CRD IV and CRR
10. ROBUSTNESS OF THE RESULTS
Figure 28: Spreads for Different Numbers of Simulations
Figure 29: Spread Ratio for Different Numbers of Simulations
Ideally, we would run 100,000 paths for each estimate. However, when using the Monte Carlo simulation, the computational burden is an important factor to con-sider. This is also a problem in our case. The computation time is not linear in the number of simulated paths: by doubling the number of paths, the computation time
50 55 60 65 70 75 80 85 90 95 100
1,000 10,000 25,000 50,000 100,000
Spread (bps)
No. Simulations
1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30
1,000 10,000 25,000 50,000 100,000
T2/T3Ratio
No. of Simulations
number of paths, we find confidence in our choice of 25,000 simulations per esti-mate.
No Jumps in the Process
A main feature of our model is that it incorporates the possibility of sudden jumps in the asset value of the bank. As discussed in the results section, this adds a realistic feature to our model, and allows us to obtain positive credit spreads for even low levels of volatility. To test the robustness of our results, however, we now calculate spreads without incorporating this possibility.
Thus, under the risk-neutral probability measure ℚ, we assume that the book value of the bank’s assets, Vt, follows a geometric Brownian motion, such that the rate of return at date t can be described by
𝑑𝑉𝑡 =𝑉𝑡 𝑟 − 𝛿(𝑡) 𝑑𝑡+𝜎𝑑𝑊𝑡ℚ
instead of equation (1). The figure below shows the obtained credit spreads on T3, as well as the T2/T3 ratio from our original jump-diffusion model and the reduced model without jumps. As previously discussed, we find that credit spreads are zero when the volatility of assets is less than 1% if we leave out the jump part, which means that the bonds are riskless. Of course, this is not the case in practice.
Figure 30:Volatility of Assets (With Jump and Without Jump)
0 100 200 300 400 500 600
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
T3 (jump) T3 (no jump)
Spread (bps)
Sigma (%)
Figure 31: Spread Ratio - Volatility of Assets (With Jump and Without Jump)
When the volatility of assets increases, we see that both the spreads and the spread ratios converge towards each other. This suggests that for higher levels of volatility, credit spreads are mainly driven by the asset volatility and not the possibility of jumps, whereas the opposite applies when volatility is low.
In our benchmark scenario, where asset volatility is set to 1.5%, spreads decrease considerably from 82bps to 18bps, and the T2/T3 ratio increases from 1.8x to 2.0x when we leave out the jump part. As the volatility of banks’ assets is relatively low in practice, these results suggest that it is indeed necessary to incorporate jumps into a pricing model of bank capital to obtain reasonable spreads at fair levels of volatility.
Jump Size and Intensity
As we have just seen, the possibility of sudden jumps in the asset value does affect credit spreads. However, we have not yet investigated how our choice of jump parameters affects our results. This is what we will do now. Figure 33 and 34 below shows the spreads that are calculated by changing the jump intensity, 𝜆, and the mean jump size, 𝜇𝑌.
Keeping the jump size fixed, we see that increasing the expected number of jumps
1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50
1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
Relative Spread (jump) Relative Spread (no jump)
T2/T3Ratio
Sigma (%)
see that spreads rise at an increasing rate when we adjust the mean jump size parameter. Most interesting is the T2/T3 spread ratio, which decreases as the jump size increases, since larger jump sizes have a greater likelihood of impacting both the T2 and T3 tranches.
Figure 32: Jump Intensity
Figure 33: Jump Size 0
50 100 150 200 250 300 350
0.0 0.2 0.4 0.6 0.8 1.0
Tier 3 Tier 2
Spread (bps)
Jump Intensity
0 50 100 150 200 250 300 350 400 450 500
0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0 -4.5 -5.0
Tier 3 Tier 2
Spread (bps)
Mean Jump Size (%)
Figure 34: Spread Ratio - Jump Size
These findings suggest that our results are, not surprisingly, somewhat sensitive to the choice of jump parameters. It is also important to note that jumps in the asset value can be incorporated into a model to capture a number of different aspects.
One application is to incorporate the possibility of large but rare jumps, which can be caused by significant events such as a financial crisis. Another application is to have jumps happen more occasionally, which could be due to the release of im-portant information regarding the bank and its assets.
