you have learned how to calculate them on the net basis, i.e., only accounting for benefits (and thus the name benefit reserves). However, in actuarial practice, you need to be also aware of all insurance company expenses, including the cost of capital, which is an expense in economic terms, and termed profit in accounting. The life insurance valuation actuary must make certain that the company s reserves and surplus are adequate, after accounting for benefits and expenses. When premiums are calculated to pay only for benefits, they are called benefit premiums, or net premiums. When expenses are also included, premiums are termed gross premiums or contract premiums. Valuation not including expenses is called net premium valuation, and if expenses are included, it is called gross premium valuation. Gross premium valuation is also called the Policy Premium Method in Canada. In the process of valuation, actuary must consider the policy cash flows. Cash flows for a life insurance policy are: premiums, dividends, benefits. They do not depend on the accounting method, or reserving method used. For educational use by Illinois State University only, do not redistribute - 52 -

Traditional life insurance products have benefits and premiums known in advance, but there are also new design dynamic products (e.g., variable universal life), where all cash flows are flexible. When analyzing cash flows, you must distinguish cash flows per unit in force (at the time of analysis) or per unit issued. For example, assume 500 policies issued, each with death benefit of $1000 and annual premium $40. Three years later 367 policies in force. Then: - Premium per unit in force = $40. - Premium per unit issued = $40 367 500 = $40 p ( τ ) t x. Note that the policies that are no longer in force are gone either because of deaths of policyholders or because of lapses, not because of just one of the two reasons. Business and regulatory considerations in life insurance valuation in the United States An important regulatory consideration in the United States is whether lapsing life insurance policyholders should receive any portion of their policy reserve, should they withdraw from the contract. Contract withdrawal can be done by notifying the insurer, but effective cancellation can also happen as a result of the policyholder not paying the premium any more. In either case, we call this event a policy lapse. Lapses form an important practical business consideration for life insurance companies, and a part of life insurance business regulation is aimed at them. The policyholder may also be entitled to receive some funds from their life insurance policy without canceling, in a form of policy loan. Withdrawal benefits are also termed nonforfeiture benefits. This term is used because those benefits cannot be lost as a result of premature cessation of premium payments. We will now present general principles adopted in the United States for the purpose of valuation of nonforfeiture benefits. The benefit that can be received, in terms of funds paid to the policyholder, is called the cash value. In general: kcv = k V k SC, (2.1) where k CV is the cash value at time k, k V is the policy benefit reserve at time k, and k SC is the surrender charge at time k. The surrender charge is simply defined as the difference between the policy benefit reserve and the cash value at a given duration. The surrender charge never exceeds the reserve, although there is no such regulatory requirement. It is simply not realistic to expect to collect additional payments from a ( 2 canceling policyholder. The withdrawal benefit can also be denoted by B ) x+ k. One theme in the regulation of cash values in the United States has been the need for direct recognition of the amount and incidence of expenses. An idea in line with this theme is to define a minimum cash value for a unit life insurance as: kcv = A( k) P a a ( k) = k V ( P a P) a ( k), (2.2) For educational use by Illinois State University only, do not redistribute - 53 -

where A k duration k, a k ( ) is the single benefit premium for the plan of insurance remaining at policy ( ) is the single benefit premium for a unit life annuity for the premium paying period remaining, P a is called an adjusted premium, and P is the regular policy annual benefit premium. In 1975, the Society of Actuaries committee studied nonforfeiture benefits and related matters. It produced a report that contained consideration of two types of expenses in defining adjusted premium: E = level annual amount per unit of insurance, incurred each year throughout the premium paying period, E 0 = additional expense for the first year. The first year additional expense component is assumed to be provided by the adjusted premium: G = P a + E (2.3) and G a = P a + E ( ) a = A + E 0 + E a. (2.4) This implies that: P a = A + E 0 a (2.5) and P a E 0 + P a a = A. (2.6) In 1980, National Association of Insurance Commissioners Standard Nonforfeiture Law used (2.2) and (2.5) to define minimum cash values required. The law states that for policies with level benefits and level contract premiums: E 0 = 1.25 min P,0.04 ( ) + 0.01. (2.7) Here, P is the benefit premium rate per unit of death benefit for the policy. Thus the first policy year expense E 0, and the corresponding adjusted premium P a, are: and E 0 = P a = 1.25P + 0.01, if P < 0.04, 0.06, if P 0.04, A +1.25P + 0.01, a if P < 0.04, A + 0.06, a if P 0.04. (2.8) (2.9) For educational use by Illinois State University only, do not redistribute - 54 -

