K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 359
ˆ Rt
≈ ˜
R
t
, ≤ t τ ∧ T ,
−∞, τ ≤ t ≤ T , 46
which means that ˆ Rt
≈ Rt as given in Eq. 40 in which case ˆSt ≈ St = S0 e
Rt
, ≤ t ≤ T ,
where St is the same as given in Eq. 43. We now make the following final approximation.
Assumption 3.
P
t
x, A = P
x
˜ R
t
∈ A, τ t = Q
t
x, A e
−λt
47 for any t
≥ 0, and Borel set A of real. This is only correct if τ is P
x
independent of ˜ R for all x. In other words, we construct a new
˜τ, by possibly enlarging the probability space, having the same probability distribution as τ and being P
x
independent of ˜ R for
all x, and consider the process ˜ R killed at
˜τ. The Feller probability transition functions of this constructed process is given by the right-hand side of 47.
Assumption 3 may be reasonable in the present model if the default probability of the company is independent of the return level of its stock. The underlying process may not strictly satisfy this property, but we shall nevertheless
work with the constructed process having transition probabilities given by the right-hand side of Eq. 47 in the sequel. We can now alter the analysis of Sections 3 and 4 using the above approximations. Conditioning on the event
{ω : τ ω T } of no failure by time T in Eq. 20, then in Eq. 21 we get µ − λ instead of µ in the exponent. In Eq. 22, we get λ
− µ instead of −µ in the exponent as well, and the equilibrium restriction between the parameters in Eq. 23 now becomes
ρ − µ + λ = δ
p ψ
− p
ψ + 2η − 2β − 1.
48 However, Eq. 24 for the equilibrium interest rate r remains unchanged.
Eq. 27 gets a µ − λ instead of µ in the expression for a, while the equations for the price of the European call
option are changed slightly. Eq. 28 gets two corrections in both the exponents of the exponential multiplication factors, where the term ρ changes to ρ
+ λ both places, and finally, in Eq. 29 the exponent of the exponential in front of the integral sign is changed from ρ to ρ
+ λ as well. All these changes are of course now approximations.
7. The survival hypothesis
Survival rates for markets were used in the paper by Brown et al. 1995 to possibly explain the equity premium puzzle. They assumed that the real risk premium µ
R
− r could be low, possibly as low as zero, and they argued that the observed risk premiums in the markets that actually survived ought to be larger. The estimates above are
really constructed on the event {τ ω 90}, and in principle we should not be able to accurately estimate the failure
rate from observing only one stretch of data that survived. However, since the parameters are related in equilibrium according to the constraint 48, we are able to estimate both the risk aversion, λ and µ
R
. Below we demonstrate how this can be done.
We intend to use the stopped version of the previous section to give a possible test of the survival hypothesis for stock markets. Let us first derive a version of the CCAPM for the stopped version. To this end, consider the process
R stopped at τ = inf{t; 1 ˜
R
t
≤ −1}, and let µ
R
t be the rate corresponding to R. We now compute EdR
t
|F
t
= Ed ˜ R
t
|τ t + dt, τ t, F
t
P τ t + dt|τ t, F
t
+Ed ˜ R
t
|τ ∈ t, t + dt, τ t, F
t
P τ ∈ t, t + dt|τ t, F
t
.
360 K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363
Because of the limited liability in the stock market, the loss can be maximum 100, so we must have E
{dR
t
|F
t
} = µ
˜ R
t dt e
−λt+dt
e
−λt
− 1λ e
−λt+dt
e
−λt
dt, or, letting E
{dR
t
|F
t
= µ
R
t dt , we obtain µ
R
t − r =
µ − r − λ +
βδ √
ψ .
49 Eqs. I–V in Section 5 will be the same in the present version. In addition, we have Eq. 49 and the stopped
version of Eq. 23, which is Eq. 48 above. The equilibrium ruin intensity λ =
R
−∞,−1
νdy, where the Lévy measure νdy is given in 15. Some easy substitutions, using 39, imply that λ can be written as
λ =
δα π
Z
∞ 1
t t
2
− 1
−12
Z
∞ αt
+β
z
−1
e
−z
dz dt.
50 This expression for λ constitutes the eighth equation, which, together with the remaining seven equations, determine
the values of all the parameters, including µ
R
, λ, RRA = ηc, and ρ, calibrated to the above data.
This system of equations has also two solutions: The first is ˆλ = 0.000005, ARA = ˆη = 1.96, RRA = 3.23,
ˆρ = 0.044, and the processconsumption-related parameters are ˆα = 4.64, ˆ β
= 2.67, ˆµ
R
= 0.0698, ˆµ = 0.02, ˆδ = 0.0695, and ˆψ = 14.41.
The changes from the previous solution are modest. Of course the most interesting new value is here the estimate for λ. It yields an estimate for the survival probability for the 100-year period of the NYSE, which is 0.9995, giving
a crash probability of 0.05 over the same period, a value somewhat smaller than conjectured by Brown et al. Thus, the data does not give much support for the crash hypothesis.
Increasing the crash probability from 0.05 to 24.42 and thus ignoring Eq. 50 lowers the estimate of RRA to 3.12 and the estimate for the impatience rate to 0.042, which are rather moderate changes. A crash probability
during the last 100 years of 25 is more in line with the conjecture of Brown et al. In other words, such an increase in the market failure probability does not seem to affect the market parameters to any significant degree according
to the model of Section 6.
8. Several primitive assets
