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BRIDGING THE GAP BETWEEN ROE AND IRR Robert W. Beal* ABSTRACT Internal rate of return (IRR) measures the level annual return over the life of an investment, whereas return on equity (ROE) measures the return over each accounting period. This paper develops the relationships between IRR and ROE by presenting and proving four algebraic theorems involving IRR and ROE. These theorems are developed using generic investment terminology that does not rely on any speciﬁc accounting basis. The relationships are then expressed using U.S. statutory and GAAP terminology. The paper demonstrates that IRR is not just a statutory concept and ROE is not just a GAAP concept. Financial projections for a hypothetical insurance product illustrate these relationships. 1. INTRODUCTION ships (in the form of theorems) between IRR and IRR is a proﬁt objective used to determine the ROE using generic investment terminology that level annual return over the life of an investment do not reference either statutory or GAAP ac- of capital. ROE measures the annual return of the counting. Section 3, Application Using Statutory investment over each accounting period. The and GAAP Accounting, applies U.S. statutory and term ROE, as used in this paper, has also been GAAP accounting terminology to the four theo- referred to as return on capital (ROC), return on rems from Section 2 and deﬁnes statutory and total capital (ROTC) and return on investment GAAP ROE, relating both of these concepts (ROI) in other papers mentioned in this paper. to IRR. Section 4 provides the concluding com- IRR has been viewed as a statutory concept; that ments. is, it is the expected level annual return over the life Appendix A presents proofs of the four theo- of an investment of statutory capital. On the other rems from Section 2. Appendix B illustrates the hand, ROE has been viewed as a generally accepted relationships from Section 3 using asset share accounting principles (GAAP) concept, (that is, the projections for a hypothetical individual disabil- return on GAAP equity over a speciﬁc accounting ity income product. period). The concept of IRR is sometimes dismissed as irrelevant when proﬁt objectives are framed in 2. ALGEBRAIC RELATIONSHIPS BETWEEN GAAP terms. In this situation, certain key relation- ROE AND IRR ships between IRR and ROE are ignored. This paper develops these relationships. This section presents four algebraic relationships Sondergeld (1975, 1982) discusses certain fun- between ROE and IRR in the form of theorems, damental characteristics of IRR and ROE. Lom- using generalized terminology to represent an- bardi (1986) and Smith (1988) further develop nual capital ﬂow, annual equity, and annual re- these concepts. All four of these papers discuss turns. Appendix A provides proofs of these theo- these concepts using statutory and GAAP termi- rems. In Section 3, these relationships are nology. expressed using U.S. statutory and GAAP termi- Section 2, Algebraic Relationships Between nology. ROE and IRR, presents four algebraic relation- 2.1 Deﬁnitions Capital Flow *Robert W. Beal, F.S.A. is a Consulting Actuary with Milliman & Robertson, Inc., 121 Middle St., Suite 401, Portland, ME 04101, Let CFt, for t 1, . . . , N, represent the annual e-mail: bob.beal@milliman.com capital ﬂow over N years for a speciﬁc investment. 