Types of reserve valuation in the United States Statutory valuation: It is required by law and submitted to insurance regulators to help them assess the financial health of the company. In the United States, statutory valuation emphasizes solvency, and generally utilizes conservative assumptions and techniques, producing relatively large reserves. Historically, the methodology of reserve calculation was prescribed by law. This was changed first in 1980, when interest rate assumption was made dependent on the value of market index of interest rate as of the date of issue, and even more so in 1991. Under the new Standard Valuation Law effective 1991, the valuation actuary is required to assure adequate provision for discharging liabilities, with consideration for the nature of company s assets. In California, the valuation actuary is also required to assure adequate provision for nonforfeiture values. Note that in many countries, e.g., Canada, United Kingdom, and Australia, all assumptions (interest rates, mortality, lapses) are set by the actuary (called the appointed actuary) and only subject to review by regulators for reasonableness. In the U.S., statutory accounting is a mixture of formulas and assumptions prescribed by law, and the overriding requirement that the actuary is supposed to make a professional judgment to assure payment of benefits and expenses. Generally Accepted Accounting Principles (GAAP): required of publicly traded companies in the United States. This valuation, and accounting is not required by any law of mutual companies in the U.S. GAAP techniques incorporate provisions for lapses, surrender benefits, etc., and are supposed to use reasonable but conservative assumption. Note that there is no distinction between statutory and regular accounting statement in Canada. Tax reserve valuation: performed in order to calculate the reserve liability for the purpose of determining taxable income. Since 1984, Federally Prescribed Tax Reserves are used in the U.S., all tax valuation assumptions are prescribed by the Internal Revenue Service (IRS), the U.S. tax collection authority. Valuation standards Valuation standards consist of interest rates, mortality and other decrement assumptions used in valuation. The standards are given in the Standard Valuation Law (SVL), created by the National Association of the Insurance Commissioners, and then adopted by each state s legislature (and signed by a governor). Only one interest rate is used for each policy in statutory valuation, and the law states how an interest rate should be determined for each policy type. Before 1980, specific numeric interest rate was used. Since 1980, SVL was adjusted and the maximum interest rate (or, as actuaries put it: minimum valuation standard, as higher interest rate corresponds to a lower reserve) is a function of the Monthly Average of the Composite Yield on Seasoned Corporate Bonds, as published by Moody s Investors Services, Inc. Resulting rates vary by guarantee duration, where the guarantee refers to the basis guaranteed in the policy. This interest rate is to be used for all calculations of reserves throughout the policy s existence. The insurance company can always use a lower rate, and increase reserve (for statutory purposes, of course, not for tax purposes). For educational use by Illinois State University only, do not redistribute - 55 -

Valuation mortality standard is stated in terms of mortality tables allowed. Different minimum standards for mortality are prescribed for males and females, except when unisex tables are allowed. But in some states, reserves for unisex products must be calculated using separate mortality for males and females. While level benefit premium reserves are allowed in the U.S., they are rarely used, virtually never for early policy durations. They simply produce too large of a reserve in the first year, and companies prefer to be able to use an expense allowance in the first year, in order to be able to pay large first year expenses. Note that in GAAP accounting, a different approach is chosen to handle the problem of large first year expense. Instead of charging the first year expense in the first year, GAAP accounting amortizes it over some period, creating an artificial asset in the balance sheet: the Deferred Acquisition Cost. Valuation Manual Codification provides a comprehensive guide for statutory accounting practices, which serves as the basis of each states statutory accounting practices SSAP 50: classifies contracts into four categories SSAP 51: establishes statutory accounting principles for the four categories in SSAP 50 Appendices: contain excerpts of SVL, actuarial opinion and memorandum regulation, and ASOP 22 Accounting Principles Statutory - Prescribed by insurance laws and regulations. - Focus is on statutory solvency (particularly statutory capital). - NAIC provides standards. GAAP - Emphasis is on the matching of revenues and expenses. - FASB is the primary accounting standards body. International - International Accounting Standards Board s mission is to develop transparent and comparable financial statements. - Large multinationals must file GAAP in the US as well as another set of financials in the other countries that they do business Tax - Life insurers are generally subject to same rules as other taxable corporations. - Since a significant deduction is the increase in reserve, tax reserves use prescribed standards. - DAC tax is meant to increase taxable income in early years (similar to GAAP). Fair Value - Uses market value of assets. - When market value is not readily available, the hierarchy for determining market value is market value when available, market value of similar instruments with appropriate adjustment, and present value of projected cash flows. For educational use by Illinois State University only, do not redistribute - 56 -

Types of Valuations Statutory - Used by regulators to assess claims paying ability of a company. - Uses conservative assumptions and expenses acquisition costs immediately. - Many techniques were developed without today s computing power, so assumptions such as lapses are allowed for via the conservative assumptions - Moving away from cookbook approach, like in Canada. GAAP - Assumptions often based on experience with margin. - Acquisition costs are deferred. Tax - Reserves are less than or equal to statutory reserves. - Gross Premium. - Uses best estimate assumptions. Embedded Value - EV = PV of Earnings, discounted at cost of capital. - Cost of capital is the rate that could be earned on a similar investment (CAPM). Effects of Statutory Valuation Requirements Gross Premium Level - GPs often set to avoid deficiency reserves. - Statutory reserve requirements must be considered in setting GPs. Product Design: guaranteed cost of insurance, premiums, and interest set according to reserve requirements. Federal Income Tax: tax reserves must be less than or equal to statutory reserve. Policyholder Dividends: reserves may be a part of dividend formula and, at the very least, will affect the amount of earnings that can be distributed as dividends. Statutory Earnings: appraisal values are based on earnings, which are based on increase in reserves. Indicators used by regulators, rating agencies, and investment analysts. Statutory Valuation Requirements In Canada Insurance Companies Act: created the appointed actuary, who - Will value and report on actuarial and other benefit liabilities. - Reports annually to board on current financial position of the company. - Will have access to all necessary records and info. - Must bring to the attention of management and the board any circumstance that may have a material impact on company s ability to meet obligations. - Must send copy of report to OSFI if satisfactory action isn t taken. - Renders opinion to the board on administration of company dividend policy. Canadian Asset Liability Method (CALM) Full gross premiums Expected experience plus margin Estimated expenses and benefits Scenario Testing For educational use by Illinois State University only, do not redistribute - 57 -

