Having determined the optimal coupon cfor each bank, we are almost ready to carry out simulation to evaluate default probabilities. A few more steps remain to be finished before the simulation.
Parameters for Investors
As for investors, we use parameters provided in Table 3.2.3 We test three levels of correlation ρ to deepen our analysis, which are ρ = 0.5, ρ = 0, and ρ = −0.5. If we consider a financial institution as a CoCo investor, the correlation should be positive.
Thus, we focus on the case of ρ= 0.5 and compare the result to the other cases.
Table 3.2: Parameters for Investor e
µ σe xe χeD ϕ ρ
0.005 0.05 100 70 0.1 -0.5/0/0.5
Parameters for Regulators: Risk Horizon and Intensity Function
We assume that the risk horizonT in (2.44) and (2.45) is set to 10 years while account-ing numbers are calculated quarterly, i.e., Tn= 0.25n, n∈N. Actually, “10 years” may be a rather long-term target for regulators. For example, the Federal Reserve Board provides a four-year scenario to conduct a stress test, so-called Comprehensive Capital Analysis and Review (CCAR). We focus on 10-year default probabilities in our study because our main purpose is to assess the impact of trigger in the long run, so that we are able to analyze the subsequent behavior of the investor.
In addition to the risk horizon, we also need to find an “appropriate” intensity function h(V(t)) which makes τR1 to follow a Cox process, as indicated in (2.8). As mentioned earlier, among functions that satisfy h(V(t)) ≥ 0 and ∫∞
0 h(V(t))dt = ∞, we should choose the function that is non-increasing in V(t). Moreover, we should select the function that generates τR1 in a manner that is likely to happen in the real world, i.e., frequencies ofτR1 andV(τR1) should be well-suited to our intuition. Among various possible functions we tested, we adopt
h(V(t)) =a(V¯ −min(V(t),V¯))2
+b (3.1)
where a = 7.5×10−4, b = 1.5×10−4 and ¯V = 90 as an intensity function. According to simulation, given intensity (3.1), probabilities of the regulatory trigger to happen
3ϕ= 0.1 is applied to the investor possessing Type 2 Bank CoCo. For Type 0 Bank and Type 1 Bank investors, we adjust the amount of ϕto ensure that the value of CoCo they possess at t = 0 would be exactly the same.
Figure 3.3: Intensity Function
Panel (a) shows a graph image of function (3.1). It clearly shows that h(V(t))≥ b > 0, and thus
∫∞
0 h(V(t))dt = ∞ is satisfied. It also shows that h is a non-increasing function with regard to V(t). Panel (b) shows the histogram of V(τR) as a result of simulation (100,000 runs). Note that τR = τR1∧τR2. Thus, it includes the frequencies of V(τR2), which is induced by the controllable valuable VR. The frequency increases as V(τR) decreases, and jumps up around 80, since we have VR≈81 in this case.
(a) Graph Image
𝑏
𝑉# 𝑉(𝑡)
0
(b) Histogram ofV(τR)
V
Frequency
80 90 100 110 120 130 140 150
020406080100
within 10 years become higher than those of the accounting trigger. It is an acceptable result provided that the regulatory trigger could happen anytime while the accounting trigger can occur only once in a quarter. More features with regard to the intensity function (3.1) is provided in Figure 3.3.
Valuation of Non-Analytic Elements
Another preparation that should be done is to calculate two expectations in advance of the main simulation, which areEQx[e−rτT] andEQx[e−rτD]. As shown in Equations (2.23), (2.24) and (2.34), calculations of these two expectations are necessary to evaluate the value of bond/CoCo, which must be known to obtain investor’s asset value Ve(t).
However, these expectations cannot be expressed analytically in the case of τT defined in (2.11). Hence, we carry out another round of simulation to find explicit functions ˆgT(x) and ˆgD(x) that satisfies ˆgT(x) ≈ EQx[e−rτT] and ˆgD(x) ≈ EQx[e−rτD].
Having done the simulation with regard to Type 2 Bank with c∗∗, we get ˆ
gT(x) = e−0.043(x−VA), gˆD(x) = 0.9103 e−0.061(x−VA), x > VA.
Note that approximations could be different if we assume a different starting date t, as shown in Figure 3.4. This is because the probability of τA to take place at the next calculation date is time-dependent. For instance, suppose that the next calculation date is approaching tomorrow, and we have V(t) << VA at that time. Then it is
Figure 3.4: Approximation by Simulation
Under the parameter set provided in Table 3.1, we obtainVA= 79.3 and VD= 71.9 for Type 2 Bank givenc∗∗. We testt= 0, t= 30 andt= 59 (days) as a starting date, and T = 60 (days) as the next calculation date. The horizontal axes showsx−VR (difference between the initial asset value and the regulatory threshold), whereVR= (VA+VD)/2 = 76.5 is assumed. The intercept of ˆgT(x) is adjusted to 1 given thatEQx[e−rτT]→1 asx→VR.
