# Hubble? Bubble? Valuation trouble?

Can Gao, Ian Martin, Arjun Mahalingam and Nicholas Vause

Since Covid-19-related crashes in March, major stock indices around the world have bounced back. This is despite little or no recovery in corporate earnings expectations. As a result, forward-looking price-to-earnings ratios have increased, rising above long-run average values in most large advanced economies and approaching record highs in the United States. Commenting on such valuations, some market participants have suggested there is ‘a great deal of optimism priced into the market’ and that stock prices ‘cannot defy economic gravity indefinitely’. This post takes a closer look at stock valuations, focusing on the UK, and drawing both on a textbook model and new research from academia.

Valuation trouble?

Major stock indices in the United Kingdom, euro area and United States fell by around one third in late February and March as Covid-19 developed into a global pandemic. Since then, however, they have bounced back, recovering partially or even fully in the case of the S&P 500, as shown in Chart 1 (blue lines).

Chart 1: Forward-looking price-to-earnings ratios and their components

Notes: Earnings are the average of 12-month ahead equity analysts’ forecasts of earnings per share. Historical ranges and averages are since the start of 2000.

Sources: Eikon and IBES by Refinitiv and Bank calculations.

This is despite equity analysts’ expectations of corporate earnings for the year ahead remaining subdued (magenta lines). Hence, forward-looking price-to-earnings (PE) ratios have increased (orange lines). In each market, they have risen above both their pre-Covid-19 values and their long-run averages, significantly so in the case of the S&P 500.

So, do these valuations look frothy?

Hubble

Not obviously, is our conclusion based on a textbook dividend discount model (DDM).

This model takes into account not only expectations of earnings that corporations will pay out to shareholders – both through dividends and share buybacks – in the year ahead, but also beyond that. And these more-distant expectations have not fallen as sharply as near-term ones, providing some support for stock valuations.

In addition, stock valuations depend on the rates at which investors discount these expected future pay-outs. Part of these discount rates reflect impatience: even if future pay-outs were perfectly certain, investors would still rather have them sooner than later. Hence, they discount relatively distant pay-outs more than relatively close ones. In major advanced economies, government bond yields are good proxies for these risk-free discount rates. And these yields have fallen sharply since before Covid-19, reaching or approach record lows. This is also consistent with higher stock valuations.

However, there is a second component of discount rates: the equity risk premium (ERP). Other things equal, this increases when future pay-outs become more uncertain. So it may have increased under the Covid-19 pandemic, pulling down on stock valuations.

Unfortunately, the ERP is not directly observable. However, the DDM can be used to infer the value of an ERP that is consistent with observed stock prices and other market variables. To do this, we take the key formula in the model,

$P_{t} = \sum_{k=1}^{\infty} \frac{E_{t}(D_{t+k}) + E_{t}(B_{t+k})}{(1 + R_{k}^{f} + ERP)^k}$, (1)

and plug in values for the current equity price ($P_{t}$), government bond yields ($R_{k}^{f}$) of varying maturity ($k$) and forecasts of future dividends ($E_{t}(D_{t+k})$) and share buybacks ($E_{t}(B_{t+k})$). Then we solve for the ERP, which is the only unknown variable.

Chart 2 shows results for the FTSE All-Share index. The ERP is currently quite high by historical standards – in the top quartile of estimates for the past two decades. This implies investors require a relatively high level of compensation for uncertainty to continue holding UK stocks.

Chart 2: Equity risk premium consistent with stock prices

Notes: The box (and whiskers) show the full (and inter-quartile) historical range since the start of 2000. The white line in the box shows the historical average.

Sources: Bloomberg Finance L.P., Eikon and IBES by Refinitiv, Tradeweb, IMF World Economic Outlook and Bank calculations.

However, it is perhaps somewhat surprising that the latest ERP estimate is only marginally higher than the one for end-2019, before Covid-19 had increased economic uncertainties. But the precise values of these ERP estimates should be taken with a pinch of salt. They depend on the accuracy of the model and the variables input into it. In particular, it is difficult to know how well average equity analyst dividend forecasts, which are used as one input in the model, represent the dividend expectations of investors. This is especially problematic at the moment because the dispersion of forecasts among individual analysts is 20%-50% higher (depending on the horizon of the forecast) than pre-Covid-19.

Overall, we think the DDM is not crying out ‘over-valuation’ of UK stocks. If the estimated ERP does reflect any valuation concerns for investors, it seems more likely they are worried about hubble (a small hump) than a bubble.

No bubble

Although the ERP is not directly observable, Gao and Martin (2020) show how to place a lower bound on its value. Moreover, they use this minimum value to develop an indicator of stock market sentiment. Specifically, the indicator shows the minimum annualised growth rate of dividends that rational investors must expect to be willing to hold stocks at prevailing prices. Other things equal, the higher the indicator the more optimistic are investors. If the indicator was so high that it was hard to rationalise such strong dividend growth expectations, investors would probably have become irrationally exuberant. Indeed, Gao and Martin apply their methodology to the S&P 500 and find the sentiment indicator peaked at almost 15% towards the end of the 1990s, a little while before the dot-com bubble burst.

Here, we apply the same methodology to the FTSE All-Share index.

