Modelling the costs and benefits of higher capital ratios

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Modelling the costs and benefits of higher capital ratios This document attempts to reproduce and extend previous work to model the costs and benefits of higher capital ratios. Previous work was undertaken by the Basel Committee on Banking Supervision and by the Reserve Bank (the “Harrison model”), but parts of this were not well documented and the Reserve Bank work appears to contain some errors. We have tried to remedy these problems by providing a more clearly documented analytical foundation for the model and identifying errors to be corrected. A decision was made not to develop a different approach from the Harrison model, because it would be a distraction from more important work (the wider capital review) and it might not materially improve estimates of net benefit. However, in the course of trying to work out how the Harrison model worked, we did find it necessary to extend the model in some ways to produce an internally consistent framework. The rest of this document proceeds as follows: firstly, the Basel capital equation is used to calculate two benefits of fewer bank failures resulting from higher capital ratios. The benefits considered are the avoidance of recessions and the avoidance of bailout costs. This is closely aligned to the framework used in past Reserve Bank work, but there are some differences which (hopefully) will result in more correct estimate. The detailed calculation of benefits in the Harrison model (i.e. the past Reserve Bank work) is discussed at the end of the first part of the document. Secondly, other costs and benefits are calculated within a supply and demand framework. One cost considered is lower GDP as a result of reduced bank lending and investment. As well, transfers to overseas residents and taxation effects are considered. While the elements which are taken into account are the same as those mentioned in past Reserve Bank work our approach differs in the detail, with a view to internal consistency. At the end of the section, the calculation of costs and benefits in the Harrison model is discussed. The earlier Reserve Bank work can be found in the following:
The Regulatory Impact Statement which describes the previous Reserve Bank work (Ian Harrison’s Tuatara model) - http://docs/webtop/drl/objectId/090000c3803c010b A Powerpoint presentation about Ian Harrison’s model - http://docs/webtop/drl/objectId/090000c380352c80 The spreadsheet with cost-benefit calculations in it (used in the Regulatory Impact Statement) - http://docs/webtop/drl/objectId/090000c3803b6929 A supporting spreadsheet - http://docs/webtop/drl/objectId/090000c3803b692e

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Using the Basel capital equation to model bank failure Assume there is single bank holding all loans. Assume that the portfolio can be modelled using the corporate capital equation from BS2B (see BS2B 4.136):
𝐾 = 𝐿𝐺𝐷 × [Φ (√
1
1−𝑅
Φ−1
(𝑃𝐷) + √
𝑅
1−𝑅
Φ−1
(0.999)) − 𝑃𝐷] ×
1 + 𝑏(𝑀 − 2.5)
1 − 1.5𝑏 where 𝑏 = [0.11852 − 0.05478 ln 𝑃𝐷] 2 , 𝑅 = 𝑅(𝐶𝑅) (in BS2B, R is a function of PD but in the Harrison model it is defined to be a different function of CR), Φ is the standard normal cumulative distribution function, and Φ−1 is its inverse. This gives the amount of capital (K) the bank needs to hold, per dollar of EAD, for there to be a 0.999 probability the bank will survive the next 12 months, unconditional on the state of the economy, assuming provisions have also been made for 𝑃𝐷 ∙ 𝐿𝐺𝐷 per dollar of exposure. 1,2,3
0.999 is the
survival probability (SP), which we generalise from this point.
Assume the average risk weight for the portfolio is RW. The following equation shows the relationship between the regulatory capital ratio and the survival probability. To be precise, it shows the minimum capital ratio necessary to achieve the desired survival probability. 𝐶𝑅 = 𝐿𝐺𝐷 𝑅𝑊 × [Φ (√ 1 1−𝑅 Φ−1 (𝑃𝐷) + √ 𝑅 1−𝑅 Φ−1 (𝑆𝑃)) − 𝑃𝐷] × 1 + 𝑏(𝑀 − 2.5) 1 − 1.5𝑏 Solving for SP allows the survival probability to be determined from a given capital ratio:
𝑆𝑃 = Φ [√
1−𝑅
𝑅
(Φ−1
[𝐶𝑅 ×
1 − 1.5𝑏
1 + 𝑏(𝑀 − 2.5)
×
𝑅𝑊
𝐿𝐺𝐷 + 𝑃𝐷] − √
1
1−𝑅
Φ−1
(𝑃𝐷))]
SP, with appropriate tweaks at the end-points, is a cumulative distribution function, and has a corresponding probability density function sp (lower case). See the Appendix for more detail. SP(CR) is the probability that all shocks over the next year will be small enough to leave some capital intact, given a capital ratio of CR (and also assuming provisions of 𝑃𝐷 ∙ 𝐿𝐺𝐷 per dollar of exposure). Equivalently, 1 – SP(CR) gives the probability that a shock will be large enough to wipe out all capital when the capital ratio is CR, causing the bank to fail. The probability of a shock that is big enough to wipe out a bank with a capital ratio between CR and CR + a (and not big enough to wipe out a bank with a capital ratio higher than CR + a) is:

1 The equation is derived from the Vasicek asymptotic single factor risk model. An accessible derivation is given in Hamerle et al (2003), “Credit Risk Factor Modeling and the II IRB Approach”, Deutsche Bundesbank Discussion Paper, No. 02/2003. 2 More properly, it is the probability of survival unconditional on the state of the economy. 3 Footnote: Strictly speaking, this equation is consistent with provisions of PD ∙ LGD ∙ EAD ∙ maturity adjustment, where maturity adjustment is the third factor in the equation for K above.

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𝑃(𝑆ℎ𝑜𝑐𝑘 𝑖𝑠 𝑏𝑖𝑔 𝑒𝑛𝑜𝑢𝑔ℎ 𝑡𝑜 𝑤𝑖𝑝𝑒 𝑜𝑢𝑡 𝐶𝑅) − 𝑃(𝑆ℎ𝑜𝑐𝑘 𝑖𝑠 𝑏𝑖𝑔 𝑒𝑛𝑜𝑢𝑔ℎ 𝑡𝑜 𝑤𝑖𝑝𝑒 𝑜𝑢𝑡 𝐶𝑅 + 𝑎) = 1 − 𝑆𝑃(𝐶𝑅) − [1 − 𝑆𝑃(𝐶𝑅 + 𝑎)] = 𝑆𝑃(𝐶𝑅 + 𝑎) − 𝑆𝑃(𝐶𝑅) Because SP is a cumulative distribution function and sp is the corresponding probability distribution function:
𝑆𝑃(𝐶𝑅 + 𝑎) − 𝑆𝑃(𝐶𝑅) = ∫ 𝑠𝑝(𝑥)𝑑𝑥
𝐶𝑅+𝑎
𝐶𝑅
Benefits of higher capital (general description) Shocks that cause the bank to fail generate two sorts of losses: direct economic losses and bank losses. Direct economic losses, caused by disruption to production, are always socially costly. In the model bank losses are socially costly if the government bails out the bank, to the extent that bailout funds are paid to non-residents. Raising the capital ratio reduces both kinds of losses, so is beneficial (in a gross sense). Reduction of economic losses causes by crises The direct economic cost of a shock depends on the size of the shock. The expected direct economic cost of shocks, for a bank with a capital ratio of CR, is given by4 ∫ 𝐸𝐶(𝑥)𝑠𝑝(𝑥)𝑑𝑥 ∞ 𝐶𝑅 where 𝐸𝐶(𝑥) is the direct economic cost of a shock that is just big enough to wipe out a bank with a capital ratio of 𝑥. This is just the ordinary expected value of EC(x), assuming economic costs are nil for shocks too small to cause the bank to fail:
∫ 𝐸𝐶(𝑥)𝑠𝑝(𝑥)𝑑𝑥 =
∞
−∞
∫ 0 ∙ 𝑠𝑝(𝑥)𝑑𝑥
𝐶𝑅
−∞

