U. Bindseil, TU Berlin, SS 2016

A simple simulation tool for the control of the operational target through monetary policy instruments

The simulation tool is based on a simple two bank, one day model. The two banks are subject to liquidity shocks before and after the trading session, whereby there are shocks of aggregate and of relative nature. Variants of it had been used in Bindseil and Würtz (2008) to simulate a TARALAC facility and in e.g. Bindseil and Jablecki (2013) to simulate the width of the corridor (on the latter see also Bindseil, 2014, section 6.2). The variant presented here allows to capture both (a) the impact of the width of the standing facilities corridor, (b) the existence and quota or a TARALAC facility, and moreover (c) the frequency and (d) tender procedure of open market operations on: (i) overnight rate volatility, (ii) interbank trading volume, and (iii) central bank balance sheet length / average recourse to the facilities. The timeline every day is as follows:

  1. Central bank open market operation. In the morning, the central bank adjusts its credit operations by means of an open market operation, such that OMO = B, i.e. the expected value of autonomous factors. The actual banknote circulation at day end is defined as B + d1 + d2 (we will also sometimes write d = d1 + d2). B is the deterministic component and level of autonomous factors in the morning, while d1, d2 are stochastic shocks hitting each bank in the course of the day, with E(d1) = E(d2) = 0 and with a symmetric density function. Since OMO = B in the morning, the total bank reserves R will be equal zero in the morning.
  2. First liquidity shock. After the central bank operation, a first stochastic component of autonomous factors realizes itself and becomes publicly known: d1. At the same time, a deposit shift shock occurs, k, which is neutral in terms of aggregate liquidity, but reflects that deposits of households move from one bank to another. For the sake of simplicity we assume here that the two shocks are normally distributed with expected value zero and standard deviation σd1and σk.
  3. Interbank trading session. At mid-day, a trading session takes place, in which the interbank rate is set as the weighted average of the two standing facility rates, the weights being the perceived probabilities of the banking system being short or long at day end. The interbank trade is Y being the amount lend from bank 1 to bank 2 (i.e. a negative value of Y indicates that bank 2 lends to bank 1)
  4. Second liquidity shock. In the afternoon, the eventual end-daylevel of autonomous factors is revealed, as the last stochastic variable d2 gets realized with expected value zero and standard deviation σd2.
  5. Day-end and recourse to standing facilities. To end the day with zero reserves, banks will take recourse
  6. first to the TARALAC facility, and if needed, i.e. if the imbalance exceeded the quota of Ψ/2 per bank, also
  7. tothe standing facilities at penalty level. In fact we will assume that the (penalty) deposit facility rate is zero, so that actual active recourse to a facility is not needed in the case that excess liquidity exceeds the capacity of the liquidity absorbing TARALAC facility.

The daily timeline is summarized in the figure below. The end of day financial accounts representation is shownafterwards. The interbank trading y is, as will be shown below, a function of (a) the width of the corridor; (b) the size of a possible TARALAC facility; (c) the parameters of the various liquidity shocks (as we assume for the sake of simplicity normally distributed shocks with expected value of zero, the unique parameter describing each of the liquidity shock stochastic variable is its standard deviation). Note that for the simplicity of the presentation in the financial accounts, interbank lending is presented as a NET item on the asset side of bank 1 and as a net liability item for bank 2. Obviously the relative liquidity shock could also induce the opposite flows.

Daily timeline of central bank operations and interbank trading

Financial accounts at end of the day (note that d = d1 + d2)

