Derivation of General Formula for Cumulative Effect

Transcription

1 Steve Malerich In my presentation, Simply Unlocking, at the 2015 Valuation Actuary Symposium I started the development of an unlocking toolkit with a formula introduced by Bruce. Darling in a 1992 issue of the Financial eporter 1. Neither the article nor my presentation showed how to derive this Cumulative ffect formula. In Simply Unlocking, I introduced a corresponding formula to measure the total cumulative effect for DAC 2 and related liabilities (UL and SOP reserves). In an article, also titled Simply Unlocking 3, I introduced formulas to estimate amounts of amortization and cumulative effect (unlocking) directly from specific sources of profit and variances therefrom. In this document, I show a derivation of the general Cumulative ffect formula, followed by a derivation of Darling s formula as a special case. I also show derivations of the formulas for source-specific cumulative effect and net amortization rates. Derivation of General Formula for Cumulative ffect To develop the general cumulative effect formula, we first define the basic inputs to a valuation. Table 1 Basic Inputs to Actuarial Valuations D Deferred expenses (ASC A) including sales inducements (ASC ) D Deferred front-end loads (ASC ) DB Deferrable benefits (benefits subject to additional reserving ASC death or other insurance benefit features and ASC annuitization benefits) GP Gross profit (ASC ) GA Gross assessments 4 (ASC & 30-2, ASC & 30-22) CP Cash profit 5 CA Cash assessments 6 With these we can define several functions. 1 Unlocking FAS 97 s Management Potential, reproduced in the 1999 Society of Actuaries Financial eporting Section Monograph. The formula appears near the bottom of page 138 in the monograph. 2 In the presentation, DAC is defined to include any sales inducement asset. 3 Simply Unlocking, The Financial eporter, June I use the term, Gross assessments, for what the accounting standards call assessments. This helps to distinguish from cash assessments, which include front-end loads but exclude amortization of deferred revenue. 5 Cash profit is the portion of FAS 97 gross profit that represents actual transactions with counterparties; it excludes the accrual of any additional reserve under to SOP See note 4. 1 P a g e

2 Table 2 Basic Functions 7 in Actuarial Valuation AV PV Prefix to any of the Table 1 variables, indicating the accumulated value of past amounts Prefix to any of the Table 1 variables, indicating the present value of future amounts A. AVGP AVCP SOP PVGP PVCP + SOP AVGP + PVGP B. AVGA AVCA UL PVGA PVCA + UL AVGA + PVGA These formulas are a consequence of the interpretation, used here, that gross profit and assessments exclude the interest component of the changes in SOP reserve and UL. (They also apply if interest is included in cash profit and assessments.) C. D. k k AVD + AVD AVD + PVD The rate at which the deferred acquisition cost (DAC) asset is amortized. The rate at which the unearned revenue liability (UL) is amortized.. k k k The net amortization rate of DAC and UL combined. F. b AVDB + PVDB The rate at which the additional (SOP) reserve accrues. G. DAC AVD k AVGP AVD k (AVCP SOP) H. UL AVD k AVGP AVD k (AVCP SOP) I. SOP b AVGA AVDB b (AVCA UL) AVDB Intangible asset for deferred expense and sales inducement. Unearned revenue liability for deferred front-end loads. Additional reserve. Throughout this section, subscripts distinguish values before (prior) and after () recognizing an experience variance. Where there is no subscript, the value is measured before any experience variance. An experience variance can be either a variance from expected current period cash profit or a change in the present value of expected cash profit. For accumulating and discounting, prior and are measured as of the same point in time the beginning of the reporting period currently under analysis. We can now define the cumulative effects of a variance. 7 Ignoring any constraints, which are beyond the scope of this document. 2 P a g e

