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May 21, 2007

How Much Do Interest Rates Affect the Fair Value of Stocks?

John P. Hussman, Ph.D.
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In recent months, I've used a wide variety of analytical methods (discounted cash flows, normalized earnings, price/peak earnings calculations, etc) to show that stocks are currently priced to deliver unusually poor long-term returns – stated simply, the U.S. stock market is more overvalued than at any point in history except during the late 1990's bubble. In general, the methods I use emphasize that stocks are a claim on some stream of cash flows that will be delivered to investors over time. To estimate fair value, it is crucial to normalize earnings (rather than using “forward operating earnings” without correcting for the level of profit margins or the position of earnings within the economic cycle, as Wall Street seems eager to do here).

But what about interest rates? Sure, S&P 500 earnings are pushing along the very top of the 6% long-term growth trend that has repeatedly connected earnings peaks across economic cycles over the past century, and the S&P 500 currently trades at over 18 times those record earnings. Sure, when earnings have been similarly elevated in the past, the P/E multiple on those “top of channel” earnings has averaged only about 10. Sure, on normalized profit margins, the S&P 500 P/E would be about 25. Sure, profit margins have repeatedly experienced mean-reverting cycles over time. But this time it's different! After all, the current interest rate on the 10-year Treasury bond is a whole 1.4% below the average level of 10-year Treasury yields since 1950, and the year-over-year inflation rate is currently a whole 1.6% below the average inflation rate since 1950. Doesn't this mean that stocks deserve to be valued 60-80% higher than the historical norms?

If you watch CNBC for a few minutes, you'll immediately hear some analyst claim that stocks are cheap because the “forward earnings yield” on the S&P 500 is higher than the 10-year Treasury yield. The next analyst will just say that “stocks look cheap compared with bonds.” The next will offer some strange convolution of the so-called "Fed Model," like “Sure, P/E multiples are above average, but bonds are trading at a P/E of 21.” After a short break from the monotony by some kind of circus clown playing with horns and buzzers, another analyst then comes on saying how the firm's “valuation model” (which is driven by forward operating earnings and interest rates) implies that stocks are 20% undervalued.

Wall Street is presently managing trillions of dollars of other people's money on the basis of a single toy model, originally discovered in a packet at the bottom of a Cracker Jack box. Despite the superficial appearance of being some sort of discounting model (with the earnings yield in the numerator and the interest rate in the denominator), the "Fed Model" doesn't actually map into any reasonable model of discounted cash flow valuation without making odd and counter-factual assumptions about the relationship between growth, payouts, interest rates, and risk premiums.

The Fed Model asserts that earnings yields and Treasury yields have a 1-to-1 relationship, that stocks are undervalued anytime the earnings yield on the S&P 500 is higher than the 10-year Treasury yield, and that the gap between earnings yields and interest rates is the prime determinant of subsequent market returns.

It speaks volumes about the shallow analysis on Wall Street these days that all of these beliefs can be dispelled in a single chart.

There is, in fact, no stable relationship between earnings yields and interest rates. The relationship is actually negative in data since 1929, is marginally positive (but statistically insignificant) in data since 1950, and is only strongly positive in data from 1980 through 2000 as a statistical artifact of the disinflationary period from 1980 to 2000.

Essentially, earnings yields were remarkably elevated in the early 1980's (a strong indication of very high subsequent total returns) and declined to historic lows by 2000. During that period, interest rates also declined sharply. But this is crucial: earnings yields started far higher than they should have been, even given the high level of interest rates. By 2000, earnings yields were far lower than they should have been, even given the low level of interest rates. In effect, the decline in earnings yields was much larger than can be attributed to a “fair value” relationship with interest rates.

Since “forward operating earnings” were not compiled prior to about 1980, we have to use some other smooth measure of earnings for historical analysis. Calculating the P/E using the highest level of earnings to-date produces a somewhat higher P/E multiple (and a lower earnings yield) than using “forward operating earnings,” but this mainly affects the “constant term” in statistical estimations. Other statistical results are unaffected.

