Fundamental bubbles in equity markets


Using an affine model to compute the price of equities based on a dataset of macroeconomic factors, we propose a measure of equity bubbles. We use a dynamic affine term structure framework to price equity and bonds jointly, and investigate how prices are related to a set of macrofactors extracted from a large dataset of economic time series. We analyze the discrepancies between market and model implied equity prices and use them as a measure for bubbles. A bubble is diagnosed over a given period whenever the discrepancies are not stationary and impact the underlying economy consistently with the literature’s findings, increasing over the shorter term economic activity before leading to a net loss in it. We perform the analysis over 3 major US and 3 major European equity indices over the 1990–2017 period and find bubbles only for two of the US equity indices, the S&P500 and the Dow Jones.

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  1. 1.

    As Gürkaynak (2008) concludes his review of literature: “For every test that “finds” a bubble, there is another paper that disputes it. The finding of a bubble, at best, suggests that the data is either consistent with a bubble or a myriad of other extensions of the standard model”

  2. 2.

    Dividend strip refers to a price of a single dividend claim k periods into the future.

  3. 3.

    One of the advantages of the above factors specification is that they are fully observable and have an approximate economic interpretation. Piazzesi (2010) reports that the model-implied dynamics for macrovariables (i.e., latent factor estimated from the model) are often disconnected from the dynamics of their historical observable counterparts. Also, latent factors require computationally burdensome estimation techniques that account for the joint distribution of yields or price to dividend ratios and the state vector which can make our task significantly more difficult (see Ang and Piazzesi 2003; Kim 2007 on those difficulties). Finally, specifications with latent factors are not always globally identifiable and this issue in itself was closely scrutinized in the academia (Collin-Dufresne et al. 2008; Christensen et al. 2011; Joslin et al. 2011; Hamilton and Wu 2012).

  4. 4.

    Technical condition are described in Harrison and Kreps (1979).

  5. 5.

    For technical details, see “Appendix D and E” in Duffie (2004). By construction, \(\xi _t\) is a strictly positive martingale under Novikov’s condition with \(\xi _0 = 1\), implying that P and Q are equivalent and Q is a well defined probability measure.

  6. 6.

    Intuitively, with the state space dynamics in Eq. (5), \(\lambda _t\) must be an affine function of \(X_t\) as in Eq. (10) for the drift term of \(X_t\) to remain affine in \(X_t\) under the risk-neutral probability measure as well. The diffusion of the state vector is the same under both measures.

  7. 7.

    Essentially affine models span the completely affine model described in Dai and Singleton (2000). The specification of the market price of risk is studied in detail in Cheridito et al. (2007).

  8. 8.

    \(r_t\) is an affine function of \(X_t\), \(X_t\) is also affine under the risk-neutral probability measure due to the specification of the \(X_t\) process as a Gaussian VAR, the SDF and the MPR specifications.

  9. 9.

    For details, see, e.g., Duffie et al. (2003), Ang and Piazzesi (2003) among others.

  10. 10.

    \(d_t\) is indexed one period ahead of the corresponding \(r_{t-1}\) due to the fact that \(r_t\) is a locally deterministic discount factor and is known at time t

  11. 11.

    See “Appendix B”.

  12. 12.

    All data used here have been obtained from Bloomberg©as a datasource.

  13. 13.

    Here, we use BEST estimates.

  14. 14.

    This type of quadratic criterion is equivalent to assuming that market quotes are a mixture between the “true” quote and an Gaussian disturbance, yielding unbiased estimates.

  15. 15.

    We acknowledge the fact that yields and price to dividend ratios have different scalings. However, we employ interior point method to minimize the objective function (38) which uses gradient methods and is relatively robust to variables scaling.

  16. 16.

    Ang and Piazzesi (2003) reports that single step maximum likelihood optimization typically produce unacceptable yields dynamics. Mönch (2008) also reports that such objective specification is better suited the recursive out-of-sample forecast routines.

  17. 17.

