Bridging the Gap Between ROE and IRR by tym76564

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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%

								
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