• ∫ 𝐸𝐶(𝑥)𝑠𝑝(𝑥)𝑑𝑥
  ∞
  𝐶𝑅
  The expected cost is reduced if the capital ratio is increased, and this is a benefit. When the capital ratio goes from a to b, the benefit is:
  ∫ 𝐸𝐶(𝑥)𝑠𝑝(𝑥)𝑑𝑥
  𝑏
  𝑎
  This can be shown, for example, using Leibniz’s rule to find the derivative of the expected cost with respect to CR then integrating over a to b. Intuitively, the benefit from moving to b is that there are some shocks big enough to cause failure with CR = a but not big enough to cause failure with CR = b. It is the cost of these shocks that is avoided. This is an annual cost; this cost is what is expected to be incurred every year, on average (in practice the cost is likely to be extremely lumpy). To find the total cost, the annual cost for each year j is divided by the appropriate discount rate for that year. Then the discounted costs are summed:

4 Direct economic and bailout costs are specified as positive quantities. In determining the net benefits of raising the capital requirement, these quantities are therefore to be subtracted.

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∑
∫ 𝐸𝐶(𝑥)𝑠𝑝(𝑥)𝑑𝑥 𝑏
𝑎
∏ (1 + 𝑟𝐸𝐶,𝑗)
𝑖
𝑗=1
∞
𝑖=1
Reduction of bailout costs
Similarly, the expected cost of bank losses, with a capital ratio of CR, is given by ∫ 𝐵𝐿(𝑥)𝑠𝑝(𝑥)𝑑𝑥 ∞ 𝐶𝑅 where BL(x, CR) is the bailout funds paid to foreigners, in the case of a shock that is just big enough to bring down a bank with a capital ratio of x. As with economic costs, bailout costs decrease when the capital ratio goes from a to b. The benefit is given by:
∫ 𝐵𝐿(𝑥, 𝑎)𝑠𝑝(𝑥)𝑑𝑥
𝑏
𝑎

• ∫ (𝐵𝐿(𝑥, 𝑎) − 𝐵𝐿(𝑥, 𝑏))𝑠𝑝(𝑥)𝑑𝑥
  ∞
  𝑏
  The first term is the saving from the cases where the larger capital buffer means the bank no longer fails. The second term reflects a lower bailout cost in the cases where there is still a bank failure, but the higher capital buffer has absorbed more losses.5

The expression above gives an annual cost. Total costs are given by discounting and summing:
∑
∫ 𝐵𝐿(𝑥, 𝑎)𝑠𝑝(𝑥)𝑑𝑥 𝑏
𝑎

• ∫ (𝐵𝐿(𝑥, 𝑎) − 𝐵𝐿(𝑥, 𝑏))𝑠𝑝(𝑥)𝑑𝑥 ∞
  𝑏
  ∏ (1 + 𝑟𝐵𝐿,𝑗)
  𝑖
  𝑗=1
  ∞
  𝑖=1
  Recognising the value of certainty
  As it stands, losses that have been avoided are all discounted at the same required rate of return, regardless of the size of the loss. But risk-averse consumers will place a higher value on the avoidance of large losses than on the avoidance of small ones, because of the concavity of their utility curves. The benefits above are therefore multiplied by a utility correction function UC(x), which can increase with the size of the loss. ∑ ∫ 𝑈𝐶(𝑥)𝐸𝐶(𝑥)𝑠𝑝(𝑥)𝑑𝑥 𝑏 𝑎 ∏ (1 + 𝑟𝐸𝐶,𝑗) 𝑖 𝑗=1 ∞ 𝑖=1 + ∫ 𝑈𝐶(𝑥)𝐵𝐿(𝑥, 𝑎)𝑠𝑝(𝑥)𝑑𝑥 𝑏 𝑎
• ∫ 𝑈𝐶(𝑥)(𝐵𝐿(𝑥, 𝑎) − 𝐵𝐿(𝑥, 𝑏))𝑠𝑝(𝑥)𝑑𝑥 ∞
  𝑏
  ∏ (1 + 𝑟𝐵𝐿,𝑗)
  𝑖
  𝑗=1

5 The need for two terms will become clearer when the precise bailout function for the Harrison model is discussed. It would be possible to specify a different function (not necessarily a sensible one) and have only the first term. The economic loss function above has only one term because additional capital held by the bank is not seen as a buffer to an economic downturn once failure has occurred.

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Benefits in the Harrison model
The Harrison model uses roughly the approach we have just described, but there are some differences which will be discussed further below. Because the documentation for the model is sparse, some of our discussion is speculative. Discount rates 𝑟𝐸𝐶,𝑗 = 𝑟𝐵𝐿,𝑗 = 3% for all j. The reason for this choice of discount rate is unclear. Discrete approximations The integrals used to calculate expected values of utility are approximated as finite sums, using constant costs and utility corrections over the interval between capital ratios (capital ratios are all expressed as whole numbers of percentage points). This allows EC, BL and UC to be taken outside the integrals for each increment, leaving only the probability density which is trivially integrated using the SP function described above. For the density function, set PD = 1.5%, LGD = 30%, M = 2.5, and RW = 0.5. R is normally calculated from PD and would be about 0.18 in this case. However, this was felt to be “too optimistic” (see paragraph 24 of RIS), so higher values were imposed:
𝑅 = {
0.2000 + 1.25(𝐶𝑅 − .02), 0.02 ≤ 𝐶𝑅 ≤ 0.14
0.21 + 𝐶𝑅, 𝐶𝑅 ≥ 0.15
Next evaluate the change in survival probability for each increment in the capital ratio, in steps of
0.01 starting from a: ∫ 𝑠𝑝(𝑥)𝑑𝑥 𝑎+1
𝑎
= 𝑆𝑃(𝑎 + 1) − 𝑆𝑃(𝑎) for 𝑎 = 0 𝑡𝑜 0.20. There is also a value calculated for ∫ 𝑠𝑝(𝑥)𝑑𝑥 ∞
0.20 , which is 1 − 𝑆𝑃(0.20).
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EC(x) is zero when there is no bank failure, and is otherwise7 𝐺𝐷𝑃 × % 𝐺𝐷𝑃 𝑙𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑠𝑖𝑛𝑔𝑙𝑒 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 = 𝐺𝐷𝑃 × { 0.1, 0.02 ≤ 𝑥 ≤ 0.03
0.12 + 2(𝑥 − 0.04), 0.04 ≤ 𝑥 ≤ 0.07
0.2, 𝑥 ≥ 0.08
BL(x) is given by the expression
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑏𝑎𝑖𝑙𝑜𝑢𝑡 𝑐𝑜𝑠𝑡𝑠 𝑓𝑜𝑟 𝑎 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝐶𝑅 = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × (0.00329 × [𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑙𝑜𝑠𝑠 + 𝑎𝑠𝑠𝑒𝑡 𝑝𝑟𝑖𝑐𝑒 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 + 𝑅𝑊 × (21 − 𝐶𝑅)]

• ∑ {[𝑎𝑠𝑠𝑒𝑡 𝑝𝑟𝑖𝑐𝑒 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 + 0.25 + 𝑅𝑊 × (𝑏 − 𝐶𝑅)][∆𝑏
  𝑏−1
  [1 − 𝑆𝑃]]}
  20
  𝑏=𝐶𝑅
  ) with terminal loss and asset price discount both set to 0.03 (3% of assets).