Households / Investors
Real Assets E-D-B
Deposits Bank 1 D/2- d/2 + k
Deposits Bank 2 D/2- d /2 - k
Banknotes B + d / Household Equity E
Bank 1
Lending to corporates D/2 + B/2
Net interbank lending y
Recourse to TARALAC dep. Fac. min(Ψ/2,max(0,k-d/2-y)
Recourse to penalty dep. Fac. max(0,k-d/2-y-Ψ/2) / Household deposits / debt D/2+k-d/2
OMO credit from central bank B/2
Recourse to TARALAC cred fac. min(Ψ/2,max(0,-(k-d/2-y))
Recourse to penalty credit facility max(0,-(k-d/2-y)-Ψ/2)
Bank 2
Lending to corporates D/2+B/2
Recourse to TARALAC dep. Fac. min(Ψ/2,max(0,-k-d/2+y)
Recourse to penalty dep. Fac. max(0,-k-d/2+y-Ψ/2) / Household deposits / debt D/2-k-d/2
Net interbank lending y
OMO credit from central bank B/2
Recourse to TARALAC cred. Fac. min(Ψ/2,max(0,-(-k-d/2+y))
Recourse to penalty cred. Fac. max(0,-(-k-d/2+y)-Ψ/2)
Central Bank
Open market operation credit operation B
TARALAC credfac min(Ψ/2,max(0,-(k-d/2-y))+min(Ψ/2,max(0,-(-k-d/2+y))
Penalty credit facility max(0,-(-k-d/2-y)-Ψ/2) + max(0,-(-k-d/2+y)-Ψ/2) / Banknotes B + d
TARALAC dep. Fac. min(Ψ/2,max(0,k-d/2-y)+ min(Ψ/2,max(0,-k-d/2+y)
Penalty dep. Fac. max(0,k-d/2-y-Ψ/2)+max(0,-k-d/2+y-Ψ/2)

How will the different policy parameters affect the interbank market and average central bank balance sheet length? If there are no interbank transaction costs, then banks would always trade in the interbank market until the interbank shock is fully offset. In contrast, if there is a cost CMM associated with transacting in the market, then the interbank trade Y will depend both on CMM and on the other parameters. If CMM > 0 and the penalty corridor has a width of zero, which is equivalent to the case that the TARLAC facility has a quasi-infinite quota, then – as we show below – there will never be any trading. In general, what the banking system needs to minimise when establishing the interbank trade is the sum of interbank transaction costs plus the expected costs of a two-sided recourse to penalty standing facilities. If banks totally reverse the deposit shift shock, then the second type of costs is certainly zero (as the two banks will end the day either both in the borrowing facility, or both in the deposit facility). But if transaction costs are material, this is sub-optimal as at the end, the incremental savings relating to the second type of cost will be lower than the interbank transaction cost. Call “CSFi” the cost of penalty standing facilities incurred by bank i. Then the optimum (and assumed equilibrium) size of the interbank trade for a given realisation of d1, k is the y∗ that minimizes the total cost function

TC = E(CSF1) + E(CSF2) + y.CMM

Where E(CSF1), E(CSF2) are defined as follows, whereby the probabilities refer to the case in which no interbank trading would take place before day end:

E(CSF1) = P(“Bank 1 short”)iB + P(“Bank 1 TARALAC”)i* + P(“Bank 1 long”)iD

E(CSF2) = P(“Bank 2 short”)iB + P(“Bank 2 TARALAC”)i* + P(“Bank 2 long”)iD

By inserting the liquidity shock variables that define the three liquidity states of the bank before day-end (note that :

E(CSF1) = P(k-d/2 < -Ψ/2)iB + P(-Ψ/2 ≤ k-d/2 ≤ Ψ/2)i* + P(k-d/2 > Ψ/2)iD

E(CSF2) = P(-k-d/2 < -Ψ/2)iB + P(-Ψ/2 ≤ -k-d/2 ≤ Ψ/2)i* + P(-k-d/2 > Ψ/2)iD

While it is obvious how to simulate the width of the corridor and the existence and quota of a TARALAC facility, the other policy choices (frequency of OMO, FRFA vs. FRFV) can be simulated in the sense that they can be considered in impacting on specific volatilities within the model.