1 2 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 4, NUMBER 4 CF0, the initial capital investment, is assumed to 2.2 Theorems be negative. CFt for t 0 may be either positive or The ﬁrst theorem states that equation (1) holds negative, where a positive value represents a re- when the IRR is replaced by the vector ROEt, for turn of interest and principal to the investor and t 1, . . . , N. Both Sondergeld (1982) and Lom- a negative value represents additional capital in- bardi (1986) discuss this relationship. vestment. Theorem 1 Equity Let Et, for t 0, . . . , N, be a vector representing N CF t EN the value of the equity at time t. The term equity, 0 CF 0 t N as used in this paper, is typically the book value t 1 1 ROE s 1 ROE s s 1 s 1 as deﬁned by some accounting standard. This (2) discussion does not limit the calculation of equity to any speciﬁc accounting standard but assumes The next theorem states that the equity at the the following two requirements are satisﬁed: (1) end of any year is the present value of the future E0 CF0, and (2) Et 0 for t 1, . . . , N 1. capital ﬂow, discounted using the vector of ROEs, Section 3 develops two deﬁnitions of equity that for the remaining years of the investment. Al- satisfy both requirements using statutory and though equity can be deﬁned using speciﬁc ac- GAAP concepts. EN is often set equal to zero. counting terminology, as it is in Section 3, equity However, a non-zero EN may represent the book (provided it satisﬁes the two criteria described in value or residual value of the investment at the the previous deﬁnition of equity) must satisfy end of the investment period. equation (3). Theorem 2 Internal Rate of Return Let IRR be a level annual interest rate that solves N CF s EN Equation (1): Et s N s t 1 1 ROE k 1 ROE k N k t 1 k t 1 CF t EN (3) 0 t N (1) t 0 1 IRR 1 IRR Unlike Theorem 1, which states that the level Although the theorems provided below hold annual IRR can be replaced by the annual ROEs true whether or not there are multiple solutions in Equation (1), Theorem 3 below states that the to Equation (1), for discussion purposes we will annual ROEs cannot be replaced by the IRR in assume that there is only one meaningful solu- equation (3) unless the ROEs are level and equal tion. Promislow (1980) provides a much deeper to the IRR. treatment of yield rates and investigates multiple- Theorem 3 valued and nonexistent yields. N Annual Return CF s EN Et s t N t Let ARt, for t 1, . . . , N represent the annual s t 1 1 IRR 1 IRR return in year t, that is, the investor’s annual proﬁts in year t, less any decrease in equity. By for t 0, . . . , N (4) deﬁnition, ARt CFt (Et 1 Et). if, and only if, IRR ROE t for t 1, . . . , N. Annual Return on Equity Unless the equity values are equal to the present The annual return on equity, ROEt, for t 1, . . . , value of future capital ﬂow, where the level an- N, is deﬁned as nual discount rate is equal to the IRR, then the annual ROEs will be nonlevel. Sondergeld (1975) AR t CF t Et 1 Et explores this idea by developing the internal rate ROE t Et 1 Et 1 of return method of accounting (IRRMA) by BRIDGING THE GAP BETWEEN ROE AND IRR 3 which the expected annual earnings related to a At the end of each year, the statutory after-tax closed block of business emerge as a uniform book proﬁt, less the increase in required capital, percentage of IRRMA surplus, where the uniform represents a ﬂow of capital between the corporate percentage is the IRR. surplus and the line’s surplus. If this amount is Theorem 4 shows that the IRR is equal to the negative, then the amount represents additional capital infusion into the line. If this amount is ratio of the present value of the annual returns to positive, then capital equal to this amount ﬂows the present value