Proponents of the Fed Model ignore pre-1980 data entirely. That period since 1980 on the above chart is the entire basis for the analysis. In effect, the Fed Model looks at the decline in earnings yields since 1980, and loads the entire explanation on the decline in interest rates. What actually happened is that you had a second factor – the move from extreme undervaluation (abnormally high earnings yields) to extreme overvaluation (abnormally low earnings yields). The true “fair value” relationship between earnings yields and interest rates is nothing close to 1-for-1.

Geek's Note

(Skip to the next section if you hate math, or come back to this bit later - it shows why the Fed Model misinterprets the 1980-2000 period). Technically, what's going on here is an “omitted variables bias.” Call interest rates R, earnings yields Y and the extent of undervaluation/overvaluation V, where higher V means stocks are cheaper (i.e. the earnings yield is higher than it “should” be, given the level of interest rates).

Suppose that the “true” model of earnings yields can be written as a function of interest rates and the extent of under- or over-valuation:

(1) Y = a0 + a1R + V

This implies stocks “should” yield a0 + a1R when stocks are fairly valued (V=0). The coefficient a1 represents the true “fair value” effect of interest rates on earnings yields.

Suppose that we estimate

(2) Y = b0 + b1R

in the belief that b1 measures the “fair value” relationship between earnings yields and interest rates.

But also suppose also that we do that estimation only over a short period where V and R trended strongly in the same direction (specifically, 1980-2000) so that estimating

(3) V = c0 + c1R

would produce a positive value for coefficient c1 during that particular period, but not over a longer span of history.

Substituting (3) into (1), it follows that:

Y = a0 + a1R + c0 + c1R = (a0+c0) + (a1+c1)R

Compare that with our estimation in (2). Clearly, the incorrectly estimated value for b1 will be the true “fair value” impact of interest rates (a1) plus a bias term (c1). So we will overestimate the importance of interest rates in determining stock yields, which is exactly and (hopefully) obviously what the Fed Model does.

Try the Pudding

As I discussed last week, the proof of the pudding for any investment model is in the eating (particularly over the long-term). One should never place much reliance on a given model without testing its validity in historical data, preferably under a variety of periods and conditions.

Statistically, the Fed Model simply doesn't work.

Yes, there are a few instances where a positive or negative “spike” in the Fed Model was followed by favorable or unfavorable stock market returns, but all of those instances (and more) would have been captured by a simple rule: buy stocks when earnings yields are high and interest rates suddenly decline; sell stocks when earnings yields are low and interest rates suddenly advance.

In fact, I discovered a few years ago that the main usefulness of the Fed Model is to identify risks in the bond market – specifically, when stock yields are low but the Fed Model gives a “buy signal” anyway, it is evidence that bond yields are unusually depressed. That signal is typically followed by poor bond market performance (with no reliable implication for stocks).

Except for the occasional spike (noted above), Fed Model signals for the stock market are useless. Since 1980, the difference between the S&P 500 earnings yield and the 10-year Treasury yield explains just 6% of the variability in subsequent 2-year total returns (leaving 94% of the variability unaccounted for), and just 3% of the variability in subsequent 10-year returns.

If we look back to 1950, we find that the difference between the S&P 500 earnings yield and the 10-year Treasury yield explains just 3% of the variability in subsequent 2-year returns, and just 6% of the variability in subsequent 10-year returns. Adherents of the Fed Model might as well toss a coin to predict whether subsequent returns will be above or below average.

In contrast, the simple, humble, raw earnings yield (again using record trailing earnings to-date) explains about 15% of the variability in subsequent 2-year returns for periods both since 1950 and since 1980. That's still low but far better than the Fed Model for 2-year horizons. For 10-year returns, the lowly earnings yield has substantial power, explaining 39% of the variation in 10-year returns since 1980, and 54% of the variation since 1950. Those are very high figures for a single indicator.

In effect, the “correction” to earnings yields afforded by the Fed Model actually has the effect of destroying the power of earnings yields to explain subsequent stock market returns. Clearly, something is awry. Specifically, the assumption that earnings yields and interest rates should move 1-for-1 wildly exaggerates the importance of interest rates on justified stock valuations.

How much do interest rates affect the fair value of stocks?

We can attack the question of how much, say, the 10-year Treasury yield should affect valuations by making a number of observations:

First, suppose that interest rates rise by, say 1% (i.e. 100 basis points) because long-term inflation rises by 1%. Suppose also that the 1% higher inflation produces 1% higher long-term earnings growth. In this situation, it's well known that the higher growth and the higher interest rate “cancel each other out” in the discounting process, with the result that long-term expected returns are 1% higher, but “justified” P/E ratios, dividend yields, and other valuation measures are actually unchanged.