In this thesis, we have chosen the mean jump size to be 𝜆=−1.5% and the jump intensity to be 𝜇𝑌 = 0.2; thus, one jump is expected every five years. The volatility of our jumps is set to 𝜎𝑌 = 2%. By choosing these parameter values, we are at the lower end of the range compared to the figure above, but we can obtain both posi-tive and negaposi-tive jumps in the asset value. A mean jump size of -1.5% could seem low, but as can be seen in the following figure, which shows the distribution for 25,000 random log-normal jumps, we do occasionally obtain jumps with a much greater magnitude, such as the ones we would see in practice. The maximum jump size observed is 6.5% and the minimum is -9.6%. It should also be noted that a jump in the asset value of a bank in the area of 10% is substantial. Such a jump could potentially wipe out the equity, T2, and T3 tranches at once.
1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50
0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0 -4.5 -5.0
Relative Spread
T2/T3Ratio
Mean Jump Size (%)
Figure 35: Distribution of Jumps
Ideally, jump parameters should be estimated more accurately, for example from a bank’s stock returns, bond prices, or CDS spreads. However, as noted by Pennacchi (2010), such estimations do not yet seem to have been performed in any study.
This is also beyond the scope of this thesis. However, considering the choices of other researchers, such as Pennacchi (2010) and Sundaresan and Wang (2015), the sensitivity of our results to the chosen parameters does not raise any concerns regarding the robustness of our results.
Payout Restrictions
In our model, we incorporate the possibility of payout restrictions being imposed on a bank if its CET1 ratio falls below a certain defined regulatory threshold, i.e.
the MDA trigger (or the MREL trigger). In practice, restrictions are imposed line-arly as the CBR is breached, and they apply to AT1 coupons, dividends, bonuses, pension rights, etc. However, a realistic implementation of this feature into our model would pose major computational issues and demand that we find both the value of AT1 and equity before we could restrict payments on these instruments in a proper way. This is beyond the scope of this thesis. For simplicity, we therefore incorporate the feature of payment restrictions by making the payout rate of the bank state-dependent, such that it decreases by a fixed amount if the restrictions are triggered. This assumption is of course stylized, but it captures some of the idea
0 100 200 300 400 500
-9.60 -8.79 -7.98 -7.17 -6.36 -5.55 -4.74 -3.93 -3.12 -2.31 -1.50 -0.69 0.12 0.93 1.74 2.55 3.36 4.17 4.98 5.79
Frequency
Jump Size (%)
of payouts being restricted by regulators16. We will now test how sensitive our results are to this assumption. The results are shown Figure 37 below.
Figure 36: Reduction in Payout Rate
On the x-axis, we have the amount by which the payout rate is reduced when restrictions are imposed on the bank, given in percentage points. We see that re-ducing the payout rate does in fact reduce spread. This coincides with the aim of the payout restrictions, namely, for banks to conserve equity capital in difficult times to restore their financial strength. However, we also see that the effect is moderate.
By leaving out the assumption completely – that is, setting the reduction to 0.00%
points – spreads on T3 and T2 increase from 80bps and 144 bps to 87bps and 161bps, respectively, compared with our benchmark scenario, where we incorporate a reduction of 0.25% points. A reduction of 0.50% points would decrease spreads slightly further, but the largest decrease is unsurprisingly achieved by cutting the payout rate by 0.75% points when restrictions are imposed. However, such a large reduction in the payout would be unrealistic, as it would effectively require the bank to raise a significant amount of equity every time restrictions are imposed.
16 It is important to note that regular debtholders, i.e. deposits, senior, T3 and T2, will always 0
25 50 75 100 125 150 175
0.00 -0.25 -0.50 -0.75
Tier 3 Tier 2
Spread (bps)
Reduction of Payout (percentage points)
We see that the spread ratio between T3 and T2 is fairly stable across the simula-tions, between 1.80x and 1.85x, and does not show any signs of a trend.
The incorporation of payout restrictions adds yet another realistic feature to our model, which we think is important to address in a model of bank capital. Given that our implementation of the feature may be somewhat stylized, we find confi-dence in the fact that the results above show that our assumptions do not have a significant impact on our conclusions.
Summary
We have in this chapter tested the robustness of our results by altering some of the assumptions we made in the previous chapter. First, we adjusted the number of simulated paths. While we found that the accuracy of our results could be improved by raising the number of simulations per estimate, we also found that the effect was diminishing above 25,000 runs. Next, we changed our assumption regarding the jump-diffusion process and found that the jump-part does indeed provide a realistic feature to our model, as the model without it calculates credit spreads of zero for low levels of volatility. Our results were however somewhat sensitive to changing the assumptions regarding jump size and intensity. Finally, we altered our assumption about the MDA restrictions. We found that the restrictions do indeed influence our results, but not considerably. Overall, our results prove to be robust against changing our assumptions. After having showed this, we are now ready to compare our results to our empirical findings in chapter 4.