Despliegue de la función de calidad (QFD)

Previously, we argued that banks try to improve their estimate of early customers based on past experience through the updating process

Wi k+1 = max  Wi k+ α(Wki − W i k), 1 − c2 R c1− c_R2  . (6.2)

Here we consider the more general case where banks combine this updating process with a learning mechanism to assess the adequacy of the estimate.

We propose that banks build an estimate for the percentage of early customers combined with an estimate of what this percentage ought to be if we believe the sample is a good representative of the society. In what follows we represent a simple model for banks to learn about their customers and show under what conditions that can motivate the establishment of an interbank market.

6.2.1

Learning from experience

Similar to individuals in the society, banks have their own targets which they try to achieve by observing their surroundings. For simplicity we mimic for banks the learning process previously used by individuals and assume that banks try to predict the adequacy of their estimate for the proportion of early (impatient) customers.

They determine the adequacy of the estimate according to two basic criteria: statistical accuracy and the current level of bank reserves. The choice of the first criterion reflects the fact that after observing the true percentage of early customers the bank judges its estimate based on its proximity to the true one using the Wilson score confidence interval [Wil27]

W +_2n1 Z2 1−α 2 ± Z1− α 2 r W(1−W ) n + Z2 1−α₂ 4n2 1 + 1_nZ2 1−α 2 , (6.3)

for Z1−α

2 is the 1 −

α

2 percentile of a standard normal distribution, and for our case

W = Wi

k and n = N i

k is the total number of clients in bank. The second criterion

reflects the bank’s worries on its available liquidity, specified as maintaining reserves per customer higher than some predetermined threshold Rmin.

The two criteria combined reflect the fact observed in the previous chapter that the two main determinants of the success of a bank are a close estimate of the actual percentage of early consumers and a sufficiently high level of reserves. We therefore say that a given estimate Wi

k is adequate if it falls within bounds of the confidence

interval in (6.3) and the amount of realized reserves per depositor Ri

k/Nki is above the

threshold Rmin. Otherwise we say that Wki is inadequate.

Banks have a memory of five periods and use the same set of seven homogeneous predictors described in Section 5.2.3, each forecasting that the estimate of early clients for the next period will be either adequate, inadequate, or neutral (i.e neither adequate nor inadequate), corresponding to banks that have just been created and do not have past experience to build their prediction. Similar to the individuals’ case, we update the predictors and their strength as times goes by. In establishing the prediction for period k + 1, banks choose the predictors with the highest strength at the end of period k, with a tie–breaking rule favouring adequate and inadequate predictions over neutral ones and setting opposite predictions to neutral.

Accordingly, banks that deem their estimates Wi

k+1 to be adequate for period

k + 1 believe that their allocation will be enough to satisfy their customers need and to maintain a minimum level of reserves. On the other hand, we argue that banks with suspicion about the adequacy of their estimates have a motive to seek each other and establish interbank links to guard against predicted losses. It follows that for interbank links to be established more than one bank should have this kind of suspicion at the same time, in what we will refer to as a “window of opportunity”.

Omneia R.H. Ismail – PhD Thesis – McMaster University – CES

6.2.2

Windows of opportunities

Applying the above predictors to our 80 × 80 society for the course of 80 periods (k = 80), we find the same type of transient behaviour observed in the previous chapter: upon establishment, banks start with neutral predictions, then through learning they experience a phase of oscillation between inadequate and adequate predictions, and finally established a persistent belief in the adequacy of their estimate.

Figure 6.1 illustrates this phenomenon, with yellow representing a neutral predic- tion, light blue representing a prediction of an inadequate estimate, and dark blue representing a prediction of an adequate estimate. We observe that a number of banks are established during the lifetime of our experiment, with columns representing time and each row represents a bank and its experiences over time. We see that a total number of 17 banks were established (each in a separate row), yet only 5 successful banks persisted untill the end. We observe that the windows of opportunity for an interbank link (that is more than one bank having yellow or light blue cell at the same time step) is less than half of the periods and occurs in the earlier life of banks. As time goes by, the frequency of such opportunities decreases as surviving banks become increasingly confident (dark blue cells) in their estimates.

Figure 6.1: Banks learning, an example .

Drawing on our earlier discussion, given a high degree of confidence in their es- timates, banks will have no motive to establish interbank links. As such, windows of opportunity for interbank links occur in the early periods of banks establishment. Later on we will introduce a mechanism for more dramatic liquidity shocks in the society which can prolong these opportunities when we introduce the idea of “com- munities” in Section 6.5. Before that, we first describe a way to determine the amount of desired deposits given that two banks want to establish a link with each other.

6.2.3

Desired deposits

As indicated above banks having confidence in their estimate for the percentage of early customers rest content that it will yield a good allocation of resources and main- tain a minimum level of reserves. On the other hand, banks predicting an inadequate estimate have the incentive to guard against possible losses by engaging in links with other banks. We assume that the interbank deposits yield the same payoffs (c1, c2) as

Omneia R.H. Ismail – PhD Thesis – McMaster University – CES

regular deposits.

So far banks have been concerned with their depositors and had no other sources of knowledge. Now we add another layer of banks’ knowledge through the assumption that banks also have some global knowledge of the society as a whole. That is, whether the society is in a state of high demand for liquidity, where most individuals are impatient, or in a state of low demand for liquidity, where most are patient. To incorporate this in our model we assume that banks can observe the realized proportion of impatient bank depositors (combined from all banks) from the current period Wk.

When, based on its predictors at period k, a bank believes that its estimate Wi k+1

for the proportion of impatient customers in the next period is inadequate, it compares it with Wk to determine whether it is overestimating or underestimating the true

percentage, leading to two possibilities:

1. Predicted underestimate: if Wi

k+1 < Wk the bank concludes that it is likely

to be underestimating the true proportion of early clients and has an incentive to raise funds by accepting deposits (in-links) from other banks up to the perceived difference of

I_k+1i := N_k+1i (Wk− W_k+1i ), (6.4)

which can be invested in the liquid asset, allowing it to pay the higher than estimated fraction of impatient clients. Instead of accepting interbank deposits, the bank might consider to reallocate its resources and invest more in the liquid asset directly, but at the risk of depleting its reserves in case its original estimate was actually correct (i.e. not an underestimate).

2. Predicted overestimate: if Wi

k+1 > Wk the bank concludes that it is likely

(out-links) the perceived difference

Oi

k+1 := Nk+1i (Wk+1i − Wk) (6.5)

with other banks and withdraw at end of the period t2k+2if the prediction comes

true. Otherwise, if its original estimate was actually correct, then the bank can withdraw its deposits at mid-period receiving c1and as such in both cases satisfy

its clients. This is again better than to reallocate resources directly and risk a bank run if the original estimate turns out to be correct.

These two cases are complementary, leading to the emergence of interbank links: banks overestimating are depositors and banks underestimating accept deposits.