Minimum Continuing Capital and Surplus Requirements Analogous to Risk-Based Capital (RBC) in the United States. Dynamic Capital Adequacy Testing Examine current financial position and ability to withstand future solvency threats. Considers in-force and new business. Look at a few adverse scenarios. Joint Policy Statement: actuary and auditor could be using each other s work. NAIC Annual Statement Financial Statements Balance Sheet assets, liabilities, and surplus. Summary of Operations revenue, costs (benefit/other), taxes, net income. Capital and Surplus Account net income, stockholder dividends, other below the line items. Cash Flow Statement cash from investments, operations, and financing. Analysis of Operations by LOB Summary of Operations but for BU. Successive Equation EOP Value = BOP Value + Increases - Decreases Reserve at time t = Reserve at time (t 1) + Net Premium + Tabular Interest Tabular Cost +/ Other Changes Surplus at time t = Surplus at time (t 1) + Net Income Shareholder Dividends +/ Other Changes Equity vs. Surplus Assets Liabilities = Equity = Surplus GAAP vs. Statutory concept Premiums at Cash vs. Accrual Basis Collected Premium = amount of cash that comes in the door (from premiums). Accrual Premium = Direct Premium + Net Premium from Reinsurance Transaction Direct Premium = Collected Premium + Increase in Deferred Premium Asset Increase in Advance Premium Liability The accrual premium formula is taking the premium collected (cash) and modifying it to reflect that reserves may assume premium frequency that does not line up with reality, to reflect that some premiums paid are not associated with the period that the check comes in the door, and to reflect that reinsurance premiums may be paid or received. For educational use by Illinois State University only, do not redistribute - 58 -

Application of modified reserves to valuation and pre-pbr valuation In the fully discrete case we have α FPT x = A 1 x:1, (2.66) β FPT x = P x+1, (2.67) as well as: EA FPT 1 = P x+1 A x:1 = P x+1 c x, (2.68) E A FPT 1 = P x A x:1 = P x c x. (2.69) For statutory valuations in the United States, there is a legal maximum on expense allowance. In the regulation, called Commissioners Reserve Valuation Method (CRVM), the expense allowance is formula-based, and it is not limited by the actual first-year expense incurred. CRVM requires FPT reserves (or larger) for durations 1 and later if the resulting FPT renewal benefit premiums are not greater than 20-pay life FPT renewal net premiums. For plans with FPT renewal net premiums greater than 20-pay life FPT renewal net premiums, the expense allowance is limited to that used for 20-pay life: 1 19 P x+1 A x:1 = 19 P x+1 c x. (2.70) This is the basic pre-pbr statutory life insurance valuation method in the United States (although reserve are still subject to cash flow testing: the valuation actuary follows the basic reserve calculation with the cash flow testing validation, and only after that issues an Actuarial Opinion). Here is the summary of CRVM methodology (for a unit benefit): If β FPT 1 A x;1 = β FPT c x < 19 P x+1 c x (2.71) then EA CRVM = β FPT c x. (2.72) Otherwise, EA CRVM = 19 P x+1 c x. (2.73) Other methodologies The most basic methodology of reserving in life insurance is the Net Level Premium (NLP) method, using the level benefit premium as the valuation premium, and calculating the reserve as the benefit reserve. FPT and CRVM methods are modifications of NLP. But mixtures of these are also used. One of the more common alternatives is a method in which reserves gradually transform ( grade, as actuaries call it) from CRVM to net level at some policy duration. The reason for such approach is to increase reserves over the CRVM required minimum so that higher cash values can be offered to customers who retain policies for such longer durations. Consider the following: Exercise You are analyzing a $1000 whole life policy in which reserves are CRVM at the end of year 1, grading to net level reserves at year 20. Issue age is 35, reserving is based on 1958 CSO, age last birthday, and the valuation interest rate is 4.0%. Then the CRVM renewal net premium is (values calculated using the table, which unfortunately, you do not have easy access to) For educational use by Illinois State University only, do not redistribute - 59 -

P CRVM = 1000 A 36 a 36 14.867. The modified reserve at time 1 is 1V 35 MOD = 0 = 1000A 36 P CRVM a 36 = 1000A 36 P G a 36:19 P NL 19 a 36. Therefore the graded net premium is P G = 1000A 36 PNL a 1000A 36 1000A 35 19 36 a 19 a 36 = 35 15.142. a 36:19 a 36:19 We have 1 35 = 0 V CRVM 10 35 = 127.28 V CRVM 20 35 = 296.25 V 1 35 = 12.24 V NL 10 35 = 137.96 V NL 20 35 = 304.85 V 1 35 = 0 V G 10 35 = 130.36 V G 20 35 = 304.85 Also note that ( P G P NL ) a 36:19 = EA CRVM a 36. a 35 Exercise What is the formula for the benefit premiums for a whole life plan with reserves that are CRVM through the end of year five, grading to the net level reserves at year twenty? Solution. Year 1: α x = c x = A 1 x:1. Years 2-5: P CRVM = A x+1 a x+1. Years 6-20: P Graded = A x+5 A x CRVM a 15 a x+5 5 V x x. a x+5:15 Year 21+: P x = A x a x. Exercise You are the valuation actuary for the Honorable Life Insurance Co., a New Mexico subsidiary of an Iceland-based conglomerate. Your company has issued a fully discrete ten-year endowment life insurance policy for which it uses the curtate terminal Commissioners Reserve Valuation Method (CRVM) reserve methodology. The policy in the amount of $1000 is issued on January 1, 2000. You are given the following for the age of issue x: For educational use by Illinois State University only, do not redistribute - 60 -