    Despite imposing highly nonlinear restriction on the parameters, in general we find that they actually improve the estimation procedure by promptly rejecting parameter regions where dividend strips diverge in the limit and thus where parameters thus do not solve the minimization problem (38).

  18. 18.

    The need for this approach is in part motivated by high sensitivity of to the initial point specification, even if following steps are equal.

  19. 19.

    Ideally, we would shift the matrix of risk premia parameters in all possible directions to generate a full set of initial values and then run the estimation. However, due to high number of parameters estimated (20), this approach is infeasible. Generating random matrices around the admissible point adheres to the same idea but reduces number of directions initiated.

  20. 20.

    Similar results have been obtained with the KPSS and Phillips and Perron tests.

  21. 21.

    An interesting reader could use this test methodology in a rolling way to try differentiating the bubble periods from the non-bubble ones. Here, given the quarterly frequency we aim at using, this rolling estimation did not lead to interesting findings.

  22. 22.

    Consistently with the previously mentioned FAVAR literature.

  23. 23.

    We refrained from doing the same with the MSCI Germany and France as their interaction with the overall European economy makes the control variable selection more complex.

  24. 24.

    E.g., see Hamilton (1994).


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Derivation of bonds recursive ODE equations

In this section, we derive a system of ODE equations that arises when pricing bonds in the ATS framework. Assuming that the price of a bond maturing in n years at time t, \(p_t^n\) is exponential affine in the state vector

$$\begin{aligned} p_t^n = \text {exp}\left( A_n + B_n'X_t \right) \end{aligned}$$

we show that the price of the bond maturity in \(n+1\) years is also exponential affine as follows

$$\begin{aligned} \begin{aligned} p_t^{n+1}&= E_t \left[ m_{t+1}p_{t+1}^n \right] \\&= E_t \left[ \text {exp}\left( -r_t - \frac{1}{2}\lambda _t'\lambda _t - \lambda _t'\varepsilon _{t+1} + A_n + B_n' X_{t+1}\right) \right] \\&=\text {exp}\left[ -r_t - \frac{1}{2}\lambda _t'\lambda _t+A_n \right] E_t \\&\quad \left[ \text {exp}\left( -\lambda _t'\varepsilon _{t+1} +B_n' X_{t+1} \right) \right] \\&= \text {exp}\left[ -r_t - \frac{1}{2}\lambda _t'\lambda _t+A_n \right] E_t \\&\quad \left[ \text {exp}\left( -\lambda _t'\varepsilon _{t+1} + B_n'\left( \mu + \Phi X_t + \Sigma \varepsilon _{t+1} \right) \right) \right] \\&= \text {exp} \left[ -\delta _0 +A_n + B_n'\mu + \left( B_n'\Phi - \delta _1' \right) X_t - \frac{1}{2}\lambda _t'\lambda _t \right] \\&\quad \times E_t \left[ \text {exp}\left( \left( - \lambda _t' +B_n'\Sigma \right) \varepsilon _{t+1} \right) \right] \\&= \text {exp} \left[ -\delta _0 +A_n + B_n'\mu + \left( B_n'\Phi - \delta _1' \right) X_t \right. \\&\qquad \left. - \frac{1}{2}\lambda _t'\lambda _t + \frac{1}{2}\left( - \lambda _t' +B_n'\Sigma \right) \left( - \lambda _t' +B_n'\Sigma \right) ' \right] \\&= \text {exp} \left[ -\delta _0 +A_n + B_n'\mu + \left( B_n'\Phi - \delta _1' \right) X_t\right. \\&\qquad \left. + \frac{1}{2}B_n' \Sigma \Sigma ' B_n- B_n' \Sigma \lambda _t \right] \\&= \text {exp} \left[ -\delta _0 +A_n + B_n'\mu + \left( B_n'\Phi - \delta _1' \right) X_t \right. \\&\qquad \left. + \frac{1}{2}B_n' \Sigma \Sigma ' B_n- B_n' \Sigma \left( \lambda _0 + \lambda _1 X_t \right) \right] \\&= \text {exp} \left[ -\delta _0 +A_n + B_n'\left( \mu - \Sigma \lambda _0\right) \right. \\&\qquad \left. + B_n'\left( \Phi - \Sigma \lambda _1 \right) X_t - \delta _1'X_t + \frac{1}{2}B_n' \Sigma \Sigma ' B_n\right] \\&= \text {exp} \left[ -\delta _0 +A_n + B_n'\mu ^Q + B_n'\Phi ^Q X_t - \delta _1'X_t \right. \\&\qquad \left. + \frac{1}{2}B_n' \Sigma \Sigma ' B_n\right] = \text {exp}\left( A_{n+1} + B_{n+1}'X_t \right) \end{aligned} \end{aligned}$$

