Single-year and multi-year insurance policies in a competitive market

Abstract

This paper examines the demand and supply of annual and multi-year insurance contracts with respect to protection against a catastrophic risk in a competitive market. Insurers who offer annual policies can cancel policies at the end of each year and change the premium in the following year. Multi-year insurance has a fixed annual price for each year and no cancellations are permitted at the end of any given year. Homeowners are identical with respect to their exposure to the hazard. Each homeowner determines whether or not to purchase an annual or multi-year contract so as to maximize her expected utility. The competitive equilibrium consists of a set of prices where homeowners who are not very risk averse decide to be uninsured. Other individuals demand either single-year or multi-year policies depending on their degree of risk aversion and the premiums charged by insurers for each type of policy.

Appendices

Appendix A: average insurer costs and prices in a competitive market

Competitive equilibrium in both the SY and MY markets occurs where insurers of each type select a BoB that minimizes their average cost, with price given by the minimum of the respective average cost curve. Thus, from (9), and noting that prices in the SY market are set after the state of the world w ∈ {d, u} is known, we have:

$$ {P_{{S1}}} = Mi{n_{{n \geqslant 0}}}\left\{ {\frac{{{C_{{SY}}}\left( {n;{r_1},\zeta } \right)}}{n}} \right\};P_{{S2}}^w = Mi{n_{{n \geqslant 0}}}\left\{ {\frac{{{C_{{SY}}}\left( {n;{r_2}(w),\zeta } \right)}}{n}} \right\},w \in \left\{ {d,u} \right\} $$

(T1)

with the optimal BoBs for the SY insurer being the corresponding solutions, ${\widehat{n}_{{S1}}},\widehat{n}_{{S2}}^d,\widehat{n}_{{S2}}^u$, to the indicated average cost minimization problems, where r₁ and r ₂(w) are reinsurance costs in periods 1 and 2, the latter being state dependent.

Similarly, from (10), the equilibrium price in the MY market is determined by the minimum of the total average cost for the two periods, so that:

$$ 2{P_M} = Mi{n_{{n \geqslant 0}}}\left\{ {\frac{{{C_{{MY}}}\left( {n;{r_1},\zeta } \right) + \varphi {C_{{MY}}}\left( {n;{r_2}(d),\zeta } \right) + \left( {1 - \varphi } \right){C_{{MY}}}\left( {n;{r_2}(u),\zeta } \right)}}{n}} \right\} $$

(T2)

with the optimal BoB for the MY insurer being the corresponding solution ${\widehat{n}_{{MY}}}$ to (T2). Note that, for all n, C _MY (n; r ₂(d), ζ) < C _MY (n; r ₁, ζ) < C _MY (n; r ₂(u), ζ), so that average costs also satisfy: $AC_{{M2}}^d < A{C_{{M1}}} < AC_{{M2}}^u$ and therefore $AC_{{M2}}^d < {P_{{M }}} < AC_{{M2}}^u$, verifying the need for imposing a cancelation fee $\psi \geqslant {P_M} - P_{{s2}}^d - \tau$ to assure expected breakeven operations for the MY insurer in period 2.

Given our assumptions a competitive equilibrium exists for both the SY and MY markets yielding the price vector $\left\{ {{P_M},{P_{{S1}}},P_{{S2}}^d,P_{{S2}}^u} \right\}$ and the BoB vector $\left\{ {{n_M},{n_{{S1}}},n_{{S2}}^d,n_{{S2}}^u} \right\}$. Some of these markets may be degenerate in the sense that there is no demand for one or other of these policies at the equilibrium prices. The assumptions on average costs imply a number of intuitively appealing results for the comparative statics of equilibrium prices and BoBs for both MY and SY insurers. For example, since reinsurance costs in period 2 increase or decrease relative to period 1 depending on the state of the world, equilibrium prices in the SY market satisfy: $P_{{S2}}^d < {P_{{S1 }}} < P_{{S2}}^u$ and the corresponding optimal BoBs satisfy: $n_{{S2}}^d > {n_{{S1}}} > n_{{S2}}^u$. Comparative statics with respect to the parameters in ζ = (μ, σ, ρ, γ ^*, φ, q, τ) can be derived using (9) and (10) together with (A3) and (A4) for SY and MY equilibrium prices. Due to the assumption with respect to average reinsurance costs equilibrium prices increase and equilibrium BoBs decrease as ρ increases for both SY and MY insurers.