3 Table 3 J. C DAC DAC prior DAC K. C UL UL prior UL L. C SOP SOP SOP prior The amount of additional DAC amortization required at the beginning of the period as a result of an experience variance. The amount of additional UL amortization required at the beginning of the period as a result of an experience variance. The amount of additional SOP reserve accrual required at the beginning of the period as a result of an experience variance. The cumulative effects of DAC and UL are expressed in terms of additional amortization positive results represent a cost from DAC and income from UL. The cumulative effect of SOP is expressed in terms of additional accrual a positive result represents a cost. Taking all three pieces together: (a) Cumulative ffect C DAC C UL + C SOP We can now move into our derivation, beginning with the three separate pieces: (b) C DAC DAC prior DAC [AVD (k (k (AVCP SOP prior )] [AVD k (AVCP SOP )] (AVCP SOP ) (AVCP SOP prior ) (AVCP SOP prior + SOP prior SOP ) (AVCP SOP prior ) ) (AVCP SOP prior ) + k (SOP prior SOP ) k (SOP SOP prior ) (c) C DAC (k k C SOP (d) C UL UL prior UL [AVD (k (k (AVCP SOP prior )] [AVD k (AVCP SOP )] (AVCP SOP ) (AVCP SOP prior ) (AVCP SOP prior + SOP prior SOP ) (AVCP SOP prior ) ) (AVCP SOP prior ) + k (SOP prior SOP ) k (SOP SOP prior ) (e) C UL (k k C SOP (f) C SOP SOP SOP prior [b (AVCA UL ) AVDB] [b prior (AVCA UL prior ) AVDB] b (AVCA UL ) b prior (AVCA UL prior ) b (AVCA UL prior + UL prior UL ) b prior (AVCA UL prior ) 3 P a g e

4 (b b prior ) (AVCA UL prior ) + b (UL prior UL ) (b b prior ) AVGA + b (UL prior UL ) (g) C SOP (b b prior ) AVGA + b C UL To solve the circularity between SOP and UL, substitute formula (e) into formula (g): (h) C SOP (b b prior ) AVGA + b [(k (b b prior ) AVGA + b (k k C SOP ] b k C SOP Adding the final term to both sides and then dividing by the resulting multiple: (i) C SOP (1 + b k ) (b b prior ) AVGA + b (k (j) C SOP (b b prior ) AVGA + b (k 1 + b k Substituting formula (j) into formula (c), establishing a common denominator, and then grouping multiples of the AVs: (k) C DAC (k (k k (b b prior ) AVGA + b (k 1 + b k (1 + b k ) 1 + b k k AVGP [k (b b prior ) AVGA + b (k 1 + b k + b (k AVGA k (b b prior ) 1 + b k k 1 + b k k k k + k k prior )] emoving offsets: (l) C DAC AVGP [(k ) b (k prior k k 1 + b k k prior )] AVGA k (b b prior ) Substituting formula (j) into formula (e), establishing a common denominator, and then grouping multiples of the AVs: 4 P a g e

5 (m) C UL (k (k k (b b prior ) AVGA + b (k 1 + b k (1 + b k ) 1 + b k k AVGP [k (b b prior ) AVGA + b (k 1 + b k + b (k AVGA k (b b prior ) 1 + b k k 1 + b k k k k + k k prior )] emoving offsets: (n) C UL AVGP (k ) AVGA k (b b prior ) 1 + b k Bringing together formulas (l), (n) and (j) and then grouping the AV terms: (o) Cumulative ffect C DAC C UL + C SOP AVGP [(k AVGP (k ) b (k prior k k 1 + b k ) AVGA k (b b prior ) 1 + b k + (b b prior ) AVGA + b (k 1 + b k AVGP [(k k + AVGA (1 k + k prior ) + b (k 1 + b k + k ) (b b prior ) 1 + b k k prior )] AVGA k (b b prior ) k k prior + k prior k )] Applying formula and grouping common multiples, we have a general formula for calculating the cumulative effect as a function of changes in amortization rates and the benefit ratio: (p) Cumulative ffect {(k ) + b [k (1 ) 1 + b k + (1 k ) (b b prior ) AVGA 1 + b k (1 k )]} AVGP 5 P a g e

