# Simple valuation question

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Hi, Let's say a car dealership company earns 100% ROE and since they are such a good and profitable company their stock is sold at 100 times price to book ratio.

So I buy 1\$ of equity for 100\$ and by year end the company added 1\$ to it's equity. since the market says the company is worth 100 P/B why isn't the stock worth now 200\$ ?

The way the math works here is that no matter the valuation the single thing that will determine the return at year end is the return on equity.

I see how absurd such a claim is but I can't figure out whats wrong with the reasoning here.

If I have bought the equity at par (1 P/B)and hold the equity forever I would still make 100% return the same as I would if I buy at 100 P/B or 100m P/B.

(assume the company puts that 1 dollar to work at the same return)

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Well, remember that it is 100% ROE, not \$1 extra per year.  Presumably, after they earned the incremental \$1, and the book value is \$2, then the next year it will earn \$4, not \$3, which could justify the \$200 price.

Or said another way, P/B is just a short-hand way of doing DCF.  Really, you are looking at a company that doubles earnings every year.  If it can always double earnings every year, then it deserves a high multiple throughout.

You also seem to be questioning whether multiples matter if they hold constant.  It is true that they don't if they are always the same; however, as I mentioned above, the multiples are just a short-hand DCF, and no company maintains high growth rates forever.  Accordingly, high multiples must contract over time.

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well I fully understand your answer but I probably didn't explain myself correctly. I understand that the equity doubles every year if ROE stays at 100%, the problem i'm having is the market assign a value to the equity, let's say on oil company has a land worth 1m on the books but it's the best land to dig oil from anywhere on the planet so the market assign a 100 P/B to that equity / assets (no liabilities so assets = equity), the company sell the oil by dusting the top soil with a small broom and the oil gushes out, so let's assume they earn 100% ROE. 12 months later the market that fully understand that such asset is worth 100 times it's book value should recognize that same value if the company is able to invest the earnings at the same price in a similar asset, so if they earn 1m during the year, they buy 1m of land and magically make it gush oil out the company is worth now 200m right ? the same would happen if we use different P/B ratio, actually ANY P/B I choose will work.

let's take the case when they market recognize the assets are worth 1 trillion times the stated book value, if the company takes the earnings reinvest them and get a similar assets at a similar price the stock is no worth 2*T*1m.

The same logic works for smaller numbers. let's assume a company is selling for 3 times book value. the earn 20% ROE. so I pay 300\$ for 100\$ of equity. at year end I own 120\$ of equity (100\$ starting equity + 20% ROE) and the market should value it at 3 times book (assuming the company reinvest the earnings at the same returns it get's on the starting equity) so my stake is worth now 120\$ * 3 = 360% or a 20% IRR. The same will stay if I use any P/B ratio.

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You seem to be asking whether, so long as the multiple the stock market applies to a given metric stays the same, changes in the stock price will be proportional to changes in that metric.  The answer, of course, is "yes," because, by assuming a static multiple, you have assumed that the stock price can always be calculated by the following equation: y=Z(x), where y is the stock price, x is whatever metric you're using, and Z is a constant, i.e., your hypothetical static multiple.

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You are confusing the current P/B with the market's valuation.  The market is not saying the company is and will continue to be worth 100x P/B, nor that the high ROE will continue.  The market is saying based on a DCF that the company is worth \$100, which happens to be 100x P/B.  Thus a year from now if everything went as the market predicted the stock would rise by approximately the discount rate.  While I don't fully ascribe to the Efficient Market Hypothesis, it is helpful in that whether a stock is valued or high low, it should, if the theory were true, result in the same return a year from now (adjusted for risk) because all information, including a high ROE, is priced in.

If the assumption you were making regarding valuation were true, growth would massively outperform value, which it doesn't.  The reason is that the growth is priced in, so there is not an excess return.  What value investors will argue is that more often than not, more growth than what is likely to occur is already priced in, and for value stocks, that less growth than what will happen is priced in.

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After sleeping about it I am convinced this is true, as long as ROE will stay constant the investor should make the same return the company underlying returns on equity are. Think about it this way, If I buy a company at 5 times book value with 100% ROE the "instant yield" I have on my investment is 20% (1\$ NI/5\$ Price). All retained earnings are invested at book value and not at the 5 times the market ascribe. that's the key point, retained earnings are reinvested at book value thus being much more valuable to investors than paying the money as dividends and then reinvesting them manually at the market rate (5 P/B). If the company would distribute the money as dividends I would make 20% on my incremental investment but as long as the company retains earnings I should make the same ROE on those earnings as the company does since they are invested at book value. for the beginning 1\$ of equity I invested 5\$, but for the added 1\$ of equity that I earned during the year I was able to invest it at a "cost" of 1\$ (book value), thus making me pay 6\$ for 2\$ of equity and earnings (now my weighted cost is 3 P/B and the yield i'm getting is 33.333%) the following year I will earn 2 extra dollars on my holding and the company reinvests the earnings so my position cost was raised by 2 more bucks and I hold 4\$ of equity. so 8\$ weighted cost for 4\$ equity (for a yield of 50%). you see how if I keep going the returns I will make will approach the company ROE of 100%.

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After sleeping about it I am convinced this is true, as long as ROE will stay constant the investor should make the same return the company underlying returns on equity are. Think about it this way, If I buy a company at 5 times book value with 100% ROE the "instant yield" I have on my investment is 20% (1\$ NI/5\$ Price). All retained earnings are invested at book value and not at the 5 times the market ascribe. that's the key point, retained earnings are reinvested at book value thus being much more valuable to investors than paying the money as dividends and then reinvesting them manually at the market rate (5 P/B). If the company would distribute the money as dividends I would make 20% on my incremental investment but as long as the company retains earnings I should make the same ROE on those earnings as the company does since they are invested at book value. for the beginning 1\$ of equity I invested 5\$, but for the added 1\$ of equity that I earned during the year I was able to invest it at a "cost" of 1\$ (book value), thus making me pay 6\$ for 2\$ of equity and earnings (now my weighted cost is 3 P/B and the yield i'm getting is 33.333%) the following year I will earn 2 extra dollars on my holding and the company reinvests the earnings so my position cost was raised by 2 more bucks and I hold 4\$ of equity. so 8\$ weighted cost for 4\$ equity (for a yield of 50%). you see how if I keep going the returns I will make will approach the company ROE of 100%.

If the company would be able to earn a ROE on equity of 100% forever it would be worth an almost infinite amount of money (assuming a discount rate lower than 100%). So if you would buy it today you would be able to buy an almost infinitesimal fraction of a share. Next year the company would still be worth an almost infinite amount of money, and your infinitesimal fraction of a share would have returned the discount rate and not the underlying return on equity.

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The problem with above example of a 100% ROE is that the company usually cannot reinvest the profits with the same ROE in the business again. It is quite obvious with the car dealership example - if a car dealer really has 100% ROE and they could reinvest with the profitability in the business, it means that they could double the business every 12 month. Realisiticaly, a car dealership is not a business that can double in 12 month - where should all the growth come from?

There are some business that can theoretically double without much capital spent (internet business like FB), but there are of course other limitations for their growth (law of large numbers, addressable market, ability to grow organization)

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