I am seeing way too many articles arguing that S&P500 is in a bubble. In this article, I will present a quantitative framework that shows that if anything, S&P500 (SPY) is actually undervalued, and most certainly not in bubble mode.

I use the Gordon Growth Model, also known as the Dividend Discount Model, for this article. Per Wikipedia (or any standard finance text book) this model states that

where is the current stock price. is the constant growth rate in perpetuity expected for the dividends. is the constant cost of equity for that company. is the value of the next year's dividends.

In this context, P is the expected value of the S&P500 index, and r is the discount factor for it. The discount factor is the sum of the risk free rate (R_{f}) for which I use the 10-year Treasury yield, and the risk premium (R_{p}), traditionally assumed to be 5%. The key input factor is the growth rate, g. Also, D_{1} is simply D_{0}*(1+g), that is this year's dividends times the annual growth rate.

Note that given historical price and dividends for S&P500 and the 10-year Treasury yield, we can calculate the historical values for g using the above equation. From then on, we can extrapolate future values of g as well as R_{f} based on our assumptions about future market conditions.

Also, given projected earnings and historical ratio of dividends of earnings, we can project future dividends as a product of the two. That is all that is needed to project out future prices for the S&P500. Here are my projections:

A few notes about this model. Any cell in black is past historical data, with S&P500 earnings, dividends, and index values coming from the S&P500 web site and 10-year Treasury yields coming from Yahoo Finance. Any cell in green are my calculations, which I will explain later. Cells in blue are projections made by S&P500 as well, and cells in red are projections made by me, which I will explain later as well.

Also, any cell in italics are projections, calculated or otherwise.

Let's explain the columns one by one.

1. The E column is taken straight from S&P500 web site, so no explanation is necessary.

2. The D column is taken from S&P500 web site as well, but the top 6 values are calculated by me. To do that I first calculated the column TTM D/E from actual S&P500 data, and then assumed that this column, which has been very stable over the past many quarters, will remain stable at 36% (last quarter's value). Then I multiplied the EPS assumptions from S&P500 to get DPS assumptions.

3. The R_{f} column assumes that the Feds start to taper by end of the year, and are fully done by end of next year. However, ZIRP will remain in place till 2014 end, hence I assumed that by the end of 2014, the 10-year Treasuries will reach the high end of what we have seen over the ZIRP era, or 3.5%. I gradually increased it every quarter at a steady rate to reach 3.5% in 5 quarters, assuming that till September, 2013, the rates remain virtually unchanged at around 2.5%.

4. Calculating the g column was the crux of this analysis.

I first back calculated the g values expected by the market over time till the end of the June, 2013. Over the last 8 quarters, the implied growth has ranged from 4.5-5.3%. In the mean time, the actual dividend growth has ranged from 17-25%. Looking forward, the dividend growth is expected to continue at the same pace.

In addition, if the Feds do start to taper as assumed in No. 3, that means one or both of real GDP growth and inflation start to accelerate, meaning that nominal GDP growth starts to accelerate at an even faster rate. Since earnings grow at a faster rate than NGDP growth as the economy comes out of a recession because of efficiency gains in the mean time, this to me indicates that the market expectation of dividend growth should go up as well (in the event of taper).

I have chosen to increase the implied dividend growth by the same amount as the 10-year Treasury yield, or 1% (from 2.5% to 3.5%), such that by end of 2014, the expected value of g is 6.3% from the current 5.3%. Then I have gradually increased the growth rate every quarter to match the final value.

5. Given No. 3 and No. 4, it is simple formula to calculate the P column using the Dividend Discount Model.

So what does that yield? My model indicates that by the end of this year S&P500 should be ~1850, and by end of 2014, it should be ~2100. To me, this indicates that S&P500 is undervalued at the current ~1700. There is almost another 10% growth left by the end of the year, and another 15-20% in 2014. Not bubble territory by any means. During that time, TTM PE should be in the 17.5-18.5 range, which is higher than historical averages, but still appropriate for the low interest rate regime that we currently live in.

Of course, the devil is always in the assumptions, when it comes to modeling the future. The most common criticism I expect to get is that S&P500 forward earnings projections are over-estimated, and will come down over time. Hence, the assumptions I made about future dividends and their growth rates (D and g) are overestimates, as is the S&P500 index level (P).

I actually have a lot of sympathy for this line of thinking, but I would like to remind those raising it that it is unlikely, then, that the economy would be improving, and hence there will be no taper and R_{f} will remain low as well.

This is the key point to make. In the Dividend Discount Model, the numerator is D, and the denominator is (r-g). If D and g don't go up, then chances are r doesn't go up either. They effectively balance each other out, as the Fed is on record that it won't start to taper until the economy improves at a solid pace.

But what if we have stagflation? What if D and g remain stable, or even goes down, but the Feds are forced to increase r because of inflation. That's the classic 70s situation. **This is the only valid argument against this model.** Yes, it can happen, but how likely is it?

The 70s stagflation was caused by the oil price shock, and not by endogenous factors (wages for employees and pricing power for employers). Given the current level of unemployment I simply can't envision wage pressures coming anytime soon to companies. By the same token, consumers have little means or inclination to allow pricing rises by companies given the poor employment and wage scenario.

Hence, the chances of raging inflation is small. As for an external shock, the USA is becoming less and less dependent on foreign oil. Quite simply, all this put together means that the chances of an inflation spike without a corresponding spike in NGDP, company revenues and earnings is small.

Hence, the calls for a bubble forming are premature. The market is quite properly valued, if only a little undervalued. There are gains to be made this year and next. I am invested in the 3x leveraged ETF on SPY (UPRO) and will keep holding it till the ZIRP regime comes to an end.

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