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In the model previously implemented there is an unusual, hardcoded value of 0.089 for the change in probability when the capital ratio goes from 0.01 (or possibly from 0.00) to 0.02. It seems this value should properly be 0.070 (for 1.00 to 0.20) or 0.203 (for 0.00 to 0.20). 7 In the practical implementation of the Harrison model in 2011, a single GDP loss is calculated for the shift from a capital ratio of 0.00 to 0.02 (or possibly 0.01 to 0.02) and then manually apportioned to CR = 0.01 and CR = 0.02. The basis for the apportionment is unclear.

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The assumption implicit in the expression is that when a bank fails, its losses depend on the size of the shock. If the shock would have just wiped out a bank with a capital ratio a, it will also wipe out the capital of a bank with capital ratio a – b and put its capital ratio b in the red. With a risk weight of 50%, each lost unit of capital below zero is worth 0.5 times the bank’s initial assets. The amount 0.5b is therefore funded by deposit-holder losses, which the government will need to pay for in the case of a bailout. In additional to any capital losses, the bank is expected to lose money through fire sales (this is our interpretation of the asset price discount) and bailout administration costs (this is our interpretation of the 0.25 figure). The expected loss from shocks which are very large indeed – big enough to wipe out a bank with a capital ratio higher than 20% – is a 21% reduction in the capital ratio, plus a further 6% of assets (the asset price discount and the terminal loss). The probability of a government bailout is set at 50%, which halves the expected bailout costs. The discrete approximation as implemented in the Harrison model in practice – losses are treated as constant over the range of capital ratios (a, a+1] – means that the losses are all greater than in the continuous case. For instance, the loss used when CR = 19 and b = 20 (ignoring terminal losses) is 3.25% + 0.5% of assets. Because of the discrete approximation, as well as applying to shocks which would just wipe out firms with CR = 20, this loss applies to all shocks which would just wipe out firms with 19 < CR <
20. The actual loss should be smaller in this second set of cases, because the shocks are smaller (less
equity is wiped out).
A potential remedy to this problem is to use a better discrete approximation. See the appendix for one possibility. Calculating UC(x) A utility adjustment factor, for a particular capital ratio, is calculated:
𝑈𝑡𝑖𝑙𝑖𝑡𝑦 𝑎𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡 =
1
5(1 − 5𝑥 × 0.6)
6 +
4
5(1 −
5
4
𝑥 × 0.4)
6 with
𝑥 = (500 × (𝐶𝑅 − 0.01))
0.47
A possible derivation for the utility adjustment (we are guessing this is how the result was arrived at) follows. Divide society into two groups – the bottom quintile by income and the rest. Before a shock which causes a bank failure, annual GDP is a + b. This is entirely consumed, a by the bottom quintile and b by the rest. Following a bank failure GDP is reduced by a fraction x, to (a + b)(1 – x). The lost GDP is x(a + b). The bottom quintile bears a fraction y of the loss and the other quintiles bear the fraction (1 – y). For the

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Ref #7877392 v1.0 purposes of the model, y is 0.6. This implies that when a shock hits, it mostly affects those on the lowest incomes (e.g. through unemployment, lack of savings etc.). We end up with:
Quintile Consumption before Consumption after
Bottom 𝑎 𝑎 − (𝑎 + 𝑏)𝑥𝑦 = 𝑎 [1 − (1 +
𝑏
𝑎
) 𝑥𝑦]
Others combined 𝑏 𝑏 − (𝑎 + 𝑏)𝑥(1 − 𝑦) = 𝑏 [1 − (1 + 𝑎 𝑏 ) 𝑥(1 − 𝑦)] If it is assumed that all quintiles initially have consumption equal to 1, then a = 1 and b = 4 and the post-shock consumption becomes [1 − 5𝑥𝑦] for the bottom quintile and [1 − 5 4 𝑥(1 − 𝑦)] for each of the other quintiles. The assumption that all quintiles have equal consumption is very unlikely to be correct and is arguably inconsistent with the idea that those on low incomes are most affected by a shock, but the assumption simplifies the later mathematics. For a given utility function U, utility for the bottom quintile would be given by 𝑈(1 − 5𝑥𝑦) and the utility for each of the other quintiles would be given by 𝑈 (1 − 5 4 𝑥(1 − 𝑦)). Weighting these together using population weights gives “society’s” utility:
𝑈𝑠𝑜𝑐𝑖𝑒𝑡𝑦 =
1
5
𝑈(1 − 5𝑥𝑦) +
4
5
𝑈 (1 −
5
4
𝑥(1 − 𝑦))
Society’s marginal utility would then be:
𝑈𝑠𝑜𝑐𝑖𝑒𝑡𝑦
′ =
1
5
𝑈
′
(1 − 5𝑥𝑦) +
4
5
𝑈
′
(1 −
5
4
𝑥(1 − 𝑦))
Next assume that the marginal utility function is given by:
𝑈
′ =
1
𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛6
This gives
𝑈𝑠𝑜𝑐𝑖𝑒𝑡𝑦
′ =
1
5(1 − 5𝑥𝑦)
6 +
4
5 (1 −
5
4
𝑥(1 − 𝑦))
6 which is the function used by the model.
The functional form of U’ is consistent with the isoelastic utility function with γ = 6, but also with other utility functions including, under certain conditions, the one specified on page 249 of Barro (2009), “Rare Disasters, Asset Prices, and Welfare Costs”, AER 99(1) (the article is referred to in the RIS). It is not clear why γ = 6 was chosen; the RIS argues for a “relatively high degree of risk aversion”, but no for a precise figure. A feature of this marginal utility function is that marginal utility is 1 when consumption is 1. Since consumption is initially assumed to be 1 for each quintile, society’s marginal utility in the absence of

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Ref #7877392 v1.0 a shock is 1. So with no shock, the utility adjustment is zero. The utility of an additional dollar of consumption increases as consumption falls below 1 (which happens when there is a shock). The model assumes that consumption losses – in the event of a bank failure – increase as the capital ratio increases. That is, the shocks that cause a loss of consumption at high capital ratios are larger, while shocks that cause a loss of consumption at low capital ratios are smaller (recall that society highly values avoiding the biggest shocks). At a detailed level, the model has two apparently conflicting formulas for consumption (GDP) losses. The first – for the % GDP loss due to single failure – appears in an earlier section. The second, used only for the utility adjustment, is the expression for x earlier in this section. The expression for x assumes losses (2-8% of GDP as far as I can discern) which are considerably smaller than those assumed elsewhere in the model. Using the losses of 10-20% of GDP that appear in other parts of the model would result in marginal utility adjustments of 50 times, rather than 1-2 times adjustment which is actually used. It is unclear how the parameters in the expression for x were decided, or why 0.01 is subtracted from CR. My guess is that the subtraction is a mistake – an off-by-one error in pasting from one spreadsheet to another. Possibly, the model could be recalibrated to use the losses assumed elsewhere in the model but still produce reasonable results (e.g. by reducing γ). The use of marginal utility seems to overstate the effect of risk aversion, particularly for larger losses. The marginal utility in the Harrison model should, strictly, apply only to the last dollar of the large loss, with progressively lower marginal utilities for earlier dollars of loss. Overall calculation of benefits In the overall calculation of benefits, it is assumed that a bank will fail when its capital ratio is 2%, rather than 0% (the Basel Committee assumed 4%). This means that the originally calculated benefits need to be shifted: a capital ratio of a implies the benefits originally calculated for a capital ratio of a-2. In the practical implementation of the model in 2011, this shift appears to have been done inconsistently. The GDP benefit is shifted by 3%, and the fiscal bail-out benefit is not shifted at all. This is assumed to unintended. Costs of higher capital (general description) Changing the capital ratio may shift the supply curve upwards in the domestic market for mortgages. This is because of corporate taxes and / or because the Modigliani Miller theorem does not hold. In response to the upward shift, the mortgage interest rate will change, affecting GDP, transfers to foreigners, and taxes. How much it changes depends on the response of bank shareholders. If bank shareholders recognise that a higher capital ratio has reduced the riskiness of their investment, the change in the mortgage rate will be
modest. If, by contrast, they maintain that the investment is just as risky, the mortgage rate will change by a greater amount.

