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A managerial approach to risk theory: Some suggestions from the theory of financial decisions
Insurance: Mathematics North-Holland
and Economics
47
8 (1989) 47-56
A managerial approach to risk theory: Some suggestions from the theory of financial decisions Flavio
PRESSACCO
*
Uniuersitci degli Studi di Udine, 33100
Udine, Italy
As suggested by the theory of financial economics, strategic decisions of an insurance company should aim to maximize the market value of the company. This simple idea could be advantageously applied to a key problem of ruin theory; that is, the choice of the optimal level of surplus of an insurance company. Keywords:
Risk theory,
Surplus-dividend
strategies,
Market
value.
1. Introduction Both in theoretical models and in practical regulation rules of nonlife insurance markets, the surplus (free reserves) of an insurance company is seen as the prominent control variable to keep the solvency of the company at a desired level. The great majority of actuarial models (and a fortiori the idea behind real-world surplus regulation) look at solvency as something that matters to policyholders granting full payments of their claims. 1 On the contrary, the so-called managerial approach to risk (ruin) theory is based on the idea that companies follow profit-oriented strategies,
I wish to thank Dr. Liviana Picech who performed, while working for her graduate thesis at University of Trieste, a lot of numerical computations, including programs for computer simulations. Indeed the so-called collective risk theory is precisely an elegant and powerful building able to give exact or proxy or at least upper bounds on the asymptotic (or over some different time horizon) ruin probability of a stylized (i.e. working now and in the future under convenient simple conditions) insurance company. We will not enter in any detail concerning this theory, signalhng that it is currently and continuously enriched of new nice results [for a good and relatively up-to-date review, see Bowers et al. (1987)]. 0167-6687/89/$3.50
0 1989, Elsevier Science Publishers
where solvency (survival) is merely a by-product that matters for shareholders if it grants future profit opportunities, but being absolutely out of direct managerial goals. This approach goes back to a path-breaking paper of de Finetti (1957). Here a dynamic surplus policy was derived according to a so-called barrier strategy; that is (given the initial surplus), choosing a constant surplus target level B, in such a way as to maximize the expected discounted value of future dividends, to be distributed to shareholders only if the surplus exceeds that level. This introductory model was later discussed and refined by many authors, most notably by Borch (1968, 1974) undoubtedly the strongest supporter of the validity of the managerial approach. His last and in some sense, testamentary paper [Borch (1984)], and summarizing contribution to the problem is based on the perhaps astonishing idea that the target surplus level should be unconditionally (except, of course, in case of ruin) restored at the end of any exercise, if necessary even through the distribution of ‘negative dividends’ (raising money from stockholders), should it, after bad periods, have been fallen below that level. In the first part of this paper (Sections 2-5) a revised version of the constant surplus model of Borch is presented and discussed, making proper use of some basic ideas of the modern theory of financial decisions, and with the aim to obtain new insights in rational surplus-dividend strategies of an insurance company. The core of the refinement is emphasized in Section 2, the choice of the net discounted expected value of the business cash flow (obtained by subtracting the initial outflow B from the discounted expected value of the future dividends) as the target of the company. In Section 3, with reference to an exponential density function for the global claim distribution
B.V. (North-Holland)
48
F. Pressacco / Theory of financial decisions
of any period, optimality conditions for the surplus are derived and discussed through the analysis of the behaviour of the first derivative of the goal function. Shortly summarizing, under the usual underwriting and financial conditions, at the optimal level B* the decreasing marginal advantage induced (by a marginal increase in surplus) on the expected underwriting profitability (through an increase in expected business life) is exactly countered by the constant marginal disadvantage induced on the financial side. After that in Section 4 a table containing the optimal values of the surplus as a function of financial and underwriting parameters is reported and discussed. The most important result is that for any given financial conditions the optimal surplus is an increasing function of underwriting profitability (at least for realistic underwriting parameters). Even without going into details, this result seems to be rather relevant for the choice of rational surplus regulation rules. At the end of the first part of the paper, Section 5 is devoted to the discussion of the economic foundations of the goal function. It is claimed that the goal function of our model may be seen as the market value of the insurance business; that is, the proper target of shareholders, only if a proper discount factor is used to actualize expected future cash flows. Precisely as suggested by the theory of financial economics [Brealey and Myers (1984)] the discount factor should be chosen in such a way as to keep account of the extra reward given under financial market equilibrium to the relevant (nondiversifiable) riskiness of the insurance business. Quite likely this implies that a mistake is made if (as usual in actuarial literature as well as in our previous chapters) the discount factor is treated as a constant with respect to the surplus. Anyway no general treatment of problems where the discount factor is seen as a function of the surplus is offered; only a numerical example of the optimal surplus for an apparently acceptable form of the discount factor function is given. All that is probably a detailed picture of the properties of the managerial approach to risk theory. The surplus problem is here treated as a particular capital budgeting problem, and solved by a constant surplus strategy whose optimality comes from assumptions of invariant operating
conditions of companies and non-frictionality of markets (without taxes and transaction costs). Unfortunately, as revealed by the smoothness of the dividend paths followed by the insurance companies, real-world behaviour seems to be very far from this type of constant surplus rationality (quite likely implying on the contrary an erratic path of dividends). To escape from this surplus-dividend puzzle we need to keep proper account of market frictions as well as of meaningful temporal changes in profit expectations of insurance companies. That is precisely what we try to do in the second part of the paper. After an informal and quick treatment of the role of taxation (Section 6), we discuss in some detail the consequences induced by uncertainty on the parameters governing the claim process. Precisely as said in Section 7, we treat the stochastic process of claims as exchangeable (no more i.i.d.); that is with an uncertain parameter h whose posterior distribution is updated from an initial prior, according to new informations (data on past claims). Just under this assumption opinions about future profit opportunities change over time. We claim that this renders no more rationale to follow a constant surplus strategy and introduce a new type of adaptive surplus strategy. After that (Section 8) with reference to some cases of priors belonging to the gamma type the performances (market values) evaluated by simulations of that adaptive strategy are compared with those of the optimal constant surplus strategies. At first glance the adaptive strategy seems to perform almost uniformly better in terms of market values, but sometimes at the cost of higher ruin probability (as detected by’ the relative frequency of ruin trajectories in the simulation) and then, of less safety for policyholders. Finally in Section 9 we try to check if adaptive surplus strategies are really successful in determining smooth (or at least smoother than the constant surplus) dividend policies. We did not obtain conclusive evidence from numerical simulation on this point. Some dividend paths coming from the adaptive strategy are really smoother, but some others are more erratic. An explanation for this apparently paradoxical behavior is offered. Despite these shortcomings of numerical simu-
F. Pressacco / Theory of financial decisions
lations we are still faithful that the joint effect of market frictions and adaptive changes in profit expectations are able to solve and explain the surplus dividend puzzle. To obtain unambiguous answers from simulations confirming this faith a better model of the dynamic working of an insurance company is needed along the directions shown in this paper.
2. A managerial model of risk theory In its broader generality a managerial model of risk theory is something based on three building blocks: _ present and future operating conditions of an insurance company including an exogenously given ruin rule specifying conditions forcing the company to cease business; _ a dynamic strategic rule that determines at any decisional instant the values to be chosen for the control variables in order to reach the above goals; - a goal (value) function specifying unambiguously the goals of the company. In what follows we will treat discrete versions of the model with decisional instants coming at the end of any underwriting period, and just one dynamic control variable: the surplus S,, (free reserves) of the company. Along this line let us discuss as a starting point a model that conveniently modifies a testamentary paper on this subject by Borch. The model is formally described by the following set of relations: S,=B>O G,, = S,_,r
(1) + P(l
+ r) - X,, if S,_, 2 0, if S,_,
G, = 0 Sn =S,_i+G,,,
S,- -S,,=D,,,
(2b) (3)
S,, = S; - G,, if S,- 2 0, s, = s,-
< 0,
(2a)
if S,- (0,
(4a) (4b) (5)
49
Equation (1) fixes the initial surplus at some non-negative level B, (2a) defines the random gain of the n th period as the sum of the financial returns and the underwriting profit, coming respectively from the investment at the sure, and constant through time, rate of return r of the resources of the company at the beginning of any period (non-negative surplus and premiums collected at the beginning and assumed constant throughout the whole life of the company), and from the algebraic excess of premiums to global claims and expenses X, paid at the end of any underwriting period, with { X,,} a stochastic process of assumed i.i.d. random variables. Alternatively, according to (2b), if the initial surplus is negative the company was previously ruined and ceased business, so that the gain of the period is zero. In (3) SHp denotes the global resources of the company at the end of the n th period just before a surplus decision is made. Equations (4) and (5) clarify how such decisions are made: if the company is not ruined (that is, the final surplus S,- is non-negative) the surplus is restored at the constant goal level B. Indeed (3) and (4a) imply that in the survival case S,, = S,_ i, so that recursively S,, = S,, = B! Put another way, the company follows a constant surplus strategy at a given level B (to be properly determined unless ruin has in the meantime happened), or formally:
S, = B for any n < h (time of ruin).
