By Wesley R. Gray, Ph.D.
In our last post, we looked at tactical allocation using valuation metrics and trendfollowing measures. Our conclusion from the analysis is that discerning robust trading signals based on market valuations is difficult at best.
This research piece attempts to dig a little deeper and addresses the following questions:

Opportunity Costs
 How do other asset classes perform during different CAPE regimes? Why go allin on equity if bonds are the asset class that outperforms?

Diversification
 Do we abandon diversification and shift heavily into equities following cheap markets?

Real Equity Premium
 CAPE ratios are absolute pricing measures, but what we care about is the equity premium. If equity is expected to earn 15% real and bonds are expected to earn 15% real, equity isn't cheap.
Strategy Background
The following 3 data series are used in our tests:
 LTR: Merrill Lynch 710 year government bond index—ML1US10 INDEX
 SPX: SP500 Total Return Index—SPXT INDEX
 Shiller P/E: Shiller’s Cyclically Adjusted PE ratio—10year average real earnings /real price
Simulation results are from January 1, 1929 through April 30, 2014.
No transaction costs are included in any of our analysis. All results are gross of any transaction fees, management fees, or any other fees that might be associated with executing the models in realtime. The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
Shiller P/E and Market Performance (1/1/1929 to 4/30/2014)
 Shiller P/E and S&P 500 Over Time
 Low Shiller P/Es, on average, are followed by bull markets over long cycles.
 The figure to the right plots the Shiller P/E over time (left axis) and the S&P 500 return series over time (right axis).
 Log scale.
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
Opportunity Costs
Given a Shiller P/E starting point, what do subsequent 3year stock and bond returns look like?
 Low valuation implies higher expected future stock returns, on average.
 Bonds are fairly stable across equity valuation regimes, but also see a boost during the cheapest market environments.
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request. The figure plots different Shiller P/E regimes (Xaxis). For example, “1516” represents all historical samples when the S&P 500 was priced between a 15 and a 16 Shiller P/E. Compound annual growth rates (OTCPK:CAGR) are presented on the Yaxis. All CAGRs are calculated over 3year periods. We test periods ranging from 1 to 5 years and find similar results.Given a Shiller P/E starting point, what do subsequent 3year Sharpe ratios look like?
 Stocks show a relationship between riskadjusted returns and valuation, but the correlation is not as strong as Shiller P/E level and 3yr CAGR (chart above).
 Bonds have strong riskadjusted returns across equity valuation regimes, to include the extreme cheap and extreme expensive equity valuation regimes (bonds actually outperform at the extremes).
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request. The figure plots different Shiller P/E regimes (Xaxis). For example, “1516” represents all historical samples when the S&P 500 was priced between a 15 and a 16 Shiller P/E. Sharpe ratios are presented on the Yaxis. All Sharpes are calculated over 3year periods. We test periods ranging from 1 to 5 years and find similar results.
Digging deeper into extremely cheap markets
The figure below plots the Shiller P/E over time (left axis) and the S&P 500 return series over time (right axis). We highlight prominent valuation troughs with red circles. In the analysis that follows, we examine the detailed performance data for stocks and bonds.
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
S&P Statistics Summary
The following table contains the statistics for various Shiller P/E regimes (each column represents a red circle above).
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
LTR Statistics Summary
The following table contains the statistics for various Shiller P/E regimes (each column represents a red circle above).
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
Conclusion on Opportunity Costs
On Sharpe ratio basis, stocks outperform in the 4952, 7477, and the 0912 bear markets.
On Sortino ratio basis (MAR = 5%), stocks outperform in the 4952, 7477, 8285, and the 0912 bear markets.
Stocks AND bonds are interesting during cheap equity markets.
Diversification
Given a trough Shiller P/E starting point, what does a diversified allocation between stocks and bonds do for the investor?
The common story is that cheap stock markets warrant a wholesale move into equities.
50/50 Statistics Summary (50% equity; 50% 10year bonds)
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
60/40 Statistics Summary (60% equity; 40% 10year bonds)
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
Conclusion on Diversification
Diversified stock/bond portfolios following trough Shiller P/E events typically outperform single position portfolios in either stocks or bonds independently.
 Diversification always matters
Real Equity Premium
Absolute valuation metrics may not precisely capture the benefit/cost to holding risk. In other words, a P/E of 20x doesn't mean anything in a vacuum.
Any firstyear finance student will tell you that expected equity returns will roughly equal the riskfree rate plus an equity premium. Why anchor on the riskfree rate and why add an equity premium? Well, the riskfree rate is what you get in your checking account  so that is the baseline we should all expect from ANY asset class. Next, ask yourself, who would hold risky equity if there was no extra kick in performance over stashing the cash in the checking account? ...that is the equity premium... the extra return we get for taking on risk.
Understanding the "RF +equity premium concept" helps us better understand why absolute valuation levels may not capture true equity market valuations. For example, a P/E of 10 (a 10% earnings yield) with a 10year yield of 15% may imply an expensive equity market (5% equity premium) relative to a situation where the P/E is 20x (5% earnings yield) and the 10year yield is 1% (+4% equity premium). The P/E in the second scenario signals "expensive," but on an equity premium basis, stocks are cheap relative to the scenario where the P/E is 10.
We test how stocks and bond perform during different Real Equity Premium periods, where Real Equity Premium = 1/CAPE [Earnings Yield]  Real 10year yields (Note: CAPE is already in "real" terms).
The figure below plots the Shiller P/E, or cyclicallyadjusted P/E (NYSEARCA:CAPE), over time (right axis) and the Real Equity Premium (1/CAPE – Real Treasury Yield) (left axis). Extreme spread events are highlighted with red circles.
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.Real Equity Premiums tend to roughly correspond with trough Shiller P/Es, but not always (e.g., 1947)
Real Equity Premium Performance Characteristics
Given a Real Equity Premium starting point, what do subsequent 3year returns look like?
 Future stocks appear to have a relationship between Real Equity Premium, but the correlation is weak.
 There is no clear pattern between Real Equity Premium and future bond returns.
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request. The figure plots different Real Yield regimes (Xaxis). For example, “77.5” represents all historical samples when the 1/CAPE – 10Year Treas was priced between a 7% and a 7.5%. Compound annual growth rates (OTCPK:CAGR) are presented on the Yaxis. All CAGRs are calculated over 3year periods. We test periods ranging from 1 to 5 years and find similar results.Given a Real Equity Premium starting point, what do subsequent 3year returns look like?
 Future stocks appear to have a relationship between Real Equity Premium, but the correlation is weak.
 There is no clear pattern between Real Equity Premium and future bond or stock returns.
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
Real Equity Premium Performance CharacteristicsDetails at Trough Real Equity Premium
S&P Statistics Summary
The following table contains the statistics for various Real Equity Premium regimes (each column represents a red circle above).
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
LTR Statistics Summary
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
50/50 Statistics Summary
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
Conclusions on Real Equity Premium
 High Real Yield periods predict high spreads in CAGRs between stocks and bonds.
 On a riskadjusted basis, stocks tend to outperform bonds
 Diversified portfolios  in contrast to the results for low absolute Shiller P/E event  do not always outperform allin stock bets.
 Diversification is less valuable and targeted bets may be warranted during high Real Equity Premium periods.
Overall Conclusions
 Bonds perform well across different valuation periods  to include extreme low valuation periods.
 We should stay diversified during low absolute valuation periods.
 We might want to tactically shift towards equity during high Real Equity Premium periods.
 The results highlight a lot of noise and variation in the data. Identifying a "silver bullet" for tactical asset allocation is impossible... but good luck trying.