of the equity, where the dis- from the line’s surplus to corporate surplus. count rate is the IRR. The following terminology will be used: Theorem 4 N number of years that the cohort of poli- Let PV( AR) and PV(E) be deﬁned as follows: cies as a whole persists; N Pt premium income in year t; AR t NIIt net investment income in year t; PV AR t t 1 1 IRR Bt beneﬁts paid in year t; Expt expenses incurred in year t; N Et 1 ResS statutory reserves and liabilities at the t and PV E . end of year t; t 1 1 IRR t G Rest GAAP reserves and liabilities at the end Then of year t; DACt unamortized deferred acquisition costs PV AR at the end of year t; IRR . (5) FIT tS statutory federal income taxes incurred PV E in year t; Using Equation (5), the IRR may be solved re- G FIT t GAAP federal income taxes incurred in iteratively given that the annual returns and eq- year t; uity values are known for each year of the invest- BPt S statutory after-tax book proﬁt in year t ment. In addition, based on Equation (5), the IRR 0; can be viewed as the weighted average annual Pt NIIt Bt Expt (Rest S return divided by the weighted average equity at S Rest 1) FITt S the beginning of each year, where the weights are BPtG GAAP after-tax book proﬁt in year t 0; deﬁned as 1/(1 IRR)t for t 1, . . . , N. Pt NIIt Bt Expt (Rest G G Rest 1) (DACt DACt 1) FITG t RCt required capital at the end of year t; 3. APPLICATION USING STATUTORY AND CFt capital ﬂow at time t; GAAP ACCOUNTING RC0 at t 0; The relationships between IRR and ROE dis- S BP t (RCt RCt 1) for t 0; cussed in Section 2 are presented below using DTRt GAAP deferred tax reserves at the end of both statutory and GAAP accounting terminol- year t; ogy. In other words, statutory and GAAP equity, G S FIT t FIT t DTRt DTRt 1 annual returns, and ROE are deﬁned and applied G Since DTRt DTRt 1 FIT t FIT S, then t to the theorems. Although the GAAP expressions DTRt can be deﬁned as follows: are more widely utilized, illustrating both sets of terms demonstrates that the relationships in Sec- t tion 2 are not limited to GAAP accounting only. DTR t G FIT s S FIT s The formulas assume the following scenario: s 1 A line of business of an insurance company issues Statutory and GAAP equity are deﬁned as fol- a cohort of policies. At issue (t 0), the company lows: S transfers statutory capital from the corporate sur- Et Statutory equity at time t plus to the line’s surplus equal to the required RCt S capital necessary to cover the risk in the ﬁrst year. E0 RC0 CFt 4 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 4, NUMBER 4 G Et GAAP equity at the end of year t N S S G BP s RC s RC s 1 RCt Rest DACt Rest DTRt RC t s (11) G S E0 RC0. s t 1 1 ROE k In this example, EN is equal to zero for both k t 1 statutory and GAAP terms, since no policies are N S BP s RC s RC s 1 assumed to persist beyond N years. G Et s (12) The following demonstrates that the statutory s t 1 1 G ROE k and GAAP annual returns are equal to the statu- k t 1 tory and GAAP book proﬁts: Theorem 3 is not true when statutory and S S S ARt CFt (Et 1 Et ) GAAP values are used, since both statutory and S BPt (RCt RCt 1) (RCt 1 RCt) GAAP equity are not equal to the present value of S BPt . capital ﬂow where the discount rate is equal to G G G ARt CFt (Et 1 Et ) the IRR. This is illustrated in the hypothetical S S BPt (RCt RCt 1) (RCt 1 Rest 1 example in Appendix B. G DACt 1 Rest 1 DTRt 1) (RCt Re Finally, Equation (5) in Theorem 4 can be ex- S G st DACt Rest DTRt) pressed in statutory and GAAP terms as follows: G G Pt NIIt Bt Expt (Rest Rest 1) G N S (DACt DACt 1) FITt BP t G BPt . 