We record here one general comparative result between SY and MY policies. Suppose there are no marketing cost advantages for MY insurers (ν = 1 in (10)). Then the equilibrium price vector $\left\{ {{P_M},{P_{{S1}}},P_{{S2}}^d,P_{{S2}}^u} \right\}$ satisfies⁸: ${P_M} \geqslant {P_{{S1}}} + \varphi P_{{S2}}^d + \left( {1 - \varphi } \right) P_{{S2}}^u$. In particular, a risk neutral homeowner would always prefer SY policies to MY policies when ν = 1 and when there are no transactions costs for the homeowner from policy cancellation (τ = 0). Of course, even when τ = 0, risk averse homeowners might still prefer MY policies to avoid the risk of price volatility in period 2. Nonetheless, the above inequality expresses clearly one advantage of SY policies, namely the ability to adjust the BoB in the face of changing reinsurance costs.

Appendix B: proofs of the CARA/Gaussian case

This appendix provides proofs of the basic properties of the CARA/Gaussian case shown in Fig. 3. We assume a fixed parameter vector ζ = (μ, σ, ρ, γ ^*, φ, q, τ) and a price vector $P = \left\{ {{P_M},{P_{{S1}}},P_{{S2}}^d,P_{{S2}}^u} \right\}$. To avoid special cases we assume that σ > τ ≥ 0 so that the uncertainty associated with the hazard is larger than the transactions costs of finding a new policy if the policy is cancelled.

Claim 1:

The solution $\widehat{a}\left( {{P_{{S1}}}} \right)$ in (12) is unique

Proof:

Consider the function arising from (12) defined as:

$$ g(a) = \mu + \frac{1}{2}a{\sigma^2} - \frac{1}{a}log\left[ {L\left( {a,\zeta } \right){e^{{a{P_{{S1}}}}}}} \right]\;{\text{where}}\;L\left( {a,\zeta } \right) = q{e^{{a\tau }}} + \left( {1 - q} \right). $$

(T3)

To establish the claim, it suffices to show the following properties for the function $h(a) = ag(a) = a\mu + \frac{1}{2}{a^2}{\sigma^2} - log\left[ {L\left( {a,\zeta } \right){e^{{a{P_{{S1}}}}}}} \right]$.

• P1: h(0) = 0; h′(0) = μ−(P _S1 + qτ);

• P2: $h(a) > 0\;{\text{for}}\;a > \underline a = \frac{{2\left( {P + \tau - \mu } \right)}}{{{\sigma^2}}};$

• P3: h′′(a) > 0 (i.e., h is strictly convex for a > 0).

Assume P1-P3. Then h(0) = 0 and h′(0) < 0. Since, by P1, h(a) > 0 for a sufficiently large (viz., for $a > \underline a$), it must be that the continuous function h(a) has a minimum in the interval [$0,\underline a$]. However, given P3, this minimum is unique and (again by P3) the value “a” at which h(a) crosses zero is also unique. We need therefore only show that P1-P3 hold (under the assumptions that σ > τ ≥ 0).

P1 is obvious from direct calculation. P2 follows by noting that log( ) is monotonic increasing and e ^aτ > 1, so that

$$ ln\left[ {q{e^{{a\tau }}} + \left( {1 - q} \right)} \right] < ln\left( {{e^{{a\tau }}}} \right) = a\tau . $$

(T4)

Thus, for $a > \underline a$, we have

$$ h(a) = ag(a) > a\left( {\mu + \frac{1}{2}a{\sigma^2} - {{\text{P}}_{{{\text{S}}1}}} - \tau } \right) > \underline a \left( {\mu + \frac{1}{2}a{\sigma^2} - {{\text{P}}_{{{\text{S}}1}}} - \tau } \right) = 0. $$

(T5)

Concerning P3, it can be calculated that

$$ h\prime \prime (a) = {{\sigma }^{2}} - \left( {1 - q} \right){{\tau }^{2}}\left[ {\frac{{q{{e}^{{a\tau }}}}}{{q{{e}^{{a\tau }}} + \left( {1 - q} \right)}}} \right]\left[ {\frac{1}{{q{{e}^{{a\tau }}} + \left( {1 - q} \right)}}} \right]. $$

(T6)

Both fractions in (T6) are clearly <1. Thus, since σ > τ ≥ 0, h′′(a) > σ ²−(1−q)τ ² ≥ σ ²−τ ² > 0, completing the proof of Claim 1.

For the next claim, we need the following property of CARA risk preferences: Let $\widetilde{Y}$ be any random variable with positive variance. The Certainty Equivalent $CE\left( {\widetilde{Y},a} \right)$ under CARA preferences is decreasing in “a”. This follows directly from Theorem 1 of Pratt (1964).