Note that this implies that negative as well as positive dividends are allowed: after bad periods (G, < 0), but not too bad to determine ruin a negative dividend (an inflow from stockholders to the company) is provided. Even if largely unrealistic, it is nevertheless an optimal (through the Bellman principle of optimality in dynamic programming) strategy, at least under our hypothesis of time invariance of operating conditions and of the goal function. Finally the goal function (6) is the net present value of the insurance business as derived by the above result, that is, the discounted expected value of the stockholders’ cash flow from the insurance business actualized at a proper given (and not dependent from B, at least for the moment) factor V.
50
F. Pressacco / Theoty of financial decisions
After that it is straightforward with u = 1+ r we obtain
to prove
u
V(B)=
(J(p+E%“(x)
dx - BE;((P
O
l-
that
+ B)u))
uF,((P+B)u)
-B.
(7)
Looking for the best strategic choice of the surplus B, we need to study the sign of the first derivative of the goal function which is the same as the following expression [numerator of the derivative hence denoted by NV’(B)]: NV’(B)
= uuf,((P + B)u)(
z$‘+~%“(x)
dx - B)
The condition of ‘financial neutrality’ means roughly that the company receives exactly the desired financial reward from its surplus, so that modifying the surplus is neither advantageous nor disadvantageous on the financial side. But from the underwriting (technical) point of view if P > 1 (the business is profitable) it is advantageous to play as long as you can , as the company (expects to) gains something at every exercise. And of course the greater the surplus, the longer the expected life of the business (exercises before ruin). Just reverse the reasoning when P < 1 (bad business): now the company expects to lose something at every exercise, so that the greater the surplus, the bigger the loss expectation on the underwriting side!
-l+u(l+u)F,((P+B)u) -u2uFX((P+
B)u).
(8)
3. The exponential case
z = P(l
Let us treat here a conveniently simple example where f,(x) = exp( - x), a special case of f,(x) = A exp( -Xx) with X = 1. Even if we realize that an exponential distribution is a poor and unrealistic proxi of an insurance company’s global claims, at least qualitatively some results seem to be quite robust to changes in the claim distribution. Anyway with this density and writing uu = 1 w it turns out that NV’(B)=exp{-(P+B)u} x{B(w2-
+w(u-1).
(9)
We will discuss the sign of NV’(B) for the meaningful cases uu = 1 (roughly ‘financial neutrality’) and uu < 1 (that is 0 < w < l), leaving unanswered the third less realistic option uu > 1. uu = 1 (w = 0).
NV’(B)sOiffP-E(X)
- w)2 + w(1 -u)
the optimal surplus conditions hold:
- 1,
is zero unless
(11) both
following
z > 0,
(12a)
zexp(-Pu)-w(l-u)>O.
(12b)
If so there is a unique optimal surplus level at some positive B *. Conditions (12) are not of easy reading, anyway to gain some insight, at least concerning (12a), note that
w)+P(1-w)2
+w(1-u)-1)
(4
uu < 1 (0
It is immediately (orP-l)$O.
seen that (10)
This means that the optimal surplus is unbounded (when P > l), or at the other extreme zero (when P < 1). Let us try to discuss the (rather simple) reasons of this ‘strategic extremism’.
>l/(l-w), so that finally z > 0 implies Puu > 1; that is, something denoting an underwriting profit adjusted to keep account of financial conditions. After that things are different at low or high level of surplus. As the marginal growth in expected business life is intuitively decreasing with surplus, the marginal adjusted underwriting profit coming from surplus is decreasing too, and only at levels lower than that of a boundary surplus B* > 0 exceeds the constant marginal financial loss deriving from a financial reward on the surplus less than the neutral one.
F. Pressacco / Theory of financial decisions
4. A numerical example and some comments Concerning the exponential case previously discussed Table 1 reports the optimal values of the surplus (four exact decimal figures) corresponding to r = 0.05as a function of the underwriting conditions contained in P [recall E(X) = 11, and of some convenient values of the discount factor. Some comments are in order: reading the columns we see that B*(P) has the same regular shape (for any u). Precisely at low levels of underwriting profitability the optimal surplus is increasing: intuitively the greater the underwriting profit the bigger the desire to increase the business life through a choice of a big enough surplus. But as the underwriting profit is high enough it becomes able to offer a self protection from ruin, thus lowering the need for surplus (remember that on the financial side you would like to shorten surplus). It is rather interesting to note that anyway at realistic underwriting conditions (say 1.10 < P <
Table 1 Optimal surplus level B * (P, 0); r = 0.05, E(X) P
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30
51
1.30) B* is an increasing function of P. We will come back to this point later in Section 7. On the other side, a reading of the rows reveals that the optimal values are for any P a decreasing function of the discount rate. This is even more intuitive: given underwriting conditions an increase in the discount rate increases the financial loss, and this forces to lower the surplus. It is clear that looking at reasonable values of the discount factor (say 1.10-l) some regulation is needed if we don’t want to allow that bad companies [with a too small ratio P/E(X)] enter in the market without a convenient surplus coverage. Indeed the difference between good and bad companies is that once exogeneously given a minimum surplus &, the good ones spontaneously work with their optimal surplus greater than _B, while some others are forced to work with the surplus imposed by regulation at a level greater than their unconstrained optimum (provided that V(B) is greater than zero), while only the really bad ones, having V(B) < 0, are forced to leave the market.
= 1.
”
5. Comments
l/(1.06)
l/(1.07)
l/(1.08)
l/(1.09)
l/(1.10)
0.0000 1.9559 3.2193 3.7713 4.1016 4.3216 4.4936 4.6207 4.7208 4.8009 4.8656 4.9182 4.9610 4.9951 5.0236 6.0456 5.0627 5.0754 5.0842 5.0897 5.0921 5.0918 5.0891 5.0842 5.0772 5.0684 5.0579
0.0000 0.3750 1.8388 2.5787 3.0093 3.2949 3.5000 3.6547 3.7153 3.8713 3.9486 4.0116 4.0630 4.1049 4.1390 4.1665 4.1882 4.2051 4.2177 4.2265 4.2319 4.2344 4.2342 4.2315 4.2260 4.2197 4.2110
0.0000 0.0000 0.9153 1.7300 2.2363 2.5740 2.8150 2.9954 3.1352 3.2460 3.3352 3.4078 3.4672 3.5159 3.5557 3.5882 3.6143 3.6351 3.6511 3.6629 3.6712 3.6761 3.6782 3.6775 3.6746 3.6695 3.6624
0.0000 0.0000 0.2999 1.0872 1.6280 2.0025 2.2728 2.4756 2.6326 2.7569 2.8570 2.9385 3.0053 3.0603 3.1055 3.1426 3.1728 3.1971 3.2163 3.2310 3.2418 3.2491 3.2533 3.2546 3.2534 3.2499 3.2443
0.0000 0.0000 0.0000 0.5997 1.1383 1.5312 1.8218 2.0423 2.2138 2.3499 2.4591 2.5492 2.6228 2.6835 2.7337 2.7751 2.8090 2.8367 2.8588 2.8762 2.8893 2.8988 2.9049 2.9080 2.9085 2.9065 2.9023
on the choice of the discount factor
This paragraph is devoted to a deeper look at the meaning and role of the discount factor in our model. At the time of his path-breaking paper de Finetti seemed not to be much troubled by the choice of the value function. After having introduced the one
E very close to ours, he himself suggests the opportunity to keep account of some higher moments than the pure expectation of discounted profits. Moreover he looks at the discount factor as something related to the decision maker’s temporal preferences and then treats it automatically as a given constant with respect to the other relevant variables of the problem. These ideas have remained prevalent in risk theory literature until now, but a careful look to the question reveals that both points are just two sides of the same coin. After Sharpe (1964), Lint-
F. Pressacco / Theory of financial decisions
52
ner (1965) and Mossin (1966) and their multiperiod followers [most notably Fama (1977)] we know that under proper equilibrium conditions on financial markets the relevant value of a business project is its market value; that is, just the net expected value of its cash flow [our goal function (6) in risk theory applications] but actualized at a discount factor that reflects the relevant (non-diversifiable) riskiness of the project (and not subjective temporal and or risk preferences of’ the decision maker). Then according to the theory of financial economics it is sound to use a discounted expectation as a value function, but realizing that the discount factor is constant with respect to other relevant variables of the problem (here the surplus) only if the relevant riskiness of the project does not depend on those variables! Without discussing at length this point involved, my opinion is that, either if you measure the riskiness of the insurance project in a naive (mean square deviation) or in a more refined (covariance with the market rate of return) way, changing the surplus quite likely changes the relevant riskiness of the project: but this means unfortunately that our surplus choice becomes much more involved as formally the discount factor u becomes some function of B, whose shape is even difficult to realize! 2 A deeper insight on this critical point is beyond the scope of this paper, thus here (just as an exercise) we want to show a table of the market values V(B) generated by a tentative choice of the following simple discount functions (see Tables 2a, b):
uh(B)=(l+ih(B))-I,
h=l, 2,3,
03)
Table
2a
Market count
values function;
V(B, oh(B)) as a function of surplus and disP =l.l, r = 0.05, E(X) = 1.