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Higher capital ratio reduces riskiness of equity cashflows (Modigliani Miller case) If the Modigliani Miller theorem holds, then end-of-period cashflows to shareholders and debtholders respectively, valued at the beginning of the period and after all taxes, are given by:
𝐶𝐹𝐸 = (
𝐴0𝑟0
𝑟𝑑𝑢
−
𝐷0𝑟𝑑
𝑟𝑑𝑑
) (1 − 𝑡𝑐
)(1 − 𝑡𝑒
) and
𝐶𝐹𝐷 =
𝐷0𝑟𝑑
𝑟𝑑𝑑
(1 − 𝑡𝑑
) with the meaning of the variables in the following table:
Variable Meaning
𝐴0 Mortgage balances
𝑟0 Mortgage interest rate
𝑟𝑑𝑢 Discount rate for unlevered bank equity 𝐷0 Bank liabilities (balance) 𝑟𝑑 Interest rate on bank’s liabilities 𝑟𝑑𝑑 Discount rate for bank liabilities 𝑡𝑐 Effective corporate tax rate (New Zealand) 𝑡𝑒 Effective tax rate on dividends and capital gains on shares (NZ or overseas) 𝑡𝑑 Effective tax rate on interest payment and capital gains on debt (NZ or overseas) Following a change in the capital ratio, 𝐴0 and 𝐷0 change to 𝐴1 and 𝐷1 and 𝑟0 changes to some value 𝑟1 which is to be determined. For tractability it is assumed that 𝑟𝑑, 𝐹, and all the discount rates are constant. The changes in cashflows to shareholders and debtholders are then given by:
∆𝐶𝐹𝐸 = (
𝐴1𝑟1 − 𝐴0𝑟0
𝑟𝑑𝑢
−
(𝐷1 − 𝐷0
)𝑟𝑑
𝑟𝑑𝑑 ) (1 − 𝑡𝑐
)(1 − 𝑡𝑒
) +
(𝐷1 − 𝐷0
)𝑟𝑑
𝑟𝑑𝑑
(1 − 𝑡𝑑
) and
∆𝐶𝐹𝐷 =
(𝐷1 − 𝐷0
)𝑟𝑑
𝑟𝑑𝑑
(1 − 𝑡𝑑
) −
(𝐷1 − 𝐷0
)𝑟𝑑
𝑟𝑑𝑑
(1 − 𝑡𝑑
) = 0
In the case where debt decreases, shareholders are effectively purchasing a stream of income from debtholders. The second term in the shareholder equation is what is paid to debtholders for the stream of income – debtholders will choose to hold on to their securities if the amount paid is any less. The first term in the shareholder equation incorporates the purchased stream of income (after the subtraction sign). This new stream of shareholder income is less risky – it is plausibly assumed – than the stream of gross income, and so the shareholder is willing to accept a lower rate of return on it. This is the Modigliani Miller theorem in action. Because 𝐴1𝑟1 − 𝐴0𝑟0 = 𝑟1 (𝐴1 − 𝐴0 ) + 𝐴0 (𝑟1 − 𝑟0 ) = 𝑟1∆𝐴 + 𝐴0∆𝑟

10
Ref #7877392 v1.0 and
𝐷1 − 𝐷0 =
𝐷1
𝐴1
(𝐴1 − 𝐴0
) + 𝐴0 (
𝐷1
𝐴1
−
𝐷0
𝐴0
) =
𝐷1
𝐴1
∆𝐴 + 𝐴0∆ (
𝐷
𝐴
) ,
∆𝐶𝐹𝐸 can be decomposed into changes due to the change in the capital ratio (with assets held constant) and changes due to movements in bank assets as a result of the interaction of supply and demand:8 ∆𝐶𝐹𝐸 = ∆𝐴 ( 𝑟1 (1 − 𝑡𝑐 )(1 − 𝑡𝑒 ) 𝑟𝑑𝑢 − 𝐷1 𝐴1 𝑟𝑑 [(1 − 𝑡𝑐 )(1 − 𝑡𝑒 ) − (1 − 𝑡𝑑 )] 𝑟𝑑𝑑 )

• 𝐴0 (
  ∆𝑟(1 − 𝑡𝑐
  )(1 − 𝑡𝑒
  )
  𝑟𝑑𝑢
  −
  ∆ (
  𝐷
  𝐴
  ) 𝑟𝑑
  [(1 − 𝑡𝑐
  )(1 − 𝑡𝑒
  ) − (1 − 𝑡𝑑
  )]
  𝑟𝑑𝑑 )
  Suppose that, with assets held constant, the shareholder wishes ∆𝐶𝐹𝐸 to be zero and will adjust the mortgage interest rate to ensure this is the case. This amounts to recovering the additional tax costs of a higher capital ratio from mortgage borrowers. Then the change in the mortgage rate must be:
  ∆𝑟 = ∆ (
  𝐷
  𝐴
  )
  𝑟𝑑
  𝑟𝑑𝑑
  [𝑟𝑑𝑢 −
  1 − 𝑡𝑑
  (1 − 𝑡𝑐
  )(1 − 𝑡𝑒
  )
  ∙ 𝑟𝑑𝑢]
  In reality, assets are likely to change. Depending on how they change – this depends on the precise specification of the supply and demand curves – some other value of ∆𝑟 might be the equilibrium outcome (in the case of a higher capital ratio, the expression for ∆𝑟 above is an upper bound). Higher capital does not reduce riskiness of equity cashflows In contrast to the previous case, suppose shareholders attribute the same riskiness to their cashflows no matter how indebted the bank. End-of-period cashflows, valued at the beginning of the period after all taxes, are then:
  𝐶𝐹𝐸
  ∗ = (
  𝐴0𝑟0 − 𝐷0𝑟𝑑
  𝑟𝑑𝑒
  ) (1 − 𝑡𝑐
  )(1 − 𝑡𝑒
  ) and
  𝐶𝐹𝐷
  ∗ =
  𝐷0𝑟𝑑
  𝑟𝑑𝑑
  (1 − 𝑡𝑑
  ) where 𝑟𝑑𝑒is the (assumed constant) discount rate for equity, whether levered or unlevered. Changes in cashflows following a change in the capital ratio are then:
  ∆𝐶𝐹𝐸
  ∗ = (
  𝑟1
  ∗ − 𝐴0𝑟0 − (𝐷1
  ∗ − 𝐷0
  )𝑟𝑑
  𝑟𝑑𝑒 ) (1 − 𝑡𝑐
  )(1 − 𝑡𝑒
  ) +
  (𝐷1
  ∗ − 𝐷0
  )𝑟𝑑
  𝑟𝑑𝑑
  (1 − 𝑡𝑑
  ) and

8 ∆𝐴 is normally correlated with ∆𝑟, so the decomposition is not pure. It is still useful, particularly in the cases where the demand curve is vertical (∆𝐴 = 0) or the supply curve is horizontal (∆𝑟 does not depend on ∆𝐴).