Matching the coefficients on the RHS and LHS in Eq. (42) leads to the system of ODE equations

$$\begin{aligned} A_n&= A_{n-1} + B_{n-1}'(\mu - \Sigma \lambda _0)+ \frac{1}{2}B_{n-1}' \Sigma \Sigma 'B_{n-1} - \delta _0 \end{aligned}$$
$$\begin{aligned} B_n'&= B_{n-1}'(\Phi - \Sigma \lambda _1)-\delta _1' \end{aligned}$$

where we used a general specification of the short rate \(r_t = \delta _0 + \delta _1' X_t\). The boundary conditions are given by

$$\begin{aligned} A_1&= -\delta _0 \end{aligned}$$
$$\begin{aligned} B_1'&= - \delta _1' \end{aligned}$$

Finally, short rate specification used throughout this work corresponds to the case when

$$\begin{aligned} \delta _0&= 0 \end{aligned}$$
$$\begin{aligned} \delta _1&= \delta ^r \end{aligned}$$

Derivation of equity recursive ODE equations

In this section, we derive a system of ODE equations, arising when pricing price to dividend ratio (or normalized dividend strips) under risk-neutral probability measure Q.

The price of a single dividend strip that pays at time \(t + \tau \) is given by Eq. (32), which we reiterate here for convenience

$$\begin{aligned} V_{t,n}^d= & {} E_t^Q\left[ e^{-\sum _{k=1}^{n}r_{t+k-1}}\frac{D_{t+n}}{D_t} \right] \nonumber \\= & {} E_t^Q \left[ \text {exp}\left( \sum _{k=1}^{n}d_{t+k} -r_{t+k-1}\right) \right] \end{aligned}$$

The derivation starts with the guess that price of dividend strip is exponential affine in the state vector \(X_t\)

$$\begin{aligned} V_{t,n}^d = \text {exp}\left( a_n + b_n' X_t \right) \end{aligned}$$

where \(X_t\) follows Gaussian VAR as given by Eq. (5). First, we notice that we can rewrite using the guess given by Eq. (50), the price of dividend strip recursively as follows

$$\begin{aligned} V_{t,n}^d&= E_t^Q \left[ \text {exp}\left( \sum _{k=1}^{n}d_{t+k} -r_{t+k-1}\right) \right] \nonumber \\&= E_t^Q \left[ \text {exp}\left( d_{t+1} -r_{t}\right) \text {exp}\left( \sum _{k=2}^{n}d_{t+k} -r_{t+k-1}\right) \right] \nonumber \\&= E_t^Q \left[ \text {exp}\left( d_{t+1} -r_{t}\right) E_{t+1}^Q \left[ \text {exp}\left( \sum _{k=2}^{n}d_{t+k} -r_{t+k-1}\right) \right] \right] \nonumber \\&= E_t^Q \left[ \text {exp}\left( d_{t+1} -r_{t}\right) V_{t+1,n-1}^d \right] \nonumber \\&= E_t^Q\left[ \text {exp}\left( d_{t+1} -r_{t}\right) \text {exp}\left( a_{n-1} + b_{n-1}'X_{t+1} \right) \right] \end{aligned}$$

where we used the law of iterated expectations in the third equality and Eq. (50) in the last equality.

