*5.1.2. Conclusions about optimal basel III LCRs*

We obtained an analytic solution to an optimal bank LCR problem with a quadratic objective function. In principle, this solution can assist in managing LCRs. Here, liquidity provisioning and HQLA allocation are expressed in terms of a reference process. To our knowledge such processes have not been considered for LCRs before. This chapter makes a clear connection between liquidity and financial crises in a numerical-quantitative framework.

An interpretation of the control laws given by (33) and (34) follows. In times of deficit, the cash injection rate, *u*2<sup>∗</sup> , is proportional to the difference between the LCR, *x*, and the reference process for this ratio, *xr*. The proportionality factor is *qt*/*c*<sup>2</sup> that depends on the relative ratio of the cost function on *<sup>u</sup>*<sup>2</sup> and the deviation from the reference ratio, (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>r*). The property that the control law is symmetric in *x* with respect to the reference process *x<sup>r</sup>* is a direct consequence of the cost function *br*(*x*) = <sup>1</sup> <sup>2</sup> *<sup>c</sup>*3(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>r*)<sup>2</sup> being symmetric with respect to (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>r*). The optimal portfolio distribution is proportional to the relative difference between the LCR and its reference process, (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup><sup>r</sup> <sup>t</sup>*)/*x*. This seems natural. The proportionality factor is *<sup>C</sup>*<sup>−</sup><sup>1</sup> *<sup>t</sup> <sup>r</sup> y <sup>t</sup>* that represents the relative rates of asset return multiplied with the inverse of the corresponding variances. It is surprising that the control law has this structure. Apparently the optimal control law is not to liquidate first all HQLAs with the highest liquidity provisioning rate, then the HQLAs with the next to highest liquidity provisioning rate, etc. The proportion of all HQLAs depend on the relative weighting in *<sup>C</sup>*<sup>−</sup><sup>1</sup> *<sup>t</sup> <sup>r</sup> y <sup>t</sup>* and not on the deviation (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup><sup>r</sup> t*).

<sup>−</sup>*x*˙*<sup>r</sup>*

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>(*xt* <sup>−</sup> *<sup>m</sup>*)*q*0/*c*<sup>2</sup> <sup>=</sup>

The interpretation for the two cases follows below.

*5.1.3. Conclusions about numerical results for LCRs*

assets and restricting cash outflows and risky activities.

*<sup>C</sup><sup>r</sup><sup>j</sup>* <sup>=</sup>

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>(*xt* <sup>−</sup> *<sup>m</sup>*) *xt*

and lim*t*↓<sup>0</sup> *<sup>x</sup><sup>r</sup>*

time becomes then,

lead to higher LCRs.

*u*2<sup>∗</sup>

*π*∗

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*k*(*x<sup>r</sup>*

*k* = (*r*

*<sup>m</sup>* <sup>=</sup> *<sup>l</sup>*

*<sup>t</sup>* <sup>−</sup> *<sup>m</sup>*), *<sup>x</sup><sup>r</sup>*

<sup>R</sup> <sup>+</sup> *<sup>r</sup><sup>e</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup>* <sup>+</sup> <sup>2</sup>((*σ<sup>e</sup>*

(*t*1) = *l r* ;

*rc*3/*q*<sup>0</sup> <sup>−</sup> (*u*<sup>1</sup> <sup>−</sup> *<sup>r</sup><sup>e</sup>* <sup>−</sup> (*σe*)2)+(*σe*)<sup>2</sup> (*r*<sup>R</sup> <sup>+</sup> *<sup>r</sup><sup>e</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup>* <sup>+</sup> <sup>2</sup>((*σe*)<sup>2</sup> + (*σi*)2)) + *<sup>c</sup>*3/*q*<sup>0</sup> .

*<sup>t</sup>* = *m* where the down arrow prescribes to start at *t*<sup>1</sup> and to let *t* decrease to 0.

*<* 0, if *xt > m*, if *π*∗ *<* 0 then set *π*∗ = 0,

Because the finite horizon is an artificial phenomenon to make the optimal stochastic control problem tractable, it is of interest to consider the long term behavior of the LCR reference trajectory, *xr*. If the values of the parameters are such that *k >* 0 then the differential equation with the terminal condition is stable. If this condition holds then lim*t*↓<sup>0</sup> *qt* <sup>=</sup> *<sup>q</sup>*<sup>0</sup>

Depending on the value of *m*, the control law for at a time very far away from the terminal

 *>* 0, if *xt < m*, *<* 0, if *xt > m*,

Case 1 (*xt > m*): Then the LCR *x* is too high. This is penalized by the cost function hence the control law prescribes not to invest in riskier HQLAs. The payback advice is due to the quadratic cost function that was selected to make the solution analytically tractable. An increase in liquidity provisioning will increase NCOs that, in turn, will lower the LCR.