B
u,(B)
uz(B)
u,(B)
0.0
1.2500
1.2466
1.2433
0.1
1.2656
1.2573
1.2496
0.2
1.2808
1.2661
1.2525
0.3
1.2953
1.2723
1.2515
0.4
1.3088
1.2756
1.2459
0.5
1.3210
1.2752
1.2350
0.6
1.3313
1.2708
1.2183
0.7
1.3396
1.2617
1.1051
0.8
1.3453
1.2475
1.1651
0.9
1.3481
1.2276
1.1279
1.0
1.3476
1.2018
1.0832
1.1
1.3433
1.1697
1.0310
1.2
1.3350
1.1312
0.9713
1.3
1.3224
1.0861
0.9043
1.4
1.3052
1.0344
0.8302
1.5
1.2832
0.9764
0.7496
1.6
1.2564
0.9121
0.6629
1.7
1.2408
0.8981
0.6094
1.8
1.2247
0.8420
0.5706
1.9
1.2254
0.8591
0.5534
2.0
1.2076
0.8176
0.4952
The idea behind this type of discount functions is very simple: it is assumed that the riskiness of the project is sharply increasing for low levels of
Table Market count
2b values function;
V( B, o*(B)) as a function of surplus and disP = 1.2, r = 0.05, E(X) = 1.
B
u,(B)
02(B)
u,(B)
0.0
1.5774
1.5728
1.5683
0.1
1.6201
1.6088
1.5981
0.2
1.6637
1.6435
1.6250
0.3
1.7077
1.6764
1.6481
0.4
1.7519
1.7066
1.6664
ir( B) = 0.061 + O.OlB
if B G 1.7,
0.5
1.7956
1.7335
1.6791
ir( B) = 0.075 + O.OOlB
if B > 1.7;
0.6
1.8383
1.7561
1.6853
0.7
1.8794
1.7739
1.6842
i2(B)
if B G 1.7,
0.8
1.9182
1.7859
1.6752
1.9542
1.7915
1.6577
i,(B)=0.089+0.002B
ifB>1.7;
0.9 1.0
1.9867
1.7901
1.6313
i3( B)=0.063 +O.O31B
if Bg 1.7,
1.1
2.0151
1.7812
1.5958
if B > 1.7.
1.2
2.0387
1.7644
1.5511
1.3
2.0571
1.7395
1.4974
1.4
2.0698
1.7064
1.4349
1.5
2.0763
1.6652
1.3641
1.6
2.0765
1.6159
1.2855
1.7
2.0701
1.5589
1.1997
1.8
2.1279
1.6698
1.2869
1.9
2.1430
1.6537
1.2486
2.0
2.1545
1.6341
1.2073
i,(B) ’
= 0.062 + 0.021B
= 0.104 + 0.003B
Formally
this produces
a modified
goal function
~?(R)(~‘~+~‘“F~(~)dx-~~~((p+B)u))
V(B)
=
I-u(B)F,((P+B)u) -B.
as follows:
F. Pressacco / Theoty of financial decisions Table 3 Optimal surplus
level B *(P,
uh( B)).
P
u,(B)
+(B)
uj(B)
1.1 1.2
0.9 2.3
0.4 0.9
0.2 0.6
the surplus (as you face some possibility of losing everything) but later slowly increasing (indeed this is just what happens to the mean square deviation of the goal function obtained by simulation). Note that the covariance with the market rate of return is here completely left out of the picture. As resumed in Table 3, here too the optimal surplus is an increasing function of the premium level and a decreasing function of the riskiness embodied in the discount function.
6. The surplus dividend puzzle The starting point of the second part of our paper is that the surplus strategy of the company determines a dividend strategy and that constant surplus strategies (as well as barrier strategies) quite likely generate erratic paths of dividends never found in the real world. Then we must face what could be called a surplus dividend puzzle: 3 rational surplus strategies give rise to unrealistic dividend paths, and real life smooth dividend policies seem to imply silly or largely suboptimal surplus decisions, thus giving up the opportunity to control the surplus evolution, that is, the very same idea of dynamic managerial rationality. How to escape from the dilemma? Leaving the easy but blind way of simple frictionless insurance markets to keep proper account of such factors as (i) taxes 4 and transaction costs, and (ii) incomplete information. Let’s give here only an informal and quick picture of the role of the first type factors; the
An entirely different approach to solve the surplus dividend puzzle was proposed by Ettle (1984). We don’t discuss here his paper. The role of the tax system in altering the optimal decision rules in a lot of economic problems is generally recognized as one of the key points in modern finance. See chapter 3 in Altman and Subrahmanyam (1985).
53
impact of imperfect information will be discussed later in some detail. What is the effect of taxation on our surplus problem? We are going here to discuss what strategic changes are induced from a different tax treatment of dividends and capital gains (leaving aside the effect of corporate taxes). Precisely suppose that some taxes are to be paid on distributed dividends. Then a strategy trying to avoid to pay dividend taxes retaining (partially or totally) realized gains, and thus accepting a tendency of the surplus to increase over time can be the best after tax strategy, as the one that maximizes the net present after-tax value of the insurance project. This destroys, even under invariant underwriting conditions, the optimality of a constant surplus strategy. But what about liquidity needs of stockholders with less or no dividends? Of course they are satisfied by selling shares and under advantageously enough tax treatment of capital gains in comparison to dividends, this gives more money than adopting the constant surplus strategy optimal in a world without taxes. This preference for a changing surplus and smooth dividends is reinforced by the existence of transaction costs to raise new capital to restore the optimal surplus after some losses. As non-proportional transaction costs determine a big cost to restore small losses it is much better to keep a safety cushion of non-distributed profits devoted to the covering of these needs, thus avoiding a too frequent recourse to the financial market.
7. The role of parameter uncertainty Let’s discuss in some detail the impact of imperfect information on surplus dividend strategies. We will consider the case where the parameter(s) governing the global claims distribution (X in our example) of any period is no more surely known (and constant over time) but carries some uncertainty. Precisely it is seen as a random number A whose probability distribution is updated according to Bayesian rules from some initial prior as long as new informations are obtained. Under this scenario and with informations deriving only from data collected about past claims, the sequence { X,, } is no more i.i.d. but exchangea-
F. Pressacco / Theory of financial decisions
54
ble or independent only conditionally to any value of the parameter. This being the situation a good information (i.e. a low level of global claims of last exercise) implies that the posterior distribution of the parameter reflects more optimism about future profit opportunities than the initial one; and conversely after bad information. This in turn implies that contrary to the stationary case (with invariant opinions) it is no more rational to follow a constant surplus strategy; we guess that better results are obtained applying a simple adaptive surplus strategy defined at any instant as the weighted average of optimal surplus for the parameter certainty, with weights given by the posterior probability distribution of the parameter, or formally at time n, by K+=(A) = j-B*(h)f,(A) A
dh
(14)
with B*(h) as the best constant surplus strategy under parameter certainty as defined in Section 3. We don’t try here to offer a theoretical rationale for choosing this strategy rather than some other adaptive strategy. 5 In the next chapter the performances of the adaptive strategy previously defined are compared with those of the constant surplus strategy defined as the one that satisfies mBax/ V(B, A
h)f,(X)
dX
(15)
with &(X) as the prior distribution of the parameter at the beginning of the story, and obviously V(B,A) as the market value of the constant strategy B for a sure value X of the parameter A.