1 IRR t t 1 As a result of the above deﬁnitions, both statutory IRR (13) and GAAP ROEs can be calculated as follows: N RC t 1 t S BP t t 1 1 IRR S ROE t for t 0; RC t 1 N G BP t G t G BP t 1 IRR t 1 ROE t G for t 0. IRR (14) Et 1 N G Et 1 Equation (1) from Section 2, which deﬁnes IRR, 1 IRR t t 1 can now be written as follows: Equation (13) provides an alternative to equa- N BP S t RC t RC t 1 tion (1) for deriving the IRR using statutory book 0 proﬁts and required capital. Equation (14) shows t 0 1 IRR t that IRR can be determined using GAAP book S proﬁts and equity values. These two equations where BP 0 0 and RC 1 0 (8) demonstrate that IRR is both a statutory and GAAP concept. Equation (2) from Theorem 1 can be expressed Appendix B illustrates the results in Section 3 in terms of statutory or GAAP ROE: using a hypothetical individual disability income N S product. BP t RC t RC t 1 0 t (9) S t 0 1 ROE s 4. CONCLUSION s 1 For an insurance company, new business involves N BP S t RC t RC t 1 the investment of statutory capital. GAAP equity 0 t (10) is the book value determined by GAAP account- G t 0 1 ROE s ing standards and represents a value placed on s 1 the outstanding investment. The IRR measures Similarly Equation (3) in Theorem 2 can be the level annual return over the life of the invest- expressed in statutory and GAAP terms as fol- ment. ROE measures the return over each ac- lows: counting period, and its deﬁnition is subject to BRIDGING THE GAP BETWEEN ROE AND IRR 5 speciﬁc accounting rules. ROE is typically non- value of the equity at the end of year t, satisfying level and might be only a rough approximation of three conditions: (1) E0 CF0, (2) Et 0 for t the IRR. Only in very speciﬁc deﬁnitions of equity 1, . . . , N 1, and (3) EN 0. will the expected annual ROE equal the expected IRR. Although GAAP parameters might be used The internal rate IRR satisﬁes the following that will generate expected ROEs that are rela- equation tively level each year, management should appre- ciate the differences between GAAP ROE and IRR N CF t EN when trying to understand the ﬁnancial results. 0 t N (A.1) t 0 1 IRR 1 IRR Although IRR and ROE are not normally ex- pected to be equal, this paper clariﬁes their close interrelationship, and in doing so, clariﬁes the Theorem 1 close interrelationship between statutory and GAAP accounting. The IRR can be derived using N CF t EN statutory values or GAAP values. The present 0 CF 0 t N value of the capital ﬂow is zero whether dis- t 1 1 ROE s 1 ROE s s 1 s 1 counted using the IRR or the vector of annual (A.2) ROEs. As a result, it is incorrect to state that IRR is only a statutory concept or ROE is only a GAAP PROOF concept. Proof is by mathematical induction. Assume N 1. By deﬁnition of ROE, REFERENCES CF 1 E0 E1 ROE 1 . ANDERSON, J.C.H. 1959. “Gross Premium Calculation and Proﬁt E0 Measurement for Nonparticipating Insurance,” TSA 11: 357– 420. Solving for E0, we arrive at the following equa- LOMBARDI, L.J. 1988. “Relationships Between Statutory and Gen- tion erally Accepted Accounting Principles (GAAP),” TSA 40: 485–508. CF 1 E1 PROMISLOW, S.D. 1980. “A New Approach to the Theory of Inter- E0 (A.3) 1 ROE 1 1 ROE 1 est,” TSA 32: 53–118. SMITH, B. 1987. “Pricing in a Return-on-Equity Environment,” Since E0 CFO0, we can derive Equation TSA 39:257–93. SONDERGELD, D.R. 1975. “Earnings and the Internal Rate of Re- (A.4), which is Equation (A.2) for N 1. turn Measurement of Proﬁt,” TSA 26: 617–36. SONDERGELD, D.R. 1982. “Proﬁtability as a Return on Total Cap- CF 1 E1 0 CF 0 (A.4) ital,” TSA 34: 415–33. 