In this paper, we study the expectation hypothesis bias for two popular particular instances of the ATS class of models.
4.1. Vasicek Model
The Vasicek instantaneous spot rate model is defined by
where
is the instantaneous spot interest rate at time
t,
k is the speed of mean reversion,
is the long-term or equilibrium level of the interest rate,
is the volatility of interest rates, and
is a Wiener process under the risk-neutral measure
Q representing the random shock to interest rates at time
t. It captures the stochastic nature of interest rate movements in the risk-neutral world.
The Vasicek model belong to the class of ATS models and thus has a closed-form solution for bond prices, making it analytically tractable. Its simplicity facilitates its ease of use and interpretation. Bond prices are given by (
20) with
From the above expressions, we know that only the parameter
k representing how quickly interest rates revert to the mean
affects
, which is what multiplies
in the bond price expression.
also depends on
, the level to which interest rates revert, and
, which represents the random fluctuations in interest rates. This is a well-known result. For further details, we refer to [
49].
From (
23) and (
24) and the forward rate result in (
21), we have the following result.
Lemma 2. Under the Vasicek model, with spot rate Q-dynamics as in Equation (22), we have What is particularly striking about the above result is the fact that only the last term is stochastic (as it multiplies ). The first two terms are deterministic. The first term established is a deterministic mean-reverting term, where is a factor that decreases exponentially with time, capturing the tendency of interest rates to revert to the long-term mean. The second deterministic term accounts for the stochastic or random component in the interest rate process. Its negative sign indicates that the volatility adjustment is subtracted from the mean-reverting term.
Here, we are primarily interested in finding closed-form expressions for the adjustments defined in Equations (
18) and (
19). Proposition 1 states our main Vasicek result.
Proposition 1. Under the Vasicek model with spot rate Q-dynamics as in Equation (22), and given the definitions of risk adjustment and stochastic adjustment in Equations (18) and (19), respectively, we find Notably, both adjustments are purely deterministic, as they do not even depend on . As it turns out, this is a feature that is specific to the Vasicek model.
The risk adjustment (RA) closed form tell us it depends on investors’ aversion to risk (). In particular, if we were to live in a risk-neutral world (), there would be no RA. In addition, RA increases with both risk aversion and volatility and thus can be interpreted as the compensation required for holding a risky asset. The term can be seen as the risk premium per unit of time, and is a factor that adjusts this premium over the investment horizon.
The stochastic adjustment (SA) does not depend on investors’ risk aversion (as we would expect), and it increases with the volatility and reduces with the speed of mean reversion k. It is a pure stochastic term because if interest rates were not stochastic, , it would be zero. The term adjusts the correction over the investment horizon, reflecting how the stochastic adjustment varies with time.
In
Table 1 we quantify both adjustments for various levels of risk aversion
and maturities
T. The Vasicek model parameters,
, are from [
50]. The results allow us to get a sense of the values of both RA and SA and how much they are responsible for the expectation hypothesis bias. As it would be expected, for the case of risk loving investors (
) RA is negative, for risk neutral (
) RA is zero and for risk averse (
), the higher the relative risk aversion coefficient
the higher is the RA, and its weight on the overall bias. SA, on the other hand, does not depend on
, consistently with the formulas in Equations (
26) and (
27), increasing with maturity
T, for fixed
and
k. For
, we observe the total adjustment starts by being negative (for
), RA dominates SA but as maturity increases (
) it becomes positive with SA dominating RA. Naturally for
there is no risk adjustment, as investors are risk neutral and the full bias can be attributed to SA. For any fixed
, in absolute terms both adjustments tend to increase with maturity
T, but in relative terms the RA tends to decrease with maturity and SA tends to increase. For the chosen parameter values the significance of the bias is relative low overall, when measured as percentage of
, it does not go beyond 2.62% of the level of interest rates.