8. Numerical results for the exponential case
gamma
We present here some numerical results for the case where the prior distribution of the random
5
Yet we found that the chosen strategy turns out to offer better performances (at least looking at results coming from some simulations) than the strategy that maximizes
exponential parameter A is r(a, with density = -Aa*-’ p” r(a)
g,(X)
j?), or, formally,
exp( -/3X).
(16)
Under this hypothesis it is easy to check that the posterior distribution after having observed a n-sequence 5 of claims is given by g,(Vx) (p+
C.X,)~+~A~+‘-~
exp-
(p+
Cxi)X. f
r(a+n)
(17) that is, a gamma again with parameters (Y+ n and p + Xx,. Moreover it could be proved that the unconditional claim distribution is Pareto ((Y, p); that is, with density
(18) and, respectively claims,
fn(x/x) =
after an observed n-vector x of
(a+n)(P+ CxJaCn. (p+
CX,+X)a+n+l ’
09)
that is, again Pareto ((CX+ n), (j3 + Xx,). After that we offer (Table 4) a comparison of the estimated market values (average discounted expected values of the dividend cash flows offered by a computer simulation) produced by the choice of the two surplus strategies previously described.
Table 4 Market values, relative frequency of ruin and initial surplus for the constant surplus (c.s.) and the adaptive surplus (a.s.) strategy; P =l, r = 0.05, u=1.07-‘.
(a. P) m.v.c.s.
m.v.a.s.
(2)/
(1)
(2)
(1)
8.5797 8.7507 8.8642 3.0489 4.2294 4.3465 1.9672 3.1006 3.5614
9.1882 9.4820 10.1452 3.6575 4.1965 4.7924 2.6259 3.4482 3.9993
1.07 1.08 1.14 1.19 0.99 1.10 1.33 1.11 1.12
fr.c.s.
fr.a.s.
B, c.s.
B, as.
0.145 0.13 0.11 0.8 0.61 0.40 0.98 0.71 0.44
0.255 0.215 0.20 0.55 0.50 0.47 0.64 0.55 0.51
2.6037 2.7271 2.7945 0.6871 1.4893 2.1574 0.3165 1.4579 2.2644
0.6978 0.6485 0.6110 1.3553 1.4267 1.4711 1.3412 1.4960 1.6018
F. Pressacco / Theory of financial decisions Table 5 Couple of values of the parameters for the prior ga( X) used for simulation, and related expected value of the unconditional claim distribution. (a. P)
(Y
P
E(X)
1 2 3 4 5 6 7 8 9
1.2 1.2 1.2 3.2 3.2 3.2 5.2 5.2 5.2
0.2 0.18 0.166 2.2 2 1.83 4.2 3.81 3.5
1 0.9 0.83 1 0.9 0.83 1 0.9 0.83
The simulation is based on 200 sequences of Monte Carlo random claims concerning 50 underwriting periods, 6 for the following nine different combinations of (a, p) values of the prior g,(h) (table 5). Some comments are in order: (1) The adaptive strategy gives almost uniformly better results as measured by market values of the respective projects (the average advantage is about 11%). (2) The highest values for both constant and adaptive strategy are found for low values of the alpha coefficient; that is, for higher uncertainty about A. This is not surprising as the uncertainty is exploited on the upper side, but in the absence of regulation only partially suffered downside. We are very near to a well-known low of option pricing theory: the price is ceteris paribus an increasing function of volatility. (3) The relative frequency of ruin is for both strategies uniformly too high. Under the assumptions of the model a surplus regulation rule is absolutely needed to grant survival at the level desired by policyholders. Alternatively, a major cut in ruin probability could be obtained forcing a change in operating conditions.
Of course, the real life of the company could be less than 50 years in case of ruin in a shorter period. In case of survival the final surplus value is discounted and added as a final dividend to the discounted sum of dividends. Note that this is not a correct procedure: we should add the (discounted) market value of the surplus instead of the same surplus.
9. Comparison
55
of dividend paths
We are now going to investigate if adaptive strategies determine dividend paths smoother than those produced by constant surplus strategies. As said in the introduction the intuition behind this idea is that as long as B*(h) is under parameter certainty an increasing function of the expected underwriting profitability, the adaptive strategies enjoy the property that the surplus level tends to be raised after good periods and lowered after bad periods. This in turn should determine a desire to retain part of the profit obtained in good periods (to reach the new upper desired surplus level) or symmetrically to restore only partially the surplus after a loss. Unfortunately, the numerical results obtained don’t give unambiguous support to this idea, as we can see from Table 6. Note that, as under constant surplus strategies the dividend distributed is (in case of survival) exactly equal to the overall exercise profit (loss), higher variability of dividends means that on average a dividend bigger than the profit (loss) is distributed (or get from) stockholders under the adaptive strategy. This is apparently surprising but happens any time a claim greater than the updated expectation is detected (thus lowering the desired surplus), but a profit is at the same time realized, either coming from the loading charges or from financial profits. Indeed, while the premium level stays constant by assumption, the past story may have produced a big departure of the updated expected claim from the one initially expected, and this opens the
Table 6 Average mean square deviation of dividend paths implied by constant and adaptive surplus strategies for some selected couples of ((Y, p). (a. B)
m.s.d.c.s.
m.s.d.a.s.
1 2 3 4 5 6 7 8 9
0.71 0.64 0.71 0.44 0.47 0.43 0.42 0.41 0.38
0.57 0.45 0.58 0.57 0.54 0.50 0.46 0.44 0.42
56
F. Pressacco / Theory of financial decisions
way to a big enough excess of dividends under the adaptive strategy. This explains why more erratic dividend paths may be found for the adaptive strategy; then a more refined model with feedbacks from changing expectations to the premium level and eventually to the business volume, keeping account of such factors as dividend taxes and transactions costs on financial markets is needed to solve the surplus dividend puzzle. We hope that the model presented in this paper could be seen just as the first step toward this direction.
References Altman, E. and M. Subrahmanyam (1985). Recent Advances in Corporate Finance. Irwin, Homewood, IL. Borch, K. (1968). The rescue of an insurance company after ruin. The Astin Bulletin, 5, 280-292. Borch, K. (1974). The Mathematical Theory of Insurance. Lexington Books, Lexington, MA. Borch, K. (1984). An alternative dividend policy for an insurance company. Mitteilungen der Vereinigung schweiz. Versicherungsmathematiker, 2 201-206. Bowers, N., H. Gerber, J. Hickman, D. Jones and C. Nesbitt (1987). Actuarial Mafhematics. Society of Actuaries.
Brealey, R. and S. Myers (1984). Principles of Corporate Finance. McGraw Hill, New York. Buhlmann, H. (1970). Mathematical Methods in Risk Theory. Springer Verlag, Heidelberg, New York. de Finetti, B. (1957). Su un’impostazione altemativa della teoria collettiva de1 rischio. Transactions of the 15th International Congress of Actuaries, Vol. 2, 433-443. Ettle, W. (1984). Optimale dividendenpolitik. Transactions of the International Congress of Actuaries - Sydney, Vol. 3, 435-445. Fama, E.F. (1977). Risk adjusted discount rates and capital budgeting under uncertainty. Journal of Financial Economics 5, 3-24. Gerber, H. (1979). An Introduction to Mathematical Risk Theory. Huebner Foundation, distributed by Irwin, Homewood, IL. Lintner, J. (1965). The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, 13-37. Markovitz, H. (1952). Portfolio selection. Journal of Finance 6, 77-91. Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica 34, 768-783. Robichek, A.E. and S.C. Myers (1966). Conceptual problems in the use of risk adjusted discount rates. Journal of Finance 21, 727-730. Seal, H. (1969). Stochastic Theory of Risk Business. Wiley, New York. Sharpe, W.F. (1964). Capital asset prices: A theory of market equilibrium under condition of risk. Journal of Finance 19, 425-442.
and Economics
47
8 (1989) 47-56
A managerial approach to risk theory: Some suggestions from the theory of financial decisions Flavio
PRESSACCO
*
Uniuersitci degli Studi di Udine, 33100
Udine, Italy
As suggested by the theory of financial economics, strategic decisions of an insurance company should aim to maximize the market value of the company. This simple idea could be advantageously applied to a key problem of ruin theory; that is, the choice of the optimal level of surplus of an insurance company. Keywords:
Risk theory,
Surplus-dividend
strategies,
Market
value.
