A Taxonomy Of Cash Flows

• The identification and taxonomy of the cash flows that can occur during the business activity of a financial institution is crucial to building effective tools to monitor and manage liquidity risk. The taxonomy focuses on two main dimensions: time and amount. Like any other classification, this one also depends on the reference point of view. The cash flows are classified by considering them from a certain point in time; for example, cash flows may fall in one of the categories from, say, today’s point of view. They can also change category when the point of view is shifted to some other date in the future.

• The first dimension in the classification of future cash flows, is time:
  • Cash flows may occur at future instants that are known with certainty at the reference time (e.g., today). In this case, according to the time of their appearance, they are deterministic.
  • Cash flows may manifest themselves at some random instants in the future. In this case they are stochastic (again, according to time).

• The second dimension to consider is the amount
  • Cash flows may occur in an amount that is known with certainty at the reference time. In this case, they are deterministic in amount.
  • Cash flows may occur in an amount cannot be fully determined with certainty at the reference time. In this case, they are stochastic in amount.
• Classification according to the amount, though, can be further broken down such that subcategories can be identified. Moreover, when the amount is deterministic, cash flows can be labelled simply as fixed as a result of being set in such a way by the terms of a contract.
• When the amount is stochastic, it is possible to recognize four possible subcategories
  • Credit related when amount uncertainty can be due to credit events, such as the default of one or more of the bank’s clients
  • Indexed/contingent, when the amount of cash flows depends on market variables, such as Libor fixings;
  • Behavioral, when cash flows are dependent on decisions made by the bank’s clients or counter-parties: these decisions can only be loosely predicted according to some rational behavior based on market variables and sometimes they are based on information the bank does not have
  • New business, in which cash flows originated by new contracts that are dealt in the future and more or less planned by the bank, so that their amount is stochastic.
• Deterministic amount/deterministic time cash flows -They are typically related to financial contracts, such as fixed rate bonds or fixed rate mortgages or loans. These cash flows are produced by payments of periodic interests (e.g., every six months) and periodic repayment of the capital instalments if the asset is amortizing. It should be noted that not only bonds or loans held in the assets of the bank generate these kinds of cash flows, but also bonds issued and loans received by the bank held in its liabilities.
• Deterministic amount/stochastic times cash flows can arise either because the contract can provide for a given sum to be paid or received by the bank, or because the amount depends on a choice made by the bank. An example of the first kind of cash flow can be represented by the payout of one-touch options. The buyer of this type of option receives a given amount of money when the underlying asset breaches some barrier level, from below or from above. The time is unknown from the reference instant, but the amount can easily be deducted from the contract’s terms. An example of cash flows that depend on the bank’s decisions can be represented by the positive cash flows originated by the withdrawals of credit lines received by the bank. In this case, the amount can safely be assumed to be some level between 0 and the limit of the credit line chosen by the bank depending on its needs, but it can occur at any  time until the expiry of the contract, since liquidity needs may arise at some stochastic time.
• Stochastic amount/deterministic time cash flows can be indexed/contingent (i.e., linked to market variables). Examples of such cash flows are, amongst others, floating rate coupons that are linked to market fixings (e.g., Libor) and the payout of European options, which also depend on the level of the underlying asset at the expiry of the contract. Stochastic amount/deterministic time cash flows can also be related to the replacement of expiring contracts and to new business activity. For example, the bank may plan to deal new loans to replace exactly the amount of loans expiring in the next two years, and this will produce a stochastic amount of cash flows since it is unsure whether new clients will want or need to close such contracts. Old contracts expire at a known maturity so that cash flows linked to their replacement are known as well. Moreover, the new issuance of bonds by the bank can be well defined under a time perspective, but the amount may not be completely in line with the plans.
• In the stochastic amount/stochastic time category, credit-related cash flows can be identified that are due to the default of one or more clients. The default time and recovery rate is obviously not known at the reference time, so that the amount and time when these cash flows occur  are stochastic. An example can be represented by the missing cash flows, after the default, of the contract stream of fixed interests and of capital repayment of the loan. A missing cash flow may be considered a negative cash flow that alters a given cash flow schedule: if a loan, for example, is fixed rate amortizing, then we have a modified cash flow schedule on the contract interest rate and capital repayment times. Moreover, the recovery value after default can be inserted in this category of cash flows.
• There may also be indexed/contingent cash flows in this category. Examples are the payout of American options, both when the bank is long or short them, because the exercise time is stochastic and depends on future market conditions that determine the optimality of early exercise
• Another type of cash flow in this category is behavioral; for example, when a bank’s clients decide to prepay the outstanding amount of their loans or mortgages. In this case the bank may compute the prepaid amount received on prepayment: the prepayment time cannot be predicted with certainty by the bank, so that the time when the cash flows occur is stochastic in this case. Other behavioral cash flows, are those originated by credit lines that are open to clients: withdrawals may occur at any time until the expiry of the contract and in an uncertain amount, although within the limits of the line. Withdrawals from sight or saving deposits belong to this category too.
• In the new business subcategory, the evolution of completely new contracts may not follow the pattern predicted in business development documents, so that related cash flows are both amount and time stochastic. For example, new retail deposits, new, mortgages and new assets in general may be only partially predicted. In general, cash flows due to new business, or to the rollover of existing business, are quite difficult to model and to manage.