1 ROE 1 1 ROE 1 Assume that equation (A.3) is true for all years APPENDIX A through N 1, that is, Appendix A provides proofs of the four theorems k 1 presented in Section 2. All theorems assume the CF t Ek 1 0 CF 0 t k 1 following deﬁnitions and conditions: t 1 1 ROE s 1 ROE s s 1 s 1 CFt, for t 1, . . . , N, is a vector representing for k 1, . . . , N 1 (A.5) the annual capital ﬂow over N years for a speciﬁc investment. CF0, the initial capital investment, is By deﬁnition, assumed to be negative. CFt for t 0 may be either positive or negative, where a positive value CF N EN 1 EN represents a return of interest and principal to the ROE N (A.6) investor and a negative value represents further EN 1 capital infusion. Et, for t 0, . . . , N, is a vector representing the Solving for EN 1, 6 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 4, NUMBER 4 CF N EN Theorem 3 EN 1 , (A.7) 1 ROE N 1 ROE N N CF s EN Et s t N t , s t 1 1 IRR 1 IRR and substituting Equation (A.7) into Equation (A.5) for k N 1, we arrive at equation for t 0, . . . , N (A.9) (A.2). if, and only if, IRR ROE t for t 1, . . . , N. Theorem 2 Proof N CF s EN If IRR ROEt for t 1, . . . , N, then by Theorem Et s N 2, Equation (A.9) is true. Assume the converse; s t 1 1 ROE k 1 ROE k that is, Equation (A.9) is true. By deﬁnition, k t 1 k t 1 (A.8) CF t Et 1 Et ROE t . (A.10) PROOF Et 1 This theorem can be proved in much the same Using Theorem 2, we get the following Equa- way as Theorem 1. tion, as shown in Equation (A.10): N CFs EN N CFs EN CFt s t 1 N t 1 s t N t s t 1 IRR 1 IRR s t 1 1 IRR 1 IRR ROEt . (A.11) Et 1 Then, 1 N CFs EN N CFs EN CFt 1 IRR s t 1 IRR s t 1 IRR N t s t 1 1 IRR s t 1 IRR N t ROEt . (A.12) Et 1 1 IRR Since 1 , Equation (A.12) 1 IRR 1 IRR becomes Equation (A.13). IRR N CFs EN N CFs EN CFt 1 s t N t s t N t 1 IRR s t 1 IRR 1 IRR s t 1 1 IRR 1 IRR ROEt . Et 1 (A.13) By rearranging the terms in Equation (A.13), we arrive at Equation (A.14). BRIDGING THE GAP BETWEEN ROE AND IRR 7 IRR N CF s EN CF t CF t s t N t 1 IRR s t 1 IRR 1 IRR ROE t . (A.14) Et 1 Equation (A.14) simpliﬁes to Equation (A.15): N N Et 1 CFt R t t ROE t t 1 1 IRR t 1 1 IRR N CF s EN N IRR s t 1 N t 1 1 Et 1 s t 1 IRR 1 IRR t 1 . 1 IRR 1 IRR Et 1 t 1 (A.15) N Et . (A.18) Because it is assumed that Et 1 t 1 1 IRR t N CFs EN ¥s , then ROEt t 1 IRR s t 1 1 IRR N t 1 1 IRR IRR. Because 1 , Equation (A.18) 1 IRR 1 IRR becomes Equation (A.19): Theorem 4 Let PV( AR) and PV(E) be deﬁned as follows: N Et 1 N CF t R t t N 1 IRR 1 IRR AR t t 1 t 1 PV AR t t 1 1 IRR N IRR Et 1 1 t 1 N Et 1 1 IRR t 1 1 IRR and PV E . t 1 1 IRR t N Et t (A.19) PV AR 1 IRR Then, IRR . t 1 PV E We recognize from Equation (A.1) that PROOF N CFt Et 1 Et N CF t EN PV AR 1 IRR t ¥ t CF 0 N , t 1 t 1 1 IRR 1 IRR Let R , then R . PV E N Et 1 1 IRR t and we rearrange the terms in Equation (A.19) to t 1 get Equation (A.20): (A.16) N Et 1 EN Rearranging Equation (A.16), we get R t CF0 N t 1 1 IRR 1 IRR N N Et 1 CF t EN R t t E0 t 1 1 IRR t 1 1 IRR 1 IRR N N N N Et 1 Et Et 1 t . (A.17) IRR . (A.20) t 1 1 IRR t 1 1 IRR t t 1 1 IRR t 8 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 4, NUMBER 4 Because CF0 E0, we can determine from proﬁt divided by required capital at the beginning Equation (A.20) that R IRR. of the year. Table 2 shows the projected GAAP asset share. As in Section 3, the GAAP ROE for each year is APPENDIX B deﬁned as the after-tax GAAP book proﬁt divided For the purpose of illustrating the theorems and by GAAP equity at the beginning of the year. results in Sections 2 and 3, a hypothetical indi- Table 