Framework and notation

Consider an insurance portfolio subject to liability payments L(i) _{≥ 0 at} times i = 1, 2, . . ., where i = 0 denotes the present. Let L(i) be a random variable and suppose that it is modified by certain forces that influence the liability over time.

For example, suppose that L(i)_t denotes the amount of liability ex- pressed in money values of time i. Then L(i)_t evolves in the sense that

L(i)_t = L(i)_t−1RLt, t = 1, . . . , i,

where the RLt are strictly positive random variables of the form

RLt= 1 + rLt,

with rLt the inflation of claims costs over interval (t− 1, t]. The liability

finally paid is

L(i) = L(i)_s .

As an example, L(i)_t−1 and RLt might be independently distributed as fol-

lows:

L(i)_{t−1 ∼ logN(ν, τ}2) and RLt ∼ logN(µ, σ2).

It is emphasized that, in this example, rLt denotes claims inflation. This

might include influences other than simple community inflation, such as the particular pressures of the legal and health care environments on claim costs.

Similarly, a holding of assets of value A_t−1 at time t− 1 accumulates at time t to

At= At−1RAt,

with

Assume that RXt, where X is either A or L, follows the capital Asset

Pricing Model (CAPM):

rXt = rF t+ βX∆t+ Xt, (2.57)

where rF tis the risk-free rate in period t, βX is the CAPM beta associated

with X, Xt is the idiosyncratic risk associated with X, and

∆t= rM t− rF t,

with rM denoting the period increase in value of the economy-wide port-

folio of assets. The distribution of ∆t is assumed independent of t. The

assumption of CAPM returns is consistent with an assumption that assets and liabilities here are marked to market.

Henceforth, it will be assumed that rF t = rF, independent of t. This

simplifies the following algebraic development considerably. It should be emphasized, however, that the whole development generalizes to the case in which rF t varies with t. The generalization is theoretically straight-

forward, but adds considerable notational baggage without yielding any deeper insight.

Assume that the At are i.i.d. and similarly the Lt. Assume that all

variables At, Lt and ∆tare stochastically independent, and that E[Xt] =

0. Let us further denote the variance of Xt with ωX2.

It follows that the RAt and RLt are independent and identically dis-

tributed. Suppose now the following distribution assumptions: L(i)_{0 ∼ logN νL0}(i), τ_L02(i)
and RXt∼ logN µX, σX2

, (2.58)

with stochastic independence between L(i)₀ and RXt for all i, t, and X =

A, L. Denote

ρ = Corr(logRAt, logRLt)

and

κ(rs)= Corr logL(r)₀ , logL(s)₀
. Define the accumulation factor

RXt:u = RX,t+1RX,t+2. . . RXu, for u = t + 1, t + 2, . . .

By relation (2.58) and the independence between distinct time inter- vals,

RXt:u∼ logN (u − t)µX, (u− t)σ2X

.

The implicit asset allocation is any that is consistent with relation (2.58). One might assume, for example, a constant allocation by asset sector, with continuous rebalancing and sector-specific returns that are constant over time. As remarked earlier in this section, the last of these assumptions could be weakened. Indeed, if the assumptions of constant returns over time were weakened, no assumption would be required with respect to asset allocation. Define the discounted liability payment

V(i) = L(i)_i R−1_A0:i = L(i)₀ RL0:iR−1_A0:i

= L(i)₀

i

Y

j=1

(RLjR−1_Aj)

∼ logN(α(i), δ2(i)), with α(i) _{= ν}(i)

L0+ i(µL− µA) and δ2(i)= τ 2(i)

L0 + i(σL2+ σA2 − 2ρσLσA). The

present value S, given by

S = n X i=1 V(i):= n X i=1 eZi_{, (2.59)}

with n the number of cash-flow liabilities in the discounted value of the total outstanding losses of the portfolio.

In Taylor (2004), the mean and variance of S are calculated and given by E[S] = n X s=1 E[V(s)] = n X i=1 E[L(s)₀ ] ¯R_¯L RA  1 + (β2_Aσ_M2 + ω2_A)/ ¯R2_A 1 + βAβLσ2M/ ¯RAR¯L s ,

Var[S] = n X r,s=1 Cov[V(r), V(s)] = n X r,s=1

E[V(r)]E[V(s)]expκ(rs)τ_L0(r)τ_L0(s)

+ min(r, s)[σ2_L+ σ_A2 _{− 2ρσ}AσL]

 − 1, with ¯RX = E[RXt] and σM2 = Var[rM t]. We will denote the variance of S

by σ2 S.

There are now three relevant values of loss reserve: • Pns=1E[L

(s)

0 ], which is the CAPM-based economic value of the lia-

bility.

• E[S], which is the expected value of the discounted liability cash flows, the discount rate taking into account the insurers asset hold- ings.

• Ap = F_S−1(p) = E[S]exp(σSΦ−1(p)− 1₂σ2_S), which is the p× 100%-

confidence loss reserve.

It may be convenient to write the last of these conditions in the form Ap= [1 + η(ρ, σS)]E[S],

where η(ρ, σS) may be regarded as a security loading. Note, however,

that the security loading in this formulation is applied to E[S] and not to the economic value of the liability. The first two of the above three possibilities for loss reserve are the ones involved in the current debate over the appropriate rate(s) at which to discount liabilities. The quantity E[S] is obtained using the expectations of discount factors that reflect the insurers expected returns. In broad (though not quite precise) terms, it may be thought of as the amount of assets which, accumulating with expected investment return, will be sufficient to meet liabilities as they are required to be paid. This value depends on the insurer-specific asset holdings, and so cannot be market or fair value of the liabilities. This is given by the first of the above three candidates for loss reserves.

Taylor (1996) pointed out for high security margins (Φ−1(p) > σS), the

for low security margins (Φ−1(p) < σS), the size of the security margin

decreases with increasing asset beta. In this latter case the additional yield expected from an increased asset risk outweighs the additional risk.

Taylor (2004) defines the security margin for confidence level p as SMp[S] := η(p, σS) = (VaRp[S]/E[S])− 1, which is based on the quan-

tile risk measure from the distribution of the discounted reserve S. In general, it is hard or even impossible to determine the quantiles of the dis- counted reserve analytically, because in any realistic model for the return process the random variable S will be a sum of strongly dependent ran- dom variables. Here, S is is a finite sum of correlated lognormal random variables. This implies that its cumulative distribution function cannot be determined exactly and is even too cumbersome to work with. An inter- esting solution to this difficulty consists of determining the lower bound Sl and the upper bound Sc as explained earlier in this chapter.