ROE x P/VP: this is one of the most powerful relationships in stock analysis, but many people use it without understanding why these two metrics are connected.

First, it's important to understand what each of these variables represents.

ROE measures return on equity—that is, how much profit the company generated relative to shareholders' equity. It shows how much equity "yielded" over a given period.

• Initial equity: $100 • Profit: $10 • ROE = 10 / 100 = 10% per year

In other words, for every $100 invested by shareholders, the company generated $10 in profit per year.

P/BV (Price to Book Value) measures how much the market is paying for a company's equity.

In other words, it shows whether the market value is above or below the book value recorded on the balance sheet.

• Equity: $100 million • Market value: $200 million • P/BV = 200 / 100 = 2x

This means the market is paying twice the book value per share.

Okay, so with that said, where do ROE and P/BV meet?

To understand this relationship, we first need to understand the concept of shareholder cost of capital.

The cost of equity represents the minimum rate of return an investor requires to invest their money in a company.

In its simplified form, it is calculated as: Risk-Free Rate + Risk Premium

The risk-free rate reflects the return on a safe investment (such as government bonds), while the risk premium represents the additional compensation required by the investor to take the risk of investing in stocks — which are naturally more volatile and uncertain.

• Expected Federal Funds Rate for the year: 4.25% • Market risk premium: 5% (This premium can be adjusted by the stock's beta or set based on perceived risk)

• Cost of capital = 9.25%

Therefore, it's clear that how much an investor is willing to pay for the Equity Value (EV) depends directly on two variables:

• the required cost of capital (minimum expected return); and • the company's effective ROE (return on equity).

• The investor demands a return of 9.25% per year, based on the calculated average cost of capital (Fed Funds Rate of 4.25% + 5% risk premium). • The company has a ROE (Return on Equity) of 9.25% per year.

In this case, if the investor pays exactly 1x the book value (P/BV = 1), their expected return will be equal to the ROE, or 9.25%—exactly what they demand as a minimum return.

If ROE is below the cost of capital, the P/BV must be less than 1x for the generated ROE to meet the shareholder's cost of capital.

If ROE is greater than the cost of capital, there is a greater margin for the P/BV to be greater than 1x and still meet the cost of capital with the generated ROE.

It's worth noting that this return to shareholders can occur in two main ways:

• Increase in Net Equity (NE), when the company reinvests profits and generates additional value for its equity; • Dividend distribution, when part of the profit is returned to shareholders as direct compensation.

Furthermore, for the relationship between ROE and P/BV to remain valid in the long term, it is necessary to assume that both the company and the investor reinvest the resources received — whether retained earnings within the company or dividends paid — at a rate equivalent to ROE.

The basic formula that can be used to find the fair P/BV in a scenario of stable ROE over time is:

P/BV = ROE/Cost of Capital.

If the ROE is 2X Cost of Capital, it means that the investor can pay twice the VP and the return on his capital will still be equivalent to what is required.

• ROE = Cost of Capital = 9.25% per year, with the investor paying 1x the Equity Value (P/BV = 1) at the beginning and selling at the same multiple at the end.

In this case, the IRR obtained is 9.25% per year, exactly equal to the return required by the investor.

Even if the company distributes all of its profits in the form of dividends and the investor reinvests these dividends at the same ROE rate, the final accumulated value will be identical to that of the scenario without distribution, which implies an annual return to the shareholder identical to that required.

The same reasoning applies to partial distribution situations, in which part of the profit is reinvested in the company itself and part is paid to the shareholder.

In all cases, as long as the reinvestment occurs at ROE, the final result demonstrates the consistency of the relationship between ROE, cost of capital, and fair P/B.

Okay, but all this reasoning only applies in a scenario of stable ROE over time.

What if ROE varies annually?

In this case, you need to project the profit (or return on equity) for each period individually and convert these values ​​to present value using the shareholder's cost of capital.

The sum of the present values ​​represents the Fair Equity Value (Fair PV)—that is, the maximum amount the investor should pay today to precisely achieve the required return.

The challenge is that, at the end of the projections, you need to define an exit P/P (terminal multiple) or calculate a perpetuity value, ensuring that the flow is complete.

Finally, by dividing the total present value obtained by the book value, you find the fair P/P, reflecting how much the market should pay for the company's capital given the projected returns.

In short, it's almost like creating a Discounted Cash Flow model, but from the perspective of value generated rather than cash generated.