We assume that earnings logarithmic growth rates are affine in the state vector \(X_t\) (which implies that earnings themselves are exponential affine in the state vector \(X_t\))

$$\begin{aligned} g_t = \gamma _0 + \gamma _1' X_t \end{aligned}$$

where \(\gamma _0\), \(\gamma _1\) are constant parameters. Next, we assume that logarithmic growth rates of the payout ratio are also affine in the state vector \(X_t\)

$$\begin{aligned} c_t = \alpha _0 + \alpha _1 g_t \end{aligned}$$

This implies that dividend (log) growth rates \(d_t\) satisfy

$$\begin{aligned} d_t= & {} \text {ln}\frac{D_t}{D_{t-1}} =\text {ln} \frac{E_t C_t}{E_{t-1}C_{t-1}} = c_t + g_t \nonumber \\= & {} \alpha _0 +\left( 1+\alpha _1 \right) \gamma _0 + \left( 1+\alpha _1 \right) \gamma _1' X_t=: \omega _0 + \omega _1' X_t \end{aligned}$$

where \(\omega _0 = \alpha _0 +\left( 1+\alpha _1 \right) \gamma _0\) and \(\omega _1 = \left( 1+\alpha _1 \right) \gamma _1\) are obtained from earnings and payout ratio growths loadings on the state vector factors. \(d_t\) is thus also affine in \(X_t\).

Having dividend growth rate process in place, we can further rewrite Eq. (51) as follows (directly under Q, using the same pricing kernel as for bonds pricing)

$$\begin{aligned} V_{t,n}^d&= E_t^Q\left[ \text {exp}\left( d_{t+1} -r_{t}\right) \text {exp}\left( a_{n-1} + b_{n-1}'X_{t+1} \right) \right] \nonumber \\&= E_t^Q\left[ \text {exp}\left( \omega _0 + \omega _1' X_{t+1} - r_t \right) \text {exp}\left( a_{n-1} + b_{n-1}'X_{t+1} \right) \right] \nonumber \\&= \underbrace{\text {exp}\left( \omega _0 - \delta _0 - \delta _1' X_t + a_{n-1} \right) }_{C}E_t^Q\left[ \text {exp}\left( \left( \omega _1' + b_{n-1}'\right) X_{t+1} \right) \right] \nonumber \\&= C \cdot \text {exp}\left[ \left( \omega _1' + b_{n-1}'\right) \left( \mu ^Q + \Phi ^Q X_{t}\right) \right. \nonumber \\&\left. \quad + \frac{1}{2}\left( \omega _1' + b_{n-1}'\right) \Sigma \Sigma ' \left( \omega _1 + b_{n-1}\right) \right] \nonumber \\&= \text {exp}\left( a_n + b_n' X_t \right) \end{aligned}$$

where we used the guess from Eq. (50) in the last equality and the fact that state vector follows conditional Gaussian VAR \(X_{t+1} = \mu ^Q + \Phi ^Q X_t + \Sigma \varepsilon _{t+1}\) under Q (under essentially affine risk premia specification used throughout this work).

Similar to bonds pricing, in order for the last equality to hold as an identity, coefficients \(a_n\) and \(b_n\) have to satisfy the following system of equations (obtained via coefficients matching in Eq. (55))

$$\begin{aligned} a_n&= a_{n-1} + \omega _0 + \left( \omega _1 + b_{n-1}\right) ' \mu ^Q \nonumber \\&\quad + \frac{1}{2}\left( \omega _1 + b_{n-1}\right) '\Sigma \Sigma ' \left( \omega _1 + b_{n-1}\right) - \delta _0 \end{aligned}$$
$$\begin{aligned} b_n&= \Phi ^{Q'}\left( \omega _1 + b_{n-1}\right) - \delta _1 \end{aligned}$$