(a) gˆT(x)≈EQx[e−rτT]
y = e-0.042x y = e-0.043x y = e-0.043x
0 0.2 0.4 0.6 0.8 1
0 20 40 60 80 100
t=0 t=30 t=59
(b) ˆgT(x)≈EQx[e−rτD]
y = 0.91e-0.061x y = 0.77e-0.049x y = 0.84e-0.048x 0
0.2 0.4 0.6 0.8 1
0 20 40 60 80 100
t=0 t=30 t=59
highly likely that the accounting trigger would happen tomorrow, unless we have a dramatic increase in V(t). On the other hand, suppose that V(t) << VA but we still have enough time until the next calculation date, then the possibility of the asset value to exceed the accounting threshold would be higher than the former case, especially when we have a positive drift.
As such, although we obtain slightly different approximations for different starting dates, we confirm that the difference does not significantly affect the default probabil-ities of both bank and investor. Thus, we proceed with the approximations obtained from the result of t= 0.
The above approximations show the case of Type 2 Bank withc∗∗, i.e., σT = 0.07.
We have conducted simulations with all the possible cases and successfully obtained similar approximations.
Controllable Variables
As defined in (2.44),VRand ¯care supposed to be controllable variables for the regulator.
If we could test all the possible choices with respect to these variables, we may be able to find a “solution” to the regulator’s problem. However, unfortunately, it requires tremendous computational costs to do so. Hence, we pick up some “sample” quantities for each variable to enable simulation and necessary analysis.
As for the regulatory threshold VR, we test three levels and name them High(H), Middle(M) and Low(L), respectively.
• Strategy H：VRis determined to satisfyVA < VR. It indicates that the regulator is “aggressive,” as they intend to trigger a CoCo at a relatively “early stage”
where the capital ratio is still above the accounting threshold.
• Strategy M：VR is set to meet VD << VR≤VA. Under this strategy, the regu-lator triggers a CoCo when the capital ratio falls below the accounting threshold yet some buffers remain until it hits the default barrier.
• Strategy L：VR is fixed to be VR=VD+ε, where ε is positive but almost zero.
In this case, we may say that the regulator is “easy-going,” as they wait until the capital ratio to become quite close to the default barrier.
The most feasible strategy to be chosen by regulators would be Strategy M, because the other strategies are impractical from the following reasons. Under Strategy H, the accounting trigger is unworkable because the regulatory trigger is expected to happen in advance.4 In the case of Strategy L, though loss-absorption of the CoCo enhances the capital ratio, it may not be enough to bring the bank back to the solvency level.
It is important to include extreme cases when we want to get some implications from just a few samples; thus, we analyze Strategy H and Strategy L although it may be unrealistic.
As for the coupon c, we test three quantities as well:
• Case 1: c = c∗. It indicates that the issuance limit ¯c is large enough to allow a bank to choose an optimal c∗. The bank finds an optimal coupon with the assumption that the trigger of CoCo has nothing to do with the asset volatility, i.e., σ =σT.
• Case 2: c = c∗∗. It also indicates that ¯c is large enough, but a bank finds an optimal coupon under the assumption that the trigger of CoCo generates the volatility hike, i.e., σ < σT.
• Case 3: c= ¯c. Regulator imposes issuance limit ¯c <min{c∗, c∗∗}to banks, thus a bank can only issue a bond/CoCo that has a coupon amount upto ¯c.
Table 3.3 summarizes the amount of cand corresponding C(x) in each case.
Scenarios
As indicated from the comparative statics, σT considerably impacts the firm value, which implies that it remarkably affects the default probability of the bank as well.
4As defined in (2.10), regulators observeV(t) +ϵ. It indicates that a regulatory trigger is induced by the noise added process, while an accounting trigger is associated with V(t) itself. Thus, the probability of the accounting trigger to happen under Strategy H is not zero.
Table 3.3: Simulation Setup ofc and C(x) Case 1 Case 2 Case 3
c∗ c∗∗ ¯c
Type 0 Bank c 0.024 0.020
C(x) 2.23 1.86
Type 1 Bank c 0.137 0.096 0.020 C(x) 10.77 8.19 1.87 Type 2 Bank c 0.099 0.037 0.020
C(x) 7.95 3.29 1.81
Thus, as we have done in the previous section, we test two scenarios with regard toσT to check its impact.
• Scenario A: σ =σT. The trigger of CoCo does not affect the volatility ofV(t).
• Scenario B:σ < σT. The volatility hike occurs due to the trigger of CoCo.
It is reasonable to assume Scenario B to happen in practice since the trigger of CoCo may weaken the bank’s credibility in current circumstances.