CAPÍTULO 3. COMPARACIÓN DE LOS MÉTODOS Y ALGORITMOS
3.4 Comparación de los algoritmos
In this section, we extend our analysis to consider the relativemagnitudeof gains and losses. The disposition e↵ect is commonly understood as the preference for selling assets that have increased in value relative to assets that have decreased in value since purchase. Researchers have recently studied how the rate of sale depends on the relative magnitude of the gain or loss. A number of authors estimate proportional hazards models to derive the hazard rate
17Although such an extension is straightforward and would clearly also be able to achieve realistic levels
for the sale of stock conditional on return since purchase, see Feng and Seasholes (2005), Seru et al. (2010) and Barber and Odean (2013). Others, in particular, Ben-David and Hirshleifer (2012) (see also Kaustia (2010)) document the probability of selling as a function of profits, whilst allowing for di↵erent prior holding periods to be taken into account.
There is broad agreement amongst researchers that the estimated hazard rate as a function of returns since purchase is higher on gains than on losses. This is an evidence in favour of a disposition e↵ect amongst investors because the higher rate over gains means that the average propensity to sell is higher for gains than losses. For all but very short holding periods, researchers consistently find the hazard rate or selling schedule on losses is fairly flat (see Ben-David and Hirshleifer (2012), Barber and Odean (2013) and Seru et al. (2010)).
There is less consensus over results concerning the overall shape of the hazard rate or selling schedule in the literature with findings depending upon the length of the holding period under consideration. For instance, over short holding periods, Ben-David and Hirsh- leifer (2012) demonstrate a strong asymmetric V shaped pattern in their empirical selling schedules. Some authors even find that when holding periods are aggregated, the selling intensity function may exhibit an inverted V shape (Odean (1998), Meng and Weng (2016)). In the remainder of this section, we will develop a model-based selling intensity for a single and for many investors, and compare to the findings of the empirical literature. Since our model gives an intensity over all holding periods, our focus is on achieving the empirical features coming from the aggregated data.
1.5.1
Model-based implied selling intensity
The empirical measure of the selling rate at price levelpcan be defined as (p) = number of sales atp
amount of time spent atp. The equivalent model-based quantity is
⇣(p) = P(P⌧2dp)
E⇥R⌧
0 du1(Pu2[p,p+dp))
⇤ (1.15)
provided this quantity is well defined. Similar to our model-based disposition ratio in (1.12), it can be shown that⇣(p) is the same forallstopping times⌧ such thatP⌧ has lawµ.
Proposition 1.9. Suppose µ is the law of the target prospect P⌧ and ⌫ is the law of the
scaled prospectX⌧ =s(P⌧). Assume a reference point ofR=P0. Ifµhas a density then
⇣(p) = P(p)
2s0(p) u⌫(s(p)) |s(p)|
Proof. Recall the notation we use in the proof of Proposition 1.8. For any arbitrary Borel function we have Z ⌧ 0 (Pu)d[P]u= Z ⌧ 0 (s 1(Xu)) 2 P(s 1(Xu)) ⇠2(X u) d[X]u= Z LX⌧(a) (s 1(a)) da [s0(s 1(a))]2 = Z LX⌧(s(u)) (u) s0(u)du. In particular if we choose (z) =1(z2[p,p+dp)) 2 P(z) then we have Z ⌧ 0 1(Pu2[p,p+dp))du= LX ⌧(s(p)) 2 P(p)s0(p) dp.
The expected value of the above expression can be computed by Tanaka’s formula, and (1.16) follows.
⇤
More generally, to allow for optimal prospects which contain atoms we set (dp) = P(p)
2s0(p) u⌫(s(p)) |s(p)|
µ(dp). (1.17)
Ifµis absolutely continuous then (dp) =⇣(p)dp. Conversely, if the optimal selling rule is a pure threshold strategy, i.e. a strategy in which it is optimal to sell the asset the first time the price process leaves an interval (whence the optimal prospect is a pair of point masses) then we find u⌫(s(p)) =|s(p)| at the ends of the interval and the measure consists of a
pair of point masses of infinite size. (This is intuitive: we must stop the price process at the first time it leaves the relevant interval, and the only way we can ensure this is to stop at an infinite rate.) We have seen an example of this when there is no probability weighting in the model.
In fact, the optimal selling rule in our asset liquidation model contains a point mass on losses, and a mixture distribution on gains consisting of a point mass and a continuous distribution above that point. The corresponding stopping rate has an atom of infinite size at the location of the point mass on losses, an atom of finite size at the location of the point mass on gains, and a continuous density above this point.
Given a non-negative functiong=g(p) then we can consider selling at a rateg(Pt)
per unit time. This is equivalent to stopping at the random time⌧g = inf
{u:R0ug(Pv)dv >
T} where T is an independent exponential random variable with unit parameter. Given a target law, for example the lawµ⇤ of the optimal prospect, we can ask if it is possible to choose a level dependent (but not explicitly time-dependent) functiong such thatP⌧g has
the desired target law. This can be done, and makes use of the measure in (1.17), as described in the following lemma where the proof is given in Appendix 1.E.