11
Ref #7877392 v1.0
∆𝐶𝐹𝐷
∗ = 0
As in the previous case, the second term on the right hand side of the first equation is the amount paid to debtholders in the case of a higher capital ratio to acquire their stream of cashflows.9

Decomposing the changes10:
∆𝐶𝐹𝐸
∗ = ∆𝐴
∗ (
𝑟1
∗
(1 − 𝑡𝑐
)(1 − 𝑡𝑒
)
𝑟𝑑𝑒
−
𝐷1
𝐴1
𝑟𝑑 [
1 − 𝑡𝑑
𝑟𝑑𝑑
−
(1 − 𝑡𝑐
)(1 − 𝑡𝑒
)
𝑟𝑑𝑒 ])

• 𝐴0 (
  ∆𝑟
  ∗
  (1 − 𝑡𝑐
  )(1 − 𝑡𝑒
  )
  𝑟𝑑𝑒
  − ∆ (
  𝐷
  𝐴
  ) 𝑟𝑑 [
  1 − 𝑡𝑑
  𝑟𝑑𝑑
  −
  (1 − 𝑡𝑐
  )(1 − 𝑡𝑒
  )
  𝑟𝑑𝑒 ])
  Holding assets constant and finding the change in mortgage rates which leaves discounted cashflows unchanged gives:
  ∆𝑟
  ∗ = ∆ (
  𝐷
  𝐴
  )
  𝑟𝑑
  𝑟𝑑𝑑
  [𝑟𝑑𝑑 −
  1 − 𝑡𝑑
  (1 − 𝑡𝑐
  )(1 − 𝑡𝑒
  )
  ∙ 𝑟𝑑𝑒]
  As in the case where the Modigliani Miller theorem holds, ∆𝑟 ∗ is an upper bound. Depending on the specification of the supply and demand curves, ∆𝑟 ∗ could be lower. Changes in income and transfers A rise in the mortgage interest rate is expected to reduce GDP growth.11 This is modelled using the Basel Committee’s approach, which is discussed later in this document. A rise in the interest rate and a changing capital ratio might also result in a transfer of money from mortgage borrowers to offshore bank shareholders and debtholders, which is a cost to New Zealand. The following tables show how the changes in cashflows are directed to various stakeholders.12 It is assumed that all shareholders are foreign and that all debtholders who sell back their debt (or take on new debt) are also foreign. Corporate taxes are imposed on the bank in New Zealand and apply to income net of interest payments. Dividend and interest taxation are imposed on foreign stakeholders only by foreign governments.13

9 This amount is determined by the debtholder’s valuation of the cashflows. The debtholder’s valuation is the same in the Modigliani-Miller case and in this one, because the discount rate for debt is assumed constant. This ignores the likelihood that the discount rate for debt might decrease because debt claims on a more highly capitalised bank are less risky. 10 Assuming the bank is always at the prescribed capital ratio, then 𝐷1 𝐴1 = 𝐷1 ∗ 𝐴1 ∗ . 11 This is the assumption which has been made in the Basel modelling and earlier RBNZ work. It is inconsistent with a fixed value of bank assets, which has also been assumed in the past when working out the increase in the interest rate, unless the demand curve is vertical. In principle, the more general specification in this document allows for a decrease in GDP and a consistent increase in interest rates. In practice, it might be difficult to know the specifications for the supply and (especially) demand curves. 12 The initial retirement of existing debt is a payment to debtholders at the market price for the debt. The market price is determined by the discounted value of future cashflows. (In the case of a falling capital ratio, the payment is from debtholders to the bank, again at the market price for the stream of future interest cash flows). This initial payment is shown as a “one-time exchange” in the table). The initial exchange is assumed to have no tax consequences. 13 At the cost of some complexity, it should be straightforward to repeat the analysis with alternative assumptions.

12
Ref #7877392 v1.0
Stakeholder Change in cash flows as the result of a change in the capital ratio Shareholder (perpetually) [𝐴0 (∆𝑟 − 𝑟𝑑∆ ( 𝐷 𝐴 )) + ∆𝐴 (𝑟1 − 𝑟𝑑 𝐷1 𝐴1 )] (1 − 𝑡𝑐 )(1 − 𝑡𝑒 ) Shareholder (one-time exchange with debtholder) 𝑟𝑑 𝑟𝑑𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴] (1 − 𝑡𝑑 ) Debtholder (perpetually) 𝑟𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴] (1 − 𝑡𝑑 ) Debtholder (one-time exchange) − 𝑟𝑑 𝑟𝑑𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴] (1 − 𝑡𝑑 ) NZ fisc (perpetually) [𝐴0 (∆𝑟 − 𝑟𝑑∆ ( 𝐷 𝐴 )) + ∆𝐴 (𝑟1 − 𝑟𝑑 𝐷1 𝐴1 )]𝑡𝑐 Foreign fisc (perpetually) [𝐴0 (∆𝑟 − 𝑟𝑑∆ ( 𝐷 𝐴 )) + ∆𝐴 (𝑟1 − 𝑟𝑑 𝐷1 𝐴1 )] (1 − 𝑡𝑐 )𝑡𝑒 + 𝑟𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴]𝑡𝑑 Total (perpetually) [𝐴0 (∆𝑟 − 𝑟𝑑∆ ( 𝐷 𝐴 )) + ∆𝐴 (𝑟1 − 𝑟𝑑 𝐷1 𝐴1 )] + 𝑟𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴] = 𝐴0∆𝑟 + ∆𝐴𝑟1 = 𝐴1𝑟1 − 𝐴0𝑟0 Stakeholder Change in discounted cash flows as the result of a change in the capital ratio Shareholder (perpetually) [𝐴0 ( ∆𝑟 𝑟𝑑𝑢 − 𝑟𝑑 𝑟𝑑𝑑 ∆ ( 𝐷 𝐴 )) + ∆𝐴 ( 𝑟0 𝑟𝑑𝑢 + ∆𝑟 𝑟𝑑𝑢 − 𝑟𝑑 𝑟𝑑𝑑 𝐷1 𝐴1 )] (1 − 𝑡𝑐 )(1 − 𝑡𝑒 ) Shareholder (one-time exchange with debtholder) 𝑟𝑑 𝑟𝑑𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴] (1 − 𝑡𝑑 ) Debtholder (perpetually) 𝑟𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴] (1 − 𝑡𝑑 ) Debtholder (one-time exchange) − 𝑟𝑑 𝑟𝑑𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴] (1 − 𝑡𝑑 ) NZ fisc (perpetually) [𝐴0 ( ∆𝑟 𝑟𝑑𝑢 − 𝑟𝑑 𝑟𝑑𝑑 ∆ ( 𝐷 𝐴 )) + ∆𝐴 ( 𝑟0 𝑟𝑑𝑢 + ∆𝑟 𝑟𝑑𝑢 − 𝑟𝑑 𝑟𝑑𝑑 𝐷1 𝐴1 )]𝑡𝑐 Foreign fisc (perpetually) [𝐴0 ( ∆𝑟 𝑟𝑑𝑢 − 𝑟𝑑 𝑟𝑑𝑑 ∆ ( 𝐷 𝐴 )) + ∆𝐴 ( 𝑟0 𝑟𝑑𝑢 + ∆𝑟 𝑟𝑑𝑢 − 𝑟𝑑 𝑟𝑑𝑑 𝐷1 𝐴1 )] (1 − 𝑡𝑐 )𝑡𝑒 + 𝑟𝑑 𝑟𝑑𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴]𝑡𝑑 Total (perpetually) [𝐴0 ( ∆𝑟 𝑟𝑑𝑢 − 𝑟𝑑 𝑟𝑑𝑑 ∆ ( 𝐷 𝐴 )) + ∆𝐴 ( 𝑟0 𝑟𝑑𝑢 + ∆𝑟 𝑟𝑑𝑢 − 𝑟𝑑 𝑟𝑑𝑑 𝐷1 𝐴1 )] + 𝑟𝑑 𝑟𝑑𝑑 [𝐴0∆ ( 𝐷 𝐴 ) + 𝐷1 𝐴1 ∆𝐴] = 𝐴0∆𝑟 + ∆𝐴𝑟1 𝑟𝑑𝑢 = 𝐴1𝑟1 − 𝐴0𝑟0 𝑟𝑑𝑢