In
Table 2, we present the same statistics for the case when we allow for higher interest rate volatility—
(instead of 0.01 in
Table 1). This increase in the “stochasticity” of interest rates had the expected effect, resulting in considerably higher SA and, thus, higher bias. This time, the bias size was as high as 44.9% when measured as a percentage of
and for
. However, even for lower relative risk aversion coefficients between 1 and 2, we obtain relative sizes of bias that range from 25.27% to 31.38%. In terms of RA, we also observed increased values for risk-loving investors (higher negative values) and risk-averse investors (higher positive values). Notice that
shows up in both formulas in Equations (
26) and (
27), i, but for RA, it always comes multiplied by
, so zero RA for risk-neutral investors is guaranteed. The relative importance of SA over the full bias is similar to that shown in
Table 1.
An interesting question we aim at answering is if the new stochastic adjustment can help explain the implicit high levels of relative risk aversion found in the literature. For that, we need to consider the full bias (RA + SA) and implicitly derive what would be the risk-aversion coefficient if one would think that only RA explains the full bias; i.e., we would like to solve
where on the l.h.s.
is as in (
26) but depends on an implicit
, and on the r.h.s.
and
are as in (
26) and (
27). Corollary 1 tell us the answer in the case of the Vasicek model.
Corollary 1. For the Vasicek with model parameters and implicit risk aversion defined as in (28), we are given From Equation (
29), it is clear the implicit risk aversion
is always bigger than the true risk aversion
, and the difference between them depends only on
k, the speed of mean reversion.
From
Table 3, we see that not considering a stochastic adjustment (SA) would lead to an increase in the implicit risk aversion coefficient
and the increase is bigger the larger the maturity we consider. In particular, for risk-loving investors (
), we observe that the increase in the implicit risk aversion
is such that for maturities higher than 2, they could be perceived as risk-averse instead of risk-lovers. Likewise, risk-neutral investors (
) are perceived as risk-averse whenever one does not take into account SA. The implicit risk-aversion coefficient for the risk averse (
) is higher for the lower mean reversion parameter
k, and we can clearly observe that this can lead to perceive investors as much more risk-averse than they actually are.
The difference between the implicit risk aversion and the true risk aversion,
, is, however, constant across the various levels of risk aversion as it only depends on
k and
T.
Figure 1 presents this difference for
and
.
In
Figure 1b we see the increase in implicit risk aversion in not higher than 2, while empirical studies suggest it to be bigger. On the other hand, from
Figure 1a, which takes the case of
, we see that even with the simple Vasicek model, it is possible to have implicit risk aversion
much higher than the true relative risk aversion coefficient
and with values in line with the empirical literature.
The model we have studied up until now has its simplicity as an advantage, but it also has some disadvantages. The Vasicek model assumes a constant volatility, which may not capture the time-varying nature of interest rate volatility observed in the real world. Market volatility tends to change over time, and models that incorporate this feature may provide a better fit. The model also assumes normally distributed interest rate changes. In reality, interest rates often exhibit fat tails and other deviations from normality, particularly during periods of financial stress. These limitations may impact the accuracy of the model in extreme market conditions. Finally, the model’s stationarity depends on the chosen parameter values. If not carefully calibrated, the model may fail to exhibit stationary behaviour, leading to unrealistic interest rate paths.
The next model we look at is the CIR model of [
19], which takes into consideration the limitations of the Vasicek model, improving the dynamics of the spot interest rate
to take them into account. This extra realism is a plus of the CIR model but comes at the cost of not so nice/harder to interpret formulas.
4.2. CIR Model
The [
19] (CIR) instantaneous spot rate model is defined by
where
is the instantaneous short-term interest rate at time
t,
k is the speed of mean reversion,
is the long-term or equilibrium mean of the interest rate to which rates revert,
is the volatility of interest rates, and
is a Wiener process under the risk-neutral measure
Q representing the random shock to interest rates at time
t.
Similarly to the Vasicek model, the CIR model also implies that interest rates mean-revert to the level with a speed determined by k. But it incorporates a volatility term, , allowing it to capture the time-varying nature of volatility in interest rates. This provides more flexibility than the constant volatility assumption of the Vasicek model. The CIR model is commonly used in interest rate modelling due to its ability to capture the observed behaviour of interest rates, such as mean reversion and volatility clustering. It also ensures that interest rates remain non-negative and belong to the ATS class, allowing for closed-form solutions for bond prices and other related financial derivatives.