To obtain boundary conditions for \(a_n\) and \(b_n\), we note that for 1-period ahead dividend strip, we have (again, using Eq. (50))

$$\begin{aligned} V_{t,1}^d&= E_t^Q \left[ \text {exp}\left( d_{t+1} - r_t \right) \right] \nonumber \\&= E_t^Q\left[ \text {exp}\left( \omega _0 + \omega _1' X_{t+1} - \delta _0 - \delta _1' X_t\right) \right] = \nonumber \\&= \text {exp}\left( \omega _0 - \delta _0 - \delta _1' X_t\right) E_t^Q\left[ \text {exp}\left( \omega _1' X_{t+1}\right) \right] \nonumber \\&= \text {exp}\left( \omega _0 - \delta _0 - \delta _1' X_t + \omega _1'\left( \mu ^Q + \Phi ^QX_t\right) \right. \nonumber \\&\qquad \left. + \frac{1}{2}\omega _1' \Sigma \Sigma ' \omega _1 \right) \nonumber \\&= \text {exp}\left( a_1 + b_1'X_t \right) \end{aligned}$$

as previously, last equality must hold as an identity. Matching the coefficients in Eq. (58), we obtain the following boundary conditions

$$\begin{aligned} a_1&= -\delta _0 + \omega _0 + \omega _1' \mu ^Q + \frac{1}{2}\omega _1' \Sigma \Sigma ' \omega _1 \end{aligned}$$
$$\begin{aligned} b_1&= - \delta _1 + \Phi ^{Q'}\omega _1 \end{aligned}$$

The short rate specification used throughout this work corresponds to the case when

$$\begin{aligned} \delta _0&= 0 \end{aligned}$$
$$\begin{aligned} \delta _1&= \delta ^r \end{aligned}$$

Parameter closed-form solutions and limiting behavior

In this section, we analyze the system of recursive ODE Eqs. (56), (57) derived in “Appendix B”. Equation (57) implies that \(b_n\) follows a simple recursive equation with initial boundary condition given by Eq. (60) and is independent of \(a_n\). Using the boundary condition, we can rewrite Eq. (57) as follows (we temporarily drop the superscript \(Q'\) from \(\Phi ^{Q'}\) to ease the notations)

$$\begin{aligned} b_n&= \Phi \left( \omega _1 + b_{n-1}\right) - \delta _1 = \Phi b_{n-1} + \Phi \omega _1 - \delta _1 \nonumber \\&= \Phi \left[ \Phi b_{n-2} + \Phi \omega _1 - \delta _1 \right] + \omega _1 - \delta _1 = \cdots \nonumber \\ \cdots&= \Phi ^{n-1} b_1 + \sum _{k = 1}^{n-1}\Phi ^{k}\omega _1 - \sum _{k = 0}^{n-2}\Phi ^{k}\delta _1 \nonumber \\&= \sum _{k = 1}^{n}\Phi ^{k}\omega _1 - \sum _{k = 0}^{n-1}\Phi ^{k}\delta _1 \end{aligned}$$

where by convention we set \(\Phi ^0 = I\). Since the terms \(\delta _1\) and \(\omega _1\) are constant, last equation in Eq. (63) represents a finite sum of a geometric series, which admits a closed-form solution and is given by

$$\begin{aligned} \begin{aligned} b_n&= \sum _{k = 1}^{n}\Phi ^{k}\omega _1 - \sum _{k = 0}^{n-1}\Phi ^{k}\delta _1 \\&= \Phi \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\delta _1 \end{aligned} \end{aligned}$$

In Eq. (64), we require that matrix \(I - \Phi \) is invertible. This condition is equivalent to absolute value of eigenvectors of matrix \(\Phi \) being different from one. This condition alone does not guarantee the convergence of \(\Phi ^n\) (and hence \(b_n\)). However, as long as absolute values of eigenvalues of matrix \(\Phi \) are also less than one, we have

$$\begin{aligned} \lim _{n \rightarrow \infty }\Phi ^n = 0 \end{aligned}$$

which implies \(I - \Phi \) is invertible and that in the limit \(n \rightarrow \infty \) parameter \(b_n\) satisfies

$$\begin{aligned} b_{\infty } = \Phi \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi \right) ^{-1}\delta _1 \end{aligned}$$

and is also known as the fixed point in Eq. (57).Footnote 24 Importantly, restriction on eigenvalues is not the only restriction that can be imposed on matrix \(\Phi \) that leads to stable results for \(b_n\). These restriction present sufficient, but not necessary condition for \(b_n\) convergence.