3 demonstrates that the sum of the dis- vidual disability product is shown here. It is based counted capital ﬂow is zero whether the discount on an insured age 45 with a 90-day elimination rates are based on the IRR, the GAAP ROEs, or period and a to-65 beneﬁt period (with a 24- the statutory ROEs. month minimum beneﬁt period). Because these projections are just illustrative, The reader can use Table 3 to calculate the it is not necessary to describe all but a few actu- GAAP equity and statutory equity (i.e., required arial assumptions: capital) at the end of any given year for each of 1. The pre-tax interest rate on assets, net of in- the three sets of discount rates by summing the vestment expenses and defaults, is a level 7% future discounted capital ﬂows and dividing the per year. sum by the discount factor for that year. For 2. The ratio of the present value of paid beneﬁts example, GAAP equity at the end of year 15 is to present value of premiums over the life of 5.6161 divided by 0.16881, which is 33.27. This the policy (that is, 22 years), discounted at 7%, agrees with GAAP equity at the end of year 15 is 51.4%. from Table 2. Similarly, required capital (i.e., 3. The federal income tax rate is 35%, and the statutory equity) at the end of year 15 is 4.1554 DAC tax is incurred in every year; except for divided by 0.16126, which is 25.77. This agrees illustration purposes, the DAC tax is fully am- with required capital at the end of year 15 from ortized at the end of year 20. Table 1. 4. The IRR is 12.53%. Table 4 demonstrates that the ratio of the 5. All noncommission acquisition expenses are present value of after-tax GAAP book proﬁts di- deferred for GAAP purposes. vided by the present value of GAAP equity (as of 6. The GAAP beneﬁt reserves are based on the the beginning of each year but discounted from expected claim costs with a 10% margin for the end of each year) is equal to the IRR when the adverse deviation and 100 basis points for present values are based on a discount rate equal margin in the valuation interest rate. to the IRR. Similarly, the present value of after- 7. The GAAP expense reserves take into account tax statutory book proﬁts divided by the present the ongoing claim management expenses and value of required capital (as of the beginning of inﬂation on the per policy maintenance ex- each year but discounted from the end of each penses. year) is equal to the IRR when the present values 8. The GAAP expenses in the GAAP income are based on a discount rate equal to the IRR. statement are equal to commissions and other expenses plus the increase in the GAAP ex- Discussions on this paper can be submitted until pense reserves. April 1, 2001. The author reserves the right to reply to Table 1 shows the projected statutory asset any discussion. Please see the Submission Guidelines share. As in Section 3, the statutory ROE for each for Authors on the inside back cover for instructions year is deﬁned as the after-tax statutory book on the submission of discussions. BRIDGING THE Table 1 GAP BETWEEN ROE Projection of Statutory Results AND Change Pre-tax Statutory After-tax After-tax IRR Net in Statutory Federal Statutory Statutory Policy Premium Investment Paid Statutory Commission Book Income Book Statutory Tax Required Capital Book Statutory Year Income Income Beneﬁts Reserves & Expenses Proﬁt Tax Proﬁt Reserves Reserves Capital Flow Proﬁt ROE 0 29.89 (29.89) 1 49.81 2.47 1.71 15.81 46.98 (12.23) (2.46) (9.77) 15.81 13.95 27.63 (7.51) 9.77 32.68% 2 43.19 5.43 3.91 15.70 10.26 18.75 8.05 10.70 31.51 27.86 26.49 11.83 