6 If there are no front-end loads, we can set the revenue amortization rates to zero: (q) Cumulative ffect (k + (b b prior ) AVGA (1 k ) Where there is no requirement for an additional SOP reserve, we can simplify formula (p) by setting the benefit ratios to zero. This is the form recognized by Darling in his 1992 article. (r) Cumulative ffect (k Derivation of Cumulative ffect Formulas by Source of Variance We now want to find formulas for estimating the Cumulative ffect directly from either: (1) deviations from expected cash activity (acquisition costs, front-end loads, other assessments, reserved benefits, and other costs); or (2) changes in their projections. Cash activity is defined broadly to include anything representing actual transactions between the insurer and relevant counterparties. It excludes changes in DAC, UL and SOP reserves but may include interest spread on some of those balances if that is included in gross profit 8 and may include some other accruals (such as changes in claims ICOS and IBN). Before continuing with formula development, we add two more definitions to those given in Table 1 and three more formulas to those given in Tables 2 and 3. Table 4 M. N. CC x h P h A AVGP AVGP + PVGP AVGA AVGA + PVGA Other cash costs (benefits and expenses not subject to reserving or deferral) Variances from expected cash activity or changes to the present value of expected cash activity. The gross profit historical ratio, one measure of the age of the business. The gross assessments historical ratio, another measure of age. O. CP (CA D) DB CC Cash profit expressed as a combination of its components. When applied to variables, Table 2 s formulas C, D and F will see the simple addition or subtraction of deltas in either the numerator or denominator. 8 Under some interpretations of SOP 03-1, gross profit and assessments include interest spread on some intangible balances. 6 P a g e

7 1. Acquisition Cost Variance We ll now evaluate the effect of a variance ( D ) from expected acquisition costs. We can see from formulas C-F that D has no effect on ratios k or b and that 1.1. k prior 1.2. k k + k prior k AVD + D + AVD D + k AVD + AVD 1.3. b prior b 1.4. b b prior 0 Multiplying formula 1.2 by AVGP and substituting formulas A and M: 1.5. (k (k Substituting these into formula (p): 1.6. Cumulative ffect {(k ) + b [k [(k ) + b k D AVGP D AVGP AVGP + PVGP D h P (1 ) 1 + b k + (1 k ) (b b prior ) AVGA 1 + b k (k 1 + b k D h P + b k D h P 1 + b k D h P 1 + b k 1 + b k )] AVGP (1 k )]} AVGP + (1 k ) 0 AVGA 1 + b k 1.7. Cumulative ffect D h P 7 P a g e

8 2. Front-nd Load Variance Next, we evaluate the effect of a variance ( D ) from expected front-end loads. We can see from formulas C- that D will have no effect on k and that: 2.1. k prior 2.2. k k k + k prior k + k prior (k ) 2.3. k AVD + D + AVD D AVD + AVD Multiplying formula 2.2 by AVGP, then substituting formulas 2.3, A and M: 2.4. (k (k D AVGP D AVGP AVGP + PVGP D h P From formula F we can determine: 2.5. b AVDB + PVDB AVCA + D + PVCA 2.6. b prior AVDB + PVDB AVDB + PVDB AVCA + D + PVCA AVCA + D + PVCA b AVCA + D + PVCA 2.7. b b prior b b AVCA + D + PVCA b (1 AVCA + D + PVCA ) 8 P a g e

9 b ( AVCA + D + PVCA ) D b b D Multiplying formula 2.7 by AVGA and substituting formulas B and N: 2.8. (b b prior ) AVGA b D AVGA b D AVGA AVGA + PVGA b D h A Substituting these into formula (p): 2.9. Cumulative ffect {(k ) + b [k { (k (1 ) 1 + b k + (1 k ) (b b prior ) AVGA 1 + b k ) + b (k 1 + b k (1 k ) b D h A 1 + b k [1 b (1 k )] [ (k [1 b (1 k ) (1 k )} AVGP (1 k )]} AVGP ] (1 k ) b D h A 1 + b k )] [ D h P ] (1 k ) b D h A 1 + b k D h P [1 b (1 k )] + h A b (1 k ) 1 + b k Using the gross profit historical ratio as an approximation to the assessments historical ratio: h A h P Cumulative ffect D h P [1 b (1 k )] + h P b (1 k ) 1 + b k D h P [1 b (1 k )+b (1 k )] 1 + b k 9 P a g e