Next, from Eq. (56), we have

$$\begin{aligned} \begin{aligned} a_n&= a_{n-1} +\omega _0 + \left( \omega _1 + b_{n-1}\right) ' \mu ^Q \\&\quad + \frac{1}{2}\left( \omega _1 + b_{n-1}\right) '\Sigma \Sigma ' \left( \omega _1 + b_{n-1}\right) - \delta _0 \\&= a_1 + \left( n-1\right) \omega _0 + \left( n-1\right) \omega _1' \mu ^Q + \left( \sum _{k=1}^{n-1}b_k'\right) \mu ^Q \\&\quad + \left( n-1\right) \frac{1}{2}\omega _1'\Sigma \Sigma ' \omega _1 + \cdots \\&\quad \cdots + \omega _1' \Sigma \Sigma '\left( \sum _{k=1}^{n-1}b_{k}\right) + \frac{1}{2}\sum _{k=1}^{n-1}b_k' \Sigma \Sigma ' b_k - \left( n-1\right) \delta _0 \end{aligned} \end{aligned}$$

Equation (67) provides an analytical expression for \(a_n\) in terms of model parameters, since the solution for \(b_n\) was derived in Eq. (64) and boundary condition \(a_1\) is given in Eq. (59).

As evident from Eq. (67), in order to analyze the limiting behavior of \(a_n\), we first have to analyze the limiting behavior of \(\sum _{k=1}^{n-1}b_{k}\). From Eq. (64), we have

$$\begin{aligned} \begin{aligned} \sum _{k=1}^{n}b_k&= \sum _{k=1}^n \Phi \left( I - \Phi ^k \right) \left( I - \Phi \right) ^{-1}\omega _1 \\&\quad - \sum _{k=1}^n \left( I - \Phi ^k \right) \left( I - \Phi \right) ^{-1}\delta _1 \\&= n\Phi \left( I - \Phi \right) ^{-1}\omega _1 - \Phi ^2 \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-2}\omega _1 - \cdots \\&\quad \cdots - n \left( I - \Phi \right) ^{-1}\delta _1 + \Phi \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-2}\delta _1 \end{aligned} \end{aligned}$$

where, as previously, we assume that matrix \(I-\Phi \) is invertible. Note that under the restriction that absolute values of eigenvalues of \(\Phi \) are less than one, using Eq. (65) in the limit we have

$$\begin{aligned}&\lim _{n \rightarrow \infty } \sum _{k=1}^{n}b_k = \lim _{n \rightarrow \infty }n\left[ \Phi \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi \right) ^{-1}\delta _1 \right] - \cdots \nonumber \\&\cdots - \Phi ^2 \left( I - \Phi \right) ^{-2}\omega _1 + \Phi \left( I - \Phi \right) ^{-2}\delta _1 \end{aligned}$$

where two last terms in Eq. (69) is constant and two first terms increase linearly with n. This implies that generally, expression \(\sum _{k=1}^{n-1}b_k\) does not have a limit as \(n \rightarrow \infty \). The relation (69) is, however, useful to determine restrictions on parameters that determine limiting the dynamics of \(a_n\).

Finally, we analyze the limiting behavior of the term \(\sum _{k=1}^{n-1}b_k' \Sigma \Sigma ' b_k\) that appears in Eq. (67). First, we note that

$$\begin{aligned} \begin{aligned} \sum _{k=1}^{n-1}b_k' \Sigma \Sigma ' b_k = \sum _{k=1}^{n-1} \left( \Sigma ' b_k \right) ' \Sigma ' b_k \end{aligned} \end{aligned}$$

where, following Eq. (64), we have

$$\begin{aligned} \Sigma ' b_n&= \Sigma ' \left[ \Phi \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\delta _1 \right] \end{aligned}$$
$$\begin{aligned} \left( \Sigma ' b_n\right) '&= \left[ \Phi \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\delta _1 \right] ' \Sigma \end{aligned}$$