A x:20 = 0.50260, A x+1 = 0.51739, A x:10 = 0.69006, A x+1:9 = 0.71452, a x:20 = 12.93, a x+1:19 = 12.55, a x:10 = 8.06, a x+1:9 = 7.42. You are also given that the probability of death is 0.011 for all lives for all years, and the valuation rate is 4%. Calculate the curtate (i.e., fully discrete) terminal (i.e., just before the second premium is paid) CRVM reserve on December 31, 2000. Solution. If Full Preliminary Term is used, reserve is zero. So the question is: Would FPT be allowed under CRVM rules? In order to determine that, we compare the FPT renewal benefit premium for this contract with renewal FPT benefit premium for 20-pay life (since this is a ten-year endowment, the answer is almost obvious): 1000 9 P x+1:9 = 714.52 7.42 = 96.30 > 1000 P = 517.39 19 x+1 12.55 = 41.23. Therefore, FPT is not allowed and the first year expense allowance (per unit of coverage) is capped at 19 P x+1 A 1 x:1. Since this policy is for $1000, we calculate 1 1000A x:1 = 1000 0.011 1.04 = 10.58, and 1000 19 P x+1 = 41.23, and the cap (equal to the this policy s first year expense allowance) is 41.23 10.58 = 30.65 per thousand. The level benefit premium for this policy is 1000 10 P x:10 = 690.06 8.06 = 85.61. The reserve at duration 1 based on the level benefit premium is 1000 1 V x = 714.52 85.61 7.42 = 79.25. The CRVM reserve is equal to the level benefit premium reserve minus the unamortized portion of the first year expense allowance, i.e., 1000 1 V x CRVM = 79.25 30.65 a x+1:9 a x:10 = 51.03. Deficiency reserves Basic prospective reserve is the present value of future benefits less the present value of future premiums. However, what if gross premiums are less than the valuation net premiums? Deficiency reserves are reserves, which may be required in addition to basic policy reserves when the gross premium is below a certain level. Before the 1976 changes, the Standard Valuation Law required deficiency reserves if the gross premium for a policy were less than the valuation net premium used. Those were subject to criticism: Suppose that a company voluntarily strengthens reserves by reducing the valuation interest rate from 3.5% to 3%, and suppose the gross premiums for a policy are greater than the net premiums at 3.5%, but less than at 3%. Reserve strengthening would cause For educational use by Illinois State University only, do not redistribute - 61 -

the basic policy reserve to increase. Yet as a result of lowering the valuation rate, deficiency reserves would be required as well. Although reserve strengthening occurs rarely in practice, this example illustrates that the prior law sometimes required companies with conservative reserve bases to hold deficiency reserves even though they would not have been necessary if a more liberal basis had been adopted. Deficiency reserves have not been allowed as a tax reserve in the U.S since the early 1980's. The 1976 amendments to the Standard Valuation Law removed any explicit reference to deficiency reserves. Instead, basic policy reserves are required to be increased under certain circumstances: if the gross premium for a policy is less than the valuation net premium calculated using the valuation method actually used, but using the minimum standards of mortality and interest, then the required total reserve is the greater of (a) or (b), defined as: (a) the reserve calculated according to the method, mortality table, and interest rate actually used in the policy, and (b) the reserve calculated by the method actually used for the policy, but using the minimum valuation standards of mortality and interest, and replacing the valuation net premium by the actual gross premium in each year that the actual gross premium is less than the valuation net premium. Exercise You are given the information below for a $1,000 whole life policy issued by a U.S. life insurance company to an insured age 35 in 2002 (the values below are based on minimum valuation standard for mortality and interest): Duration t 1000A 35+t a 35+t a 35+t:5 t 0 363.24 21.86 4.69 1 372.52 21.59 3.61 2 382.02 21.22 2.90 3 391.71 20.85 1.97 4 401.59 20.54 1.00 5 411.65 20.20 ----- 6 421.86 19.85 ----- The contract premium G is $16.65, and there is no policy fee. Reserves are curtate graded from CRVM at the end of the first policy year to benefit level premium reserves at the end of the fifth policy year. Calculate the minimum terminal reserve required on the policy anniversary in 2004. Solution. Note that since this is a whole life policy, FPT is used for CRVM reserves. We have: 1000P 35 = 1000 A 35 = 363.24 a 35 21.86 = 16.6167. The first year terminal reserve is zero, because it is an FPT reserve. The graded premium is: For educational use by Illinois State University only, do not redistribute - 62 -