Case 2 (*xt < m*): The LCR *x* is too low. The cost function penalizes and the control law prescribes to invest more in riskier HQLAs. In this case, more funds will be available and credit risk on the balance sheet will decrease. Thus higher valued HQLAs should be held. On the other hand, when banks hold less HQLAs, they should decrease their NCOs that may

We approximate the Basel III standard, LCR, that is a measure of asset liquidity for global EMERG banking data mentioned earlier This is a challenging task given the nature of the data available and the ever-changing nature of Basel III liquidity regulation. In the light of the determined results, our analysis gives us a new understanding of the problem of approximating liquidity risk measures. From Table 1, we observe that from Q209 to Q412 there was a steady increase in the LCR. This is probably due to banks holding more liquid

In this paragraph, we highlight how our research on approximating Basel III and traditional liquidity risk measures has advanced the knowledge in this field of endeavor. For both Class I and II banks, our research approximates LCRs for a large diversity of banks for an extended

*>* 0, if *xt < m*,

)<sup>2</sup> + (*σ<sup>i</sup>*

)2)) + *c*3/*q*0;

Optimizing Basel III Liquidity Coverage Ratios

http://dx.doi.org/10.5772/58395

87

The novel structure of the optimal control law is the *LCR reference process*, *<sup>x</sup><sup>r</sup>* : *<sup>T</sup>* <sup>→</sup> R. The differential equation for this reference function is given by (31). This equation is new for the area of LCR control and therefore deserves discussion. The differential equation has several terms on its right-hand side that will be discussed separately. Consider the term

$$u\_t^1 - r\_t^\varepsilon - (\sigma^\varepsilon)^2.$$

This represents the difference between normal rate of liquidity provisioning per monetary unit of the bank's NCOs and NCO outflows, where *r<sup>e</sup> <sup>t</sup>* is the rate of outflow per monetary unit of NCOs. Note that if [*u*<sup>1</sup> *<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup> <sup>t</sup>* <sup>−</sup> (*σe*)2] *<sup>&</sup>gt;* 0, then the reference LCR function can be increasing in time due to this inequality so that, for *t > t*1, *xt < l <sup>r</sup>*. The term *c*3(*x<sup>r</sup> <sup>t</sup>* − *l <sup>r</sup>*)/*qt* models that if the reference LCR function is smaller than *l <sup>r</sup>*, then the function has to increase with time. The quotient *c*3/*qt* is a weighting term that accounts for the running costs and for the effect of the solution of the Riccati differential equation. The term

$$\mathbf{x}\_t^r[r^\mathbf{R}(t) + r\_t^\varepsilon - r\_t^i + (\sigma^\varepsilon)^2 + (\sigma^i)^2]\_\prime$$

accounts for two effects. The difference *r<sup>e</sup> <sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup> <sup>t</sup>* is the nett effect of the rate of outflows per monetary unit of the bank's NCOs, *re*, and rate of NCO increase before outflows per monetary unit of NCOs, *r<sup>i</sup> <sup>t</sup>*. The term *<sup>r</sup>*R(*t*)+(*σe*)<sup>2</sup> + (*σ<sup>i</sup>* )<sup>2</sup> is the effect of NCO increase due to the reserves and the variance of riskier liquidity provisioning. The last term

$$(\mathfrak{x}\_t^r - 1)((\sigma^\varepsilon)^2 + (\sigma^i)^2) - (\sigma^i)^2 \mathsf{a}$$

accounts for the effect on HQLAs and NCOs. More information is obtained by streamlining the ODE for *xr*. In order to accompolish this it is necessary to assume the following.

**Assumption 5.1. (Liquidity Parameters):** *Assume that the parameters of the problem are all time-invariant and also that q has become constant with value q*0.

Then the differential equation for *x<sup>r</sup>* can be rewritten as

$$\begin{aligned} -\dot{\mathbf{x}}^r{}\_t &= -k(\mathbf{x}^r\_t - m), \ \mathbf{x}^r(t\_1) = l^r; \\ k &= (r^\mathsf{R} + r^\varepsilon - r^i + 2((\sigma^\varepsilon)^2 + (\sigma^i)^2)) + c^3/q^0; \\ m &= \frac{l^r c^3/q^0 - (u^1 - r^\varepsilon - (\sigma^\varepsilon)^2) + (\sigma^\varepsilon)^2}{(r^\mathsf{R} + r^\varepsilon - r^i + 2((\sigma^\varepsilon)^2 + (\sigma^i)^2)) + c^3/q^0}. \end{aligned}$$

Because the finite horizon is an artificial phenomenon to make the optimal stochastic control problem tractable, it is of interest to consider the long term behavior of the LCR reference trajectory, *xr*. If the values of the parameters are such that *k >* 0 then the differential equation with the terminal condition is stable. If this condition holds then lim*t*↓<sup>0</sup> *qt* <sup>=</sup> *<sup>q</sup>*<sup>0</sup> and lim*t*↓<sup>0</sup> *<sup>x</sup><sup>r</sup> <sup>t</sup>* = *m* where the down arrow prescribes to start at *t*<sup>1</sup> and to let *t* decrease to 0. Depending on the value of *m*, the control law for at a time very far away from the terminal time becomes then,

$$\begin{aligned} \mu\_t^{2\*} &= -(\mathbf{x}\_t - m)q^0 / c^2 = \begin{cases} > 0, \text{ if } \mathbf{x}\_t < m, \\ < 0, \text{ if } \mathbf{x}\_t > m, \end{cases} \\ \pi\_t^\* &= -\frac{(\mathbf{x}\_t - m)}{\mathbf{x}\_t} \widetilde{C} \overline{r}^j = \begin{cases} > 0, \text{ if } \mathbf{x}\_t < m, \\ < 0, \text{ if } \mathbf{x}\_t > m, \text{ if } \pi^\* < 0 \text{ then set } \pi^\* = 0, \end{cases} \end{aligned}$$

