|
Figure 1 |
Figure 2 |
Figure 3 | |||||||||
|
Y1 |
5 |
δ 1 |
4.5 |
Y1 |
20 |
δ 1 |
20 |
Y1 |
20 |
δ 1 |
19 |
|
γ2 |
3 |
δ2 |
3 |
Y2 |
3 |
δ2 |
12 |
Y2 |
18 |
δ2 |
6 |
|
γ3 |
1 |
δ 3 |
1.2 |
Y3 |
1 |
δ 3 |
2 |
Y3 |
1 |
δ 3 |
2 |
|
T^ |
α1 |
T |
ʌ ' |
10 - 3 |
α1 |
Γ |
ʌ ' |
10 |
α1 |
1 | |
|
λ2-1 |
.6 |
α2 |
1 |
ʌ - 1 |
20 |
α2 |
1 |
ʌ - 1 |
6 |
α2 |
0.001 |
|
ʌ 3 1 |
___1 |
α 3 |
.2 |
ʌ - 1 |
______1 |
α 3 |
.01 |
ʌ - 1 |
___1 |
α 3 |
0.0001 |
Table 1: For each figure the table shows the RRA of the three investors
(γ) and their relative expected marginal utility (λ-1). The parameters used
in the polynomial approximation of the stochastic discount factor are the
exponents (δ) and the weights (α).
with αi, δi ∈ R. This specification is quite general. Since the δi,s are not
required to be integers, this approximation is at least as good as a Taylor-
series approximation. We use polynomials with N = 3 terms. Table 1
displays for each figure of the previous section the parameters of the investors
and the parameters δi and αi used in the generalized polynomial.
The table shows that the exponents δ1 and δ3 used in the polynomial approx-
imation of the stochastic discount factor correspond closely to the RRA γ1
resp. γ3 . The quality of the approximation can be seen in the lower graphs
of Figures 1 to 3 depicting aggregate RRA derived from the social planner
model (fat curve) and aggregate RRA derived from the approximation (thin
curve). The approximation appears to be quite good. It could be further
improved by using more than three power functions (see also Diking and
LUders, 2005).
The simulation of the process of the price of the market portfolio can be
facilitated strongly if the price at any time can be derived analytically. To
achieve this, we approximate the infinite horizon setting by a finite horizon
setting. The date t-price of the market portfolio is the present value of future
dividends. We approximate this value by the present value of dividends until
a given horizon t + h and a suitable approximation for the horizon wealth
generated by subsequent dividends. The horizon h is constant over time so
that t + h moves over time. The asset price at the horizon is a deterministic
function of the dividend paid at the horizon, St+h = St+h(Dt+h) consistent
with the infinite horizon model. The approximation used here is that at
the horizon the elasticity of the asset price with respect to the dividend is
assumed to be constant, i.e. St+h = dpD%+ h. dp is a kind of price dividend
ratio. The exponent iï is assumed to be greater or equal to 1 indicating
declining or constant aggregate RRA (see Proposition 1). Since we use a
very long horizon of 240 months, we expect the impact of this approximation
on our simulation results to be very small.13
13 Further numerical simulations based on a finite horizon model that are not shown in
15