Bond prices are given by
, where
with the notation
Note that CIR is affine as in Equation (
20) with
. Under the CIR model, we also obtain a closed-form expression for forward rates that depends only on model parameters and the spot rate
.
Lemma 3. Under the CIR model, with spot rate Q-dynamics as in Equation (30), the forward rate is given bywhere θ and k are as defined in Equation (30) and η and x are as defined in Equation (33). The deterministic term grows with , and (through ), and the stochastic term depends only on k and (through ).
Proposition 2. Under the CIR model, with spot rate Q-dynamics as in Equation (30) and given the definitions of risk adjustment and stochastic adjustment in Equations (18) and (19), respectively, we find The first thing to notice concerning both adjustments is that they are no longer deterministic (as was the case for Vasicek) as they depend on . That is, both adjustments have a deterministic component and a stochastic component.
As would be expected, RA depends on the level of risk aversion, , and in particular, if we have risk neutrality, we obtain zero RA. Recall that the risk adjustment RA formula captures the difference between the expectations of the instantaneous short-term interest rate under the objective measure P and the risk-neutral measure Q in the CIR model. It involves terms related to mean reversion, risk aversion, and volatility, providing a comprehensive adjustment over the investment horizon .
The stochastic adjustment SA, on the other hand, does not depend on the risk aversion ; instead, it depends on the volatility through . Note that when we have , . Unfortunately, it is not possible to simplify the SA expression any further.
Table 4 and
Table 5 present the computations for the CIR model when we consider value parameters that are similar to those considered for the Vasicek model. The main difference is in the volatility term because now, the volatility is given by
, so we consider
(
Table 4) and
(
Table 5), which, when multiplied by
, give volatilities of about
and
with the same order of magnitude as the Vasicek volatilities of 1% and 5%.
Comparing both tables, the first conclusion is that, in the higher volatility scenario, the biases are considerably more relevant when measured as a percentage of
. In
Table 4, the values range from −0.107% to 5.340%, while in
Table 5, the values range from −2.710% to 81.596%.
The negative values are associated only with the risk-loving investor case () and for maturities lower than five years (). For risk lovers, just as in the case of the Vasicek model, SA is always positive, while RA is always negative. For lower maturities, RA dominates, while for longer maturities, SA dominates.
For risk-neutral and risk-averse investors, the bias is always positive, and its relevance increases with risk aversion and maturity T. The relative weight of SA over the full bias, on the other hand, is more relevant for than . This happens because relative risk aversion in the model always shows up together with volatility, so the higher the volatility, the higher is RA.
For the exercise of determining the implicit risk aversion, if we would consider the bias to be explained only by risk aversion, as before, we have
but given the formulas in Equations (
35) and (
36), we need to find the implicit risk aversion
numerically.
Table 6 presents the implicit risk aversion for combinations of
and
. For a low speed of mean reversion
and low volatility
, it seems the difference between the implicit and true risk aversion coefficient,
, is almost constant across risk aversion coefficients and increasing with maturity. If we however consider
but
, we note that the implicit risk aversion is no longer increasing with maturity, having a humped-shape with higher values in between the maturities
and
depending on the true risk aversion.
Even when considering a higher speed of mean reversion , the same humped shape result happens in the case of . A surprising result is that the difference does not increase with maturity; instead, it increases up to and it then decreases.
Figure 2 and
Figure 3 allow for a visualisation of the difference
. As in the Vasicek model, the lower the speed of mean reversion, the higher the difference level tends to be.
For , the differences increase with maturities, but it is rather stable across risk aversion coefficients. For lower mean reversion, one can explain an increase in implicit risk aversion as high as 4 from long maturities, while for high mean reversion, the distortion does not go beyond 2.
For , we always obtain humped-shape differences with maximal points between and and risk-aversion coefficients close to . The curve is at a slightly higher level for than for , but the difference between implicit risk aversion and true risk aversion does not go beyond 1.