10 D h P [1+b ( 1 + k + 1 k )] 1 + b k D h P [1+b (k k )] 1 + b k D h P [1+b (k k 1 + b k D h P (1+b k ) 1 + b k + k )] Cumulative ffect D h P 3. Cash Profit Variance Generally We have three more possible deviations to consider other assessments, deferrable benefits, and other costs. All are components of cash profit and change the denominator of the expense and revenue amortization rates. Using x to represent expense, revenue, or the net of the two, for any deviation ( CP ) from expected cash profit: x 3.1. k x 3.2. k prior x 3.3. k x AVDx + PVDx AVCP + CP + PVCP AVDx + PVDx AVDx + PVDx AVCP + CP + PVCP AVCP + CP + PVCP AVDx + PVDx AVCP + CP + PVCP AVCP + CP + PVCP x x x x AVCP + CP + PVCP x k AVCP + CP + PVCP (1 AVCP + CP + PVCP ) ( AVCP + CP + PVCP k x CP ) Multiplying formula 3.3 by AVGP and substituting formulas A and M, we see: 10 P a g e

11 x 3.4. (k x k x CP AVGP k x CP AVGP AVGP + PVGP x k CP h P Next, we use this to evaluate a piece of formula (p) for any variance from expected cash profit b [k b [k [(k b (k (1 ) k k prior ) k k b ( k CP h P ) b k Inserting these into formula (p): 3.6. Cumulative ffect CP h P (1 k )] AVGP {(k ) + b [k + k prior k ] AVGP AVCP + CP + PVCP + k (1 ) 1 + b k + (1 k ) (b b prior ) AVGA 1 + b k ( k CP h P ) b k 3.7. Cumulative ffect AVCP + CP + PVCP (1 k )]} AVGP k ] AVGP CP h P + (1 k ) (b b prior ) AVGA 1 + b k CP h P (k + b k ) + (1 k ) (b b prior ) AVGA 1 + b k To further evaluate formula 3.7, we need to know the specific type of variance other assessment, deferrable benefit, or other cost. 4. Other Assessment Variance Beginning with a variance ( CA 9) from expected other assessments, we can see from formula O that (ignoring any change in other variables) CP CA and from formula F that CA will alter the denominator of b. 9 CA is defined here to exclude any variance from expected deferrable revenue. 11 P a g e

12 4.1. b AVDB + PVDB AVCA + CA + PVCA 4.2. b prior AVDB + PVDB AVDB + PVDB AVCA + CA + PVCA AVCA + CA + PVCA b AVCA + CA + PVCA 4.3. b b prior b b AVCA + CA + PVCA b (1 AVCA + CA + PVCA ) b ( AVCA + CA + PVCA ) CA b b CA Multiplying by AVGA and applying formulas B and N: 4.4. (b b prior ) AVGA b CA AVGA b CA AVGA AVGA + PVGA b CA h A Inserting these into formula 3.7: 4.5. Cumulative ffect CP h P (k + b k ) + (1 k ) (b b prior ) AVGA 1 + b k CA h P (k + b k ) (1 k ) b CA h A 1 + b k CA h P (k + b k ) + h A (1 k ) b 1 + b k Applying the formula 2.10 approximation: 12 P a g e

13 4.6. Cumulative ffect CA h P (k + b k ) + h P (1 k ) b 1 + b k CA h P k + b [k + (1 k )] 1 + b k 4.7. Cumulative ffect CA h P k + b k b + b k 1 + b k To explain results of the current valuation in terms of variables known prior to valuation, we need a suitable approximation for the k-factors and benefit ratio. If we think of formula 4.7 as a curve dependent on CA, we can approximate it with a line tangent to the curve at the point where CA 0. We get that line by substituting the prior k-factors and benefit ratio for the ratios Cumulative ffect CA h P k prior + b prior b prior + b prior k prior 1 + b prior k prior 5. Deferrable Benefit Variance With a variance ( DB ) from expected deferrable benefits, we can see from formula O that (ignoring any change in other variables) CP DB and from formula F that DB will alter the numerator of b b AVDB + DB + PVDB 5.2. b b prior AVDB + DB + PVDB AVDB + PVDB DB Multiplying by AVGA and applying formulas B and N: 5.3. (b b prior ) AVGA DB AVGA DB AVGA AVGA + PVGA DB h A Inserting these into formula 3.7: 13 P a g e

