You hear a lot about the “Fed Model” when discussions break out about valuing equities versus bonds. Larry Kudlow loves talking about this strategy on CNBC to back up his never-ending bullish stance on stocks. The model, simply speaking, compares the earnings yield on the stock market to the 10-year treasury bond yield.

If the S&P 500 trades at a 20 P/E, it’s earnings yield is 5% (100/20). Since this compares favorably with the 4% treasury yield, Kudlow will argue that stocks are more attractive than bonds, and therefore should be bought. The thinking goes that if both investments were fairly valued, relative to each other, then their yields would be equal.

However, there is a problem with this so-called model. The equity market and the bond market hardly ever “yield” the same amount. As a result, any model that aims to make them equal is inherently flawed. If stocks and bonds were supposed to be relatively equal in value, it would make sense to base investment decisions on any discrepency, such as 4% versus 5% yields. If such discrepencies are extremely common, there is little reason to conclude they signal relative attractiveness or unattractiveness.

Let’s show some evidence to further conclude that the Fed Model shouldn’t be used as a powerful investment tool. When bond yields are historically low, as they are today, the Fed Model would predict a high level of attractivenness for equities as an investment class. After all, the lower the yield on bonds, the more likely stock earnings yields would surpass them.

Below you will see various ranges for the 10-year Treasury bond yield since 1965, along with the actual total return that the S&P 500 achieved over the following 10-year period. As you can see, low bond yields have not resulted in attractive stock price returns. In fact, one can argue the exact opposite.

10-Year Bond Yield —–Subsequent S&P 500 returns

——— 0-5% ——————— 4.3%

——— 5-6%———————- 5.5%

——— 6-7% ——————– 10.5%

——— 7-8% ——————– 13.2%

——— 8-9% ——————– 16.7%

——— 9-10% ——————- 17.5%

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The Fed Model is not about the relationship between future appreciation of the stock index relative to bond yield, but about the relationship of earnings yield (1/PE) to bond yield. It doesn’t suggest that they are, or should be, equal. In fact, the Fed references the “equity premium” which is the additional yield desired to be in stocks rather than bonds, and the equity premium varies according to perception of the volatility of stocks.

1/PE = 10YT + EquityPremium

Today’s “as reported” PE for the S&P500 is 16.89, and the inverse of that is 5.92%. Today’s 10YT is 4.14%. This means the equity premium is 1.79%, which is low, historically speaking.

Your chart is cute, but consistent with what the Fed model predicts. Follow me here: low 10YT implies high PE, when you invest in the index at high PE you get low returns. high 10YT implies low PE, when you invest in the index at low PE you get high returns.

I don’t watch Kudlow, but if he’s using it as you describe, he’s actually MISusing it.

True, the Fed Model is about the relationship between the treasury bond yield and the earnings yield on equities. That is what I said in the last sentence of the opening paragraph.

The fact that the equity premium right now is low by historical standards is the statistic that users of the Fed Model are using to make the “stocks are undervalued relative to bonds” argument.

They feel the gap should be higher. In order for the equity risk premium to rise, we would need to see multiple expansion for equities, which implies solid share price appreciation from stocks.

However, if the level of the 10 year bond is signaling equity returns of 4.3% annually (based on the chart), then you will not get meaningful outperformance from stocks, given the current 4.1% yield on the 10 year treasury bond.

Chad,

I think youâ€™ve got the math exactly reversed.

1/PE = 10YT + EquityPremium

5.92 = 4.14 + 1.79

PE = 16.89

If the equity premium increases, then so does the inverse of the PE, which means the PE goes down. Imagine instead of todayâ€™s 1.79% equity premium, the equity premium was closer to the historic 3%. That makes the calculation

1/PE = 4.14 + 3.00 = 7.14

1/PE = 7.14%

PE = 14.01

Which implies a SELLOFF of stocks. Todayâ€™s low equity premium means that stocks are OVERvalued relative to bonds. Of course, one could say that the low VIX means the equity premium SHOULD be low â€¦

A high equity premium means that investors are demanding much higher returns from stocks than bonds, relatively speaking, which implies UNDERvaluing.

Like I said, if â€œusers of the Fed modelâ€ are using it as you describe, theyâ€™re MISusing it. Or perhaps you are misunderstanding or misrepresenting their usage of it.

Well, Since I didn’;t run any numbers when I wrote the original entry I apologize for the reversal errors.