1000P Graded = 1000A 36 1000P 35 ( a 36 a 36:4 ) = a 36:4 372.52 16.6167 ( 21.59 3.61 ) = 20.4298 > 16.65. 3.61 Since this graded premium ends up being more than the contract premium, to find the minimum reserve required, we must use the contract premium instead of graded premium for policy durations 2-5. But the benefit level premium is less than the contract premium, so we use it after duration 5. Thus, minimum reserve required is: 1000A 37 1000P Contract a 37:3 1000P 35 3 a 37 = ( ) = = 1000A 37 1000P Contract a 37:3 1000P 35 a 37 a 37:3 ( ) = 29.25. = 382.02 16.65 2.90 16.6167 21.22 2.90 Approximations Terminal reserve: Reserve as we have been calculating it, at the end of a policy year. Initial reserve: Reserve at the beginning of a policy year. Mean reserve: Mid-year reserve, commonly approximated as the average of the two. Mid-terminal reserve is the average of the terminal reserve of the last year and the terminal reserve of this year. For educational use by Illinois State University only, do not redistribute - 63 -

Immediate Payment of Claims Reserve o Used with curtate reserves that assume BOP premium and EOP death benefit o If claims are paid at death, without interest from the date of death, the IPC reserve = (i/3) * death portion of reserve (e.g. A x 1 :n for n-year term) o If interest is paid from the date of death to the date of payment, the IPC reserve = (i/2) * death portion of reserve (e.g. A x 1 :n for n-year term) Continuous Reserves o Semicontinuous assumes BOP premium and moment of death DBs o Eliminates the need for an IPC reserve o The semicontinuous NLP reserve is m! t V(!) [x]:n =! [x]+t:n-t m P(!) [x]:n * ä [x]+t:m-t!! [x]+t:n-t = (i /!) " A [x]+t:n-t o Fully continuous assumes continuous premiums moment of death DBs o No need for IPC reserve m o t V [x]:n =! [x]+t:n-t m P(!) [x]:n * " [x]+t:m-t o Discounted continuous assumes BOP premiums, refund of the unearned premium, and moment of death DBs o Mean reserves are used with discounted continuous premium = m tp [x]:n * " 1 o Expense Allowance under Continuous Assumptions: EA = (! [x]+1 / " [x]+1) c [x] Non-deduction Reserve o Reflects that some modal premiums will not be collected in year of death o Reserve = [(m -1)/2m ] * P (m ) [x]:n * t V x:n Refund Reserve o Required if the company refunds unearned premiums in the year of death o Terminal reserve factor = P {m } [x]:n * t V x:n where P (m ) [x]:n = m t P [x]:n 1 ((m -1)/2m )*d m t P [x]:n Exercise: Society of Actuaries Course 150 Sample Examination 150-81-99, Problem No. 30 For a fully discrete whole life insurance of 1000 on ( x), you are given: (i) i = 0.03. (ii) Reserves are determined using a modified reserve method, where the modification period is the entire policy period, the modified premium for each of the first three years is 17.72, and modified premiums are level thereafter. (iii) a x:3 = 2.9083. (iv) a x+ 3 = 25.2094. (v) 3 E x = 0.9101. (vi) 1000A x = 247.020. (vii) 1000A x+20 = 406.585. Calculate the modified reserve at the end of year 20. ACE Manuals Page 102 For educational use by Illinois State University only, do not redistribute - 64 -

Solution. We have A x = α a x:3 + β 3 a x, so that β = A x α a x:3 = A x α a x:3 a E a 3 x 3 x x+ 3 The modified reserve sought is = 0.24702 17.72 2.9083 0.9101 25.2094 8.52. 1000 20 V Mod x = 1000A x+20 β a x+20 = 1000A x+20 β 1 A x+20 i 1 + i 406.585 8.52 1.03 1 0.406585 233. 0.03 Exercise: Society of Actuaries Course 150 Sample Examination 150-81-99, Problem No. 35 For a fully discrete insurance of 1 on ( x), you are given: (i) Reserves are determined using a modified reserve method, where the modification period is the entire policy period, the modified premium for the first year, α, is given by α = α FPT, if β FPT α FPT 0.04, β 0.04, otherwise, and modified premiums, β, are level thereafter. (ii) d = 0.06. (iii) a x = 13. (iv) a x:10 = 7.55. (v) A x:10 1 = 0.507. 1 (vi) A x:1 = 0.006. For a fully discrete whole life insurance of 1 on ( x), calculate β. Solution. For a fully discrete whole life insurance of 1 on x ( ), we have the following α FPT 1 = A x:1 = 0.006, 1 d a x = 1 0.06 13 = 0.22 = A x = α FPT + β FPT ( a x 1) = 0.006 +12β FPT, and this results in 0.22 0.006 β FPT = 0.01783333. 12 Therefore β FPT α FPT 0.01783333 0.006 = 0.01183333 < 0.04, For educational use by Illinois State University only, do not redistribute - 65 -