10.70 38.72 3 38.46 6.10 5.75 21.84 9.40 7.56 3.73 3.83 53.35 48.39 26.50 3.83 3.83 14.47 4 35.42 7.38 7.59 19.35 8.87 6.99 3.25 3.74 72.69 66.72 26.92 3.32 3.74 14.11 5 33.18 8.57 9.35 16.30 8.46 7.63 3.21 4.43 88.99 82.35 27.44 3.90 4.43 16.44 6 31.38 9.58 11.10 13.88 8.16 7.82 3.04 4.78 102.88 95.87 28.04 4.18 4.78 17.43 7 29.94 10.44 12.97 11.21 7.92 8.27 3.00 5.28 114.08 106.97 28.54 4.79 5.28 18.83 8 28.59 11.11 14.83 8.58 7.71 8.59 2.92 5.67 122.67 115.67 28.90 5.30 5.67 19.85 9 27.35 11.60 16.71 6.00 7.51 8.73 2.80 5.93 128.67 122.01 29.22 5.62 5.93 20.52 10 26.35 11.91 18.63 3.44 7.37 8.82 2.69 6.13 132.10 125.93 29.29 6.06 6.13 20.98 11 25.31 12.07 20.61 0.72 6.57 9.48 2.87 6.61 132.82 127.23 29.04 6.85 6.61 22.56 12 24.16 11.96 22.68 1.46 6.42 5.55 2.52 3.03 134.29 126.47 28.67 3.39 3.03 10.42 13 23.11 11.89 24.90 (0.15) 6.29 3.96 1.57 2.39 134.14 125.34 28.07 3.00 2.39 8.33 14 22.12 11.70 27.24 (3.26) 6.16 3.68 0.81 2.86 130.89 123.06 27.11 3.82 2.86 10.20 15 21.12 11.26 29.68 (7.04) 6.01 3.73 0.80 2.93 123.85 117.20 25.77 4.28 2.93 10.82 16 20.17 10.53 32.18 (12.18) 5.82 4.88 1.18 3.71 111.67 106.38 23.91 5.57 3.71 14.38 17 19.33 9.41 34.64 (17.33) 5.65 5.77 1.50 4.27 94.34 90.49 21.48 6.70 4.27 17.85 18 18.48 7.89 37.18 (23.47) 5.47 7.19 2.05 5.15 70.86 68.46 18.36 8.26 5.15 23.97 19 17.63 5.88 39.83 (28.02) 5.37 6.34 1.86 4.48 42.85 41.66 14.82 8.02 4.48 24.37 20 16.89 3.53 42.42 (31.77) 5.33 4.44 1.35 3.09 11.07 10.74 1.25 16.66 3.09 20.84 21 0.00 0.58 8.07 (7.94) 0.19 0.26 (2.54) 2.80 3.13 3.05 0.35 3.70 2.80 224.92 22 0.00 0.13 3.13 (3.13) 0.07 0.07 0.02 0.05 0.00 0.00 0.00 0.40 0.05 14.27 9 10 Table 2 Projection of GAAP Results Change in Pre-tax GAAP After-tax Net GAAP GAAP Federal GAAP GAAP GAAP Deferred Deferred Policy Premium Investment Paid Beneﬁt GAAP Change Book Income Book Beneﬁt Expense Tax Acquisition GAAP GAAP Year Income Income Beneﬁts Reserves Expense in DAC Proﬁt Tax Proﬁt Reserve Reserve Reserves Costs Equity ROE 0 29.89 1 49.81 2.47 1.71 29.19 47.85 (32.88) 6.40 2.24 4.16 29.19 0.87 4.70 (32.88) 41.55 13.92% 2 43.19 5.43 3.91 23.95 11.01 2.13 7.61 2.66 4.94 53.15 1.62 (0.69) (30.75) 34.67 11.90 3 38.46 6.10 5.75 20.04 10.08 1.81 6.88 2.41 4.47 73.18 2.29 (2.01) (28.93) 35.31 12.89 4 35.42 7.38 7.59 17.18 9.47 1.63 6.93 2.42 4.50 90.36 2.89 (2.83) (27.30) 36.49 12.75 5 33.18 8.57 9.35 14.94 8.97 1.52 6.97 2.44 4.53 105.30 3.40 (3.60) (25.79) 37.12 12.41 6 31.38 9.58 11.10 12.81 8.60 1.44 7.02 2.46 4.56 118.10 3.85 (4.18) (24.35) 37.50 12.29 7 29.94 10.44 12.97 10.68 8.29 1.38 7.05 2.47 4.58 128.78 4.21 (4.71) (22.97) 37.30 12.22 8 28.59 11.11 14.83 8.52 8.00 1.34 7.02 2.46 4.57 137.30 4.50 (5.17) (21.63) 36.57 12.24 9 27.35 11.60 16.71 6.28 7.72 1.30 6.94 2.43 4.51 143.58 4.71 (5.54) (20.33) 35.46 12.33 10 26.35 11.91 18.63 4.02 7.49 1.29 6.83 2.39 4.44 147.60 4.84 (5.84) (19.04) 33.84 12.52 11 25.31 12.07 20.61 1.59 6.61 1.93 6.64 2.33 4.32 149.18 4.88 (6.39) (17.11) 31.30 12.76 12 24.16 11.96 22.68 (1.25) 6.38 1.91 6.39 2.24 4.16 147.93 4.84 (6.68) (15.20) 32.06 13.28 13 23.11 11.89 24.90 (4.30) 6.16 1.90 6.35 2.22 4.12 143.63 4.70 (6.02) (13.30) 33.19 12.86 14 22.12 11.70 27.24 (7.56) 5.92 1.89 6.33 2.22 4.11 136.07 4.46 (4.62) (11.41) 33.49 12.40 15 21.12 11.26 29.68 (11.08) 5.65 1.88 6.24 2.18 4.06 125.00 4.11 (3.23) (9.52) 33.27 12.11 16 20.17 10.53 32.18 (14.66) 5.32 1.88 5.98 2.09 3.88 110.34 3.61 (2.31) (7.64) 31.59 11.68 17 19.33 9.41 34.64 (18.39) 5.02 1.89 5.57 1.95 3.62 91.95 2.97 (1.86) (5.75) 28.51 11.46 18 18.48 7.89 37.18 (22.32) 4.67 1.90 4.94 1.73 3.21 69.63 2.18 (2.18) (3.85) 23.46 