13
Ref #7877392 v1.0
The first table shows undiscounted cash flows. The second table shows discounted cash flows. The discount rates used are “NZ Inc”’s rates. These are assumed to be always the same as the discount rates used by investors in the case where Modigliani Miller holds. The change in discounted cashflows that go offshore, given the assumptions above, is the sum of the shaded rows in the second table:
𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟𝑠 = [𝐴0 (
∆𝑟
𝑟𝑑𝑢
−
𝑟𝑑
𝑟𝑑𝑑
∆ (
𝐷
𝐴
)) + ∆𝐴 (
𝑟0
𝑟𝑑𝑢
+
∆𝑟
𝑟𝑑𝑢
−
𝑟𝑑
𝑟𝑑𝑑
𝐷1
𝐴1
)] (1 − 𝑡𝑐
)
+
𝑟𝑑
𝑟𝑑𝑑
[𝐴0∆ (
𝐷
𝐴
) +
𝐷1
𝐴1
∆𝐴]
Assuming bank shareholders recognise changes in risk (the Modigliani Miller theorem holds), substitute in ∆𝑟 = ∆ ( 𝐷 𝐴 ) 𝑟𝑑 [ 𝑟𝑑𝑢 𝑟𝑑𝑑 − 1 − 𝑡𝑑 (1 − 𝑡𝑐 )(1 − 𝑡𝑒 ) ∙ 𝑟𝑑𝑢 𝑟𝑑𝑑 ] and rearrange to obtain 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟𝑠 = 𝐴0∆ ( 𝐷 𝐴 ) 𝑟𝑑 𝑟𝑑𝑑 (1 − 1 − 𝑡𝑑 1 − 𝑡𝑒 )

• ∆𝐴 (
  𝑟0
  𝑟𝑑𝑢
  (1 − 𝑡𝑐
  ) + ∆ (
  𝐷
  𝐴
  )
  𝑟𝑑
  𝑟𝑑𝑑
  [1 − 𝑡𝑐 −
  1 − 𝑡𝑑
  1 − 𝑡𝑒
  ] +
  𝑟𝑑
  𝑟𝑑𝑑
  𝐷1
  𝐴1
  𝑡𝑐)
  The first term has two components: the decrease in pre-tax interest payments to non-resident debtholders (in the case of a higher capital ratio), and the increase in post-New Zealand-tax dividends to non-resident shareholders. The non-resident shareholder wants the purchased stream of debtholder income to generate the same after-tax return as it would if it was held as debt. This means that the dividend is grossed up to reverse the effect of New Zealand company taxes (which do not apply to debt). A further grossing up (or, quite plausibly, down) is required to reflect any difference between foreign dividend and interest taxation. If bank shareholders do not recognise changes in risk (the Modigliani Miller theorem does not hold even in part), then instead substitute in ∆𝑟 ∗ and obtain 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟𝑠 = 𝐴0 (∆ ( 𝐷 𝐴 ) 𝑟𝑑 𝑟𝑑𝑑 [ 𝑟𝑑𝑑 𝑟𝑑𝑢 (1 − 𝑡𝑐 ) − 1 − 𝑡𝑑 1 − 𝑡𝑒 ∙ 𝑟𝑑𝑒 𝑟𝑑𝑢
• 𝑡𝑐
  ])
• ∆𝐴 (
  𝑟0
  𝑟𝑑𝑢
  (1 − 𝑡𝑐
  ) + ∆ (
  𝐷
  𝐴
  )
  𝑟𝑑
  𝑟𝑑𝑑
  [
  𝑟𝑑𝑑
  𝑟𝑑𝑢
  (1 − 𝑡𝑐
  ) −
  1 − 𝑡𝑑
  1 − 𝑡𝑒
  ∙
  𝑟𝑑𝑒
  𝑟𝑑𝑢
  ] +
  𝑟𝑑
  𝑟𝑑𝑑
  𝐷1
  𝐴1
  𝑡𝑐)

14
Ref #7877392 v1.0
By way of illustrative example, assign some values to the parameters:
Parameter Value Notes
𝑨𝟎 $1.000 Arbitrary
𝑨𝟏 $1.000 For the MMI case (noting ∆𝑟 is negligible) or 𝑨𝟏 $0.999 For the case without MM1 (arbitrarily less than 𝐴0) ∆𝑨 -$0.001 Calculated directly 𝑫𝟎 𝑨𝟎
0.960 Rough estimate of current ratio
𝑫𝟏
𝑨𝟏
0.955 Assumed lower ratio
∆ (
𝑫
𝑨
)
-0.005 Calculated directly
𝒓𝟎 0.05500 Rough estimate of current interest rate on loan book 𝒓𝒅 0.03000 Rough estimate of current interest rate on bank debt 𝒓𝒅𝒖 0.02975 Rough estimate of current required return after all taxes 𝒓𝒅𝒅 0.02310 Rough estimate of current required return after all taxes 𝒓𝒅𝒆 0.18935 Calculated from numbers above using Modigliani Miller II 𝒕𝒄 0.30 Approximate corporate tax rate 𝒕𝒅 0.30 Guessed foreign tax rate 𝒕𝒆 0.15 Guessed foreign tax rate The results are:
With MMI Without MMI
∆𝒓 0.341bp 13.0bp
Transfers (per $ of initial assets)
Assets held constant -0.001 0.029
Due to asset change 0.000 -0.002
Total -0.001 0.027
Changes in income and transfers in the Harrison model The Harrison model incorporates three effects of a higher capital ratio on income and transfers:

1. A reduction in GDP because of an increase in mortgage lending interest rates
2. An increase in transfers abroad because of a switch from debt to (higher-cost) equity
3. An increase in tax paid in New Zealand, owing to reduced interest deductions and increased
   company profits.
   GDP effect
   The GDP effect is taken directly from earlier work by the Basel Committee on Banking Supervision, which found that a one percentage point increase in the capital ratio would increase mortgage lending interest rates by 13 basis points and decrease GDP by 0.09%. In calculating the interest rate effect, the Basel Committee assumed that total bank assets would be unchanged. The Committee then relied on a suite of models to assess the likely effect on GDP. 𝐿𝑜𝑠𝑡 𝐺𝐷𝑃 𝑝𝑒𝑟 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑝𝑜𝑖𝑛𝑡 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑟𝑎𝑡𝑖𝑜 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐺𝐷𝑃 × 0.0009 = $200 𝑏𝑖𝑙𝑙𝑖𝑜𝑛 × 0.0009 = $180 𝑚𝑖𝑙𝑙𝑖𝑜𝑛