The interpretation for the two cases follows below.

with respect to (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>r*). The optimal portfolio distribution is proportional to the relative

with the inverse of the corresponding variances. It is surprising that the control law has this structure. Apparently the optimal control law is not to liquidate first all HQLAs with the highest liquidity provisioning rate, then the HQLAs with the next to highest liquidity provisioning rate, etc. The proportion of all HQLAs depend on the relative weighting in

The novel structure of the optimal control law is the *LCR reference process*, *<sup>x</sup><sup>r</sup>* : *<sup>T</sup>* <sup>→</sup> R. The differential equation for this reference function is given by (31). This equation is new for the area of LCR control and therefore deserves discussion. The differential equation has several

> *<sup>t</sup>* <sup>−</sup> (*σ<sup>e</sup>* )2.

This represents the difference between normal rate of liquidity provisioning per monetary

with time. The quotient *c*3/*qt* is a weighting term that accounts for the running costs and for

*<sup>t</sup>* + (*σ<sup>e</sup>*

*<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup>*

per monetary unit of the bank's NCOs, *re*, and rate of NCO increase before outflows per

)<sup>2</sup> + (*σ<sup>i</sup>*

accounts for the effect on HQLAs and NCOs. More information is obtained by streamlining

**Assumption 5.1. (Liquidity Parameters):** *Assume that the parameters of the problem are all*

the ODE for *xr*. In order to accompolish this it is necessary to assume the following.

)<sup>2</sup> + (*σ<sup>i</sup>*

)2) <sup>−</sup> (*σ<sup>i</sup>*

)2,

)2],

*t*).

terms on its right-hand side that will be discussed separately. Consider the term

*u*1 *<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup>* *<sup>t</sup>*)/*x*. This seems natural. The

*<sup>t</sup>* is the rate of outflow per monetary

*<sup>t</sup>* is the nett effect of the rate of outflows

)<sup>2</sup> is the effect of NCO increase due

*<sup>r</sup>*. The term *c*3(*x<sup>r</sup>*

*<sup>r</sup>*, then the function has to increase

*<sup>t</sup>* − *l*

*<sup>r</sup>*)/*qt*

*<sup>t</sup>* <sup>−</sup> (*σe*)2] *<sup>&</sup>gt;* 0, then the reference LCR function can be

*<sup>t</sup>* that represents the relative rates of asset return multiplied

difference between the LCR and its reference process, (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup><sup>r</sup>*

*<sup>t</sup> <sup>r</sup> y*

86 Dynamic Programming and Bayesian Inference, Concepts and Applications

unit of the bank's NCOs and NCO outflows, where *r<sup>e</sup>*

models that if the reference LCR function is smaller than *l*

*xr t* [*r*

(*x<sup>r</sup>*

*time-invariant and also that q has become constant with value q*0.

Then the differential equation for *x<sup>r</sup>* can be rewritten as

accounts for two effects. The difference *r<sup>e</sup>*

*<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup>*

increasing in time due to this inequality so that, for *t > t*1, *xt < l*

the effect of the solution of the Riccati differential equation. The term

<sup>R</sup>(*t*) + *r<sup>e</sup>*

*<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup>*

*<sup>t</sup>*. The term *<sup>r</sup>*R(*t*)+(*σe*)<sup>2</sup> + (*σ<sup>i</sup>*

to the reserves and the variance of riskier liquidity provisioning. The last term

*<sup>t</sup>* <sup>−</sup> <sup>1</sup>)((*σ<sup>e</sup>*

*<sup>t</sup>* and not on the deviation (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup><sup>r</sup>*

proportionality factor is *<sup>C</sup>*<sup>−</sup><sup>1</sup>

unit of NCOs. Note that if [*u*<sup>1</sup>

monetary unit of NCOs, *r<sup>i</sup>*

*<sup>C</sup>*<sup>−</sup><sup>1</sup> *<sup>t</sup> <sup>r</sup> y*

> Case 1 (*xt > m*): Then the LCR *x* is too high. This is penalized by the cost function hence the control law prescribes not to invest in riskier HQLAs. The payback advice is due to the quadratic cost function that was selected to make the solution analytically tractable. An increase in liquidity provisioning will increase NCOs that, in turn, will lower the LCR.

> Case 2 (*xt < m*): The LCR *x* is too low. The cost function penalizes and the control law prescribes to invest more in riskier HQLAs. In this case, more funds will be available and credit risk on the balance sheet will decrease. Thus higher valued HQLAs should be held. On the other hand, when banks hold less HQLAs, they should decrease their NCOs that may lead to higher LCRs.