14 5.4. Cumulative ffect CP h P (k + b k ) + (1 k ) (b b prior ) AVGA 1 + b k DB h P (k + b k ) + (1 k ) DB h A 1 + b k DB h P (k + b k ) + (1 k ) h A 1 + b k Applying the formula 2.10 approximation: 5.5. Cumulative ffect DB h P (k + b k ) + (1 k ) h P 1 + b k DB h P k + b k + 1 k 1 + b k DB h P 1 + b k 1 + b k 5.6. Cumulative ffect DB h P 6. Other Cash Cost Variance With a variance ( CC ) from expected other costs, we can see from formula O that (ignoring any change in other variables) CP CC and from formula F that there will be no effect on b b b prior 0 Inserting these into formula 3.7: 6.2. Cumulative ffect CP h P (k + b k ) + (1 k ) (b b prior ) AVGA 1 + b k CC h P (k + b k ) 1 + b k 6.3. Cumulative ffect CC h P k + b k 1 + b k To explain results of the current valuation in terms of variables known prior to valuation, we need a suitable approximation for the k-factors and benefit ratio. If we think of formula 6.3 as a curve dependent on CC, we can approximate it with a line tangent to the curve at the point where CC 0. We get that line by substituting the prior k-factors and benefit ratio for the ratios. 14 P a g e

15 6.4. Cumulative ffect CC h P k prior + b prior k prior 1 + b prior k prior Derivation of General Formula for Normal Amortization To derive a general formula for normal amortization 10, we add some equations to the earlier definitions, notation and basic functions as above. Table 5 P. Am k GP The amount of DAC amortization. Q. Am k GP The amount of UL amortization.. Ac B b GA DB The amount of SOP reserve accrual. S. GP CP Ac B These are a consequence of the interpretation that T. GA CA D + Am gross profit and assessments exclude the interest component of the change in SOP and UL. 11 In this formulation, amortization of DAC and accrual of SOP reserves are both costs and amortization of UL is revenue, such that: i. Amortization + Accrual Am Am + Ac B From formulas P to T: ii. Am k GP k (CP Ac B ) iii. Am k GP k (CP Ac B ) iv. Ac B b GA DB b (CA D + Am ) DB b [CA D + k (CP Ac B )] DB b (CA D + k CP) b k Ac B DB Adding the reserve accrual term to both sides of the equation, dividing by its combined multiple, and then inserting into formulas ii and iii: v. Ac B (1 + b k ) b (CA D + k CP) DB vi. Ac B [b (CA D + k CP) DB] (1 + b k ) 10 xcluding interest accrual and the cumulative effect of any experience variance or assumption change. 11 These can also be achieved by including interest in cash profit and cash assessments. 15 P a g e

16 vii. Am k (CP Ac B ) viii. Am k (CP Ac B ) k [CP b (CA D + k CP) DB 1 + b k ] k CP + b k CP b (CA D + k CP) + DB 1 + b k k CP + b CP (k k ) b (CA D) + DB 1 + b k k CP b (CA D) + DB 1 + b k k [CP b (CA D + k CP) DB 1 + b k ] k CP + b k CP b (CA D + k CP) + DB 1 + b k k CP + b CP (k k ) b (CA D) + DB 1 + b k CP b (CA D) + DB k 1 + b k Inserting formulas vi, vii and viii into formula i, then substituting formula and regrouping terms: ix. Amortization + Accrual Am Am + Ac B (k k ) [CP b (CA D) + DB] + b (CA D + k CP) DB 1 + b k k [CP b (CA D) + DB] + b (CA D + k CP) DB 1 + b k x. Amortization + Accrual (k + b k ) CP + [b (CA D) DB] (1 k) 1 + b k Where there is no front-end load, we can simplify by setting deferred revenue and the revenue amortization rate to zero in formula x: xi. Amortization + Accrual k CP + (b CA DB) (1 k) Where there is no requirement for an additional SOP reserve, we can simplify by setting deferrable benefits and the benefit ratio to zero in formula x: xii. Amortization + Accrual k CP 16 P a g e