So an increase in interest rates that is driven by inflation that fully passes through to earnings growth will not affect valuation multiples. An increase in interest rates that doesn't pass through to earnings growth will affect valuation multiples. Still, in both cases, higher interest rates imply that stocks will need to be priced to deliver higher long-term returns.

Let's look at the effect of interest rate fluctuations on inflation and earnings growth first, and then address the question of how much a change in interest rates should affect the long-term return on stocks.

Do changes in 10-year Treasury yields imply changes in future inflation and earnings growth? In data since 1950, neither effect is significant. A 1% increase in the 10-year Treasury yield has been associated with only a 0.20% increase in the subsequent 10-year average inflation rate, but this effect does not explain much of the variation in inflation over time. Moreover, a 1% increase in the 10-year Treasury yield has been associated with an increase of just 0.09% in the subsequent 10-year growth rate of S&P 500 earnings (even after filtering out earnings volatility from recessions, which otherwise destroys the entire statistical relationship). Regardless of the level of interest rates, the long-term growth of S&P 500 earnings, measured from peak-to-peak, is virtually constant at about 6% annually.

So the 10-year Treasury yield does not signal much about subsequent long-term inflation or earnings growth. Instead, the 10-year Treasury yield is strongly correlated with the average inflation rate over the past 10-year period. In data since 1950, the prior 10-year inflation rate has explained 79% of the variation in 10-year yields. The intercept of that relationship (i.e. the average “real” spread over the past inflation rate) has been 2.40%.

Now, even though higher interest rates don't imply higher future inflation or earnings growth, it's clear that higher rates increase the long-term return that stocks have to deliver to remain competitive, and that lower interest rates decrease the hurdle. Accordingly, we can expect that higher interest rates should be accompanied by higher earnings yields and lower P/E ratios, while lower interest rates should be accompanied by lower earnings yields and higher P/E ratios.

This point is entirely sound. The issue, however, is how much impact a 1% increase in the 10-year Treasury yield should have on the long-term return priced into stocks.

See, if a 1% increase in the 10-year Treasury yield can be expected to be permanent – so that the higher interest rate will persist over the effective life of stocks – then it's reasonable that the expected return on stocks should reflect that full 1% increase. But the data immediately argue that this isn't the case. Since 1950, if the 10-year Treasury yield has been 1% above average, the yield 5-years later has only been about 0.67% above average. At longer horizons, we find that only 0.36% of that yield fluctuation persists after 10 years, and that there is literally no relationship between the 10-year Treasury yield at one date and the level 15 years later. Far from interest rate fluctuations being permanent, we observe that they are damped out fairly quickly.

Moreover, a 10-year Treasury bond has a short duration compared with U.S. stocks, which currently have an effective duration of over 50 years*. If a 1% increase in the Treasury yield completely damps out after about 15 years, we can calculate that the interest rate change should have an impact of only about 0.10% on the long-term return demanded on stocks with far longer durations. Assuming companies pay out about 50% of earnings to shareholders (and reinvest the remainder), the earnings yield would have to increase by about 0.20% (20 basis points) in response to the 1% increase in the Treasury yield.

In fact, we observe a small relationship in historical data since 1950, but it's such a weak factor in practice that cyclical bull and bear swings overwhelm the effect. The interest rate impact generally implies a change of less than half of one point in the P/E for every 100 basis point change in the 10-year Treasury yield. Note that in the chart at top, the raw earnings yield is strongly correlated with the subsequent long-term return on stocks, regardless of the level of 10-year Treasury yields.

Again, it's perfectly valid to expect interest rates to affect stock market valuations, but 10-year Treasury bonds have durations that are far too short to assume anything remotely close to a 1-to-1 relationship with earnings yields.

In short, interest rates affect “justified” stock valuations when 1) the interest rate changes do not pass through to equal changes in earnings growth, and either; 2) the change in interest rates can be expected to be permanent, rather than damping out over time, or; 3) the change in interest rates is based on a security with the same effective duration as stocks. Presently, stocks are so overvalued that you would have to use a 50-year zero-coupon bond (even a 30-year Treasury bond has a duration of only about 14 years). If such a security existed, you can be sure it would have a higher and far less volatile yield than a 10-year Treasury.