1. Introduction Both in theoretical models and in practical regulation rules of nonlife insurance markets, the surplus (free reserves) of an insurance company is seen as the prominent control variable to keep the solvency of the company at a desired level. The great majority of actuarial models (and a fortiori the idea behind real-world surplus regulation) look at solvency as something that matters to policyholders granting full payments of their claims. 1 On the contrary, the so-called managerial approach to risk (ruin) theory is based on the idea that companies follow profit-oriented strategies,
I wish to thank Dr. Liviana Picech who performed, while working for her graduate thesis at University of Trieste, a lot of numerical computations, including programs for computer simulations. Indeed the so-called collective risk theory is precisely an elegant and powerful building able to give exact or proxy or at least upper bounds on the asymptotic (or over some different time horizon) ruin probability of a stylized (i.e. working now and in the future under convenient simple conditions) insurance company. We will not enter in any detail concerning this theory, signalhng that it is currently and continuously enriched of new nice results [for a good and relatively up-to-date review, see Bowers et al. (1987)]. 0167-6687/89/$3.50
0 1989, Elsevier Science Publishers
where solvency (survival) is merely a by-product that matters for shareholders if it grants future profit opportunities, but being absolutely out of direct managerial goals. This approach goes back to a path-breaking paper of de Finetti (1957). Here a dynamic surplus policy was derived according to a so-called barrier strategy; that is (given the initial surplus), choosing a constant surplus target level B, in such a way as to maximize the expected discounted value of future dividends, to be distributed to shareholders only if the surplus exceeds that level. This introductory model was later discussed and refined by many authors, most notably by Borch (1968, 1974) undoubtedly the strongest supporter of the validity of the managerial approach. His last and in some sense, testamentary paper [Borch (1984)], and summarizing contribution to the problem is based on the perhaps astonishing idea that the target surplus level should be unconditionally (except, of course, in case of ruin) restored at the end of any exercise, if necessary even through the distribution of ‘negative dividends’ (raising money from stockholders), should it, after bad periods, have been fallen below that level. In the first part of this paper (Sections 2-5) a revised version of the constant surplus model of Borch is presented and discussed, making proper use of some basic ideas of the modern theory of financial decisions, and with the aim to obtain new insights in rational surplus-dividend strategies of an insurance company. The core of the refinement is emphasized in Section 2, the choice of the net discounted expected value of the business cash flow (obtained by subtracting the initial outflow B from the discounted expected value of the future dividends) as the target of the company. In Section 3, with reference to an exponential density function for the global claim distribution
B.V. (North-Holland)
48
F. Pressacco / Theory of financial decisions
of any period, optimality conditions for the surplus are derived and discussed through the analysis of the behaviour of the first derivative of the goal function. Shortly summarizing, under the usual underwriting and financial conditions, at the optimal level B* the decreasing marginal advantage induced (by a marginal increase in surplus) on the expected underwriting profitability (through an increase in expected business life) is exactly countered by the constant marginal disadvantage induced on the financial side. After that in Section 4 a table containing the optimal values of the surplus as a function of financial and underwriting parameters is reported and discussed. The most important result is that for any given financial conditions the optimal surplus is an increasing function of underwriting profitability (at least for realistic underwriting parameters). Even without going into details, this result seems to be rather relevant for the choice of rational surplus regulation rules. At the end of the first part of the paper, Section 5 is devoted to the discussion of the economic foundations of the goal function. It is claimed that the goal function of our model may be seen as the market value of the insurance business; that is, the proper target of shareholders, only if a proper discount factor is used to actualize expected future cash flows. Precisely as suggested by the theory of financial economics [Brealey and Myers (1984)] the discount factor should be chosen in such a way as to keep account of the extra reward given under financial market equilibrium to the relevant (nondiversifiable) riskiness of the insurance business. Quite likely this implies that a mistake is made if (as usual in actuarial literature as well as in our previous chapters) the discount factor is treated as a constant with respect to the surplus. Anyway no general treatment of problems where the discount factor is seen as a function of the surplus is offered; only a numerical example of the optimal surplus for an apparently acceptable form of the discount factor function is given. All that is probably a detailed picture of the properties of the managerial approach to risk theory. The surplus problem is here treated as a particular capital budgeting problem, and solved by a constant surplus strategy whose optimality comes from assumptions of invariant operating
conditions of companies and non-frictionality of markets (without taxes and transaction costs). Unfortunately, as revealed by the smoothness of the dividend paths followed by the insurance companies, real-world behaviour seems to be very far from this type of constant surplus rationality (quite likely implying on the contrary an erratic path of dividends). To escape from this surplus-dividend puzzle we need to keep proper account of market frictions as well as of meaningful temporal changes in profit expectations of insurance companies. That is precisely what we try to do in the second part of the paper. After an informal and quick treatment of the role of taxation (Section 6), we discuss in some detail the consequences induced by uncertainty on the parameters governing the claim process. Precisely as said in Section 7, we treat the stochastic process of claims as exchangeable (no more i.i.d.); that is with an uncertain parameter h whose posterior distribution is updated from an initial prior, according to new informations (data on past claims). Just under this assumption opinions about future profit opportunities change over time. We claim that this renders no more rationale to follow a constant surplus strategy and introduce a new type of adaptive surplus strategy. After that (Section 8) with reference to some cases of priors belonging to the gamma type the performances (market values) evaluated by simulations of that adaptive strategy are compared with those of the optimal constant surplus strategies. At first glance the adaptive strategy seems to perform almost uniformly better in terms of market values, but sometimes at the cost of higher ruin probability (as detected by’ the relative frequency of ruin trajectories in the simulation) and then, of less safety for policyholders. Finally in Section 9 we try to check if adaptive surplus strategies are really successful in determining smooth (or at least smoother than the constant surplus) dividend policies. We did not obtain conclusive evidence from numerical simulation on this point. Some dividend paths coming from the adaptive strategy are really smoother, but some others are more erratic. An explanation for this apparently paradoxical behavior is offered. Despite these shortcomings of numerical simu-
F. Pressacco / Theory of financial decisions
lations we are still faithful that the joint effect of market frictions and adaptive changes in profit expectations are able to solve and explain the surplus dividend puzzle. To obtain unambiguous answers from simulations confirming this faith a better model of the dynamic working of an insurance company is needed along the directions shown in this paper.
2. A managerial model of risk theory In its broader generality a managerial model of risk theory is something based on three building blocks: _ present and future operating conditions of an insurance company including an exogenously given ruin rule specifying conditions forcing the company to cease business; _ a dynamic strategic rule that determines at any decisional instant the values to be chosen for the control variables in order to reach the above goals; - a goal (value) function specifying unambiguously the goals of the company. In what follows we will treat discrete versions of the model with decisional instants coming at the end of any underwriting period, and just one dynamic control variable: the surplus S,, (free reserves) of the company. Along this line let us discuss as a starting point a model that conveniently modifies a testamentary paper on this subject by Borch. The model is formally described by the following set of relations: S,=B>O G,, = S,_,r
(1) + P(l
+ r) - X,, if S,_, 2 0, if S,_,
G, = 0 Sn =S,_i+G,,,
S,- -S,,=D,,,
(2b) (3)
S,, = S; - G,, if S,- 2 0, s, = s,-
< 0,
(2a)
if S,- (0,
(4a) (4b) (5)
49
Equation (1) fixes the initial surplus at some non-negative level B, (2a) defines the random gain of the n th period as the sum of the financial returns and the underwriting profit, coming respectively from the investment at the sure, and constant through time, rate of return r of the resources of the company at the beginning of any period (non-negative surplus and premiums collected at the beginning and assumed constant throughout the whole life of the company), and from the algebraic excess of premiums to global claims and expenses X, paid at the end of any underwriting period, with { X,,} a stochastic process of assumed i.i.d. random variables. Alternatively, according to (2b), if the initial surplus is negative the company was previously ruined and ceased business, so that the gain of the period is zero. In (3) SHp denotes the global resources of the company at the end of the n th period just before a surplus decision is made. Equations (4) and (5) clarify how such decisions are made: if the company is not ruined (that is, the final surplus S,- is non-negative) the surplus is restored at the constant goal level B. Indeed (3) and (4a) imply that in the survival case S,, = S,_ i, so that recursively S,, = S,, = B! Put another way, the company follows a constant surplus strategy at a given level B (to be properly determined unless ruin has in the meantime happened), or formally:
S, = B for any n < h (time of ruin).