Liquidity Options

• Some of the categories of cash flows described previously are connected with the exercise of so- called liquidity options. These kinds of options are conceptually no different from other options, yet the decision to exercise them depends on their particular nature. More specifically, a liquidity option can be defined as the right of a holder to receive cash from, or to give cash to, the bank at predefined times and terms. Exercising a liquidity option does not directly entail a profit or a loss in financial terms, rather it is as a result of a need for or a surplus of liquidity of the holder. This does not mean that the exercising of a liquidity option cannot be linked to financial effects; on the contrary, such a link is sometimes quite strong: this is clear from the examples of liquidity options we provide below.
• Comparing these options with standard options usually traded in financial markets, the major difference is that standard options are exercised when there is a profit, independently of the cash flows following exercise, although typically they are positive. The decision to exercise liquidity options is dependent on the cash flows occurring after the exercise, which may or may not be independent from profit in financial terms. For example, consider the liquidity option that a bank sells to a customer when the bank opens a committed credit line: the obligor has the right to withdraw whatever amount up to the notional of the line whenever she wants under  specified market conditions, typically a floating rate (say, 3-month Libor) plus a spread, that will be mostly determined only by the obligor’s default risk. The option to withdraw can be exercised when it makes sense under a financial perspective; for example, if the spread widened due to worsening of the obligor’s credit standing: in this case funds can be received under the contract’s conditions (which are kept fixed until expiry) and hence there is a clear saving of costs in terms of fewer paid interests on the line’s usage rather than opening a new line. On the other hand, the line can be used even if the credit spread shrinks, so that it would be cheaper for the obligor to get new funds in the market with a new loan, but maybe for convenience, she chooses to withdraw the needed amounts from the credit line, even though it doesn’t make sense from a financial perspective.
• Another example of a liquidity option is given by sight and saving deposits: the bank’s clients can typically withdraw all or part of the deposited amount with no or short notice.
• Exercising a liquidity option can also work the other way round: the bank’s client has the right to repay the funds before the contract ends. Although the bank would benefit in this case from the greater amount of liquidity available, economic effects are usually negative and thus cause a loss.

An example is given by the prepayment of fixed rate mortgages or loans: they can be paid back before the expiry for exogenous reasons, often linked to events in the life of the client such as divorces or retirements. In this case the bank would not suffer any loss if market rates rose or stayed constant, on the contrary it could even reinvest at better market conditions those funds received earlier than expected.

More often prepayment is triggered by a financial incentive to close the contract and reopen it under current market conditions if the interest rate falls. In this case, prepayment would cause a loss since replacement of the mortgage or closure of the loan before maturity would be at rates lower than those provided for by old contracts.