The historical data simply do not support the notion that fluctuations the 10-year Treasury yield should cause wide swings in “justified” stock valuations.

But that doesn't mean that interest rate fluctuations aren't important…

The importance of trends in interest rates

When we think about the effect of interest rates on the stock market, we have to make a major distinction between the level of interest rates, and the trend of interest rates.

What we've established so far is only that the level of interest rates (at least on the basis of short or intermediate term bonds) does not justify large changes in the “fair” multiple applied to stocks. Within a wide range of movement, changes in the level of interest rates normally warrant only one or two points of variation in “fair” P/E multiples. The remainder of the actual fluctuations in P/E multiples reflect changes in the long-term attractiveness of stocks.

Having addressed the question of "fair value," we can ask a different question – what happens when stocks become significantly overvalued or undervalued? Regardless of the level of interest rates, does the trend of interest rates affect how quickly valuations revert toward more normal levels? The answer here is a resounding “yes.”

Note the distinction. Interest rate levels on conventional short- or intermediate-term Treasury bonds don't have a huge effect on the “fair” or “justified” value of stocks, but when stocks are far from fair value, interest rate trends (regardless of the level) do affect the speed at which stocks revert back toward normal valuations.

When valuations are cheap and risk premiums are wide, falling interest rates contribute to downward pressure on risk premiums and upward pressure on stock prices (especially when interest rates are falling across a range of maturities, coupled with improved breadth, oversold conditions, expanding trading volume and other factors). In contrast, when valuations are high and risk premiums are already thin, rising interest rates contribute to upward pressure on risk premiums and downward pressure on stock prices (especially when interest rates are rising across a range of maturities, coupled with deteriorating breadth, overbought conditions, dull trading volume and other factors).

Consider a very basic cross-section of valuations and interest rate trends using data since 1950. Specifically, we'll place P/E ratios into 3 classes: below 12, 12-18, and above 18. We'll classify the 10-year Treasury yield as “falling” if it is below its level of 6 months earlier, and “rising” otherwise. Clearly, this is an extremely simplistic classification setup, but even here we can see the combined impact of valuations and interest rate trends. The following table provides the annualized total return and volatility for the S&P 500 based on each set of conditions.

 

Treasury yield falling

Treasury yield rising

 

Annualized return

Annualized volatility

Annualized return

Annualized Volatility

PE < 12

26.32%

11.86%

10.99%

13.74%

PE 12-18

18.88%

13.70%

4.90%

16.03%

PE > 18

8.94%

16.53%

-0.41%

18.57%

Notice that rising interest rate trends are invariably accompanied by weaker returns and greater volatility, regardless of the level of valuations.

So as noted at the outset, we often observe very good buying opportunities in stocks when valuations are cheap and interest rates suddenly decline. Conversely, we often see deep market declines when valuations are rich and interest rates suddenly advance. One need not use an inept tool such as the “Fed Model” to identify these instances.

Capitulating at the highs

Of course, there are numerous ways to improve the classification of market conditions by including additional relevant factors. The problem is that even an approach that performs remarkably well on a total return and risk-adjusted basis over the long-term, and even over repeated market cycles, may produce fairly dull returns over extended periods shorter than a full market cycle. At times like the present (as in 1999 and 2000) when the market advances despite conditions that have historically produced poor returns, it is easy to capitulate at the highs, and buy into a market that seems to be “running away.”

It may help to realize that if an investor had sold stocks and moved to Treasury bills in September 1996, missing the last 4 years of the bull market advance to the 2000 peak (during which time the S&P 500 gained an additional 120%), that investor would still have performed better than a buy-and-hold on the S&P 500 by the time that the subsequent bear market was complete in 2002. I cannot emphasize enough how profoundly a large loss can damage long-term returns. Bear markets routinely erase more than half of the preceding bull market advance, but that's not easy to remember when every marginal new high is celebrated as if there can't possibly be any end in sight.