and therefore

$$\begin{aligned}&\left( \Sigma ' b_n \right) ' \Sigma ' b_n\nonumber \\&\quad = \left[ \Phi \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\delta _1 \right] ' \Sigma \times \nonumber \\&\quad \times \Sigma ' \left[ \Phi \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi ^n \right) \left( I - \Phi \right) ^{-1}\delta _1 \right] \end{aligned}$$

which implies that

$$\begin{aligned} \begin{aligned}&\sum _{k=1}^{n}\left( \Sigma ' b_k \right) ' \Sigma ' b_k\\&\quad =\underbrace{\sum _{k=1}^{n} \left[ \Phi \left( I - \Phi ^k \right) \left( I - \Phi \right) ^{-1}\omega _1 \right] ' \Sigma \Sigma ' \Phi \left( I - \Phi ^k \right) \left( I - \Phi \right) ^{-1}\omega _1}_{1} + \\&\qquad +\underbrace{ \sum _{k=1}^{n} \left[ \left( I - \Phi ^k \right) \left( I - \Phi \right) ^{-1}\delta _1 \right] ' \Sigma \Sigma ' \left( I - \Phi ^k \right) \left( I - \Phi \right) ^{-1}\delta _1}_{2} - \\&\qquad - 2\underbrace{\sum _{k=1}^{n} \left[ \Phi \left( I - \Phi ^k \right) \left( I - \Phi \right) ^{-1}\omega _1 \right] ' \Sigma \Sigma ' \left( I - \Phi ^k \right) \left( I - \Phi \right) ^{-1}\delta _1}_3 \\ \end{aligned} \end{aligned}$$

Since we are first and foremost interested in the limiting behavior of pricing parameters, in what follows we will only focus on terms that growth linearly with maturity n (and we will denote these terms with a subscript n for clarity). In fact, in the limit, terms of (74) either growth linearly with n or converge to constant values (under the assumption 65). We, therefore, focus on former terms, which characterize the limiting behavior of \(a_n\) and are necessary to derive the constraint that guarantees convergence of prices of dividend strips to zero.

$$\begin{aligned} \begin{aligned} 1_{n \rightarrow \infty }&= n \left[ \left( \Phi \left( I - \Phi \right) ^{-1}\omega _1 \right) ' \Sigma \Sigma ' \Phi \left( I - \Phi \right) ^{-1}\omega _1 \right] \\ 2_{n \rightarrow \infty }&= n \left[ \left( \left( I - \Phi \right) ^{-1}\delta _1 \right) ' \Sigma \Sigma '\left( I - \Phi \right) ^{-1}\delta _1 \right] \\ 3_{n \rightarrow \infty }&= n \left[ \left( \Phi \left( I - \Phi \right) ^{-1}\omega _1 \right) ' \Sigma \Sigma ' \left( I - \Phi \right) ^{-1}\delta _1 \right] \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} \left( \sum _{k=1}^{n}\left( \Sigma ' b_k \right) ' \Sigma ' b_k\right) _{n \rightarrow \infty } = 1_{n \rightarrow \infty } + 2_{n \rightarrow \infty } - 2 \cdot 3_{n \rightarrow \infty }\nonumber \\ \end{aligned}$$

This allows us to analyze the limiting properties of \(a_n\) as \({n \rightarrow \infty }\)

$$\begin{aligned} \begin{aligned} a_{n \rightarrow \infty }&= n\omega _0 + n\omega _1' \mu ^Q \\&\quad + n\left[ \Phi \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi \right) ^{-1}\delta _1 \right] '\mu ^Q \\&\quad + \frac{n}{2}\omega _1'\Sigma \Sigma ' \omega _1 + \cdots \\&\cdots + n \omega _1' \Sigma \Sigma '\left[ \Phi \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi \right) ^{-1}\delta _1 \right] \\&\quad + \frac{1}{2}\left[ 1_{n \rightarrow \infty } + 2_{n \rightarrow \infty } - 2 \cdot 3_{n \rightarrow \infty } \right] - n \delta _0 \end{aligned} \end{aligned}$$