At any rate, I have heard many people (not just Kudlow — who is an economist, not a investment guy) cite the Fed Model as a way to support their theory that stocks are undervalued relative to bonds at current levels.

If they are all misusing the model, and it actually shows that stocks should fall from here, based on historical averages, so be it. However, I have yet to hear someone cite the Fed Model as a reason to be bearish on equities.

Since I don’t use the Fed Model, it really doesn’t matter to me. The main point of the piece I wrote was to show that 10 year treasury bond yields of 5% or less indicate similar returns from stocks, which runs counter to those who are arguing that stocks are undervalued relative to bonds that are currently yielding 4.1%.

Thank you for the chart. It is another way of saying that most stock market returns are a result of P/E expansion. P/E’s are not likely to expand if long term rates go up.

Based on consensus earnings estimates the S&P forward earnings yield is at a historically high spread above the 10-year. The estimates give the S&P a forward P/E of 15.1 or an earnings yield of 6.6%. The spread of 2.4% is almost off the chart.

I believe the risk premium is a different number. After all, do we expect a to buy a 10 year bond today and sell it two or three years from now and actually achieve a 4.1% return? Indeed, a holder for 5 years will only break even if rates move to 5.1%.

The implication is that you must be very bearish on the stock market or you must believe 10 year rates will go lower than 4.1%.

I am a market bull and it seems to me that the wide spread gives one a nice cushion. Rates can rise to any number less than 6.6% while stocks retain an earnings yield advantage over bonds. If earnings suprise again and grow by another 13%, stocks could trade at the same PE ratio and make investors 13% even though rates may have moved to 5% while punishing the bond buyer.

“The main point of the piece I wrote was to show that 10 year treasury bond yields of 5% or less indicate similar returns from stocks..”

On that point I agree, but just feel compelled to add a couple of caveats:

(1) the line of causality is low yields on the 10YT imply that the stock indices currently have a high PE, and investing in the stock indices when the PE is high generally leads to low returns. It’s two steps removed.

(2) one should look at volatility (whether one uses the VIX or whatever) and compare that, historically, to the current “equity premium” generated by the Fed model to get an even better read on whether the stock indices are over/undervalued.

Yep, moving from 4.1 to 5.1 yield on a new 10YT and selling after 5/6 years is breakeven. That is why most bond investors hold until expiration and use laddering techniques to mitigate interest rate risk. Holding to expiration guarantees your return.

From 1962 on, comparing 10YT to subsequent 10 year SPX returns with weekly data points, I performed the following regression:

SPXreturn = -0.1817*(10YT^2) + 4.4572*(10YT) – 14.592. The R2 = 0.5565. The fascinating thing about this is, for very high 10YT yields (above 10%), the subsequent returns on the SPX are lower than they are for relatively high 10YT yields (say 8-10%). This is obviously the time to invest in bonds!

no doodahs, Can you give me the details on your regression, including where you pulled the data.

By the way, the problem with the comparisons we are making is that we have to go back to the 50’s to find such low inflation expectations. If inflation turns out to be as low as market is forecasting, the 10 year bond and the S&P will offer wonderful returns.

Jack,

I downloaded the 10-Year Treasury Constant Maturity Rate (WGS10YR) from http://www.stlouisfed.com and their “FRED II” database. I then got weekly closing values for the SPX from Yahoo!Finance. I arranged the 10YT for each week with the gain on the SPX from that week going 520 weeks forward and did the regression in Excel ’97. You could also use the 10YT data from Yahoo!Finance but I don’t know if it goes back to 1962, I know the Fed gives an average yield for the week while you could get an exact Friday close from Yahoo.

The [tools>data analysis>regression] menu only allows for linear regression, but if you graph the two variables with 10YT as the X variable, you can right-click on the graph and [add trendline]. Select type polynomial [2nd degree] and also select [display equation on chart] and [show r-squared]. Using the chart function and trendlines from there allows linear, polynomial, exponential, power, and logarithmic charts.

The history (since ’62 in any event) has been that when treasuries are extremely high or low yeilding, it’s better to invest in them and hold to maturity than it is to hold the SPX. That being said, I’m not so sure about today’s situation and if you can do a better job of picking stocks than the SPX, that skews the situation as well.

No Doodahs, I would appreciate your contacting me at jackmiller@triad.rr.com. I would like to ask you more questions about your regression techniques.