17 Derivation of Amortization and Accrual by Source of Profit Formula x, though precise, does little to really help us understand the effects of cash profit on earnings. To enhance understanding, we want to calculate amortization for each component of cash profit. Cash activity is defined broadly to include anything representing actual transactions between the insurer and relevant counterparties. It excludes changes in DAC, UL and SOP reserves but may include interest spread on some of those balances if that is included in gross profit 12 and may include other accruals (such as changes in claims ICOS and IBN). Substituting formula O into formula x and then combining multiples of the cash profit components: xiii. Amortization + Accrual (k + b k ) [(CA D) DB CC] + [b (CA D) DB] (1 k) 1 + b k (CA D) [k + b k + b (1 k)] DB [k + b k + 1 k] CC (k + b k ) 1 + b k (CA D) [k + b k b + b k ] DB [1 + b k ] CC (k + b k ) 1 + b k k + b k b + b k (CA D) 1 + b k 1 + b k DB k + b k b + b k (CA D) 1 + b k DB CC k + b k CC 1 + b k 1 + b k k + b k 1 + b k In formula xiii, we see three principal terms, each a multiple of one component of cash profit. Those multiples are the amortization and accrual rates, introduced in Simply Unlocking, applicable to the respective components. xiv. Amortization ate for Non Deferred Cash Assessments k + b k b + b k k CA 1 + b k xv. xvi. Amortization ate for Cash Deferrable k CD 1 Amortization ate for Other Cash Costs k CC k + b k 1 + b k 12 Under some interpretations of SOP 03-1, gross profit and assessments include interest spread on some intangible balances. 17 P a g e

18 Alternative Derivation of Amortization and Accrual Amortization and accrual formulas for different components of cash profit can also be constructed from separate derivations for each type of asset and liability (DAC, UL, and SOP reserve). In the following sections, we derive amortization rates for each balance and each component of cash profit. To help visualize the common alignments among the three types of amortization and accrual, these formulas are structured in tabular form. We have three rows DAC amortization, UL amortization, and SOP reserve accrual. The columns represent sequential terms for each amortization and accrual formula. In each derivation, we begin in column (a) with the basic accrual the applicable rate times the cash component. Column (b) has any additional accrual caused by the column (a) accruals negative expense and revenue amortization on the reserve accrual and positive reserve accrual on the revenue amortization. Column (c) represents the additional accrual caused by the column (b) accruals, and so on, ad infinitum. 18 P a g e

19 Non-Deferred Cash Assessments (A) (a) Basic Accrual (b) Add l Accrual on (a) (c) Add l Accrual on (b) (d) Add l Accrual on (c) (e) Add l Accrual on (d) Am CA k (CA D) k b (CA D) k b k (CA D) +k b 2 k (CA D) +k b 2 (k ) 2 (CA D) Am CA k (CA D) k b (CA D) k b k (CA D) +k b 2 k (CA D) +k b 2 (k ) 2 (CA D) Ac B CA b (CA D) +b k (CA D) b k b (CA D) b 2 (k ) 2 (CA D) +b 3 (k ) 2 (CA D) Seeing common factors among the columns (the product of the cash profit component and the respective amortization and accrual rates), we factor them out of the infinite sums (and extend the formulas for a few more terms). (B) (a) (b) (c) (d) (e) (f) (g) (h) Am CA (CA D) k [ 1 b b k +b 2 k +b 2 (k ) 2 b 3 (k ) 2 b 3 (k ) 3 +b 4 (k ) 3 ] Am CA (CA D) k [ 1 b b k +b 2 k +b 2 (k ) 2 b 3 (k ) 2 b 3 (k ) 3 +b 4 (k ) 3 ] Ac B CA (CA D) b [ 1 +k b k b (k ) 2 +b 2 (k ) 2 +b 2 (k ) 3 b 3 (k ) 3 b 3 (k ) 4 ] In this table, we can see some commonalities among successive pairs of terms. Added together, each pair in each amortization formula becomes the product of (1 b) and some power (from zero to infinity) of ( b k ). Added together, each pair of the reserve accrual formula becomes the product of (1 + k ) and some power (from zero to infinity) of ( b k ). This combining of terms is shown in Table (C). (C) (a+b) (c+d) (e+f) (g+h) Am CA (CA D) k [ (1 b) +(1 b) ( b k ) +(1 b) ( b k ) 2 +(1 b) ( b k ) 3 ] Am CA (CA D) k [ (1 b) +(1 b) ( b k ) +(1 b) ( b k ) 2 +(1 b) ( b k ) 3 ] Ac B CA (CA D) b [ (1 + k ) +(1 + k ) ( b k ) +(1 + k ) ( b k ) 2 +(1 + k ) ( b k ) 3 ] To move towards solving the infinite sums, we multiply everything in Table (C) by ( b k ). 19 P a g e