and full preliminary term applies. Hence, β = β FPT = 0.214 12 = 107 6000 0.01783333. Universal Life Insurance In December of 1983, the National Association of Insurance Commissioners adopted the Universal Life Model Regulation that sets forth minimum reserve standards for universal life policies. These standards represent an effort to fit universal life into traditional valuation methodologies. An assumption was made regarding future premium payments and a factor was developed to adjust for actual policy performance. The minimum reserve standards specified in this regulation are rather involved. The following is a summary of the steps to be performed for a flexible premium universal life policy under this regulation: 1. A Guaranteed Maturity Premium is calculated using policy guarantees (i.e., guaranteed expense charges, cost of insurance charges, credited interest rates, etc.). This Guaranteed Maturity Premium is the level gross premium that provides for an endowment for the face amount at the latest permissible maturity date under the contract. 2. A set of Guaranteed Maturity Funds is calculated. Guaranteed Maturity Funds (GMFs) are the projected fund values calculated as of the issue date using policy guarantees and assuming that Guaranteed Maturity Premiums are paid. 3. The actual or current fund value at the valuation date must be known. 4. The policy fund is projected forward from the valuation date, on a guaranteed basis, using the larger of the current fund or the Guaranteed Maturity Fund at each future valuation date, and assuming that Guaranteed Maturity Premiums are paid. This projection produces a set of "guaranteed death benefits" and a "guaranteed endowment benefit" for valuation purposes. 5. A net level premium is calculated based on the plan of insurance produced at issue on a guaranteed basis assuming Guaranteed Maturity Premiums are paid. 6. The present value of future guaranteed benefits as of the valuation date is calculated. The guaranteed benefits are the set of "guaranteed death benefits" and "guaranteed endowment benefit" calculated in the fourth step. 7. The ratio, r-ratio, of the current fund value to the Guaranteed Maturity Fund at the valuation date is calculated. The r-ratio is never allowed to exceed 1. 8. A net level reserve is calculated as r-ratio times the difference between the present value of guaranteed benefits and the present value of net level premiums. 9. The Commissioners Reserve Valuation Method reserve is calculated as the difference between the net level reserve determined in step 8, and the r-ratio times the unamortized Commissioners Reserve Valuation Method expense allowance for the plan of insurance generated at issue on a guaranteed basis and assuming Guaranteed Maturity Premiums are paid. Alternative minimum reserves may be required for flexible premium universal life plans if the Guaranteed Maturity Premium is less than the valuation net premium for the plan of insurance produced at issue, on a guaranteed basis, assuming that Guaranteed Maturity Premiums are paid. For educational use by Illinois State University only, do not redistribute - 66 -

premiums, the charges actually assessed, investment performance, and other items o 7702 defines limitations on relationship of benefits & AV UL Model Regulation Companies once held CV as the reserve for lack of anything better The Here following is a summary steps are of used the methodology: in calculating the UL Model Reg reserve: 1) Guaranteed Maturity Premium (GMP) is the level gross premium that will endow the policy at the contract s latest possible maturity date o Calculated using the policy s guaranteed mortality charges, expense charges, and interest rate 2) Guaranteed Maturity Fund (GMF) is the vector of projected guaranteed (uses policy guarantees) fund value (as of issue date), assuming that GMPs are paid 3) At each valuation date, the max(current fund, GMF) is projected forward, using policy guarantees, assuming that GMPs are paid, producing set of guaranteed death benefits and guaranteed endowment benefits to be used in step #4 and #6 4) Net level premium (NLP), based on the plan of insurance produced at issue, is calculated on a guaranteed basis, assuming GMPs are paid o NLP t = (PVFB 0 / PVGMP 0 ) * GMP o NLP is same for all t (for CRVM) since GMP is level for all t 5) r = Min(current fund value / GMF, 1) 6) NLP reserve = r * [PV(future guaranteed benefits) PV(future NLPs)] 7) CRVM reserve = NLP reserve r * (unamortized CRVM expense allowance) *Note Note: that EA the calculation EA calculation is the is same the same as in EA CRVM. CRVM calc from chapter 5 ACE Manuals Example Page 113 Assume r = 0.5 at t = 4, what is the CRVM reserve for this UL product? Age 1000q x ä x :4 1000 A x :4 ä x:5 1000 A x:5 ä x:19 1000 A x:19 ä x 1000 A x 25 1.2330 3.7421 4.6669 4.5763 5.8291 12.9975 23.6187 20.0237 137.7373 26 1.2735 3.7419 4.8645 4.5759 6.0847 12.9889 25.0240 19.9043 142.8787 27 1.3288 3.7416 5.0914 4.5753 6.3752 12.9793 26.5716 19.7801 148.2235 28 1.3895 3.7412 5.3403 4.5748 6.6936 12.9687 28.2643 19.6514 153.7691 29 1.4560 3.7408 5.6131 4.5741 7.0426 12.9572 30.1153 19.5178 159.5208 30 1.5289 3.7404 5.9121 4.5734 7.4251 12.9446 32.1388 19.3793 165.4842 Solution o Let s continue on a per-unit basis o NP = A 25 / ä [25] = 137.7373 / 20.0237 = 6.8787 o EA CRVM = A 26 / ä [25]+1 c [x] = 142.8787 / 19.9043 1.233 / 1.045 = 5.9983 OR = A 26 / ä [25]+1:19 c [x] = 142.8787 / 12.9889 1.233 / 1.045 = 9.8202 o 4V NLP = r * [1000 * A 29 NP * ä 29 ] = 0.5 * [159.5208 6.8787 * 19.5178] = 12.6317 o 4V CRVM = 4 V NLP r * EA unamortized = r * [1000A 29 (NP + EA CRVM / ä 25 ) * ä 29 ] = 0.5 * [159.5208 6.8787 * 19.5178 (5.9983 / 20.0237) * 19.5178] = 9.7085 For educational use by Illinois State University only, do not redistribute - 67 -