Note that this implies that negative as well as positive dividends are allowed: after bad periods (G, < 0), but not too bad to determine ruin a negative dividend (an inflow from stockholders to the company) is provided. Even if largely unrealistic, it is nevertheless an optimal (through the Bellman principle of optimality in dynamic programming) strategy, at least under our hypothesis of time invariance of operating conditions and of the goal function. Finally the goal function (6) is the net present value of the insurance business as derived by the above result, that is, the discounted expected value of the stockholders’ cash flow from the insurance business actualized at a proper given (and not dependent from B, at least for the moment) factor V.
50
F. Pressacco / Theoty of financial decisions
After that it is straightforward with u = 1+ r we obtain
to prove
u
V(B)=
(J(p+E%“(x)
dx - BE;((P
O
l-
that
+ B)u))
uF,((P+B)u)
-B.
(7)
Looking for the best strategic choice of the surplus B, we need to study the sign of the first derivative of the goal function which is the same as the following expression [numerator of the derivative hence denoted by NV’(B)]: NV’(B)
= uuf,((P + B)u)(
z$‘+~%“(x)
dx - B)
The condition of ‘financial neutrality’ means roughly that the company receives exactly the desired financial reward from its surplus, so that modifying the surplus is neither advantageous nor disadvantageous on the financial side. But from the underwriting (technical) point of view if P > 1 (the business is profitable) it is advantageous to play as long as you can , as the company (expects to) gains something at every exercise. And of course the greater the surplus, the longer the expected life of the business (exercises before ruin). Just reverse the reasoning when P < 1 (bad business): now the company expects to lose something at every exercise, so that the greater the surplus, the bigger the loss expectation on the underwriting side!
-l+u(l+u)F,((P+B)u) -u2uFX((P+
B)u).
(8)
3. The exponential case
z = P(l
Let us treat here a conveniently simple example where f,(x) = exp( - x), a special case of f,(x) = A exp( -Xx) with X = 1. Even if we realize that an exponential distribution is a poor and unrealistic proxi of an insurance company’s global claims, at least qualitatively some results seem to be quite robust to changes in the claim distribution. Anyway with this density and writing uu = 1 w it turns out that NV’(B)=exp{-(P+B)u} x{B(w2-
+w(u-1).
(9)
We will discuss the sign of NV’(B) for the meaningful cases uu = 1 (roughly ‘financial neutrality’) and uu < 1 (that is 0 < w < l), leaving unanswered the third less realistic option uu > 1. uu = 1 (w = 0).
NV’(B)sOiffP-E(X)
- w)2 + w(1 -u)
the optimal surplus conditions hold:
- 1,
is zero unless
(11) both
following
z > 0,
(12a)
zexp(-Pu)-w(l-u)>O.
(12b)
If so there is a unique optimal surplus level at some positive B *. Conditions (12) are not of easy reading, anyway to gain some insight, at least concerning (12a), note that
w)+P(1-w)2
+w(1-u)-1)
(4
uu < 1 (0
It is immediately (orP-l)$O.
seen that (10)
This means that the optimal surplus is unbounded (when P > l), or at the other extreme zero (when P < 1). Let us try to discuss the (rather simple) reasons of this ‘strategic extremism’.
>l/(l-w), so that finally z > 0 implies Puu > 1; that is, something denoting an underwriting profit adjusted to keep account of financial conditions. After that things are different at low or high level of surplus. As the marginal growth in expected business life is intuitively decreasing with surplus, the marginal adjusted underwriting profit coming from surplus is decreasing too, and only at levels lower than that of a boundary surplus B* > 0 exceeds the constant marginal financial loss deriving from a financial reward on the surplus less than the neutral one.
F. Pressacco / Theory of financial decisions
4. A numerical example and some comments Concerning the exponential case previously discussed Table 1 reports the optimal values of the surplus (four exact decimal figures) corresponding to r = 0.05as a function of the underwriting conditions contained in P [recall E(X) = 11, and of some convenient values of the discount factor. Some comments are in order: reading the columns we see that B*(P) has the same regular shape (for any u). Precisely at low levels of underwriting profitability the optimal surplus is increasing: intuitively the greater the underwriting profit the bigger the desire to increase the business life through a choice of a big enough surplus. But as the underwriting profit is high enough it becomes able to offer a self protection from ruin, thus lowering the need for surplus (remember that on the financial side you would like to shorten surplus). It is rather interesting to note that anyway at realistic underwriting conditions (say 1.10 < P <
Table 1 Optimal surplus level B * (P, 0); r = 0.05, E(X) P
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30
51
1.30) B* is an increasing function of P. We will come back to this point later in Section 7. On the other side, a reading of the rows reveals that the optimal values are for any P a decreasing function of the discount rate. This is even more intuitive: given underwriting conditions an increase in the discount rate increases the financial loss, and this forces to lower the surplus. It is clear that looking at reasonable values of the discount factor (say 1.10-l) some regulation is needed if we don’t want to allow that bad companies [with a too small ratio P/E(X)] enter in the market without a convenient surplus coverage. Indeed the difference between good and bad companies is that once exogeneously given a minimum surplus &, the good ones spontaneously work with their optimal surplus greater than _B, while some others are forced to work with the surplus imposed by regulation at a level greater than their unconstrained optimum (provided that V(B) is greater than zero), while only the really bad ones, having V(B) < 0, are forced to leave the market.
= 1.
”
5. Comments
l/(1.06)
l/(1.07)
l/(1.08)
l/(1.09)
l/(1.10)
0.0000 1.9559 3.2193 3.7713 4.1016 4.3216 4.4936 4.6207 4.7208 4.8009 4.8656 4.9182 4.9610 4.9951 5.0236 6.0456 5.0627 5.0754 5.0842 5.0897 5.0921 5.0918 5.0891 5.0842 5.0772 5.0684 5.0579
0.0000 0.3750 1.8388 2.5787 3.0093 3.2949 3.5000 3.6547 3.7153 3.8713 3.9486 4.0116 4.0630 4.1049 4.1390 4.1665 4.1882 4.2051 4.2177 4.2265 4.2319 4.2344 4.2342 4.2315 4.2260 4.2197 4.2110
0.0000 0.0000 0.9153 1.7300 2.2363 2.5740 2.8150 2.9954 3.1352 3.2460 3.3352 3.4078 3.4672 3.5159 3.5557 3.5882 3.6143 3.6351 3.6511 3.6629 3.6712 3.6761 3.6782 3.6775 3.6746 3.6695 3.6624
0.0000 0.0000 0.2999 1.0872 1.6280 2.0025 2.2728 2.4756 2.6326 2.7569 2.8570 2.9385 3.0053 3.0603 3.1055 3.1426 3.1728 3.1971 3.2163 3.2310 3.2418 3.2491 3.2533 3.2546 3.2534 3.2499 3.2443
0.0000 0.0000 0.0000 0.5997 1.1383 1.5312 1.8218 2.0423 2.2138 2.3499 2.4591 2.5492 2.6228 2.6835 2.7337 2.7751 2.8090 2.8367 2.8588 2.8762 2.8893 2.8988 2.9049 2.9080 2.9085 2.9065 2.9023
on the choice of the discount factor
This paragraph is devoted to a deeper look at the meaning and role of the discount factor in our model. At the time of his path-breaking paper de Finetti seemed not to be much troubled by the choice of the value function. After having introduced the one
E very close to ours, he himself suggests the opportunity to keep account of some higher moments than the pure expectation of discounted profits. Moreover he looks at the discount factor as something related to the decision maker’s temporal preferences and then treats it automatically as a given constant with respect to the other relevant variables of the problem. These ideas have remained prevalent in risk theory literature until now, but a careful look to the question reveals that both points are just two sides of the same coin. After Sharpe (1964), Lint-
F. Pressacco / Theory of financial decisions
52
ner (1965) and Mossin (1966) and their multiperiod followers [most notably Fama (1977)] we know that under proper equilibrium conditions on financial markets the relevant value of a business project is its market value; that is, just the net expected value of its cash flow [our goal function (6) in risk theory applications] but actualized at a discount factor that reflects the relevant (non-diversifiable) riskiness of the project (and not subjective temporal and or risk preferences of’ the decision maker). Then according to the theory of financial economics it is sound to use a discounted expectation as a value function, but realizing that the discount factor is constant with respect to other relevant variables of the problem (here the surplus) only if the relevant riskiness of the project does not depend on those variables! Without discussing at length this point involved, my opinion is that, either if you measure the riskiness of the insurance project in a naive (mean square deviation) or in a more refined (covariance with the market rate of return) way, changing the surplus quite likely changes the relevant riskiness of the project: but this means unfortunately that our surplus choice becomes much more involved as formally the discount factor u becomes some function of B, whose shape is even difficult to realize! 2 A deeper insight on this critical point is beyond the scope of this paper, thus here (just as an exercise) we want to show a table of the market values V(B) generated by a tentative choice of the following simple discount functions (see Tables 2a, b):
uh(B)=(l+ih(B))-I,
h=l, 2,3,
03)
Table
2a
Market count
values function;
V(B, oh(B)) as a function of surplus and disP =l.l, r = 0.05, E(X) = 1.
