Why does P/B = 1 when ROE = r?

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SpareTime

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I’m hoping somebody can help me out not with the mathematics of this, because I’m sure if I tried I could get through the derivation…but of understanding the concept.
The B in P/B is accounting book value, not market book value (as I understand it). So that should mean there’s lots of stuff at historical cost on there, and lots of intangibles that aren’t listed in shareholder’s equity (like strength of brand and all that). So how can it be that when ROE = r that P/B = 1? Because when P/B = 1 that means that the price of the security is exactly the same as book value per share, which means it’s priced accurately. Except it’s priced accurately against accounting standards, which are not a true representation of the strength of the company. But if that is true, then most P/B should be higher than 1, which would mean most ROE is higher than r. And that’s what doesn’t make sense to me. In aggregate, shouldn’t ROE = r?
Sorry for the long convoluted question. I want to understand this conceptually, rather than just memorize the underlying formulae.
 
SpareTime wrote:
I’m hoping somebody can help me out not with the mathematics of this, because I’m sure if I tried I could get through the derivation…but of understanding the concept.
The B in P/B is accounting book value, not market book value (as I understand it). So that should mean there’s lots of stuff at historical cost on there, and lots of intangibles that aren’t listed in shareholder’s equity (like strength of brand and all that). So how can it be that when ROE = r that P/B = 1? Because when P/B = 1 that means that the price of the security is exactly the same as book value per share, which means it’s priced accurately. Except it’s priced accurately against accounting standards, which are not a true representation of the strength of the company. But if that is true, then most P/B should be higher than 1, which would mean most ROE is higher than r. And that’s what doesn’t make sense to me. In aggregate, shouldn’t ROE = r?
Sorry for the long convoluted question. I want to understand this conceptually, rather than just memorize the underlying formulae.
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A P/B=1 does not mean the security is accuratley priced, rather it means that price accurately reflects book value as it naturally implies. If only you assume that book value reflects fair value, then you would be correct. P/B ratios are more useful for companies with more liquid assets like financial insitiutions.
Why would ROE=r? if you’re return on equity is the same as the required return on equity, then you aren’t creating shareholder value, or in other words, the company isn’t growing economically. ROE tends to be overestimated because of off-balance sheet variables, and the limitaions of book value as a measure of fair value, among other reasons and accounting records. This is why an intrinsic valuation, like DCFs, are more reliable than relative valuations.
 
Thanks for your answers.
I’m still having issues here, and I am probably making an incorrect assumptions. Here are my assumptions:
1) That if a security were priced at a P/B of 1, that would most likely be an undervalued stock since the true value of the majority of securities is higher than its acccounting book value.
and
2) That in aggregate, all stocks ROEs should be equal to r. Since, like you said MrSmart, an ROE greater than r results in economic gains, all stocks in aggregate should not make economic gains.
But these two assumptions don’t seem like they can fit together. If all stocks in aggregate make no economic gains and ROE = r, then the average P/B should be equal to 1. Yet that would mean that the average stock is valued only on its accounting book value, which is too low.
I did read the link S2000Magician (so you know I’m not being lazy), but I I still can’t reconcile these two assumptions and it’s driving me crazy.
 
Suppose, to make things simple, that a company has $100 in assets and $100 in (book value of) equity. The required return on equity is 10%.
If the company’s ROE is 9%, then the (book value of) equity in one year is $109. Investors needed a 10% return, which would happen only if the original market value of equity were $109/1.1 = $99.09. Thus, P/B = $99.09 / $100 = 0.9909 < 1.0.
If the company’s ROE is 11%, then the (book value of) equity in one year is $111. Investors needed a 10% return, which would happen only if the original market value of equity were $111/1.1 = $100.91. Thus, P/B = $100.91 / $100 = 1.0091 > 0.
 