15
Ref #7877392 v1.0
The approach appears to be internally inconsistent because the decrease in GDP would be driven by lower investment and – logically – reduced bank assets. However, it has the advantage that nothing needs to be known about the precise specification of supply and demand in the mortgage market. The Basel-determined effect is an annual effect. The Harrison model discounts the infinite stream of annual effects (assumed to be the same at all initial capital ratios) using a compounding 3% rate to determine the net present value of GDP losses. The Harrison model also reduces the discounted result by 85%, because the Basel Committee estimate assumes the Modigliani Miller theorem does not hold at all, which results in over-stated estimates. The Harrison model doubles the Basel-determined effect, apparently because of an erroneous assumption that the effect of an increase in the leverage ratio – and not the capital ratio – was wanted. Transfers effect The transfers effect is calculated by assuming that when a dollar of equity is substituted for a dollar of debt, the interest rate on debt is saved and the rate of return on equity is instead paid. The margin between the interest rate on debt and the return on equity (a risk premium) is assumed to be 10 percentage points, regardless of the level of capital. The total amount of equity substituted for debt should be calculated as 1% of risk-weighted assets. Risk weighted assets are calculated indirectly, using tier 1 capital and the current capital ratio. 𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟𝑠 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑏𝑡 × 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑒𝑞𝑢𝑖𝑡𝑦 − 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑏𝑡 × 𝑑𝑒𝑏𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑏𝑡 × (𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑒𝑞𝑢𝑖𝑡𝑦 − 𝑑𝑒𝑏𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒) = .01 × 𝑟𝑖𝑠𝑘 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑎𝑠𝑠𝑒𝑡𝑠 × (𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑒𝑞𝑢𝑖𝑡𝑦 − 𝑑𝑒𝑏𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒) = .01 × 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑡𝑖𝑒𝑟 𝑜𝑛𝑒 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑟𝑎𝑡𝑖𝑜 × .10 But as with the GDP effect, the Harrison model apparently doubles the transfers effect by determining the change using the current leverage ratio rather than the current capital ratio. The transfer effect is annual. The endless stream of transfers is discounted by a compounding 3% rate to determine a total cost. The Harrison model also reduces the discounted result by 85%, because assumption that the risk premium does not vary with the capital ratio is unrealistic and results in overstated estimates. Tax effect It is unclear exactly how the tax effect was calculated, but we can roughly reproduce the numbers. That basic rationale is that as debt decreases there are fewer tax deductions allowed to the bank, and New Zealand tax increases. New Zealand does not capture all the benefit because foreign shareholders seek to recoup some of the additional tax costs they incur by raising mortgage interest rates for bank customers (an earlier presentation of the model indicated that half of the
benefits are lost in this way).

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𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑎𝑥 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑏𝑡 × 𝑑𝑒𝑏𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 × 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑡𝑎𝑥 𝑟𝑎𝑡𝑒 × 1 2 = 𝑟𝑖𝑠𝑘 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑎𝑠𝑠𝑒𝑡𝑠 × 0.01 × 0.05 × 0.28 × 1 2 We have assumed an interest rate of 5% and a company tax rate of 28%. The effect is an annual one. Annual effects are summed and discounted to establish the net present value of tax increases. The net real discount rate on debt is assumed to be 5%, “consistent with a risky debt investment”. We arrived at an estimate of $266 million, which is close to the $250 million the Harrison model should produce. As in the cases of the GDP and transfer effects, the Harrison model appears to have incorrectly assumed that the effect should be calculated for a one percentage point change in the leverage ratio, rather than the capital ratio (it is a little more difficult to tell if this is so for the tax effect because there is less documentation, but we are assuming it is and that it erroneously doubles the size of the tax effect). Comparing the Harrison approach and the new approach in this document The combined effect of the tax and transfer effects after correcting for the doubling which we think is unintended is a net cost of roughly 0.2% of total bank assets. Using the illustrative numbers from the approach presented earlier produces a cost of about 0.4% of total bank assets (this uses the numbers from the case where Modigliani Miller does not hold, but then reduces them by 85% as Harrison has done).
Appendix
SP is a cumulative distribution function
From earlier results:
𝑆𝑃 = Φ [√
1−𝑅
𝑅
(Φ−1
[𝐶𝑅 ×
1 − 1.5𝑏
1 + 𝑏(𝑀 − 2.5)
×
𝑅𝑊
𝐿𝐺𝐷 + 𝑃𝐷] − √
1
1−𝑅
Φ−1
(𝑃𝐷))]
As it stands, this is not defined everywhere, because the domain of Φ−1 is only [0,1]. So define SP to be zero when 𝐶𝑅 × 1 − 1.5𝑏 1 + 𝑏(𝑀 − 2.5) × 𝑅𝑊 𝐿𝐺𝐷 + 𝑃𝐷 ≤ 0 or, equivalently when 𝐶𝑅 ≤ −𝑃𝐷 × 1 + 𝑏(𝑀 − 2.5) 1 − 1.5𝑏 × 𝐿𝐺𝐷 𝑅𝑊 = 𝑙𝑜𝑤 That is, the best the bank can possibly do is to suffer no losses at all, in which case it can convert its provisions into equity. If it has negative equity which exceeds provisions in magntitude, the bank fails with certainty. (This explanation is slightly hand-wavy; I am ignoring the second factor, the term involving b).

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Also, define SP to be one when
𝐶𝑅 ×
1 − 1.5𝑏
1 + 𝑏(𝑀 − 2.5)
×
𝑅𝑊
𝐿𝐺𝐷 + 𝑃𝐷 ≥ 1 or
𝐶𝑅 ≥ (1 − 𝑃𝐷) ×
1 + 𝑏(𝑀 − 2.5)
1 − 1.5𝑏
×
𝐿𝐺𝐷
𝑅𝑊 = ℎ𝑖𝑔ℎ
That is, if the bank holds enough equity, together with provisions, to cover the loss of its entire portfolio, at the given LGD, then it is certain to survive. (Again, hand-wavy). Then lim 𝐶𝑅→𝑙𝑜𝑤+ 𝑆𝑃(𝐶𝑅) = 𝑆𝑃(𝑙𝑜𝑤) = lim 𝐶𝑅→𝑙𝑜𝑤− 𝑆𝑃(𝐶𝑅) = lim 𝐶𝑅→−∞ 𝑆𝑃(𝐶𝑅) = 0 (where the left hand side follows, at least intuitively, from the properties of the cumulative normal and inverse cumulative normal distributions) and lim 𝐶𝑅→ℎ𝑖𝑔ℎ− 𝑆𝑃(𝐶𝑅) = 𝑆𝑃(ℎ𝑖𝑔ℎ) = lim 𝐶𝑅→ℎ𝑖𝑔ℎ+ 𝑆𝑃(𝐶𝑅) = lim 𝐶𝑅→+∞ 𝑆𝑃(𝐶𝑅) = 1 From this, and the continuity of the cumulative normal and inverse cumulative normal functions over their respective domains, SP is continuous in CR. (Actually, continuity depends on the functional form of R as well. R is assumed to be continuous or, if it is not, the lack of continuity is assumed to have negligible effects). By the chain rule and the inverse function theorem:
𝑑𝑆𝑃
𝑑𝐶𝑅 =
1 − 1.5𝑏
1 + 𝑏(𝑀 − 2.5)
×
𝑅𝑊
𝐿𝐺𝐷
× √
1−𝑅
𝑅
×
𝜙 [√
1−𝑅
𝑅
(Φ−1
[𝐶𝑅 ×
1 − 1.5𝑏
1 + 𝑏(𝑀 − 2.5)
×
𝑅𝑊
𝐿𝐺𝐷 + 𝑃𝐷] − √
1
1−𝑅 Φ−1
(𝑃𝐷))]
𝜙 [Φ−1 [𝐶𝑅 ×
1 − 1.5𝑏
1 + 𝑏(𝑀 − 2.5)
×
𝑅𝑊
𝐿𝐺𝐷 + 𝑃𝐷]]