We now recall that deterministic maturity-varying parameters \(a_n\) and \(b_n\) are used to price-dividend strips via Eq. (50). In order to ensure that prices of dividend strips converge to zero as time to maturity n approaches infinity, we impose the following restriction on the model parameters

$$\begin{aligned} C&= \omega _0 + \omega _1'\mu ^Q + \left[ \Phi \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi \right) ^{-1}\delta _1 \right] '\mu ^Q \nonumber \\&\quad + \frac{1}{2}\omega _1'\Sigma \Sigma ' \omega _1 + \cdots \nonumber \\ \cdots&+ \omega _1' \Sigma \Sigma '\left[ \Phi \left( I - \Phi \right) ^{-1}\omega _1 - \left( I - \Phi \right) ^{-1}\delta _1 \right] + \cdots \nonumber \\ \cdots&+ \frac{1}{2}\left[ \left( \Phi \left( I - \Phi \right) ^{-1}\omega _1 \right) ' \Sigma \Sigma ' \Phi \left( I - \Phi \right) ^{-1}\omega _1 \right] + \cdots \nonumber \\ \cdots&+\frac{1}{2}\left[ \left( \left( I - \Phi \right) ^{-1}\delta _1 \right) ' \Sigma \Sigma '\left( I - \Phi \right) ^{-1}\delta _1 \right] - \cdots \nonumber \\ \cdots&-\left[ \left( \Phi \left( I - \Phi \right) ^{-1}\omega _1 \right) ' \Sigma \Sigma ' \left( I - \Phi \right) ^{-1}\delta _1 \right] - \delta _0 < 0 \end{aligned}$$

which implies that in the limit

$$\begin{aligned} \lim _{n \rightarrow \infty }a_{n} = - \infty \end{aligned}$$

and since \(b_n\) and \(X_t\) are finite, as seen from Eq. (66), this implies for any time t the price of a (normalized) dividend strip satisfies

$$\begin{aligned} \lim _{n \rightarrow \infty }V_{t,n}^d = 0 \end{aligned}$$

The price of equity (normalized by current dividend) is an infinite sum of individual dividend strips and is given by Eq. (37)

$$\begin{aligned} \frac{V_t}{D_t} = \sum _{n=1}^{\infty }\text {exp}\left( a_n + b_n'X_t \right) = \sum _{n=1}^{\infty }V_{t,n}^d \end{aligned}$$

Finally, we note that condition (78) is also sufficient for equity-dividend ratio (81) to converge.

Tables and figures

See Tables 67891011 and Figs. 456789.

Table 6 Correlations between the macrofactors and the underlying time series over the 1990–2017 period
Table 7 List of the dataseries used to build the macro factors. Period: 1990–2017
Table 8 Continued list of the dataseries used to build the macrofactors. Period: 1990–2017
Table 9 Continued list of the dataseries used to build the macrofactors. Period: 1990–2017
Table 10 Regression analysis between the macroeconomic factors and dividend growth, earning growth and dividend payout ratio in the case of the US and Europe
Table 11 Regression outcome of the dividend payout ratio on the growth in earnings
Fig. 4

Time-series evolution of the Principal Components factors over the 1990–2017 period

Fig. 5

First rescaled first 20 eigenvalues associated to the Principal Component Analysis of the macro dataset

Fig. 6

Impulse responses of US GDP, Liquidity, Investment and VIX to the bubble in S&P500 equity index

Fig. 7

Impulse responses of US GDP, Liquidity, Investment and VIX to the bubble in Dow Jones equity index

Fig. 8

Impulse responses of US GDP, Liquidity, Investment and VIX to the bubble in Russell equity index

Fig. 9

Impulse responses of Europe GDP, Liquidity, Investment and VIX to the bubble in MSCI Europe equity index

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Ielpo, F., Kniahin, M. Fundamental bubbles in equity markets. Soft Comput 24, 13769–13796 (2020).

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  • Bubble
  • Affine model
  • Principal component analysis
  • Data-rich
  • Stationarity

JEL Classification

  • G12
  • C58
  • E44