As always, the objective is to accept risk in conditions that have historically produced a strong return/risk profile on average, and to avoid, hedge or diversify away risk in conditions that have historically produced a poor return/risk profile on average. That phrase “on average” is what separates us from market timers. A market timer believes that the next draw out of a hat can be predicted in advance. We just believe there are different hats, and once you identify the hat (the “Market Climate”), the best one can know is the average return/risk profile of the returns that will be drawn from that hat. It is not possible to tell whether the next short-term return produced by the market will be positive or negative.

For that reason, we can't have any assurance that the market will roll over into a bear market next week, next month, next quarter or even next year. But we can have reasonable confidence that markets will continue to cycle between great optimism and great despair. With stocks at over 18 times top-of-channel earnings on record profit margins, with the major indices at multi-year highs, with short-term trends easily through the upper Bollinger bands at the monthly, weekly and daily frequencies, with 54.3% of advisors bullish and only 19.6% bearish according to the latest Investors Intelligence figures, and with 10-year Treasury yields higher than they were 6 months ago, it should be reasonably easy to determine which extreme the market more closely resembles at present.

Overvalued, overbought, overbullish, yields rising. o-v-o-b-o-b-y. Hmm. That sounds familiar.

Market Climate

As of last week, the Market Climate for stocks was characterized by unfavorable valuations, reasonably positive market action, and overall, a combination of overvalued, overbought and overbullish conditions that has historically been associated with stock market returns below Treasury bill yields, on average.

Our discipline is to align the portfolio at every point in time with the prevailing Market Climate, rather than trying to forecast the market, “catch” short-term advances and declines, or “call” market turns. In practice, that discipline can be frustrating at times, because specific short-term movements during a particular set of market conditions can look very different from what occurs “on average.” At present, we're observing incremental but persistent advances to new highs, despite an unusually extreme combination of overvalued, overbought, overbullish conditions that has historically left the market vulnerable to deep and abrupt declines.

Presently, the market's returns look very different from what prevailing conditions have produced on average. Then again, a deep and abrupt market decline would suddenly make things look very much like the average (or even produce market performance worse than what we've observed on average). For our purposes, what is important here is that prevailing conditions have generally produced total returns on the S&P 500 that have been below Treasury yields. In the instances most closely resembling current conditions, the losses have often been unusually abrupt. Valuations do not provide sufficient “investment merit” to accept market risk, and despite reasonably good technical “internals” such as market breadth, the present overbought and overbullish conditions remove the “speculative merit” that good market action might otherwise provide.

If we can clear these overextended conditions with a moderate market decline (perhaps 4%) without a substantial deterioration in the quality of market internals, we might obtain enough evidence to accept at least a moderate speculative exposure to market risk (say, allowing 20-30% our stock holdings in the Strategic Growth Fund to be unhedged).

Here and now, however, one would only be compelled to take speculative risk if one believes not only that stocks warrant an entirely new “standard of value”, but also if one believes that the short-term dynamics of the market have changed, so that overextended conditions are no longer likely to produce even a period of consolidation.

For my part, I believe neither. My view remains that we are in something of a speculative blowoff that has the potential to result in very hostile market conditions ahead. As usual, we needn't make forecasts of upcoming market direction, or make predictions about whether or when a bear market will start. It is sufficient to recognize that prevailing market conditions have historically produced a weak return/risk profile, on average. Even if the “standard of value” has changed, we can also expect that there will be many periods where valuations will be relatively more favorable, where market action will be relatively good, and even where the market may occasionally be relatively oversold. Any combination of those conditions will be better points than today at which to accept market exposure.

In bonds, the Market Climate remained characterized last week by unfavorable valuations and moderately unfavorable market action, holding the Strategic Total Return Fund to a short 2-year duration, mostly in TIPS, with about 20% of assets in precious metals shares, where the Market Climate remains favorable on our measures.

* Hyper-Geek's Note: The duration of equities is essentially the elasticity of the price to changes in the implied rate of return. Consider the dividend discount model: P = D/(k-g) = D(k-g)^(-1)

dP/dk = -D(k-g)^(-2) = -P/(k-g)
(dP/P)/dk = -P/D (kind of an elegant result)
Duration = -(dP/P)/(dk/(1+k)) = (1+k)P/D

So the duration of stocks is effectively the price/dividend ratio times 1.something, say 1.07-1.08. That P/D is currently at about 55, producing a duration of nearly 60 years.

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