One bank, one rate shock, one repricing decision: two betas a factor of 1.54 apart, and a break-even that is 17.32 points away or 5.00 depending on the denominator.
A deposit beta is the change in the rate a bank pays on its deposits divided by the change in market rates over the same period. The number only means something with its denominator attached: on an illustrative bank, a 35 per cent beta on the savings book is 22.68 per cent measured on all non-maturity deposits, and a 35 per cent beta read across all of them would mean 54.01 per cent on savings and would cut the year's pre-tax income from 36,800 to 10,620.
Worked in full in Bank Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Sagebrook National Bank funds itself mainly with non-maturity deposits: current accounts that pay nothing and savings accounts that pay a managed rate. Over one year the market rate rises from 1.55 to 5.80 per cent, a shock of 425 basis points. Treasury raises the savings rate by 35 per cent of that move. Every figure is illustrative and in thousands of dollars.
| Input | Value |
|---|---|
| Non-interest-bearing deposits | 1,760,000 |
| Savings and interest-bearing deposits | 3,240,000 |
| Non-maturity deposits | 5,000,000 |
| Market rate, start of year | 1.55% |
| Rise in market rate | 4.25% |
| Cost of savings, start of year | 0.18% |
| Beta applied to the savings book | 35% |
| Pre-tax income in the shock year | 36,800 |
The pre-tax income is the shock year as the bank's own model has it: net interest income of 169,800, plus 41,000 of fees, less 148,000 of operating expenses and 26,000 of provisions.
The savings rate rises by 35 per cent of 4.25 points, which is 1.4875 points, from 0.18 to 1.67 per cent. On 3,240,000 of savings that costs 48,195 a year. That dollar figure is the fact. The beta is that fact divided by something, and there are two candidates.
Deposit beta = change in deposit interest expense / (deposit balance × change in market rate)
On the savings book: 48,195 / (3,240,000 × 4.25%) = 48,195 / 137,700 = 35.00%
On all non-maturity deposits: 48,195 / (5,000,000 × 4.25%) = 48,195 / 212,500 = 22.68%
Excel: =(Sav*(CostSav1-CostSav0))/((NIB+Sav)*Shock)
Both are correct and both describe this bank. The first answers how the bank prices its savings product. The second answers what the deposit franchise as a whole costs when rates move, because the current accounts are part of the franchise and their beta is zero. The two differ by a factor of 1.54, which is simply the ratio of all non-maturity deposits to the interest-bearing part.
The trouble starts when a beta measured on one base is applied to the other. Suppose the 35 per cent is a peer figure, or a figure from an asset and liability committee paper, and it was meant across all non-maturity deposits. Applied that way, the bank's deposits cost 35 per cent of 212,500, or 74,375. All of it falls on the savings book, because the current accounts still pay nothing, so the savings beta is in fact 74,375 / 137,700 = 54.01 per cent.
| Reading | Savings beta | All-deposit beta | Repricing cost | Pre-tax income |
|---|---|---|---|---|
| 35% on the savings book | 35.00% | 22.68% | 48,195 | 36,800 |
| 35% on all non-maturity deposits | 54.01% | 35.00% | 74,375 | 10,620 |
The difference is 26,180 a year, and on the second reading net income after 25 per cent tax is 7,965, below the 18,000 dividend. Nothing about the bank has changed between the two rows. Only the word "beta" has been read differently.
The more useful number for a committee is the beta at which the year's profit disappears. Each point of beta costs the deposit balance times the shock divided by 100: 1,377 on the savings basis, 2,125 on the all-deposit basis. Pre-tax income of 36,800 is used up after 26.72 more points on savings, or 17.32 on all deposits.
Break-even beta = current beta + pre-tax income / (balance × shock)
Savings basis: 35.00% + 36,800 / 137,700 = 61.72%
All-deposit basis: 22.68% + 36,800 / 212,500 = 40.00%
The break-even is the same number of dollars either way: 84,995 of repricing. What changes is how far away it looks. If the 35 per cent was meant on the savings book, the bank is 17.32 points of all-deposit beta from a loss. If it was meant on all deposits, it is 5.00 points away.
| Savings beta | All-deposit beta | Repricing cost | Pre-tax income |
|---|---|---|---|
| 25.00% | 16.20% | 34,425 | 50,570 |
| 35.00% | 22.68% | 48,195 | 36,800 |
| 45.00% | 29.16% | 61,965 | 23,030 |
| 54.01% | 35.00% | 74,375 | 10,620 |
| 61.72% | 40.00% | 84,995 | 0 |
A ten point error in the beta is worth 13,770 of pre-tax income here, more than a third of the year. That is why the beta, together with the assumed life of the deposits, is one of the two inputs that decide most of a bank's interest rate answer, and why both are judgements rather than measurements.
Mix moves the all-deposit beta even when pricing does not. If a tenth of the current accounts, 176,000, migrates into savings during the year at the new 1.67 per cent, the all-deposit beta rises from 22.68 to 24.06 per cent and pre-tax income falls to 33,865, with the savings beta still at 35 per cent.
The frequent error is comparing betas across banks, or across years, without checking the denominator. One bank reports a beta on interest-bearing deposits, another on total deposits, a third on total deposits including time deposits, and a model somewhere applies whichever number is to hand to whichever balance is to hand. A second, related error is measuring the beta over a period too short for deposit pricing to catch up, which produces a low figure that then sits in the model for the whole cycle.
The fix is mechanical. State every beta with three things attached: the balances in the denominator, the period, and the size of the market move. Then compute the break-even on the same base, so that the distance to a loss is read in the units the beta was measured in. The same discipline shows up in valuation: the deposit franchise a buyer pays for, set out in what a buyer actually pays for a bank, rests on exactly this beta.
Both readings, the margin before and after the shock and the break-even are in the free workbook for this case, with the beta as an input you can move.
There is no single figure, because the answer depends on the deposits in the denominator and the length of the rate cycle. In the worked case the same repricing reads 35 per cent on the savings book and 22.68 per cent on all non-maturity deposits. Any beta quoted without its denominator, its horizon and the size of the rate move cannot be compared with another bank's figure.
Include them if the question is what the whole deposit franchise costs when rates move, exclude them if the question is how the bank prices its interest-bearing products. Both are legitimate. The error is mixing them: in the example, a 35 per cent beta meant on savings but applied to all 5,000,000 of non-maturity deposits overstates the cost by 26,180 a year.
It is the beta at which the year's pre-tax income falls to zero, holding everything else. Each point of beta on all non-maturity deposits costs 2,125 in the example, so 36,800 of pre-tax income is used up at a beta of 40.00 per cent. On the savings book alone the same break-even reads 61.72 per cent.
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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