ROE and Price to Book Value Geometric Relationship

The geometric (non-linear) relationship between Return on Equity (ROE) and Price-to-Book Value (PBV) is deeply tied to the long-term compounding of earnings and value creation.

💡 The ROE-PBV Relationship is Exponential Because of Compounding

The reason PBV expands geometrically with ROE is:

• A higher ROE allows the company to compound book value faster

• That higher compounding justifies a higher multiple today, because:

  • Future earnings are much larger

  • Market discounts future cash flows; higher ROE implies better reinvestment opportunities

• A 30% ROE business can double book value in ~2.5 years, while a 10% ROE business takes ~7 years

This difference in earnings growth and reinvestment efficiency leads to an exponential divergence in intrinsic value over 5, 10, 15 years — hence, the PBV market assigns also diverges non-linearly.

🧮 Market Assumptions: ROE + Longevity + Reinvestment Rate

When the market pays 10× PBV for a 30% ROE business, it isn't just paying for current ROE, but for:

• Longevity of that ROE (i.e., can it sustain 30% for 10+ years?)

• Reinvestment ability (can it redeploy all profits at that ROE?)

• Competitive advantage / moat (can competitors erode that ROE?)

The market is pricing in the entire stream of future economic profits, discounted to today — which explains the geometric relationship.

🔁 Diminishing PBV at Lower ROEs

At low ROEs (< Cost of Equity), the company is destroying value by retaining capital. That’s why:

• A 5% ROE business may get 0.5× PBV or lower — the market wants it to return capital rather than reinvest.

🔚 Conclusion

The geometric rise in PBV with ROE is directly tied to the long-term compounding of earnings and book value, assuming reinvestment and longevity.

• High ROE businesses, if they can reinvest at the same high rates, are extremely valuable — hence get exponential valuation premiums.

• Markets assign PBV not based on this year’s ROE, but the expected compounded value 5–15 years down the line.

Here’s a clear summary of how different ROE levels translate into intrinsic Price-to-Book Value (PBV) based on your 12% required return and assuming 100% reinvestment of profits for 15 years:

ROE	Final Book Value	PV of Earnings (15 yrs)	Implied PBV
10%	₹417.72	₹118.42	1.18×
15%	₹813.71	₹243.31	2.43×
20%	₹1,540.70	₹453.70	4.54×
25%	₹2,842.17	₹806.26	8.06×
30%	₹5,118.59	₹1,391.91	13.92×

🔍 Key Takeaways:

• The PBV multiple rises exponentially with ROE because the company’s ability to compound book value (and thus earnings) grows exponentially.

• At a 10% ROE, the company barely beats your required return (12%), so the fair value is close to book (1.18×).

• But at 30% ROE, the company massively outperforms your return hurdle, justifying a PBV of nearly 14× — despite the same starting book value.

Here's a detailed table showing the implied Price-to-Book Value (PBV) multiples for businesses with different ROE levels (10% to 70%) over 10, 15, and 20 years, assuming full reinvestment of profits and a required return of 12% per annum:

ROE	PBV (10 yrs)	PBV (15 yrs)	PBV (20 yrs)
10%	0.82×	1.18×	1.51×
15%	1.51×	2.43×	3.48×
20%	2.48×	4.54×	7.44×
25%	3.84×	8.06×	15.37×
30%	5.73×	13.92×	31.17×
35%	8.33×	23.54×	62.25×
40%	11.88×	39.17×	122.48×
45%	16.67×	64.24×	237.25×
50%	23.11×	103.95×	452.26×
55%	31.68×	166.06×	848.23×
60%	43.00×	262.04×	1,565.32×
65%	57.83×	408.62×	2,842.95×
70%	77.13×	629.94×	5,083.76×

---

ROE	PBV (5 yrs)	PBV (10 yrs)	PBV (15 yrs)	PBV (20 yrs)
8%	0.33×	0.61×	0.84×	1.03×
9%	0.38×	0.71×	1.00×	1.26×
10%	0.43×	0.82×	1.18×	1.51×
11%	0.48×	0.94×	1.38×	1.81×
12%	0.54×	1.07×	1.61×	2.14×
13%	0.59×	1.21×	1.85×	2.53×
14%	0.65×	1.36×	2.13×	2.97×
15%	0.71×	1.51×	2.43×	3.48×

🧠 Interpretation:

• Even modest increases in ROE dramatically increase the fair PBV, especially over longer time horizons.

• At 20 years and 30% ROE, the fair PBV is 31×, vs just 1.5× at 10% ROE.

• At very high ROEs (e.g., 60%+), PBVs explode — but such businesses are rare, and sustaining these ROEs over long periods is extremely difficult without a massive moat.

🔍 Key Insights:

• A business must earn at least your required return (12%) to justify a PBV > 1.

• Even a 1% difference in ROE near your hurdle rate (12%) has a significant effect on PBV over time.

• This table can help you judge whether a stock trading at 2× or 3× PBV is expensive or justified based on its ROE and how long it can sustain it.

The visual chart shown above explains how Implied Price-to-Book Value (PBV) increases with ROE over 10, 15, and 20 years:

• The y-axis is on a log scale to capture the exponential growth in PBV for high-ROE businesses.

• You can clearly see how longer horizons amplify the impact of ROE on valuation.

The chart shown above is a powerful tool to evaluate what kind of PBV multiples are justified, based on ROE and your holding horizon.