• In the end, although liquidity options can be triggered by factors other than financial convenience, the effect on the bank can be considered twofold:

1. A liquidity impact on the balance sheet, given by the amount withdrawn or repaid.
2. A (positive or negative) financial, impact, given by the difference between the contract’s interest rates and credit spread and the market level of the same variables at the time the liquidity option is exercised, applied on the withdrawn or repaid amount.

Sometimes the second impact is quite small, as, for example, when a client closes a savings account: the bank’s financial loss is given in this case by the missing margin between the contract deposit rate and the rate it earns on the reinvestment of received amounts (usually considered risk-free assets), or by the cost to replace the deposit with a new one that yields a higher rate. The liquidity impact, on the other hand, can be quite substantial if the deposit has a big notional.

• The financial effects of liquidity options can be directly, although partially, hedged by a mixture of standard and statistical techniques, but the liquidity impact can only be managed by the tools that involve either cash reserves or a constrained allocation of the assets in liquid assets or easy access to credit lines (i.e., a long position in liquidity options for the bank). All of these imply costs that have to be properly accounted for when pricing contracts to deal with clients, so that models to price long and short liquidity options have also to be designed.

Liquidity Risk

• Definition: Liquidity risk – The event that in the future the bank receives smaller than expected amounts of cash flows to meet its payment obligations.
• This definition encompasses both funding liquidity risk and market liquidity risk. In fact, if a bank is not able to fund its future payment obligations because it is receiving less funds than expected from clients, from the sale of assets, from the interbank market or from the central bank, this risk may produce an insolvency situation if the bank is absolutely unable to settle its obligations, even by resorting to very costly alternatives. Market liquidity risk according to the definition above is the result of the inability to sell assets, such as bonds, at fair price and with immediacy, and leads to a situation in which the bank receives smaller than expected amounts of positive cash flows.
• Another risk dimension for liquidity should be considered: the cost of liquidity, or cost of funding, and the related risk can be defined as follows.
• Definition : Funding cost risk – The event that in the future the bank has to pay greater than expected cost (spread) above the risk-free rate to receive funds from sources of liquidity that are available. Not too much attention has been paid to the modelling of funding cost risk, although in   the past it was always recognized as such but deemed to have little impact on banking activity. This figure shows the rolling series of the CDS Itraxx Financial spread, which can be used as a proxy for funding spread over the risk-free curve for top European banks on a 5-year maturity. The spread was almost constant and low until 2007 (the outbreak of the subprime crises in the US). This means that banks were considered virtually default risk free and funding costs, meant as the spread over the risk-free curve, had a very limited effect on the profitability of banks.
• Since the middle of 2007, financial institutions have no longer been able to always raise funds at low spreads: the volatility of spreads even over interbank market rates (e.g., Libor)  dramatically increased, and the amount of funding available in the capital market declined, at least for the banking sector. As a consequence of these two reasons, the funding policy of banks is now subject to constraints so as to abate the average funding cost, reduce rollover risk (regarding both the quantitative and risk dimension) and hence make credit intermediation activity still a profitable business.
• The conclusion that can be drawn this is that modern liquidity risk management must consider both the quantitative dimension and the cost dimension as equally important and robust modelling must be developed for the two dimensions. Both dimensions of liquidity risk can be incorporated in the following comprehensive definition –
• Definition: Liquidity risk – The amount of economic losses due to the fact that on a given date the algebraic sum of positive and negative cash flows and of existing cash available at that date, is different from some (desired) expected level.
• This definition includes a manifestation of liquidity risk as:
  • Inability to raise enough funds to meet payment obligations, so that the bank is forced to sell its assets, thus causing costs related to the non-fair level at which they are sold or to suboptimal asset allocation. The complete inability to raise funds would eventually  produce an insolvency state for the bank. These costs refer to the quantitative dimension of liquidity risk.
  • Ability to raise funds only at costs above those expected. These costs refer to the cost dimension of liquidity risk.
  • Ability to invest excess liquidity only at rates below those expected. We are in the opposite situation to point 2, and it is a rarer risk for a bank since business activity usually hinges on assets with longer durations than liabilities. These (opportunity) costs also refer to the cost dimension of liquidity risk.