20 (D) Am CA ( b k ) (CA D) k [ (1 b) ( b k ) +(1 b) ( b k ) 2 +(1 b) ( b k ) 3 +(1 b) ( b k ) 4 ] Am ( b k ) (CA D) k [ (1 b) ( b k ) +(1 b) ( b k ) 2 +(1 b) ( b k ) 3 +(1 b) ( b k ) 4 ] Ac B ( b k ) (CA D) b [ (1 + k ) ( b k ) +(1 + k ) ( b k ) 2 +(1 + k ) ( b k ) 3 +(1 + k ) ( b k ) 4 ] Next, we subtract Table (D) from Table (C). Noting that, except for (a+b), every term in (C) has an identical term in (D), all that s left after the subtraction are the common factors and that first term in column (a+b) of Table (C). ()(C-D) Am CA (1 + b k ) (CA D) k (1 b) Am CA (1 + b k ) (CA D) k (1 b) Ac B CA (1 + b k ) (CA D) b (1 + k ) Finally, we divide by (1 + b k ) to get net amortization rates multiplied by non-deferred cash assessments. The net amortization rate for DAC, UL and SOP reserves combined can then be determined by subtracting the revenue amortization rate from the expense amortization rate and then adding the reserve accrual rate. Am CA (F) Am CA Am CA (CA D) k (1 b) 1 + b k Am CA (CA D) k (1 b) 1 + b k Ac B CA (CA D) b (1 + k ) 1 + b k + Ac B CA (CA D) (k k ) (1 b) + b (1 + k ) k + b k b + b k 1 + b k (CA D) 1 + b k This is the same as the first term in formula xiii. 20 P a g e

21 Other Cash Costs We now use the same approach to find net amortization rates applicable to other cash costs (non-deferrable benefits and expenses). (G) (a) Basic Accrual (b) Add l Accrual on (a) (c) Add l Accrual on (b) (d) Add l Accrual on (c) (e) Add l Accrual on (d) (f) Add l Accrual on (e) Am CC k CC +k b k CC k b 2 (k ) 2 CC Am CC k CC +k b k CC k b 2 (k ) 2 CC Ac B CC 0 b k CC +b 2 (k ) 2 CC b 3 (k ) 3 CC Seeing common factors among the columns, we factor them out of the infinite sums. (H) emove Commons (a) (b) (c) (d) (e) (f) (g) (h) Am CC CC k [ 1 b k +b 2 (k ) 2 b 3 (k ) 3 ] Am CC CC k [ 1 b k +b 2 (k ) 2 b 3 (k ) 3 ] Ac B CC CC b k [ 0 1 b k +b 2 (k ) 2 b 3 (k ) 3 ] Dropping all of the zeros and empty columns, every term is some power (from zero to infinity) of ( b k ). (I) (a+b) (c+d) (e+f) (g+h) Am CC CC k [ 1 +( b k ) +( b k ) 2 +( b k ) 3 ] Am CC CC k [ 1 +( b k ) +( b k ) 2 +( b k ) 3 ] Ac B CC CC b k [ 1 +( b k ) +( b k ) 2 +( b k ) 3 ] To move towards solving the infinite sums, we multiply everything in Table (I) by ( b k ). 21 P a g e

22 (J) Am CC ( b k ) CC k [ ( b k ) +( b k ) 2 +( b k ) 3 +( b k ) 4 ] Am CC ( b k ) CC k [ ( b k ) +( b k ) 2 +( b k ) 3 +( b k ) 4 ] Ac B CC ( b k ) CC b k [ ( b k ) +( b k ) 2 +( b k ) 3 +( b k ) 4 ] Next, we subtract Table (J) from Table (I). Noting that, except for (a+b)1, every term in (I) has an identical term in (J). All that s left on the right in Table (K) are the common factors. (K)(I-J) Am CC (1 + b k ) CC k Am CC (1 + b k ) CC k Ac B CC (1 + b k ) CC b k Finally, we divide by (1 + b k ) to get net amortization rates multiplied by other cash costs. The net amortization rate for DAC, UL and SOP reserves combined can then be determined by subtracting revenue amortization from expense amortization rte and then adding reserve accrual. (L) Am CC Am CC k CC 1 + b k k CC 1 + b k Am CC Am CC B b k CC 1 + b k Ac CC + Ac B CC CC k k + b k k + b k 1 + b k CC 1 + b k This is the same as the third term in formula xiii. 22 P a g e