B
u,(B)
uz(B)
u,(B)
0.0
1.2500
1.2466
1.2433
0.1
1.2656
1.2573
1.2496
0.2
1.2808
1.2661
1.2525
0.3
1.2953
1.2723
1.2515
0.4
1.3088
1.2756
1.2459
0.5
1.3210
1.2752
1.2350
0.6
1.3313
1.2708
1.2183
0.7
1.3396
1.2617
1.1051
0.8
1.3453
1.2475
1.1651
0.9
1.3481
1.2276
1.1279
1.0
1.3476
1.2018
1.0832
1.1
1.3433
1.1697
1.0310
1.2
1.3350
1.1312
0.9713
1.3
1.3224
1.0861
0.9043
1.4
1.3052
1.0344
0.8302
1.5
1.2832
0.9764
0.7496
1.6
1.2564
0.9121
0.6629
1.7
1.2408
0.8981
0.6094
1.8
1.2247
0.8420
0.5706
1.9
1.2254
0.8591
0.5534
2.0
1.2076
0.8176
0.4952
The idea behind this type of discount functions is very simple: it is assumed that the riskiness of the project is sharply increasing for low levels of
Table Market count
2b values function;
V( B, o*(B)) as a function of surplus and disP = 1.2, r = 0.05, E(X) = 1.
B
u,(B)
02(B)
u,(B)
0.0
1.5774
1.5728
1.5683
0.1
1.6201
1.6088
1.5981
0.2
1.6637
1.6435
1.6250
0.3
1.7077
1.6764
1.6481
0.4
1.7519
1.7066
1.6664
ir( B) = 0.061 + O.OlB
if B G 1.7,
0.5
1.7956
1.7335
1.6791
ir( B) = 0.075 + O.OOlB
if B > 1.7;
0.6
1.8383
1.7561
1.6853
0.7
1.8794
1.7739
1.6842
i2(B)
if B G 1.7,
0.8
1.9182
1.7859
1.6752
1.9542
1.7915
1.6577
i,(B)=0.089+0.002B
ifB>1.7;
0.9 1.0
1.9867
1.7901
1.6313
i3( B)=0.063 +O.O31B
if Bg 1.7,
1.1
2.0151
1.7812
1.5958
if B > 1.7.
1.2
2.0387
1.7644
1.5511
1.3
2.0571
1.7395
1.4974
1.4
2.0698
1.7064
1.4349
1.5
2.0763
1.6652
1.3641
1.6
2.0765
1.6159
1.2855
1.7
2.0701
1.5589
1.1997
1.8
2.1279
1.6698
1.2869
1.9
2.1430
1.6537
1.2486
2.0
2.1545
1.6341
1.2073
i,(B) ’
= 0.062 + 0.021B
= 0.104 + 0.003B
Formally
this produces
a modified
goal function
~?(R)(~‘~+~‘“F~(~)dx-~~~((p+B)u))
V(B)
=
I-u(B)F,((P+B)u) -B.
as follows:
F. Pressacco / Theoty of financial decisions Table 3 Optimal surplus
level B *(P,
uh( B)).
P
u,(B)
+(B)
uj(B)
1.1 1.2
0.9 2.3
0.4 0.9
0.2 0.6
the surplus (as you face some possibility of losing everything) but later slowly increasing (indeed this is just what happens to the mean square deviation of the goal function obtained by simulation). Note that the covariance with the market rate of return is here completely left out of the picture. As resumed in Table 3, here too the optimal surplus is an increasing function of the premium level and a decreasing function of the riskiness embodied in the discount function.
6. The surplus dividend puzzle The starting point of the second part of our paper is that the surplus strategy of the company determines a dividend strategy and that constant surplus strategies (as well as barrier strategies) quite likely generate erratic paths of dividends never found in the real world. Then we must face what could be called a surplus dividend puzzle: 3 rational surplus strategies give rise to unrealistic dividend paths, and real life smooth dividend policies seem to imply silly or largely suboptimal surplus decisions, thus giving up the opportunity to control the surplus evolution, that is, the very same idea of dynamic managerial rationality. How to escape from the dilemma? Leaving the easy but blind way of simple frictionless insurance markets to keep proper account of such factors as (i) taxes 4 and transaction costs, and (ii) incomplete information. Let’s give here only an informal and quick picture of the role of the first type factors; the
An entirely different approach to solve the surplus dividend puzzle was proposed by Ettle (1984). We don’t discuss here his paper. The role of the tax system in altering the optimal decision rules in a lot of economic problems is generally recognized as one of the key points in modern finance. See chapter 3 in Altman and Subrahmanyam (1985).
53
impact of imperfect information will be discussed later in some detail. What is the effect of taxation on our surplus problem? We are going here to discuss what strategic changes are induced from a different tax treatment of dividends and capital gains (leaving aside the effect of corporate taxes). Precisely suppose that some taxes are to be paid on distributed dividends. Then a strategy trying to avoid to pay dividend taxes retaining (partially or totally) realized gains, and thus accepting a tendency of the surplus to increase over time can be the best after tax strategy, as the one that maximizes the net present after-tax value of the insurance project. This destroys, even under invariant underwriting conditions, the optimality of a constant surplus strategy. But what about liquidity needs of stockholders with less or no dividends? Of course they are satisfied by selling shares and under advantageously enough tax treatment of capital gains in comparison to dividends, this gives more money than adopting the constant surplus strategy optimal in a world without taxes. This preference for a changing surplus and smooth dividends is reinforced by the existence of transaction costs to raise new capital to restore the optimal surplus after some losses. As non-proportional transaction costs determine a big cost to restore small losses it is much better to keep a safety cushion of non-distributed profits devoted to the covering of these needs, thus avoiding a too frequent recourse to the financial market.
7. The role of parameter uncertainty Let’s discuss in some detail the impact of imperfect information on surplus dividend strategies. We will consider the case where the parameter(s) governing the global claims distribution (X in our example) of any period is no more surely known (and constant over time) but carries some uncertainty. Precisely it is seen as a random number A whose probability distribution is updated according to Bayesian rules from some initial prior as long as new informations are obtained. Under this scenario and with informations deriving only from data collected about past claims, the sequence { X,, } is no more i.i.d. but exchangea-
F. Pressacco / Theory of financial decisions
54
ble or independent only conditionally to any value of the parameter. This being the situation a good information (i.e. a low level of global claims of last exercise) implies that the posterior distribution of the parameter reflects more optimism about future profit opportunities than the initial one; and conversely after bad information. This in turn implies that contrary to the stationary case (with invariant opinions) it is no more rational to follow a constant surplus strategy; we guess that better results are obtained applying a simple adaptive surplus strategy defined at any instant as the weighted average of optimal surplus for the parameter certainty, with weights given by the posterior probability distribution of the parameter, or formally at time n, by K+=(A) = j-B*(h)f,(A) A
dh
(14)
with B*(h) as the best constant surplus strategy under parameter certainty as defined in Section 3. We don’t try here to offer a theoretical rationale for choosing this strategy rather than some other adaptive strategy. 5 In the next chapter the performances of the adaptive strategy previously defined are compared with those of the constant surplus strategy defined as the one that satisfies mBax/ V(B, A
h)f,(X)
dX
(15)
with &(X) as the prior distribution of the parameter at the beginning of the story, and obviously V(B,A) as the market value of the constant strategy B for a sure value X of the parameter A.