An intuitive understanding of the methodology can be derived from the Gordon growth model of stock valuation. This is a simple version of equation (1), with no share buybacks and expected dividends that grow at a constant rate. It can be written as

$D/P = E(R) - E(G)$, (2)

where $D/P$  is the dividend yield (dividend-to-price ratio), $E(R)$  is the expected return, which is the sum of a risk-free rate and an equity risk premium, and $E(G)$ is the expected growth rate of dividends. We use a government bond yield as a proxy for the risk-free rate (component 1 of the indicator) and derive a lower bound for the equity risk premium from option prices (component 2). The sum of these two components gives a lower bound for $E(R)$. Finally, we compute a lower bound for $E(G)$ as the difference between that and $E(R) - E(G)$ (component 3), which depends on dividend yields.

Further detail on the methodology is available below in the technical appendix.

At first glance, the sentiment indicator for the FTSE All-Share, which is shown in Chart 3, appears very far from suggesting any exuberance or valuation concerns. It is near an all-time low, not far off values seen during the recession that followed the 2007-08 Global Financial Crisis (GFC). The risk premium component (magenta shading) did spike in March and has remained somewhat elevated, suggesting investors require more compensation to hold equity risk in light of the Covid-19 pandemic. But these levels are far short of previous spikes, notably those around the time of LTCM’s collapse and during the GFC. Moreover, this upward push on the indicator was more than offset by expectations of weak dividend growth relative to overall stock returns (orange shading), which reflect the high level of dividend yields.

Chart 3: FTSE All-Share sentiment indicator

Sources: Bank of England, Bloomberg Finance L.P., Eikon by Refinitiv, Global Financial Data and Bank calculations.

But remember it is not the level of the sentiment indicator that matters, but whether rational investors expect the dividend growth rate to exceed it. The latest average of equity analysts’ forecasts of dividend growth over the next year is subdued in light of the Covid-19 pandemic at 2.4%. But this is still comfortably above the -4.8% value of the sentiment indicator on the same date. So it is quite plausible that rational investors are expecting dividend growth in excess of the indicator, which suggests that UK stock prices are not buoyed at present by irrational exuberance.

In conclusion, although forward price-earnings ratios and some market commentary might suggest equity valuations have become stretched, we find no clear support for that for UK stocks either in a well-established dividend discount model or the new sentiment indicator.

Technical appendix

The second component is based on a theoretical relationship between the equity risk premium and option prices derived in Gao and Martin (2020). This is

$ERP_{t} = E_{t}(r_{t+1}) - r_{t+1}^{f} \geq \frac{1}{P_{t}}\bigg\{\int_{0}^{F_{t}} \frac{put_{t}(K)}{K}dK + \int_{F_{t}}^{\infty} \frac{call_{t}(K)}{K}dK \bigg\}$, (3)

where $r_{t+1}=\mathrm{ln}((P_{t+1}+D_{t+1})/P_{t})$, $r_{t+1}^{f}=\mathrm{ln}(1+R_{t+1}^{f})$, $put_{t}$ and $call_{t}$ are put and call option prices with strike price $K$ and $F_{t}$ is the futures price. The term on the right-hand side of the inequality is referred to as LVIX because of its similarity to the formula for the well-known VIX index. We collect data on option prices and compute this term.

The third component is based on a first-order approximation derived by Gao and Martin (2020) that relates the current dividend yield to a weighted average of future returns and dividend growth. This is

$y_{t} = (1-\rho) \sum_{i=0}^{\infty}\rho^{i}(r_{t+1+i} - g_{t+1+i})$, (4)

where $y_{t} = \mathrm{ln}(1+D_{t}/P_{t})$, $g_{t+1} = \mathrm{ln}(D_{t+1}/D_{t})$, $\rho = e^{-E(y)}$ and $(1-\rho) \sum_{i=0}^{\infty}\rho^{i}=1$.

From (4), it follows that

$E_{t}(r_{t+1} - g_{t+1}) = \frac{1}{(1-\rho)}y_{t} - \frac{\rho}{(1-\rho)}E_{t}(y_{t+1})$. (5)

If, as we assume (and confirm), the evolution of $y_{t}$ over time is well described by an autoregressive model with three lags, then

$E_{t}(r_{t+1} - g_{t+1}) = a_{0} + a_{1}y_{t} + a_{2}y_{t-1} + a_{3}y_{t-2}$. (6)

After estimating the coefficients $a_{0}$, $a_{1}$, $a_{2}$ and $a_{3}$ by applying regression analysis to annual data since the end of the Second World War, we use the fitted equation to generate $E_{t}(r_{t+1} - g_{t+1})$ using current and past $y_{t}$.

Finally, we combine the three components to get the sentiment indicator, $B_{t}$:

$E_{t}(g_{t+1}) \geq B_{t} = \underbrace{r_{t+1}^{f}}_\text{Component 1} + \underbrace{LVIX_{t}}_\text{Component 2} - \underbrace{E_{t}(r_{t+1} - g_{t+1})}_\text{Component 3}$. (7)

Can Gao works at the Imperial College London, Ian Martin works at the London School of Economics and Arjun Mahalingam and Nicholas Vause work in the Bank’s Capital Markets Division.

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## 2 thoughts on “Hubble? Bubble? Valuation trouble?”

1. Michael Kearns says:

It seems to me that the risk for foreign investors in Sterling denominated shares is a significant element in the current value of shares and will continue to depress values for some time.

2. Donald Wilkinson says:

Interesting perspective thanks for sharing such a detailed explanation! “The second component is based on a theoretical relationship between the equity risk premium and option prices derived in Gao and Martin” very interesting point well made! Bravo, will be waiting for your next post.

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