A new stock market valuation measure with application to retriement planning

Andrey Sarantsev

doi:10.1515/strm-2022-0009

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A new stock market valuation measure with application to retriement planning

Andrey Sarantsev

Published/Copyright: April 23, 2025

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From the journal Statistics & Risk Modeling Volume 42 Issue 1-2

Abstract

We generalize the classic Shiller cyclically adjusted price-earnings ratio (CAPE) used for prediction of future total returns of the stock market. We treat earnings growth as exogenous. The difference between log wealth and log earnings is modeled as an autoregression of order 1 with linear trend 4.6 and Gaussian innovations. Detrending gives us a new valuation measure. Our results disprove the Efficient Market Hypothesis. Therefore, long-run total returns equal long-run earnings growth plus 4.6 Ẇe apply results to retirement planning. A withdrawal process governs how a retired capital owner withdraws a certain fraction of wealth annually. We study the long-term behavior of such processes.

Keywords: Autoregression; total returns; earnings growth; ergodic process

MSC 2020: 60F05; 60G10; 60J20; 62J05; 62M10; 62P05; 62P20; 91B84

A Appendix

A.1 Proof of Lemma 1

Equation (3.7) follows from (3.5) and the following computations:

Q ¯ ⁢ ( T ) = 1 T ⁢ ( ln ⁡ V ⁢ ( T ) - ln ⁡ F ⁢ ( T ) + G ⁢ ( 1 ) + … + G ⁢ ( T ) )

= c + H ⁢ ( T ) T - H ⁢ ( 0 ) T + 1 T ⁢ ( G ⁢ ( 1 ) + G ⁢ ( 2 ) + … + G ⁢ ( T ) ) .

Equation (3.8) follows from (3.3) and (3.7).

A.2 Proof of Theorem 1

Let us show (3.9). We can rewrite

1 T ⁢ ( ln ⁡ V ⁢ ( t ) - ln ⁡ F ⁢ ( t ) ) = 1 T ⁢ ( c ⁢ T + H ⁢ ( T ) - H ⁢ ( 0 ) ) = c + H ⁢ ( T ) - H ⁢ ( 0 ) T .

It suffices to show that almost surely,

(A.1) H ⁢ ( T ) - H ⁢ ( 0 ) T → 0 , T → ∞ .

The process H is ergodic in the sense of classic ergodic theory; see for example [20, Chapter 6]. The mean of the stationary distribution for H is equal to c. By the classic Birkhoff ergodic theorem, see [20, Theorem 6.2.7], we have almost sure convergence

(A.2) H ⁢ ( 1 ) + … + H ⁢ ( T ) T → c , T → ∞ .

Immediately from (A.2), we have

(A.3) H ⁢ ( 0 ) + … + H ⁢ ( T - 1 ) T → c , T → ∞ .

We get (A.1) immediately from (A.2) and (A.3). This completes the proof of (3.9). Similarly, (3.10) follows from (3.9) and (A.1).

A.3 Proof of Theorem 2

Define the density of 𝒩 ⁢ ( 0 , σ 2 ) :

(A.4) φ ⁢ ( z ) = 1 2 ⁢ π ⁢ σ ⁢ e - z 2 2 ⁢ σ 2 .

Then we can express ε ⁢ ( t ) = H ⁢ ( t ) - h - b ⁢ ( H ⁢ ( t - 1 ) - h ) using (2.1). Therefore, the transition density of ( G , H ) : condition density of ( G ⁢ ( t ) , H ⁢ ( t ) ) at ( g 1 , h 1 ) given G ⁢ ( t - 1 ) = g 0 and H ⁢ ( t - 1 ) = h 0 , is given by

(A.5) 𝐩 ⁢ ( g 0 , h 1 - h - b ⁢ ( h 0 - h ) , g 1 ) ⁢ φ ⁢ ( h 1 - h - b ⁢ ( h 0 - h ) ) .

A.4 Proof of Theorem 3

Step 1.

The function 𝐩 from Assumption 3 is strictly positive for any argument values. The Gaussian density (A.4) is strictly positive too. Thus, the same is true for the function (A.5). Therefore, the Markov process ( G , H ) has the positivity property: For a subset A ⊆ ℝ 2 of positive Lebesgue measure, and for any g 0 , h 0 ∈ ℝ 2 ,

(A.6) ℙ ( ( G ( 1 ) , H ( 1 ) ) ∈ A ∣ G ( 0 ) = g 0 , H ( 0 ) = h 0 ) > 0 .