Example What would change in the prior problem if AV 4 > 40? Solution Since GMF accumulates to 1,000, AV would accumulate to a larger amount, causing A 29 to become bigger. Variable Alternative Life Insurance Minimum Reserves Variable Universal Life Insurance (often shortened to VUL) is a type of life insurance that These builds are cash essentially value. In deficiency a VUL, the reserves cash value for UL can policies. be invested in a wide variety of separate Net Premium: accounts, similar to mutual funds, and the choice of which of the available separate o accounts P CRVM = to m PB use [x]:n is + entirely m PE [x]:n up to the contract owner. The 'variable' component in the name o refers P CRVM to = (PVFB this ability 0 / ä x ) to + invest EA / ä x in separate accounts whose values vary they vary Gross because Premium: they are invested in stock and/or bond markets. The 'universal' component in the name o GMP refers to the flexibility the owner has in making premium payments. The premiums o Book can vary doesn t from refer nothing to AMR in a as given deficiency month reserves up to maxima directly, defined but rather by says the Internal that the reserve held must be the greater of (a) and (b), where: Revenue Code for life insurance. This flexibility is in contrast to whole life insurance that (b) is the reserve according to the actual method and has fixed premium (a) is the reserve payments using that minimum typically standards cannot be of missed mortality without and interest lapsing & (GP the policy if < (although one NP) may exercise an Automatic Premium Loan feature, or surrender dividends to pay a Whole Life premium). Variable universal life is a type of permanent life insurance, because the death benefit will be paid if the insured dies any time as long as there is sufficient cash value to pay the costs Guaranteed of insurance in Maturity the policy. Premium With most Method if not all VULs, unlike whole life, there is no endowment The GMP age method (the age is not at meant which to the solve cash the value problems equals associated the death with benefit Off-Anniversary amount, which for Reserves, whole life but, is typically rather, it is 100). a way This to simplify is yet another reserve key advantage calculation of since VUL benefits over Whole do not Life. need With to be a typical projected whole and discounted life policy, back the death to the benefit valuation is date limited each to reporting the face period. amount specified in the policy, and at endowment age, the face amount is all that is paid out. The reserve is equal to Thus with either death or endowment, the insurance company keeps any cash value built o r * traditional CRVM reserve, if fund value <= GMF up over the years. However, some participating whole life policies offer riders which o traditional CRVM reserve + (fund value GMF), if fund value > GMF specify that any dividends paid on the policy be used to purchase "paid up additions" to the policy which increase both the cash value and the death benefit over time. If investments California made UL in Regulation the separate accounts out-perform the general account of the insurance company, a higher rate-of-return can occur than the fixed rates-of-return typical for Allows whole life. companies The combination to calculate over reserves the as years follows: of no t Vendowment x = (AV + CV) age, / 2 continually increasing NY 127 death only allows benefit, this and method if a high for policies rate-of-return issued before is earned 2000 in the separate accounts of a VUL Note policy, that UL this Model could Regulation result in Reserves higher value will to still the be owner needed or for beneficiary tax reserve than calculation that of a whole life policy with the same amounts of money paid in as premiums. This type of life insurance may require reserve for Minimum Death Benefit Guarantee. By allowing the contract owner to choose the investments inside the policy the insured takes on the investment risk, and receives the greater potential return of the investments in return. If the investment returns are very poor this could lead to a policy lapsing ACE Manuals Page 115 For educational use by Illinois State University only, do not redistribute - 68 -