Quantitative Liquidity Risk Measures

• Cash flows classified according to the taxonomy given earlier are produced by two classes of factors.

1. Causes of liquidity – All factors referring to existing and forecast future contracts originated by the ordinary business activity of a financial institution can be considered as the causes of liquidity risk. Cash flows generated by the causes of liquidity can be both positive and negative.
2. Sources of liquidity – All factors capable of generating positive cash flows to manage and hedge liquidity risk and can be disposed of promptly by the bank determine the liquidity generation capacity (defined in the following) of the financial institution.

• Let \( cf_{e}^{+}(t_0, t_i) = E[cf^{+}(t_0, t_i)] \) be the sum of expected positive cash flows occurring at time \(t_0\) from the reference time \(t_0\).
• Let \( cf_{e}^{-}(t_0, t_i) = E[cf^{-}(t_0, t_i)]\) be the sum of expected negative cash flows occurring on the same date.
• The positive and negative cash flows for the set of contracts and/or securities \(\{d_1, d_2, \dots, d_n\}\) can be defined as:

\( cf_{e}^{+}(t_0, t_i) = E \left[ \sum_{j=1}^{N} cf^{+}(t_0, t_i; d_j) \right] \)

and

\( cf_{e}^{-}(t_0, t_i) = E \left[ \sum_{j=1}^{N} cf^{-}(t_0, t_i; d_j) \right] \)

• From a reference time 𝑡 , the cumulative amount of all cash flows starting from date 𝑡 up to time 𝑡 :

\( CF(t_0, t_a, t_b) = \sum_{i=a}^{b} \left( cf_{e}^{+}(t_0, t_j) + cf_{e}^{-}(t_0, t_j) \right) \)

• The term structure of expected cash flows (𝑇𝑆𝐸𝐶𝐹) can be defined as the collection, ordered by date, of positive and negative expected cash flows, up to expiry referring to the contract with the longest maturity, say \(t_b\) –

\( TSECCF(t_0, t_b) = \{ cf_{e}^{+}(t_0, t_0), cf_{e}^{-}(t_0, t_0), cf_{e}^{+}(t_0, t_1), cf_{e}^{-}(t_0, t_1), \dots, cf_{e}^{+}(t_0, t_b), cf_{e}^{-}(t_0, t_b) \} \)

At the end of the 𝑇𝑆𝐸𝐶𝐹, with an indefinite expiry corresponding to the end of business activity, there is reimbursement of the equity to stockholders. 𝑇𝑆𝐸𝐶𝐹 is often referred to as the maturity ladder. It is also standard practice to identify short-term liquidity, up to one year, and structural liquidity, beyond one year.

• The term structure of cumulated expected cash flows (𝑇𝑆𝐸𝐶𝐶𝐹) is similar to the 𝑇𝑆𝐸𝐶𝐹 . It is the collection of expected cumulated cash flows, starting at 𝑡₀ and ending at 𝑡_b , ordered by date

\( TSECCF(t_0, t_b) = \{ CF(t_0, t_0, t_1), CF(t_0, t_0, t_2), \dots, CF(t_0, t_0, t_b) \} \)

The 𝑇𝑆𝐸𝐶𝐶𝐹 is useful because banks are interested not only in monitoring the net balance of cash flows on a given date, but also how the past dynamic evolution of net cash flows affects its total cash position on that date.

Assets and Liabilities Reclassified According to Maturity

Expiry	Assets	Liabilities
1	20	10
2		10
5	50
7		70
10	30
>10		20
Total	100	100

The previous example was overly simplified and can be made a bit more realistic if the interest payments are considered that assets and liabilities yield. Assume a yearly period for coupon payments and an average yield common to all contracts expiring on a given date: the interest yielded by each contract is shown in this table.