-The case of an open market operation less than every day allows to be captured by simply increasing the volatility σd1 of the pre-MM aggregate AF shock d1 in a way that this increase reflects the estimated increase of autonomous factor uncertainty and volatility from the longer period between the conduct of the OMO and the actual day end. Conceptually this shock captures also what has cumulated in terms of autonomous factor imbalance for some days, i.e. back to the last OMO. More precisely, for example a weekly OMO calibrated according to the average expected autonomous factors should be reflected with two extra volatility components within d1: one reflecting the fact that the OMO volume was set in a way to match the average AF over the one week period (if the AF does not follow a martingale), and second, one reflecting the fact that forecasts of the central bank have higher forecast errors for the more distant future (also valid in case the AF follows a Martingale). The latter implies that the volatility of the AF shock d1 (seen within a one day model capturing separately the different days) increases in the course of the period covered by the OMO. In any case, all this can conceptually be captured with our one day simulation tool.

-FRFA versus VRFV. The choice between these two options reflects the beliefs one has with regard to the noise terms each of the tender procedures injects, relative to the actual stochastic autonomous factor process. Views can again be reflected in additional elements of the AF shock d1. For example, if one believes that the FRFA suffers from the problem (in normal, liquidity-neutral times and under the symmetric corridor approach) of noisy bid aggregation, and if one believes that the VRFV approach is the effective response to it, then one would simply attribute to the FRFA a higher extra volatility add on to the AF shock d1.

Now we describe the technicalities of the EXCEL tool to simulate in the most basic and transparent way the impact of policy choices and of exogenous variables. The Excel tool can be found under: insert URL.

The key exogenous variables that the user of the tool can specify are in fields B5-B10:

-B5: “AF shock preMMStd” = σd1 =The standard deviation of the pre-money market aggregate autonomous factor shock

-B6: “AF shock postMMStd” = σd2 = The standard deviation of the post-money market aggregate autonomous factor shock

-B7: “RelLiqu shock Std” = σk = The standard deviation of liquidity shock between the two banks

-B8: MM TAC = transaction costs in the interbank market

-B9: Width of corridor (pc) = Policy variable

-B10: “Taralac quota” =Ψ/2 = The size of the TARALAC quota. Note that the TARALAC facility is specified as two sided, i.e. in both directions banks have a standing facility at the target rate with half of the variable “Taralac quota”.

The first four variables can be regarded as exogenous, or environmental variables, while the latter two are policy choices. However, as explained above, in particular the pre-MM AF shock’s standard deviation σd1can also be regarded as capturing policy choices, such as how often OMOs are conducted, and whether FRFA or VRFV is used.

The excel spreadsheet produces 5000 simulations in rows 23 to 522. Consider the columns describing each simulation draw:

  1. Number of draw (1-5000)
  2. AF1, i.e. the estimate of the AF level at day end estimated during the trading session (=NORM.INV(RAND();1;$B$5)). AF1 is a normally distributed random variable with expected value 1 and standard deviation $B$5. A normally distributed random variable can be generated in excel by applying the Norm.Inv() function, i.e. the inverted cumulative normal distribution, to RAND() which is a uniform random variable in [0,1] that Excel allows to generate.
  3. AF2 is the actual end of day AF level and is equal to AF1 plus an N(0, σd2) innovation (=B23+NORM.INV(RAND();0;$B$6))
  4. Relative LiquShock is the liquidity shock between the two banks, i.e. which on aggregate is zero. It is N(0, σk) (=NORM.INV(RAND();0;$B$7)).
  5. OMO is the open market operation, which actually can be set always equal to 1, which is the expected value of autonomous factors and (as we assume no reserve requirement) also the neutral liquidity need.
  6. ResPreMM provides the reserves of Bank 1 with the central bank just before the MM takes place, but after the morning aggregate and relative liquidity shocks. Note that the two banks share the aggregate liquidity factors half-half (=$E23/2-$B23/2+$D23).
  7. Prob DF calculates the probability that Bank 1 would end up in the Deposit facility if it would not trade in the MM. This, together with the identically defined probabilities to end in the marginal lending facility and in the TARALAC facility allows to calculate the actual marginal value of reserves – as the weighted rate of the three facilities, the weights being the probabilities. The weights are determined by the reserves at this point in time, and the uncertainty stemming from the still outstanding second aggregate AF shock. The TARALAC quota matters as only if the excess end of day reserve position (without trading) cannot be absorbed by the TARALAC facility, then really the DF rate becomes relevant (=(NORM.DIST($F23-$B$10/2;0;$B$6;TRUE))).
  8. Prob MLF is like Prob DF, but calculates the corresponding probability that bank needs to take recourse to the borrowing facility offered by the CB. (=1-(NORM.DIST($F23+$B$10/2;0;$B$6;TRUE)))
  9. Prob TARALAC is the probability that the bank can correct its liquidity imbalance at day end (assuming no MM) just with recourse to the TARALAC facility, i.e. without recourse to a facility at penalty rate level. It is calculated here as 1 minus the previous two probabilities. (=1-(G23+H23))
  10. MargValue Reserves is the weighted average of the three relevant interest rates, the weights being as calculated in the preceding three columns. (=H23*$B$9+I23*$B$9/2). The rate of the deposit facility actually is ignored as it is set to 0. Therefore only the weighted MLF rate and the weighted TARALAC rate need to be summed up.
  11. Bank 2 – see F above
  12. Bank 2 – see G above
  13. Bank 2 – see H above
  14. Bank 2 – see I above
  15. Bank 2 – see J above
  16. MM volume. This column calculates the Money market volume, reflecting: the initial difference in marginal valuation of reserves between the two banks, as well as the transaction costs. For the sake of simplicity, the problem is linearized. The two banks equalize through trading their marginal valuation until the difference in valuation is equal to the transaction costs. As it is not a priori clear which bank has the high and which one has the low valuation (while it is clear that the one with the high valuation has less reserves than the one with the low valuation), the formula has to rely on max() and min() functions offered by Excel, in order to order the two positions. (=IF(MAX(O23;J23)-MIN(24;J23)>$B$8; (MAX(K23;F23)-MIN(K23;F23))-$B$8*( MAX(K23;F23)-MIN(K23;F23 ))/(MAX(O23;J23)-MIN(24;J23));0)/2)
  17. ONR. The overnight rate is set simply as the mid-point between the two banks’ valuations before trading and before end of day aggregate shocks. (=(O23+J23)/2)
  18. End day reserves bank 1. This is the pre MM reserve position of bank 1, plus the reserves gained or lost by the interbank transaction with bank 2, plus or minus the impact from the afternoon AF aggregate shock. (=F23+SIGN(K23-F23)*P23+(C23-B23)/2)
  19. Same for bank 2
  20. Recourse MLF bank 1. Bank 1 may have to take recourse to the MLF – this is the case if its reserves are in negative territory even after taking maximum recourse to the liquidity providing of the two Taralac facilities. (=IF((R23+$B$10/2)<0;-(R23+$B$10/2);0))
  21. Same for bank 2
  22. Recourse DF bank 1: same as t but now for Deposit facility. (=IF((R23-$B$10/2)>0;(R23+$B$10/2);0))
  23. Same for Bank 2.
  24. Double recourse penalty level. This column provides a non-zero value only if one bank took recourse to the deposit facility, and the other tool recourse to the marginal lending facility. (=MAX(MIN(T23;W23);MIN(U23;V23)))

In rows 17 and 18, averages and standard deviations, respectively, are shown for all of these 24 simulation output variables. Moreover, key output appears also in column J, row 4 – 11. To produce functional relationships between parameters of output variables (e.g. volatility of overnight rates) and parameters of input variables (e.g. policy variables such as the width of corridor, or exogenous factors such as the volatility of relative liquidity shocks), I vary the input variables and then do each time a “copy-paste value” of the fields J4-J11 to the right, so that at the end I have one array of key variables for each variation of the input variable.

Obviously, to achieve really smooth curves of the functional relationships, one should either solve the relevant equations analytically, or program the simulations either in VBA or a programming language like e.g. MatLab and run many more simulation runs per parameter constellation.

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