8. Numerical results for the exponential case
gamma
We present here some numerical results for the case where the prior distribution of the random
5
Yet we found that the chosen strategy turns out to offer better performances (at least looking at results coming from some simulations) than the strategy that maximizes
exponential parameter A is r(a, with density = -Aa*-’ p” r(a)
g,(X)
j?), or, formally,
exp( -/3X).
(16)
Under this hypothesis it is easy to check that the posterior distribution after having observed a n-sequence 5 of claims is given by g,(Vx) (p+
C.X,)~+~A~+‘-~
exp-
(p+
Cxi)X. f
r(a+n)
(17) that is, a gamma again with parameters (Y+ n and p + Xx,. Moreover it could be proved that the unconditional claim distribution is Pareto ((Y, p); that is, with density
(18) and, respectively claims,
fn(x/x) =
after an observed n-vector x of
(a+n)(P+ CxJaCn. (p+
CX,+X)a+n+l ’
09)
that is, again Pareto ((CX+ n), (j3 + Xx,). After that we offer (Table 4) a comparison of the estimated market values (average discounted expected values of the dividend cash flows offered by a computer simulation) produced by the choice of the two surplus strategies previously described.
Table 4 Market values, relative frequency of ruin and initial surplus for the constant surplus (c.s.) and the adaptive surplus (a.s.) strategy; P =l, r = 0.05, u=1.07-‘.
(a. P) m.v.c.s.
m.v.a.s.
(2)/
(1)
(2)
(1)
8.5797 8.7507 8.8642 3.0489 4.2294 4.3465 1.9672 3.1006 3.5614
9.1882 9.4820 10.1452 3.6575 4.1965 4.7924 2.6259 3.4482 3.9993
1.07 1.08 1.14 1.19 0.99 1.10 1.33 1.11 1.12
fr.c.s.
fr.a.s.
B, c.s.
B, as.
0.145 0.13 0.11 0.8 0.61 0.40 0.98 0.71 0.44
0.255 0.215 0.20 0.55 0.50 0.47 0.64 0.55 0.51
2.6037 2.7271 2.7945 0.6871 1.4893 2.1574 0.3165 1.4579 2.2644
0.6978 0.6485 0.6110 1.3553 1.4267 1.4711 1.3412 1.4960 1.6018
F. Pressacco / Theory of financial decisions Table 5 Couple of values of the parameters for the prior ga( X) used for simulation, and related expected value of the unconditional claim distribution. (a. P)
(Y
P
E(X)
1 2 3 4 5 6 7 8 9
1.2 1.2 1.2 3.2 3.2 3.2 5.2 5.2 5.2
0.2 0.18 0.166 2.2 2 1.83 4.2 3.81 3.5
1 0.9 0.83 1 0.9 0.83 1 0.9 0.83
The simulation is based on 200 sequences of Monte Carlo random claims concerning 50 underwriting periods, 6 for the following nine different combinations of (a, p) values of the prior g,(h) (table 5). Some comments are in order: (1) The adaptive strategy gives almost uniformly better results as measured by market values of the respective projects (the average advantage is about 11%). (2) The highest values for both constant and adaptive strategy are found for low values of the alpha coefficient; that is, for higher uncertainty about A. This is not surprising as the uncertainty is exploited on the upper side, but in the absence of regulation only partially suffered downside. We are very near to a well-known low of option pricing theory: the price is ceteris paribus an increasing function of volatility. (3) The relative frequency of ruin is for both strategies uniformly too high. Under the assumptions of the model a surplus regulation rule is absolutely needed to grant survival at the level desired by policyholders. Alternatively, a major cut in ruin probability could be obtained forcing a change in operating conditions.
Of course, the real life of the company could be less than 50 years in case of ruin in a shorter period. In case of survival the final surplus value is discounted and added as a final dividend to the discounted sum of dividends. Note that this is not a correct procedure: we should add the (discounted) market value of the surplus instead of the same surplus.
9. Comparison
55
of dividend paths
We are now going to investigate if adaptive strategies determine dividend paths smoother than those produced by constant surplus strategies. As said in the introduction the intuition behind this idea is that as long as B*(h) is under parameter certainty an increasing function of the expected underwriting profitability, the adaptive strategies enjoy the property that the surplus level tends to be raised after good periods and lowered after bad periods. This in turn should determine a desire to retain part of the profit obtained in good periods (to reach the new upper desired surplus level) or symmetrically to restore only partially the surplus after a loss. Unfortunately, the numerical results obtained don’t give unambiguous support to this idea, as we can see from Table 6. Note that, as under constant surplus strategies the dividend distributed is (in case of survival) exactly equal to the overall exercise profit (loss), higher variability of dividends means that on average a dividend bigger than the profit (loss) is distributed (or get from) stockholders under the adaptive strategy. This is apparently surprising but happens any time a claim greater than the updated expectation is detected (thus lowering the desired surplus), but a profit is at the same time realized, either coming from the loading charges or from financial profits. Indeed, while the premium level stays constant by assumption, the past story may have produced a big departure of the updated expected claim from the one initially expected, and this opens the
Table 6 Average mean square deviation of dividend paths implied by constant and adaptive surplus strategies for some selected couples of ((Y, p). (a. B)
m.s.d.c.s.
m.s.d.a.s.
1 2 3 4 5 6 7 8 9
0.71 0.64 0.71 0.44 0.47 0.43 0.42 0.41 0.38
0.57 0.45 0.58 0.57 0.54 0.50 0.46 0.44 0.42
56
F. Pressacco / Theory of financial decisions
way to a big enough excess of dividends under the adaptive strategy. This explains why more erratic dividend paths may be found for the adaptive strategy; then a more refined model with feedbacks from changing expectations to the premium level and eventually to the business volume, keeping account of such factors as dividend taxes and transactions costs on financial markets is needed to solve the surplus dividend puzzle. We hope that the model presented in this paper could be seen just as the first step toward this direction.
References Altman, E. and M. Subrahmanyam (1985). Recent Advances in Corporate Finance. Irwin, Homewood, IL. Borch, K. (1968). The rescue of an insurance company after ruin. The Astin Bulletin, 5, 280-292. Borch, K. (1974). The Mathematical Theory of Insurance. Lexington Books, Lexington, MA. Borch, K. (1984). An alternative dividend policy for an insurance company. Mitteilungen der Vereinigung schweiz. Versicherungsmathematiker, 2 201-206. Bowers, N., H. Gerber, J. Hickman, D. Jones and C. Nesbitt (1987). Actuarial Mafhematics. Society of Actuaries.
Brealey, R. and S. Myers (1984). Principles of Corporate Finance. McGraw Hill, New York. Buhlmann, H. (1970). Mathematical Methods in Risk Theory. Springer Verlag, Heidelberg, New York. de Finetti, B. (1957). Su un’impostazione altemativa della teoria collettiva de1 rischio. Transactions of the 15th International Congress of Actuaries, Vol. 2, 433-443. Ettle, W. (1984). Optimale dividendenpolitik. Transactions of the International Congress of Actuaries - Sydney, Vol. 3, 435-445. Fama, E.F. (1977). Risk adjusted discount rates and capital budgeting under uncertainty. Journal of Financial Economics 5, 3-24. Gerber, H. (1979). An Introduction to Mathematical Risk Theory. Huebner Foundation, distributed by Irwin, Homewood, IL. Lintner, J. (1965). The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, 13-37. Markovitz, H. (1952). Portfolio selection. Journal of Finance 6, 77-91. Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica 34, 768-783. Robichek, A.E. and S.C. Myers (1966). Conceptual problems in the use of risk adjusted discount rates. Journal of Finance 21, 727-730. Seal, H. (1969). Stochastic Theory of Risk Business. Wiley, New York. Sharpe, W.F. (1964). Capital asset prices: A theory of market equilibrium under condition of risk. Journal of Finance 19, 425-442.