Why does the V_e = NGDP × S_k × P/E identity matter for forecasting long-run equity returns?

Short answer: the identity decomposes the aggregate market value of equity into three growth-rate components, each with a clear long-run constraint. Two of the three components ($S_k$ and P/E) CANNOT grow indefinitely, which forces long-run equity capital appreciation to track nominal GDP. This gives you a rigorous, theory-grounded way to project long-run equity returns from forecasts of GDP growth and inflation alone.

Reading the symbols: $V_e$ = aggregate equity market value; $\text{NGDP}$ = nominal GDP; $S_k$ = capital share (= earnings/GDP); $\text{P/E}$ = price-to-earnings ratio; $g_X$ = growth rate of variable $X$; $D/P$ = dividend yield.

The identity in growth form

Take logs of $V_e = \text{NGDP} \times S_k \times \text{P/E}$ and differentiate:

$$g_{V_e} = g_{\text{NGDP}} + g_{S_k} + g_{\text{P/E}}$$

This is just an accounting identity — it must hold by definition. The power comes from imposing LONG-RUN ECONOMIC CONSTRAINTS on each piece.

The long-run constraints

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• $g_{\text{NGDP}}$ is real growth + inflation. Forecast directly from growth accounting.
• $g_{S_k}$ must average to zero in the very long run. If earnings continually grew as a share of GDP, eventually all GDP would be profit — impossible.
• $g_{\text{P/E}}$ must also average to zero in the very long run. P/E ratios cannot rise without bound.

Therefore, in the very long run:

$$g_{V_e} \approx g_{\text{NGDP}} = g_Y + \pi$$

Total equity return adds the dividend yield:

$$r_e^{\text{long run}} \approx g_Y + \pi + \frac{D}{P}$$

Why CFA candidates love this framework

It gives you a small set of inputs (real growth, inflation, dividend yield) that produce a defensible long-run equity return estimate. It does not require you to forecast anything that is structurally impossible (rising profit shares forever, rising multiples forever).

Finite-horizon adjustments

Over finite horizons (say, the next 10 years), $g_{S_k}$ and $g_{\text{P/E}}$ can be nonzero. The framework lets you EXPLICITLY model expected changes:

Adjustment	Effect on equity return
P/E expected to fall (mean revert from elevated level)	Subtract the per-year drag
Profit margin expected to compress	Subtract the per-year drag
Profit margin expected to expand (cyclical recovery)	Add the per-year boost

This is exactly what the Bjornsdottir example does. She projects baseline 6.75% then subtracts 2.2% for expected P/E reversion, ending at 4.55%.

The exam-day pattern

When given Bjornsdottir-style decomposition data, you should:

1. Decompose historical return into $g_Y$, $\pi$, $g_{S_k}$, $g_{\text{P/E}}$, $D/P$
2. Identify which components are unsustainable (typically P/E expansion or margin expansion)
3. Forecast new $g_Y$, $\pi$, $D/P$ based on growth accounting and current yields
4. Add finite-horizon adjustments for expected $g_{S_k}$ and $g_{\text{P/E}}$ changes

For the worked example see our CME application article.