Topics: Topics: Topics: o o Unitary Unitary vs. vs. Term Term Method Method - Problem - Problem o o Determination Unitary vs. Term of Method Contract of Contract - Segments Problem Segments o o Calculation Determination of Segmented of Contract Reserves Segments Reserves o o Calculation of Unitary Segmented of Unitary Reserves Reserves (ceasing o o Calculation to exist as a of valid Basic Unitary of Basic policy). Reserves Reserves To avoid this, many insurers offer guaranteed death benefits o o Deficiency Calculation up to a certain Reserves of Basic age as Reserves long as a given minimum premium is paid. o Deficiency Reserves Term insurance reserving Unitary vs. vs. Term Term Method - Problem - Unitary o o In the In vs. the good Term good old Method old days days insurers insurers - Problem could could include include high high future future gross gross premiums premiums in their in their o reserve In reserve the good calculations, old days sometimes insurers could resulting resulting include negative high in negative future reserves! gross reserves! premiums in their o o Graph reserve Graph out calculations, out the the premium premium sometimes stream! stream! resulting in negative reserves! Example Example o o Regulators Graph out were the were premium ok with ok with stream! this this since since gross gross premiums premiums were were thought thought to be to sufficient be sufficient o o In Regulators the In the 1970s, 1970s, were competition ok with this drove since drove rates gross rates down, premiums down, causing causing were regulators thought regulators consider be to sufficient consider both both o In the 1970s, competition drove rates down, causing regulators to both Given Given the chart unitary the unitary chart below, (all below, (all GPs determine GPs considered) determine considered) the segment and the and term segment term lengths. (only (only lengths. GPs GPs renewal in renewal period period considered) considered) unitary (all GPs considered) and term (only GPs in renewal period considered) t GP t 1000qx GP 1000qx 1 Determination 20 1 20 1.233 1.233 Determination of of of Contract Contract Contract Segments 2 20 2 20 1.274 1.274 Segments Segments 3 40 3 40 1.329 1.329 4 The 40 The The 4 length 40 length length 1.390 of 1.390 a of a of segment segment a segment is determined is determined is determined by comparing by comparing by comparing the ratio the ratio the ratio of gross of gross of gross premiums premiums premiums from from from 5 period 60 period period 5 to 60 1.456 period to to 1.456 period period as well as well as well as the as the as the ratio ratio ratio of valuation of valuation of valuation mortality mortality mortality from from from period period period to period to period to period Solution A Solution new A A new new segment segment segment begins begins begins when: when: when: (GP (GP (GP [x]+t / [x]+t [x]+t GP / GP [x]+t-1 ) > [x]+t-1 [x]+t-1 ) Max(1, Max(1, > Max(1, q [x]+t [x]+t q / q [x]+t-1 / [x]+t [x]+t-1 / q ) t GP t 1000qx 1000qx r GP r GP r Mort r Mort Segment Segment [x]+t-1 ) ) 1 20 1 20 1.233 1.233 1 1 where GP[x]+t = guaranteed gross premium per 1000 2 20 2 20 1.274 where where 1.274 1.00 GP[x]+t 1.00 03285 = 1. guaranteed 032851 gross 1gross premium premium per 1000 per 1000 q[x]+t = valuation mortality rate, excluding X factors 3 40 3 40 1.329 q[x]+t q[x]+t 1.329 2.00 = valuation 2.00 1. 04342 1. 04342 mortality 2rate, 2rate, excluding excluding factors X factors 4 40 4 40 1.390 1.390 1.00 1.00 04568 1. 045682 2 5 60 5o Policy fees are only excluded from the GP if they re level during the premium o 60 1.456 Policy Policy 1.456 fees 1.50 fees are 1.50 1.04786 are only only 1.04786 excluded excluded 3 from 3 from the GP the if GP they re if they re level level during during the premium the premium payment period (in which case, they won t affect) payment period period (in (in which which case, case, they they won t won t affect) o The mortality ratio can be changed by up to 1% affect) Time Time 1 to avoid breaks due to rounding! o 120 The! The / 20 mortality mortality < / 20 1.274 < ratio 1.274 / ratio 1.233 can / can 1.233 be! changed be No! changed segment No by segment up by break to up 1% to break to 1% avoid to avoid breaks breaks due to due rounding to rounding Time Time 2! 240!/ 20 40 > / 20 1.329 > 1.329 / 1.274 / 1.274! New! New segment segment Calculation Calculation of Unitary of Unitary Reserves Reserves The unitary The unitary reserve reserve includes includes all future all future gross gross premiums premiums (like (like in chapter in chapter 5) 5) o Unitary o Unitary reserve reserve is based is based on CRVM on CRVM m m o o t V [x]:n = A [x]+t:n-t ( m P [x]:n + EA CRVM t V [x]:n = A [x]+t:n-t ( m P [x]:n + EA CRVM / ä [x]:m / ä ) [x]:m * ä [x]+t:m-t ) * ä [x]+t:m-t ACE Manuals Page 106 ACE Manuals Page 106 Calculation of Segmented of Reserves Reserves Segmented Segmented reserves reserves are calculated are calculated according according to the to benefits, the benefits, net premiums, net premiums, and unusual and unusual guaranteed guaranteed cash cash value value within within each each segment segment CV is CV unusual is unusual if (GCV if (GCV t GCV t GCV t-1 ) > t-1 (1.1 ) > *(1.1 GP* t + GP 1.1 t + * 1.1 (GCV * (GCV t-1 + GP t-1 + t ) GP *i nf t ) + *i 5% nf + * 5% SC) * SC) o Example o Example would would include include return return of premium of premium rider rider on a term on a term policy policy For educational use by Illinois State University only, do not redistribute - 69 - m m 1 1 t VB [x]:n = AB [x] +t:n1-t + v n1-t * n1-t p [x]+t *BW u t VB [x]:n = AB [x] +t:n1-t + v n1-t * n1-t p [x]+t *BW u [x]+n1 m1 P [x]+t:n1 * ä [x]+t:m1-t, for the 1 st [x]+n1 m1 P [x]+t:n1 * ä [x]+t:m1-t, for the 1 st segment segment m m 1 1 t VB [x]:n = AB [x] +ki+t:ni-t + v ni-t * ni-t p [x]+ki *BW u [x]+ki+ni - BW u t VB [x]:n = AB [x] +ki+t:ni-t + v ni-t * ni-t p [x]+ki *BW u [x]+ki+ni - BW u [x]+ki [x]+ki mi PB [x]+t:ni mi PB [x]+t:ni *ä [x]+t:mi-t, *ä [x]+t:mi-t,

The unitary reserve includes all future gross premiums (like in chapter 5) o Unitary reserve is based on CRVM m o t V [x]:n = A [x]+t:n-t ( m P [x]:n + EA CRVM / ä [x]:m ) * ä [x]+t:m-t Calculation of Segmented Reserves Segmented reserves are calculated according to the benefits, net premiums, and unusual guaranteed cash value within each segment CV is unusual if (GCV t GCV t-1 ) > (1.1 * GP t + 1.1 * (GCV t-1 + GP t ) *i nf + 5% * SC) o Example would include return of premium rider on a term policy m t VB [x]:n = AB [x] 1 +t:n1-t + v n1-t * n1-t p [x]+t *BW u [x]+n1 m1 P [x]+t:n1 * ä [x]+t:m1-t, for the 1 st segment m t VB [x]:n = AB [x] 1 +ki+t:ni-t + v ni-t * ni-t p [x]+ki *BW u [x]+ki+ni - BW u [x]+ki mi PB [x]+t:ni *ä [x]+t:mi-t, otherwise ACE Manuals Page 107 For educational use by Illinois State University only, do not redistribute - 70 -