SpareTime wrote:
Thanks for your answers.
I’m still having issues here, and I am probably making an incorrect assumptions. Here are my assumptions:
1) That if a security were priced at a P/B of 1, that would most likely be an undervalued stock since the true value of the majority of securities is higher than its acccounting book value.
and
2) That in aggregate, all stocks ROE’s should be equal to r. Since, like you said MrSmart, an ROE greater than r results in economic gains, all stocks in aggregate should not make economic gains.
But these two assumptions don’t seem like they can fit together. If all stocks in aggregate make no economic gains and ROE = r, then the average P/B should be equal to 1. Yet that would mean that the average stock is valued only on its accounting book value, which is too low.
I did read the link S2000Magician (so you know I’m not being lazy), but I I still can’t reconcile these two assumptions and it’s driving me crazy.
Click to expand...
1) Not nessecarily, that completely depends on whether the equity value reflects going concern value-in-use assets, or the fair value of tangible, more liquid assets. Graham first mentioned that if net tangible assets per share is 80% or less of price per share, then the stock is most likely undervalued. A better representation would be calculating the market value of NTA and comparing it to the stock price, that may indicate the money you’ll get back if the company is liquidated. The book value of equity may be over-estimated or under-estimated, so it would be better to look at more than just one ratio to arrive at relative value. P/B is just one item from a toolbox in a shed, it probably means nothing alone.
2) Don’t you mean that all ROE’s should equal to the average r for a group of firms operating in the same mature industry on the same consumer base? It’s perfectly fine for the aggregate stock market to make positive economic gain, especially when you consider multinational operations. First, you are assuming that stock prices cannot deviate from intrinsic value, while it’s true when you consider the long term aggregate, it’s rarely true at any point in time. Second, you’re assuming that earnings and book value are free from manipulation or reflecting fair value, while this is far from the truth. Third, you are assuming that the measure of r is accurate, while again, it’s more difficult than it seems. While theoratically, P/B should equal to one in the aggregate stock market, there are a number of reasons why this does not happen, especially when you consider the short term, international trade, accounting standards, and market inefficency.
 
S2000magician wrote:
Suppose, to make things simple, that a company has $100 in assets and $100 in (book value of) equity. The required return on equity is 10%.
If the company’s ROE is 9%, then the (book value of) equity in one year is $109. Investors needed a 10% return, which would happen only if the original market value of equity were $109/1.1 = $99.09. Thus, P/B = $99.09 / $100 = 0.9909 < 1.0.
If the company’s ROE is 11%, then the (book value of) equity in one year is $111. Investors needed a 10% return, which would happen only if the original market value of equity were $111/1.1 = $100.91. Thus, P/B = $100.91 / $100 = 1.0091 > 0.
Click to expand...
Ok I am getting closer. I realize now that one of my issues is that I was forgetting that ROE is return off book value of equity, not market value of equity.
However I am still a bit confused. Taking your second example, let’s say that the next year the company has an ROE = r = 10%. That would make the book value $111 x 1.1 = 123.21, and since ROE = r, P/B should = 1. But what if the price/market value of equity isn’t $123.21? Couldn’t it be any value? If it was a different value, would that mean that r is actually a different value as well?
Thanks for your patience with me guys….
EDIT* I just realized that this is for JUSTIFIED P/B. In reality, the P/B could be anything.
And I think ROE does not have to = r in aggregate because r is (in aggregate) the return on market value of equity (because theoretically investors would demand what the market returns), which does not have to be the same as the return on book value
 
Do you understand what r is? It is the rate required by investors. If the original price was $100 and P/B was 1, and the company made $11 = 11% on that E (equity, i.e. book value); then the investors would bid up the price until the price was $110. Then P/B would rise to 1.1. Because the investors only needed 10% (for whatever reason, CFAI doesn’t tell you why a particular discount rate is a good one. No one does, in fact ) So they would bid up anything that yielded 11% initialiy, until that thing yields 10%.
Similarly if the company only makes $9 = 9%, investors would be disheartened and sell it off until the price fell to $90 and P/B to 0.9. That is what S2000’s initial example is saying.
You are assuming that price is an independent variable, but in fact (if you believe the textbook) it is B*ROE/r. And B*ROE is nothing but return (i.e. earnings); so price = earnings / required discount rate. That’s all it is.
Not to confuse you, but you can carry this concept for real estate as well, price = income / cap rate.
 
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