Appendix: Explanation of the Math behind the Aggregate Liquidity Management Model of the Interbank Rate

This appendix explains the mathematical foundations for the "aggregate liquidity management model" of the interbank rate. All that's required to understand this a basic familiarity with probability and calculus.

Here's the basic mathematical representation of the model:

The upside down “A” just means “for every.” In this context, it means for every market session t, from session 1 to T, the interest rate at any given time equals the right side of the equation. The right side of the equation says that “the overnight rate on any day will correspond to the weighted expected rate of the two standing facilities, the weights being the respective probabilities that the market will be ‘short’ or ‘long’ of reserves at the end of the maintenance period for having recourse to the standing facilities. Equation 3.2 may be considered as the fundamental equation of monetary policy implementation.” 

The E[.|I_t] notation just means the expected values of i_B and i_D given the information I available at time t. The integrals represent the calculation of the probability weights.

To understand the equation, you first have to understand what a probability density function (pdf) is. A probability density function is a function that provides the probability that a continuous random variable takes on values between negative infinity to positive infinity. The picture below illustrates various potential probability density functions (pdfs) of a random variable X. The values of X are on the x-axis, and probability is on the y-axis.  You’re probably familiar with the standard normal distribution, which his depicted by the red line in the picture.

The way probability density functions work is that the area of the curve between two values for X equals the probability that the random variable falls between those two values. Additionally, the total area under the whole curve must sum to 1, since a valid probability density function must assign a probability to all possible values for the random variable X.

Believe it or not, this is enough knowledge to understand equation 3.2. Pretend one of the pdfs in the graph (for example, the red line) is the probability density function “f” for the continuous random variable (M – A), that that the x-axis represents the quantity (M – A), and that the x-axis can range from negative infinity to infinity. The left-most integral in equation 3.2 is calculating the area under the probability density function for M – A from negative infinity to 0. This represents the probability that M – A will be less than zero, and is thus the probability weight that the interest rate will be i­_B at the end of the maintenance period. Taking the red curve for example, the area to the left of 0 corresponds to 50% (i.e., it’s equally likely X will be less than or greater than 0). The integral on the right represents the area under the curve from 0 to positive infinity. This represents the probability that M – A will be greater than zero and is thus the probability weight that the interest rate will be i_D at the end of the maintenance period. Since the area under the whole curve must sum to 1, Bindseil gets this quantity by subtracting the probability that (M – A) is less than zero from 1. Alternatively, one could take an integral of the pdf from 0 to positive infinity, but Bindseil’s method requires less computation.

And that’s it!