23 Deferrable (eserved) Benefits We now use the same approach to find net amortization rates applicable to deferrable benefits. (M) (a) Basic Accrual (b) Add l Accrual on (a) (c) Add l Accrual on (b) (d) Add l Accrual on (c) (e) Add l Accrual on (d) (f) Add l Accrual on (e) Am DB k DB +k DB +k b k DB k b k DB k b 2 (k ) 2 DB +k b 2 (k ) 2 DB Am DB k DB +k DB +k b k DB k b k DB k b 2 (k ) 2 DB +k b 2 (k ) 2 DB Ac B DB DB b k DB +b k DB +b 2 (k ) 2 DB b 2 (k ) 2 DB +b 3 (k ) 3 DB Seeing common factors among the columns, we factor them out of the infinite sums. (N) emove Commons (a) (b) (c) (d) (e) (f) (g) (h) Am DB DB k [ 1 1 b k +b k +b 2 (k ) 2 b 2 (k ) 2 b 3 (k ) 3 +b 3 (k ) 3 ] Am DB DB k [ 1 1 b k +b k +b 2 (k ) 2 b 2 (k ) 2 b 3 (k ) 3 +b 3 (k ) 3 ] Ac B DB DB [ 1 +b k b k b 2 (k ) 2 +b 2 (k ) 2 +b 2 (k ) 2 b 3 (k ) 3 b 3 (k ) 4 ] In this table, we can see some commonalities among successive pairs of terms. Added together, each pair in the amortization formulas are zero. Added together, each pair of the accrual formula becomes the product of (1 + b k ) and some power (from zero to infinity) of ( b k ). This combining of terms is shown in Table (O). (O) (a+b) (c+d) (e+f) (g+h) Am DB DB k [ 0 Am DB DB k [ 0 ] ] Ac B DB DB [ 1 + b k +( b k ) (1 + b k ) +( b k ) 2 (1 + b k ) +( b k ) 3 (1 + b k ) ] To move towards solving the infinite sums, we multiply everything in Table (O) by ( b k ). Since the expense and revenue amortization has dropped to zero, that s all that we have left in the first two formulas. 23 P a g e

24 (P) Am DB ( b k ) 0 Am DB ( b k ) 0 ] ] Ac B DB ( b k ) DB [ ( b k ) (1 + b k ) +( b k ) 2 (1 + b k ) +( b k ) 3 (1 + b k ) ] Next, we subtract Table (P) from Table (O). Noting that, except for (a+b), every term in (O) has an identical term in (P), all that s left on the right in Table (Q) is the common factor and that first term inside brackets of Table (O). (Q)(O-P) Am DB (1 + b k ) 0 Am DB (1 + b k ) 0 Ac B DB (1 + b k ) DB (1 + b k ) Finally, we divide by (1 + b k ) to get net amortization rates multiplied by deferrable benefits. The net amortization rate for DAC, UL and SOP reserves combined can then be determined by subtracting the revenue amortization rate from the expense amortization rate and then adding the reserve accrual rate. () Am DB 0 Am DB 0 Am DB Am DB B 1 + b k DB 1 + b k Ac DB + Ac B 1 + b k DB DB DB b k This is the same as the middle term in formula xiii. 24 P a g e

25 Cash Profit Combining the Pieces Bringing together the net amortization rates for each component of cash profit, we find the formulas for determining each component of amortization and accrual. In table (S), we separate the components from the applicable rates. The three drivers of profit are shown at the top of each column, above their corresponding amortization and accrual rates. (S) (CA D) DB CC xpense amortization, Am : evenue amortization, Am : eserve accrual, Ac B : Am Am + Ac B : k (1 b) 1 + b k 0 k (1 b) 1 + b k 0 b (1 + k ) 1 + b k 1 k + b k b + b k 1 + b k 1 k 1 + b k k 1 + b k b k 1 + b k k + b k 1 + b k In the last row of table (S), we see the same results as the earlier derivation, in formulas xiv, xv and xvi. 25 P a g e