Step 2.

Solve the recurrent equation (2.1): H ⁢ ( t ) - h = b t - 1 ⁢ ε ⁢ ( 1 ) + … + ε ⁢ ( t ) + b t ⁢ ( H ⁢ ( 0 ) - h ) . This random variable has the same distribution as

H ′ ⁢ ( t ) = ∑ s = 1 t b s - 1 ⁢ ε ⁢ ( s ) + b t ⁢ ( H ⁢ ( 0 ) - h ) + h .

The following series converges in L p :

(A.7) H ′ ⁢ ( ∞ ) := ∑ s = 1 ∞ b s - 1 ⁢ ε ⁢ ( s ) .

Indeed, the p-norm of ε ⁢ ( s ) is constant (does not depend on s), and b ∈ ( 0 , 1 ) . Therefore, the series (A.7) converges in L p . We apply that the space L p is Banach, that is, every fundamental sequence converges. Thus the sequence H ′ ⁢ ( t ) → H ′ ⁢ ( ∞ ) + h as t → ∞ in L p . Therefore, the distribution of H ⁢ ( t ) (the same as that of H ′ ⁢ ( t ) ) converges in law to the distribution of H ′ ⁢ ( ∞ ) + h . In addition, the pth moments of H ⁢ ( t ) converges to the pth moment of H ′ ⁢ ( ∞ ) + h as t → ∞ . Since convergence in 𝒲 p is equivalent to weak convergence plus convergence of pth moments, the above results implies convergence in 𝒲 p . Finite pth moments of the stationary distribution Π are implied by (A.8) and Fatou’s lemma. We get the marginal distribution of Π for H from the observation that H is an autoregression of order 1 with Gaussian innovations; see any classic text on time series, for example [4].

Step 3.

Finally, let us show uniqueness of the stationary distribution and convergence in total variation. This follows from the two observations. First, as mentioned above, the process ( G , H ) has the positivity property (A.6). Second, we have

(A.8) sup t ⁡ 𝔼 ⁢ [ | G ⁢ ( t ) | p + | H ⁢ ( t ) | p ] < ∞ .

For G, this is in Assumption 4; and for H, this was shown above, since H ′ ⁢ ( t ) converges in L p as t → ∞ . From (A.8), we get that the sequence ( G ⁢ ( t ) , H ⁢ ( t ) ) is tight:

(A.9) sup t = 0 , 1 , 2 , … [ ℙ ( | G ( t ) | > u ) + ℙ ( | H ( t ) | > u ) ] → 0 , u → ∞ .

In [11], the property (A.9) is called boundedness in probability. This process ( G , H ) is also irreducible (that is, it does not have two or more disconnected components of the state space ℝ 2 ), since the transition density is strictly positive everywhere. The function φ from (A.4) is continuous. The function 𝐩 from Assumption 3 is continuous in the first two arguments. Therefore, the Markov process ( G , H ) is a T -chain in the sense of [11, p. 548]. Applying Corollary (ii) in [11, p. 550], we conclude that the Markov process H is positive Harris recurrent in the sense of [11]. Therefore, it has a unique stationary distribution. Applying [11, Theorem 3.4] to Lebesgue measure ψ (since ( G , H ) is irreducible with respect to this Lebesgue measure), we get: all compact sets are petite. Apply [11, Theorem 5.2 (iii)]. Use the aperiodicity ( m = 1 in the notation on the cited theorem), which follows from positivity of the transition density. This completes the proof of Step 3.

A.5 Proof of Theorem 4

We rewrite

Q ⁢ ( 1 ) + … + Q ⁢ ( T ) - ( c + g ) ⁢ T T = G ⁢ ( 1 ) + … + G ⁢ ( T ) - g ⁢ T T + H ⁢ ( T ) - H ⁢ ( 0 ) T .

Recall Assumption 2. By the Slutsky theorem, see for example [7, Theorem 5.5.17], it suffices to show convergence in distribution (or, equivalently, in probability):

(A.10) H ⁢ ( t ) - H ⁢ ( 0 ) t → 0 , t → ∞ .

But H ⁢ ( t ) → H ⁢ ( ∞ ) ∼ π H weakly converges to the stationary distribution. Applying the Slutsky theorem again,

H ⁢ ( t ) t → 0

in probability. This proves (A.10).