Claim 2:

The functions H ₁(a, NI−MY), H ₁(a, SY−MY) are increasing in a ∈ A.

Proof:

Consider H ₁(a, NI−MY). Divide both sides of (13) by 2. Then, Claim 2 is equivalent to the assertion that the solution P_M to the following equation is increasing in a ∈ A:

$$ U\left( { - {P_M},a} \right) = 0.5U\left( {CE\left( {NI,a} \right),a} \right) + 0.5\varphi Max\left[ {U\left( { - P_{{S2}}^d,a} \right),U\left( {CE\left( {NI,a} \right),a} \right)} \right] + 0.5\left( {1 - \varphi } \right)Max\left[ {U\left( { - P_{{S2}}^u,a} \right),U\left( {CE\left( {NI,a} \right),a} \right)} \right]. $$

(T7)

With an eye on (12), there are three relevant intervals in A associated with (T7): $0 < a < \widehat{a}_2^d:\widehat{a}_2^d \leqslant a < \widehat{a}_2^u:\widehat{a}_2^u \leqslant a$. Observe that the solution P _M (a) = H ₁(a, NI−MY) to (T7) is unique and continuous (since the solution to U(−P _M , a) = K(a) is ${P_M} = \frac{{log\left[ { - K(a)} \right]}}{a}$). Thus, it suffices to show the Claim for each of the three relevant intervals separately.

Consider the first interval, $0 < a < \widehat{a}_2^d$. In this interval, NI is always superior to SY in period 2, so that (T7) can be expressed as

$$ U\left( { - {P_M},a} \right) = 0.5E\left\{ {U\left( { - {{\widetilde{X}}_1},a} \right)} \right\} + 0.5\varphi E\left\{ {U\left( { - {{\widetilde{X}}_2},a} \right)} \right\} + 0.5\left( {1 - \varphi } \right)E\left\{ {U\left( { - {{\widetilde{X}}_2},a} \right)} \right\} $$

(T8)

where ${\widetilde{X}_1},{\widetilde{X}_2}$are the loss distributions for periods 1 and 2 respectively. We see that the RHS of (T8) is the expected utility of the random variable which yields ${\widetilde{X}_1}$ with probability 0.5, and ${\widetilde{X}_2}$ with probability 0.5. The solution P_M to (T8) is clearly just the negative of the CE of this random variable, so that Pratt’s result cited above, establishes the claim for this interval.

Consider the second interval $\widehat{a}_2^d \leqslant a < \widehat{a}_2^u$. In this interval, SY is always superior to NI in period 2, when w = d and inferior to NI when w = u, so that (T7) can be expressed as

$$ U\left( { - {P_M},a} \right) = 0.5E\left\{ {U\left( { - {{\widetilde{X}}_1},a} \right)} \right\} + 0.5\varphi U\left( { - P_{{S2}}^d,a} \right) + 0.5\left( {1 - \varphi } \right)E\left\{ {U\left( { - {{\widetilde{X}}_2},a} \right)} \right\}. $$

(T9)

From (T9) P _M is the negative of the CE of the random variable equal to ${\widetilde{X}_1}$ with probability 0.5, $- P_{{S2}}^d$ with probability 0.5φ, and ${\widetilde{X}_2}$with probability 0.5(1-φ). Pratt’s result implies that P_M is increasing in “a”.

Finally, consider the interval $\widehat{a}_2^u \leqslant a$, for which SY is superior to NI for all states of the world w ∈ {u, d}. In this case, (T7) is equivalent to

$$ U\left( { - {P_M},a} \right) = 0.5E\left\{ {U\left( { - {{\widetilde{X}}_1},a} \right)} \right\} + 0.5\varphi U\left( { - P_{{S2}}^d,a} \right) + 0.5\left( {1 - \varphi } \right)5\varphi U\left( { - P_{{S2}}^u,a} \right). $$

(T10)

Following the same logic as above establishes the claim for this interval. Thus,H ₁(a, NI−MY) is strictly increasing in “a” as asserted in Claim 2.

A similar argument establishes Claim 2 for H ₁(a, SY−MY).