Interest Yield of the Assets and Liabilities of the Simplified Balance Sheet Reclassified

Expiry	Interest Yield
	Assets (%)	Liabilities (%)
1	5.00%
2		4.00%
5	6.00%
7		4.50%
10	6.50%
>10

The Term Structure of Cash Flows and of Cumulated Cash Flows

Expiry	cf+		cf–		TSECCF
	Notional	Interest	Notional	Interest
1	20	5.95	0	-3.55	22.40
2	0	4.95	-10	-3.55	13.80
3	0	4.95	0	-3.15	15.60
4	0	4.95	0	-3.15	17.40
5	50	4.95	0	-3.15	69.20
6	0	1.95	0	-3.15	68.00
7	0	1.95	-70	-3.15	-3.20
8	0	1.69	0	0	-1.51
9	0	1.69	0	0	0.18
10	30	1.69	0	0	31.87
>10	0	0	-20	0	11.87

• When interest payments are added, the TSECF is built as in this table, where there is also the TSECCF.

In this figure, cumulated cash flows are shown.

• Some important points to be noted are
  • The 𝑇𝑆𝐸𝐶𝐹 includes the cash flows from all existing contracts that comprise the assets and liabilities: in many cases cash flows are stochastic because they can be linked to market indices, such as Libor or Euribor fixings (interest rate models are needed to compute expected cash flows).
  • The cash flows are adjusted to consider credit risks: credit models have to be used to take into account defaults on an aggregated basis by also considering the correlation existing amongst the bank’s counterparties.
  • Cash flows are adjusted to account for liquidity options: behavioural models are used for typical banking products like sight deposits, credit link usage and prepayment of mortgages.
  • The cash flows originated, by new business increasing the assets should be included: they are typically stochastic in both the amount and time dimensions, so they are treated by means of models that consider all related risks.
  • The rollover of maturing liabilities, by similar or different contracts, and new bond issuances (which could also be included in the new business category) to fund the increase in assets, have to be taken into account. The risks related to the stochastic nature of these flows have to be properly measured as well resulting in a need to set up proper liquidity buffers.
• The 𝑇𝑆𝐸𝐶𝐹, and hence the 𝑇𝑆𝐸𝐶𝐶𝐹, do not include the flows produced by the sources of cash flows. The sources of cash flows are tools to manage the liquidity risk originated by the causes of cash flows. The task of the Treasury Department is to monitor the 𝑇𝑆𝐸𝐶𝐹 and the 𝑇𝑆𝐸𝐶𝐶𝐹.The perfect condition is reached if the 𝑇𝑆𝐸𝐶𝐶𝐹 is positive at all times. This means that positive cash flows are able to cover negative cash flows, both of which are generated by usual business activity. Although this is the ideal situation it cannot be verified for two reasons:
• Many of the cash flows belong to categories that are stochastic in the amount and/or the time dimension, such that the 𝑇𝑆𝐸𝐶𝐹 always forecasts just the expected value of a distribution of flows. As a result the 𝑇𝑆𝐸𝐶𝐶𝐹 contains only expected values as well : if it is distribution of flows. As a result the 𝑇𝑆𝐸𝐶𝐶𝐹 contains only expected values as well: if it is positive on average most of the time, the distribution of cumulated cash flows at a given time can also actually envisage negative outcomes with an assigned probability.
• The temporal distribution of the maturities of the assets and the liabilities could produce periods of negative cumulated cash flows. These periods may be accepted if they are short and can be managed effectively with the tools to be introduced later. When the TSECCF shows negative values, on an expected basis, this means that the bank may become insolvent and eventually go bankrupt.
• The perfect condition is reached if the 𝑇𝑆𝐸𝐶𝐶𝐹 is positive at all times. This means that positive cash flows are able to cover negative cash flows, both of which are generated by usual business activity. Although this is the ideal situation it cannot be verified for two reasons:
  • Many of the cash flows belong to categories that are stochastic in the amount and/or the time dimension, such that the 𝑇𝑆𝐸𝐶𝐹 always forecasts just the expected value of a distribution of flows. As a result the 𝑇𝑆𝐸𝐶𝐶𝐹 contains only expected values as well : if it is distribution of flows. As a result the 𝑇𝑆𝐸𝐶𝐶𝐹 contains only expected values as well: if it is positive on average most of the time, the distribution of cumulated cash flows at a given time can also actually envisage negative outcomes with an assigned probability.
  • The temporal distribution of the maturities of the assets and the liabilities could produce periods of negative cumulated cash flows. These periods may be accepted if they are short and can be managed effectively with the tools to be introduced later. When the TSECCF shows negative values, on an expected basis, this means that the bank may become insolvent and eventually go bankrupt.
• Liquidity generation capacity (𝐿𝐺𝐶) is the main tool a bank can use to handle the negative entries of the 𝑇𝑆𝐸𝐶𝐶𝐹. It can be defined as follows.
• Definition : Liquidity generation capacity – The ability of a bank to generate positive cash flows, beyond contractual ones, from the sources of liquidity available in the balance sheet and off the balance sheet at a given date.
• The LGC manifests itself in two ways:
  • Balance sheet expansion with secured or unsecured funding.
  • Balance sheet shrinkage by selling assets.