11.25 19 17.63 5.88 39.83 (26.94) 4.44 1.91 4.27 1.50 2.78 42.69 1.25 (2.55) (1.94) 18.21 11.84 20 16.89 3.53 42.42 (31.87) 4.29 1.94 3.65 1.28 2.37 10.82 0.20 (2.62) 0.00 3.92 13.01 21 0.00 0.58 8.07 (7.75) 0.04 0.00 0.22 0.08 0.14 3.07 0.06 (0.01) 0.00 0.36 3.57 22 0.00 0.13 3.13 (3.07) 0.02 0.00 0.06 0.02 0.04 0.00 0.00 0.00 0.00 0.00 11.31 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 4, NUMBER 4 BRIDGING THE GAP BETWEEN ROE AND IRR 11 Table 3 Present Value of Capital Flow Using IRR, GAAP ROEs, and Statutory ROEs (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Discounted Discounted Discounted Policy Capital Discount Capital GAAP Discount Capital Statutory Discount Capital Year Flow IRR Factors Flow ROE Factors Flow ROE Factors Flow 0 (29.89) 1.00000 29.8880 1.00000 29.8880 1.00000 29.8880 1 (7.51) 12.534% 0.88862 6.6692 13.919% 0.87782 6.5882 32.681% 1.48546 11.1487 2 11.83 12.534 0.78964 9.3400 11.900 0.78447 9.2788 38.723 1.07081 12.6657 3 3.83 12.534 0.70169 2.6860 12.893 0.69487 2.6600 14.471 0.93544 3.5808 4 3.32 12.534 0.62353 2.0696 12.748 0.61631 2.0457 14.107 0.81979 2.7211 5 3.90 12.534 0.55408 2.1637 12.411 0.54826 2.1410 16.439 0.70405 2.7493 6 4.18 12.534 0.49236 2.0587 12.295 0.48823 2.0414 17.432 0.59954 2.5068 7 4.79 12.534 0.43752 2.0941 12.224 0.43505 2.0823 18.825 0.50455 2.4150 8 5.30 12.534 0.38879 2.0592 12.240 0.38761 2.0529 19.853 0.42098 2.2296 9 5.62 12.534 0.34549 1.9414 12.331 0.34506 1.9390 20.524 0.34929 1.9628 10 6.06 12.534 0.30700 1.8600 12.519 0.30667 1.8580 20.978 0.28872 1.7492 11 6.85 12.534 0.27281 1.8698 12.761 0.27197 1.8640 22.563 0.23557 1.6145 12 3.39 12.534 0.24242 0.8229 13.278 0.24009 0.8149 10.417 0.21335 0.7242 13 3.00 12.534 0.21542 0.6456 12.864 0.21272 0.6375 8.334 0.19693 0.5902 14 3.82 12.534 0.19143 0.7307 12.397 0.18926 0.7224 10.201 0.17870 0.6821 15 4.28 12.534 0.17010 0.7277 12.112 0.16881 0.7222 10.816 0.16126 0.6899 16 5.57 12.534 0.15116 0.8414 11.676 0.15117 0.8414 14.385 0.14098 0.7847 17 6.70 12.534 0.13432 0.8999 11.461 0.13562 0.9086 17.848 0.11963 0.8014 18 8.26 12.534 0.11936 0.9859 11.254 0.12190 1.0069 23.971 0.09650 0.7971 19 8.02 12.534 0.10607 0.8509 11.838 0.10900 0.8744 24.374 0.07759 0.6224 20 16.66 12.534 0.09425 1.5703 13.015 0.09645 1.6069 20.838 0.06421 1.0698 21 3.70 12.534 0.08375 0.3096 3.570 0.09312 0.3442 224.923 0.01976 0.0730 22 0.40 12.534 0.07442 0.0300 11.315 0.08366 0.0337 14.267 0.01729 0.0070 Total 0.0000 Total 0.0000 Total 0.0000 Table 4 Calculating IRR as Ratio of PV of Annual Returns to PV of Equity (1) (2) (3) (4) (5) IRR Discounted A-tax Discounted Discounted A- Discounted Policy Discount GAAP Book GAAP Equity tax Statutory Required Capital Year Factors Proﬁt (BOY) Book Proﬁt (BOY) 1 0.88862 3.697 26.559 8.680 26.559 2 0.78964 3.905 32.812 8.447 21.814 3 0.70169 3.137 24.327 2.690 18.591 4 0.62353 2.807 22.018 2.331 16.524 5 0.55408 2.510 20.221 2.452 14.916 6 0.49236 2.247 18.276 2.355 13.511 7 0.43752 2.006 16.408 2.310 12.270 8 0.38879 1.775 14.501 2.203 11.095 9 0.34549 1.558 12.634 2.050 9.986 10 0.30700 1.363 10.886 1.882 8.970 11 0.27281 1.178 9.231 1.803 7.990 12 0.24242 1.008 7.588 0.733 7.041 13 0.21542 0.889 6.907 0.515 6.177 14 0.19143 0.788 6.354 0.548 5.373 15 0.17010 0.690 5.697 0.499 4.612 16 0.15116 0.587 5.029 0.560 3.895 17 0.13432 0.486 4.243 0.573 3.211 18 0.11936 0.383 3.403 0.614 2.563 19 0.10607 0.294 2.488 0.475 1.948 20 0.09425 0.223 1.716 0.291 1.397 21 0.08375 0.012 0.328 0.235 0.104 22 0.07442 0.003 0.027 0.004 0.026 Total P.V. 31.543 251.652 24.890 198.574 Ratio (2)/(3) 12.53% (4)/(5) 12.53%