Since the value of a stock today is its future earnings or shareholder dividends discounted back to the present, then price-to-earnings ratios should be in part a function of changes in the interest rate. Looking at the last fifty years of the S&P 500 (NYSEARCA:SPY) index's closing price to trailing twelve month earnings ratio relative to the year-end yield on the ten-year U.S. Treasury note produces interesting observations. Examining the linear relationship of the P/E ratio and the ten-year Treasury yield would signal that either the ten-year Treasury yield is too low, the price-to-earnings multiple of the S&P 500 is too low, or a combination thereof.

Holding either the earnings multiple or the interest rate constant and solving for the other variable using the linear relationship that has been observed over the trailing fifty years demonstrates the large magnitude of the current deviation from this historical relationship. If the ten-year Treasury is appropriately valued, then the linear relationship between the yield on this security and the P/E ratio of the S&P 500 would signal that the earnings multiple should be at 20.98x rather than its current 14.2x. This earnings multiple applied to the trailing twelve months earnings of the S&P 500 of $96.89 per share would equate to an index level of 2032, presaging a 48% rise in the S&P 500. Conversely, if the P/E ratio is correctly valued, and the ten-year Treasury yield is understated by the market, then the ten-year Treasury yield predicted by the historic linear relationship would be 8.73%.

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Both numbers seem laughingly unattainable in the near-term against the current market backdrop, but perhaps a dive into this relationship could help investors better understand where equity multiples and interest rates could head, and the implications for returns in the stock and bond markets. Retrospectively, the earnings multiple predicted by the historic linear relationship between earnings multiples and the benchmark domestic interest rate was the most understated in 1999 as the model would have predicted a multiple of 16.5x, 81% lower than the actual trailing earnings multiple at year-end of 29.8x. This earnings multiple, which was abnormally high relative to the prevailing ten-year Treasury yield of 6.45%, collapsed in the next three years as the S&P 500 returned -9.1% in 2000, -11.9% in 2001, and -22.1% in 2002. Conversely, the earnings multiple predicted by this mathematical relationship with interest rates was the most overstated in 1974 when the prevailing interest rate of 7.4% would have predicted a multiple of 15.54x. The actual prevailing multiple of 7.33x was 52% lower than the predicted multiple. The next two years, 1975 and 1976, saw stock market returns of 37.2% and 23.9% respectively.

Twenty-seven of the last fifty years have seen the actual year-end earnings multiple at lower levels than predicted by the historic relationship; the market has risen on average by 12.2% over the next year, posting positive returns in 21 of those 27 years. The four years with earnings multiples more understated than the current deviation returned an average of 23.7% over the next year. In the other twenty three years, the actual earnings multiple was higher than predicted, and the market rose by only 8.9% on average. Sorting the percentage difference in the predicted earnings multiple and the actual year-end earnings multiple in ascending fashion (e.g. ranking low multiple 1974 first and high multiple 1999 last), and then dividing the fifty years into quintiles yields a greater divergence in forward returns. The ten years with earnings multiples that were the lowest relative to what was predicted by the prevailing interest rate returned 13.6% over the next one year on average, but the ten years with the highest earnings multiples versus predicted values returned only 4.4% over the next year.

A number of factors are keeping the connection between interest rates and equity multiples below its historical relationship. The Fed's anchoring of the front-end of the yield curve near zero is also weighing down the intermediate part of the Treasury curve. The expectations hypothesis of the term structure of interest rates states that long-term rates are a function of the market's expectation of short-term rates in the future plus a risk premium. With the Federal Reserve telling the market that interest rates will remain low through 2014, the ten-year Treasury yield is likely artificially low. The tremendous liquidity created in the marketplace through the collective actions of global central banks is also placing upward pressure on low-risk asset prices, which is driving down Treasury yields.

The uncertainty over the macroeconomic backdrop that has given rise to the extraordinarily accommodative global monetary policy could also be driving the high risk premium added to the interest rate that discounts forward earnings. The years that saw the actual earnings multiple depressed relative to the earnings multiple predicted by the historic relationship with interest rates were years of relative economic malaise (1974-1980, 2009 and 2011). Equity market investors in those years and in the present day are demanding a higher risk premium relative to low-risk assets as demonstrated by the gap between earnings yields and financing yields, which is at a thirty year high. This economic uncertainty could also be leading some market participants to forecast lower forward earnings for the market. A decline in the denominator of the P/E ratio would also increase the multiple back towards its historic average. However, in the current earnings cycle 467 of the S&P 500 constituents have reported, and 317 (68%) have seen positive quarter over quarter earnings growth.

Over the longer-term, demographic trends in the United States could lead to generationally low equity multiples. As the large Baby Boomer generation leaves their peak earning years and enters retirement many prognosticators are forecasting a dramatic and prolonged shift of assets from the stock market to fixed income markets. This potential downward pressure on equity multiples, coupled with the potential downward pressure on bond yields, may not allow the equity multiple/interest rate interrelationship to revert to historical norms.

While continued economic uncertainty and demographic headwinds in developed markets could mute multiple expansion and keep rates low, the most likely outcome is a slow expansion of the earnings multiple and a concomitant increase in the ten-year Treasury yield. The earnings multiple to trailing twelve months earnings at year-end 2011 was at its lowest level since 1988. Multiples this low have typically portended higher equity returns over the next year. Stronger economic data from the United States coupled with a clearer picture around the fiscal situation in the government sector across the Eurozone could lessen the uncertainty discount in risky assets.

Yesterday, the United States Bureau of Economic Analysis released the personal consumption expenditure deflator. This measure of year-over-year change in domestic household consumption expenditures on goods and services, the Federal Reserve's primary measure of inflation, rose 2.4%. This level inflation relative to the ten-year Treasury yield of just over 2% means that holders of these securities are earning a negative real return. The coupon paid on Treasury notes does not offset the annual decline in purchasing power at this level of inflation. Dollars rotating from the U.S. Treasury market to the equities market would boost both the equity multiple and Treasury yield.

A Bloomberg survey of the year-end 2012 ten-year Treasury yield by seventy-nine private market economists and academics forecasts an average yield of 2.5%. Combining this Treasury yield with the average earnings multiple over the last fifty years of 16.56x would revert the deviation of the earnings multiple/Treasury yield to -19% from its historical relationship (currently -32%). Deviations from the predicted multiple of greater than -19% have seen an average 13.6% return over the next one year. Of course, the returns in 2012 to expand the earnings multiple to its historic average of 16.56x by year-end would be even larger. Analysts are forecasting $104.14 of earnings for the S&P 500 constituents, which would translate to an S&P level of 1725 at year-end, a further 25% increase from yesterday's close. Maybe Lazlo Birinyi's prediction of a year-end S&P at 1700 yesterday is not as far fetched as it may seem at first blush. Your job as an investor is to put today's uncertain economic outlook into context, and judge whether the risk premium over the interest rate that discounts future earnings is sufficient.

**Disclosure: **I am long SPY.

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