A.6 Proof of Theorem 5

(a) We can rewrite

V W ⁢ ( t ) = V W ⁢ ( t - 1 ) ⁢ e Q ⁢ ( t ) - w * , t = 1 , 2 , … ,

where w * = - ln ⁡ ( 1 - w ) < g + c . By induction,

V W ⁢ ( t ) = exp ⁡ [ Q ⁢ ( 1 ) + … + Q ⁢ ( t ) - t ⁢ w * ] .

We can represent this exponent as

ln ⁡ V W ⁢ ( t ) = G ⁢ ( 1 ) + … + G ⁢ ( t ) + H ⁢ ( t ) - H ⁢ ( 0 ) + c ⁢ t - w * ⁢ t .

Using an earlier result (A.1), we get

1 t ⁢ ln ⁡ V W ⁢ ( t ) = 1 t ⁢ ∑ k = 1 t G ⁢ ( k ) + 1 t ⁢ H ⁢ ( t ) - 1 t ⁢ H ⁢ ( 0 ) + c - w * → g + c - w * > 0 .

Thus V W ⁢ ( t ) → ∞ a.s. as t → ∞ .

(b) Similarly, we can represent

ln ⁡ V W ⁢ ( t ) = G ⁢ ( 1 ) + … + G ⁢ ( t ) + H ⁢ ( t ) - H ⁢ ( 0 ) + c ⁢ t + ln ⁡ ( 1 - W ⁢ ( 1 ) ) + … + ln ⁡ ( 1 - W ⁢ ( t ) ) .

Note that - ln ⁡ ( 1 - w ) ≥ w for w ∈ ( 0 , 1 ) . Thus

ln ⁡ V W ⁢ ( t ) ≤ G ⁢ ( 1 ) + … + G ⁢ ( t ) + H ⁢ ( t ) - H ⁢ ( 0 ) + c ⁢ t - W ⁢ ( 1 ) - … - W ⁢ ( t ) .

Dividing by t and letting t → ∞ , we have a.s. convergence of the right-hand side:

1 t ⁢ ln ⁡ V W ⁢ ( t ) ≤ 1 t ⁢ ∑ k = 1 t G ⁢ ( k ) + H ⁢ ( t ) t - H ⁢ ( 0 ) t + c → g + c - w < 0 .

Thus lim ¯ t → ∞ ⁡ V W ⁢ ( t ) ≤ 1 , and this withdrawal rate is not sustainable.

A.7 Proof of Lemma 2

The corresponding wealth process satisfies the recurrent relation

V ⁢ ( t ) = V ⁢ ( t - 1 ) ⁢ e Q ⁢ ( t ) ⁢ ( 1 - W ⁢ ( t ) ) = V ⁢ ( t - 1 ) ⁢ e Q ⁢ ( t ) - G ⁢ ( t ) - u .

Applying this recurrent relation multiple times, we get:

(A.11) V ⁢ ( t ) = exp ⁡ [ Q ⁢ ( 1 ) + … + Q ⁢ ( T ) - G ⁢ ( 1 ) - - ⁡ ⋯ G ⁢ ( T ) - T ⁢ u ] = exp ⁡ [ H ⁢ ( T ) - H ⁢ ( 0 ) + ( c - u ) ⁢ T ] .

Combine (A.11) with an earlier result (A.1), we get the following:

If c > u , we get H ⁢ ( T ) + ( c - u ) ⁢ T → + ∞ as T → ∞ , and therefore V ⁢ ( t ) → + ∞ . The withdrawal process is sustainable.
If c < u , then H ⁢ ( T ) + ( c - u ) ⁢ T → - ∞ as T → ∞ , therefore V ⁢ ( T ) → 0 , and the withdrawal process is not sustainable.
Finally, if c = u , then the autoregressive process H ⁢ ( t ) does not have an almost sure limit, so the same is true for V ⁢ ( t ) . The withdrawal process is not sustainable.

Acknowledgements

The author is thankful to UNR students Akram Reshad, Taran Grove, Erick Luerken, Tran Nhat, and Michael Reyes for useful discussion. The author thanks the Department of Mathematics and Statistics at the University of Nevada, Reno, for welcoming atmosphere for research. The author thanks referees and editors for useful comments.

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Received: 2022-03-23

Revised: 2025-03-11

Accepted: 2025-03-27

Published Online: 2025-04-23

Published in Print: 2025-05-01

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Articles in the same Issue

https://doi.org/10.1515/strm-2022-0009

Keywords for this article

Autoregression; total returns; earnings growth; ergodic process