Where:
P = stock price
E = Earnings
g = long-term earnings growth expectation
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r = discount rate (risk-free rate + equity risk premium)
This equation explains much of the US equity market’s historical movements. Consider the period since 2008. Besides higher earnings, stocks were driven by:
1. Falling discount rates;
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2. Higher perceived durability of growth;
3. Higher confidence in US corporate dominance; and
4. Lower macro volatility
Artificial intelligence is now extending the growth narrative even further.
Meaning: (r-g) has being falling, leading to higher valuations.
The formula can also be written as:
(r-g) = E/P
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Where E/P = earnings yield
For instance, if PE is 20 times, then the earnings yield is 5%. Which means r-g = 5%.
Therefore, if investors think the long-term growth rate is 2%, then the discount rate implied is 7%.
Let’s see if the above simple formula works for over the past 20 years.
Post-dotcom (2003-2007)
S&P 500 PE was 15-18 times.
Therefore, earnings yield = 6%-7% (equals to r-g).
US 10-Year Treasury yield (risk-free rate) was 4%-5%.
Valuation was comparatively normal — normal inflation, higher real rates.
If (r-g) = 7% and g = 2%, risk-free is 5%, then the risk premium is 4%.
Post-global financial crisis (after 2008)
The US Federal Reserve cut policy rate to near zero.
QE (quantitative easing) suppressed bond yields.
Global savings glut intensified.
Investors desperately searching for yields.
US 10-Year yields collapsed to 1%-2%.
Meaning r was massively compressed. And (r-g) became very small.
Even if g stayed mediocre, the result was that PE grew mathematically.
Example:
If r = 9% and g = 2%, then r-g = 7%.
Implying PE should be 1/0.07 or 14.3 times
If r compresses to, say, 6% and g remains at 2%, then r-g = 4%.
PE = 1/0.04 or 25 times
Even with no change in earnings or long-term earnings growth expectations, stock prices or value could surge by 75% (from 14.3 times to 25 times PE) — driven only by a sharp fall in r (either the risk free or the risk premium or both).
This largely explains the 2010-2021 bull market rally.
But critically, to understand the US stock market, the US also became structurally stronger.
Remember, the formula is not just about rates.
The US market increasingly believed:
1. US firms have global monopolistic dominance;
2. Big Tech has ultra-high margins (remember the economics of platforms);
3. Asset-light models scale indefinitely;
4. Software economics are superior; and
5. US firms were more resilient than Europe/Japan/emerging markets
In other words, long-term g rose (while r was falling). Thus r-g compressed sharply. That’s why PE exploded.
Think Apple, Microsoft, Amazon, Meta.
Post-pandemic (from 2022-current)
Why did valuations stay high even when rates rose?
From 2022:
1. Treasury yields rose sharply;
2. Discount rates should rise; and
3. Valuations should compress severely. It did initially … then, AI emerged!
With AI, the market started assuming:
1. Higher future productivity;
2. Higher corporate margins;
3. Another technology super cycle; and
4. Winner-takes-most economics
In other words, higher g more than offset the higher r (even though the 10-Year yield is now above 4%).
The present implied math:
Forward PE is 21-22 times, or earnings yield of 4.5% to 4.8% (say 4.7%)
The US 10-Year yield is 4.4%.
That is: risk-free (4.4%) + risk premium (?) - g (?) = 4.7%
Meaning: Investors are implicitly assuming:
a. Very low equity risk premium;
b. Durable and high earnings growth;
c. AI-led productivity expansion; and
d. US corporate exceptionalism continues
In other words, while the valuation is high, it is not irrational if:
a. AI materially boosts profits;
b. Mega-cap tech companies remain globally dominant;
c. Inflation stays contained; and
d. Real rates eventually moderate
What the formula teaches us about risk
The danger lies when (r-g) gets very small; valuation sensitivity becomes extreme!
Example:
If r-g = 2%, then PE = 50 times
But if r-g = 4%, then PE = 25 times
A small change in assumptions of rates, equity risk premium and long-term earnings growth expectations can halve valuations (stock prices). This is why high-multiple markets are fragile (riskier).
Summary of the last 20 years for US equity market
US market’s rerating was not mainly about earnings growth alone, but was also driven by:
a. Falling interest rates (lower r);
b. Global liquidity/QE (lower required returns);
c. Big Tech dominance (higher g); and
d. AI expectations (higher g)
Together, they compressed (r-g) to unusually low levels.
The deeper implication
The market today is effectively making a gigantic bet — that AI-driven productivity and corporate concentration can outrun:
1. Higher debt levels;
2. Geopolitical fragmentation;
3. Deglobalisation;
4. Ageing demographics; and
5. Structurally higher interest rates
If AI succeeds, then current valuations may prove reasonable.
If AI disappoints, and r-g expands again, and valuations will compress violently.
That is, we think the present-day US equity market story is in the one equation we just articulated.
Implication for investors
Risk premium is individual to the beholder. Even if the market is right on the risk-free rate and the g, ultimately, your own valuation is also dependent on what your risk appetite is.
For us, we would:
a. Avoid any borrowing on stock investments; and
b. Hold part cash (even if the interest rates are low)
