The justified multiple from return on tangible equity, cost of equity and growth, and why the same price can read 1.14 or 1.50 times.
A bank should trade at a price to tangible book value of (ROTCE − g) / (cost of equity − g): the return it earns on tangible equity, less growth, over the return shareholders require, less growth. An illustrative bank earning a 12.0 per cent return on tangible common equity, with an 11.0 per cent cost of equity and 4.0 per cent growth, is worth 1.14 times tangible book, 571,429 on 500,000 of tangible equity. A bank earning 8.0 per cent on the same terms is worth 0.57 times.
Worked in full in Bank Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Price to tangible book is the multiple bank analysts, acquirers and boards argue over, and it is often quoted as if a level were a matter of market fashion: banks "should" trade at 1.5 times, or are "cheap" below book. The level is not fashion. It follows from three numbers, and one of them, the return on tangible equity, is the bank's own. A bank that earns its cost of equity is worth exactly its tangible book; one that earns more is worth a premium, one that earns less a discount.
| Input | Value |
|---|---|
| Book common equity | 560,000 |
| Goodwill and other intangibles | 60,000 |
| Net income | 58,420 |
| Amortisation of intangibles, pre-tax | 2,000 |
| Tax rate | 21% |
| Cost of equity | 11.0% |
| Long-run growth in tangible equity and earnings | 4.0% |
| Shares in issue | 40,000 |
Step 1. Tangible common equity
560,000 − 60,000 = 500,000, 12.50 a share. How to build it from a balance sheet is in how to calculate the TCE ratio.
Step 2. Tangible net income and ROTCE
Add back the after-tax amortisation of intangibles, which reduces earnings but not tangible equity: 2,000 × (1 − 21%) = 1,580. Tangible net income is 58,420 + 1,580 = 60,000, and ROTCE = 60,000 / 500,000 = 12.0 per cent. Return on book equity is only 10.43 per cent, because the denominator carries the goodwill.
Step 3. The justified multiple
P/TBV = (12.0% − 4.0%) / (11.0% − 4.0%) = 8.0 / 7.0 = 1.14. In Excel, with ROTCE computed in B10 below the inputs: =(B10-B8)/(B7-B8).
Step 4. Value
1.143 × 500,000 = 571,429, 14.29 a share, 9.78 times earnings and 1.02 times book equity.
The formula is a dividend discount model in disguise. To grow tangible equity at 4.0 per cent while earning 12.0 per cent, the bank retains a third of its earnings, 20,000, and pays out 66.7 per cent, 40,000. Discounting 40,000 growing at 4.0 per cent at 11.0 per cent gives 40,000 / 7.0% = 571,429, the same answer.
| ROTCE | COE 10% | COE 11% | COE 12% | COE 13% |
|---|---|---|---|---|
| 8% | 0.67 | 0.57 | 0.50 | 0.44 |
| 10% | 1.00 | 0.86 | 0.75 | 0.67 |
| 12% | 1.33 | 1.14 | 1.00 | 0.89 |
| 14% | 1.67 | 1.43 | 1.25 | 1.11 |
| 16% | 2.00 | 1.71 | 1.50 | 1.33 |
Every cell where ROTCE equals the cost of equity reads exactly 1.00: a bank that earns what its shareholders require is worth its tangible book, whatever it grows at. Where ROTCE is higher, growth adds value; where it is lower, growth destroys it. At 8.0 per cent ROTCE and an 11.0 per cent cost of equity, the multiple is 0.67 with 2.0 per cent growth and 0.40 with 6.0 per cent growth: a bank earning below its cost of equity is worth less the faster it expands. That is the arithmetic behind the discount at which low-return banks trade, and it is why "cheap to book" is not an argument on its own.
Suppose the same bank also carries 120,000 of after-tax unrealised losses on held-to-maturity securities, not deducted from its reported equity. Marked, tangible equity is 380,000, and the same 571,429 of value reads as 1.50 times tangible book and a 15.8 per cent ROTCE. Nothing about the bank has changed. The loss is already in the price, because the underwater securities earn low coupons and those low coupons are already in the 60,000 of earnings. Two analysts can quote 1.14 and 1.50 times for the same share price and both be right on their own denominator. The book this article draws on is built around exactly that problem: four defensible readings of one balance sheet, and which one the depositor reads.
Mixing return on book equity with tangible book. Feed the 10.43 per cent ROE into the formula and the multiple is 0.92; apply it to tangible book and the bank is worth 459,439, 20 per cent too little. The numerator and the denominator must describe the same equity: ROTCE with tangible book, ROE with book. The second mistake is to take the cost of equity from a market average while the bank's earnings are cyclically high or low. A ROTCE flattered by an unusually low credit cost year will carry its error straight into the multiple, one for one.
The formula assumes a steady state: constant ROTCE, constant growth, no excess capital. A bank holding capital well above what it needs should be valued as the steady-state bank plus the excess, which can be distributed or used to buy back shares.
The free workbook for this case sets the four readings of one balance sheet side by side and bridges them to the dollar, and the buyer's view of the same bank is in what a buyer actually pays for a bank.
Because they earn less on tangible equity than shareholders require. With a cost of equity of 11.0 per cent and 4.0 per cent growth, a bank earning an 8.0 per cent ROTCE is worth (8 - 4) / (11 - 4) = 0.57 times tangible book. Faster growth makes it worse: at 6.0 per cent growth the multiple is 0.40.
Add the after-tax amortisation of intangibles back to net income, then divide by tangible common equity, which is common equity less goodwill and intangibles. On an illustrative bank, 58,420 of net income plus 1,580 of after-tax amortisation is 60,000, over 500,000 of tangible equity: 12.0 per cent, against a 10.43 per cent return on book equity.
Exactly 1.0 times, whatever its growth rate, because each retained dollar earns what shareholders could earn elsewhere. In the formula (ROTCE - g) / (COE - g), equal ROTCE and cost of equity make numerator and denominator identical. At 11.0 per cent for both, the multiple is 1.00 with 2.0, 4.0 or 6.0 per cent growth.
This article is one calculation from Bank Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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