Lemma 1.10. Let⇤P = (⇤P
t)t 0be the increasing additive functional⇤Pt =
R
`P
t(p) (dp),
where`P = (`P
t(p))p>0,t 0 is an appropriately scaled version of local time process ofP.
If T is an independent, exponentially distributed, unit rate random variable and if
⌧= inf{u:⇤P
u > T}, thenP⌧⇠µ⇤.
This lemma gives a second interpretation of the quantity (dp): if the investor sells at a level-dependent rate per unit time given by , then he will attain the optimal prospect. Note that we are not arguing that our investors must follow this stopping rule, but rather this kind of randomised strategy provides a convenient way to interpret the model-based quantities given in (1.15) and (1.17).
1.5.2
Mixing over heterogeneous investors
We have so far discussed the case of a single investor implementing a stopping rule to generate a target prospect. However, the typical empirical selling rate estimated from market data is an amalgamation of liquidation strategies enacted contemporaneously by multiple investors who may have di↵erent risk preferences. We consider the implied selling intensity function when we average across individuals.
Let⇥denote the space of risk and probability weighting parameters. If the parame- ters of the typical investor are distributed with prior law⌘on⇥, then we find a model-based selling rate at price levelpof
⇣(p)dp= (dp) =µ(dp) P(p)
2s0(p)
u⌫(s(p)) |s(p)| (1.18)
whereµ(dp) =R⌘(d✓)µ✓(dp) and⌫ is given byF⌫(x) =Fµ(s 1(x)).
To illustrate the idea, we suppose the price processP follows a geometric Brownian motion and the expected return is such thatP is a martingale. We are free to choose⇥the space of parameters and⌘, the distribution over this parameter space. For this example, we suppose investors have a common pair of probability weighting functions (given by TK with ± = 0.7) and TK value functions (with reference level R=P0= 1). We assume ↵+= 0.5
and↵ = 0.9 are fixed across investors but that the loss aversion parameterk varies. We identify ⇥with a subset of R+ corresponding to the value of the loss aversion parameter.
Once we have specified ⌘ we can calculate the implied selling density. Di↵erent choices of⌘ will lead to di↵erent model-based predictions for the selling density. We make use of
the earlier empirical observation that the selling rate on losses is approximately constant across di↵erent returns. We design⌘ such that the model-based selling intensity at a loss is constant, and then consider the implications for the model-based rate of selling at gains. The construction of such a prior density is given in Appendix 1.F. Barber and Odean (2013) find hazard ratios of around one for losses in their analysis of the LDB and Finnish datasets. This motivates our choice of a unit rate of selling on losses, which corresponds to a daily probability of a loss of 1/250 or 0.4%.
Figure 1.6 displays our model’s implied stopping rate⇣(p) against price. The model implied sales rate on gains, given in (1.18) is plotted. Our first goal is to demonstrate that the disposition e↵ect holds. The implied sales rates are indeed consistent with the disposition e↵ect as the rate for gains is higher than that for losses and thus implies a higher propensity to sell at gains than losses. The second goal is to show the model implied sales rate captures some of the features of the empirical data. Since our focus is on results for longer holding periods or aggregate data, we compare with the estimated hazard ratios of Barber and Odean (2013) and the graphs of the probability of selling shares at Day 60, 125, and 250 in Figure 1 of Ben-David and Hirshleifer (2012).
In particular, the graphs in Figure 1 of Ben-David and Hirshleifer (2012) for Day 60 and 250 have very similar qualitative features to our Figure 1.6. Over losses, the probability of sale is relatively flat. Over gains, the sales probability is slightly humped - first rising and then falling with the magnitude of returns since purchase. This is also true (perhaps to a lesser extent) in the graph of the hazard ratio for the Finnish dataset in Barber and Odean (2013).
The rate of ⇣ between 3.5-5 in Figure 1.6 equates to a daily probability of selling between 1.4% and 2%. The implied sales rate in our example is slightly higher than those in Barber and Odean (2013) and Ben-David and Hirshleifer (2012) and thus our disposition e↵ect is on the strong side.
The presence of probability weighting in our model has had a dramatic impact on the PT model’s ability to produce realistic sale intensity schedules across di↵erent return magnitudes. Without probability weighting, recall that the optimal distribution for an individual is a two point distribution with weight on a single gain and a single loss threshold. Mixing over investors can improve the fit, but, as Ingersoll and Jin (2013) find, in the absence of probability weighting, PT investors need to be mixed with random Poisson traders to yield a reasonable calibration.
p 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 ζ ( p ) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Figure 1.6: Implied stopping rate ⇣(p) against price for a set of heterogeneous investors. Parameters are ± = 0.7, ↵+ = 0.5, ↵ = 0.9, = 0.5. Loss aversion k varies across
investors. The reference level isR=P0= 1.