We finally note that H ₁(a, NI−MY) < H ₁(a, SY−MY) for “a” sufficiently small since from (T8) H ₁(a, NI−MY) → μ as a → 0, whereas H ₁(a, SY−MY) > μ since the price P_M that would make a homeowner indifferent between an MY and SY policy is certainly no lower than the lowest SY policy price $P_{{S2}}^d$, which is clearly greater than the mean of the loss distribution μ. We see, therefore, that H ₁(a, NI−MY) is below H ₁(a, SY−MY) for “a” small. On the other hand, as explained in the text, H ₁(a, NI−MY) and H ₁(a, SY−MY) have a unique intersection at $a = \widehat{a}\left( {{P_{{S1}}}} \right)$. As both functions are monotonic increasing, it must be that H ₁(a, NI−MY) > H ₁(a, SY−MY) for $a > \widehat{a}\left( {{P_{{S1}}}} \right)$. These facts establish the basic geometry of Fig. 3.

Appendix C: reinsurance costs for the CARA/Gaussian case

Reinsurance costs in period 1 for the CARA/Gaussian case are given by:

$$ {{C}_{{s1}}}\left( {n;{{r}_{1}},\zeta } \right) = \int_{{{{L}_{1}}\left( n \right)}}^{{{{L}_{2}}\left( n \right)}} {\left[ {{{\lambda }_{1}} - 1 + {{\xi }_{1}}x} \right]} \left[ {1 - F\left( {x,n,\zeta } \right)} \right]dx $$

(T11)

where r ₁ = (λ ₁, ξ ₁) with λ ₁ > 1, ξ ₁ > 0, and where F(x, n, ζ) is the cdf of the normal distribution with mean nμ and variance σ ²[n + n(n−1)ρ] corresponding to the loss distribution $\widetilde{X}(n) = \sum\nolimits_{{i = 1}}^{n} {\widetilde{X}} \left( {{{a}_{i}}} \right)$ for a BoB of size n.

Period 2 reinsurance costs are state dependent and are given by:

$$ {C_{{s2}}}\left( {n;{r_2}(w),\zeta } \right) = \int\limits_{{{L_1}(n)}}^{{{L_2}(n)}} {\left[ {{\lambda_2}(w) - 1 + {\xi_2}(w)x} \right]\left[ {1 - F\left( {x,n,\zeta } \right)} \right]dx.} $$

(T12)

The reinsurance costs (T11) and (T12) are net of expected reinsurance payments (this is the effect of subtracting 1 from the respective loading factors, λ ₁ and λ ₂(w), in the first term under the integral). Thus, the total expected reinsurance payouts and premiums for this XoL treaty are given in a standard linearly increasing loading factor form by:

$$ Payou{{t}_{{s1}}}\left( {n;{{r}_{1}},\zeta } \right) = \int_{{{{L}_{1}}\left( n \right)}}^{{{{L}_{2}}\left( n \right)}} {\left[ {1 - F\left( {x,n,\zeta } \right)} \right]} dx; $$

(T13)

$$ Payou{t_{{s2}}}\left( {n;{r_2}(w),\zeta } \right) = \!\! \int_{L_{1} {\left( n \right)}}^{L_{2} {\left( n \right)}} {\left[ {1 - F\left( {x,n,\zeta } \right)} \right]dx;} $$

(T14)

$$ Premiu{m_{{s1}}}\left( {n;{r_1},\zeta } \right) = \!\!\! \int_{L_{1} {\left( n \right)}}^{L_{2} {\left( n \right)}} {\left[ {{\lambda_1} + {\xi_1}x} \right]\left[ {1 - F\left( {x,n,\zeta } \right)} \right]dx;} $$

(T15)

$$ Premiu{m_{{s2}}}\left( {n;{r_2}(w),\zeta } \right) = \!\!\! \int_{L_{1} {\left( n \right)}}^{L_{2} {\left( n \right)}} {\left[ {{\lambda_2}(w) + {\xi_2}(w)x} \right]\left[ {1 - F\left( {x,n,\zeta } \right)} \right]dx;} $$

(T16)

where r ₂(w) = (λ ₂(w), ξ ₂(w)) with λ ₂(w) > 1, ξ ₂(w) > 0, w ∈ {d, u}.

In line with our assumption that the state d (respectively u) represents a decrease (respectively increase) in capital cost relative to period 1, we assume:

$$ {\lambda_2}(d) \leqslant {\lambda_1} \leqslant {\lambda_2}(u);{\xi_2}(d) \leqslant {\xi_1} \leqslant {\xi_2}(u). $$

(T17)

Note that the cdf F(x, n, ζ) is identical in both periods, since we assume that the hazard distribution is identical in both periods (of course the BoB may change for the SY insurer between periods 1 and 2). For the same reason, for any fixed BoB of size n, the attachment points L ₁(n), L ₂(n) are also unchanged between periods 1 and 2.