0
where 𝜙 is the standard normal probability distribution function and b is assumed to be less than 2 3 . From the properties above, SP(CR) is a cumulative distribution function. 𝑑𝑆𝑃 𝑑𝐶𝑅 = 𝑠𝑝 (lower case) is the corresponding probability density function (it is 0 where SP(CR) is 0 or 1). A better discrete approximation to the integral This is one possibility for approximating the integral used to calculate bailout costs. Recall the expected bailout costs at a (discrete) capital ratio of CR:

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𝐵𝐶 = 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑏𝑎𝑖𝑙𝑜𝑢𝑡 𝑐𝑜𝑠𝑡𝑠 𝑓𝑜𝑟 𝑎 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝐶𝑅 = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × (0.00329 × [0.06 + 𝑅𝑊 × (21 − 𝐶𝑅)]

• ∑ {[0.0325 + 𝑅𝑊 × (𝑥 − 𝐶𝑅)][∆𝑥
  𝑥−1
  [1 − 𝑆𝑃]]}
  20
  𝑥=𝐶𝑅
  )
  That formulation assumes the capital ratios change in whole percentage point increments. But suppose instead that we allow any change ε:
  𝐵𝐶 = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠
  × (0.00329 × [0.06 + 𝑅𝑊 × (21 − 𝐶𝑅)]
• ∑ {[0.0325 + 𝑅𝑊 × (𝑥 − 𝐶𝑅)][∆𝑥
  𝑥−𝜀
  [1 − 𝑆𝑃]]}
  20
  𝑥=𝐶𝑅
  )
  It is implicit that the summation is over 𝑥, 𝑥 + 𝜀, 𝑥 + 2𝜀, … ,20. In the limit as 𝜀 → 0 we obtain bailout costs for a continuous capital ratio CR:
  𝐵𝐶 = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠
  × (0.00329 × [0.06 + 𝑅𝑊 × (21 − 𝐶𝑅)]
• ∫ [0.0325 + 𝑅𝑊 × (𝑥 − 𝐶𝑅)]𝑠𝑝(𝑥)
  20
  𝑥=𝐶𝑅
  𝑑𝑥)
  We would like to see how the bailout costs change as CR changes from some ratio a to another ratio b. We will first work how the costs change for an infinitesimal change in CR. Then we will sum up the effects of the infinitesimal changes as CR changes from a to b. To work out the effect of an infinitesimal change in CR, we calculate the derivative, using Liebniz’s rule for differentiating under the integral:
  𝑑𝐵𝐶
  𝑑𝐶𝑅 = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × (−0.00329 × 𝑅𝑊 − 𝑅𝑊 ∫ 𝑠𝑝(𝑥)𝑑𝑥 20 𝑥=𝐶𝑅 − 0.0325𝑠𝑝(𝐶𝑅)) = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × (−0.00329 × 𝑅𝑊 − 𝑅𝑊[𝑆𝑃(20) − 𝑆𝑃(𝐶𝑅)] − 0.0325𝑠𝑝(𝐶𝑅))

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Integrating this expression gives the change in the bailout costs when moving from a CR of a to a CR of b:
𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝑚𝑜𝑣𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝑎 𝑡𝑜 𝑏 = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × ∫ −0.00329 × 𝑅𝑊 − 𝑅𝑊[𝑆𝑃(20) − 𝑆𝑃(𝐶𝑅)] − 0.0325𝑠𝑝(𝐶𝑅)𝑑𝐶𝑅 𝑏 𝐶𝑅=𝑎 = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × (−0.00329 × 𝑅𝑊 (𝑏 − 𝑎) − 0.0325[𝑆𝑃(𝑏) − 𝑆𝑃(𝑎)] − 𝑅𝑊 ∫ 𝑆𝑃(20) − 𝑆𝑃(𝐶𝑅)𝑑𝐶𝑅 𝑏 𝐶𝑅=𝑎 ) The integral inside the parentheses is an area on the graph of 𝐶𝑅 vs 𝑆𝑃(𝐶𝑅) (the cumulative distribution function). Specifically, it is the area below 𝑆𝑃(20), above the cumulative distribution function, to the left of b and to the right of a. For reasonably small values of b-a, a trapezoidal approximation should be adequate and will certainly be better than a rectangular one. The approximation is:
∫ 𝑆𝑃(20) − 𝑆𝑃(𝐶𝑅)𝑑𝐶𝑅
𝑏
𝐶𝑅=𝑎
≈
1
2
[𝑆𝑃(20) − 𝑆𝑃(𝑎) + 𝑆𝑃(20) − 𝑆𝑃(𝑏)](𝑏 − 𝑎) = [𝑆𝑃(20) − 1 2 (𝑆𝑃(𝑎) + 𝑆𝑃(𝑏))] (𝑏 − 𝑎)

20
Ref #7877392 v1.0 which gives:
𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝑚𝑜𝑣𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝑎 𝑡𝑜 𝑏 = = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × (−0.00329 × 𝑅𝑊 (𝑏 − 𝑎) − 0.0325[𝑆𝑃(𝑏) − 𝑆𝑃(𝑎)] − 𝑅𝑊 [𝑆𝑃(20) − 1 2 (𝑆𝑃(𝑎) + 𝑆𝑃(𝑏))] (𝑏 − 𝑎)) This form is inconvenient for finding the effect of capital ratios in excess of 20%. If we had taken the summation in the original (discrete) equation to infinity and dropped the terminal term, we would have instead ended up with:
𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝑚𝑜𝑣𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝑎 𝑡𝑜 𝑏 = = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × (−0.0325[𝑆𝑃(𝑏) − 𝑆𝑃(𝑎)] − 𝑅𝑊 [𝑆𝑃(∞) − 1 2 (𝑆𝑃(𝑎) + 𝑆𝑃(𝑏))] (𝑏 − 𝑎)) SP is not defined for an infinite capital ratio (or, indeed, any fairly high capital ratio). But if we set SP to 1 when the capital ratio is too high to allow for it to be calculated, we get, finally:
𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝑚𝑜𝑣𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝑎 𝑡𝑜 𝑏 = = 𝑏𝑎𝑛𝑘 𝑎𝑠𝑠𝑒𝑡𝑠 × (−0.0325[𝑆𝑃(𝑏) − 𝑆𝑃(𝑎)] − 𝑅𝑊 [1 − 1 2 (𝑆𝑃(𝑎) + 𝑆𝑃(𝑏))] (𝑏 − 𝑎)) Note that this is a cost, so a negative result is a negative cost (a benefit). If b is higher than a (moving from a low to a high capital ratio) then the cost is negative (there is a benefit).