• Balance sheet expansion can be achieved via
  • borrowing through an increase of deposits, typically in the interbank market (retail or wholesale unsecured funding);
  • withdrawal of credit lines the financial institution has been granted by other financial counterparties (wholesale unsecured funding);
  • issuance of new bonds (wholesale and retail unsecured funding).
• New debt is not same as one which is planned to roll over existing contracts or to fund new business. In this case, related cash flows would be included in the 𝑇𝑆𝐸𝐶𝐹 and in the 𝑇𝑆𝐸𝐶𝐶𝐹.
• Balance sheet shrinkage is operated by selling assets, starting from more liquid ones such as Treasury bonds, corporate bonds and stocks: they are traded in the market actively and can be sold within a relatively short period. Reduction may also include the sale of less liquid assets, such as loans or even buildings owned by the bank, within a more extended time horizon.
• Repo transactions can also be considered separately from the other cases and labelled as “balance sheet neutral”.
• The bank may prefer to consider only liquidity that can be generated without relying on external factors, such as clients or other institutional counterparties. It is easy to recognize that LGC related to balance sheet expansion is dependent on these external factors, whereas balance sheet reduction, or “balance sheet neutral” repo transactions, are not. So it is possible to present an alternative distinction within LGC; namely –
  • Balance sheet liquidity (BSL), or liquidity that can be generated by the assets existing in  the balance sheet. BSL is tightly linked to balance sheet reduction LGC and it is also the ground on which to build liquidity buffers.
  • Remaining liquidity, originated by the other possible ways mentioned above, which relates to balance sheet expansion.
• A similar classification within LGC is based on the link between the generation of liquidity and the assets in the balance sheet
• Security-linked liquidity including –
  • secured withdrawals of credit fines received from other financial institutions;
  • secured debt issuance;
  • selling of assets and repo.
• Security -unlinked liquidity including
  • unsecured borrowing from new clients through new deposits;
  • withdrawals of credit lines received from other financial institutions;
  • unsecured bond issuance.
• It is immediate clear that security-linked liquidity is little more than BSL liquidity, or the liquidity obtained by balance sheet reduction.
• To sum up, three types of sources of liquidity can be identified and included in the equations which will be discussed after this:

1. Selling of assets (AS).
2. Secured funding using assets as collateral and via repo transactions (RP).
3. Unsecured funding (USF) via withdrawals of committed credit lines available from other financial institutions and via deposit transactions in the interbank market.

The first two sources generate security-linked BSL by reducing the balance sheet or keeping it constant. The third source generates security-unlinked non-BSL by expanding the balance sheet. It is worth stressing that the unsecured funding of point 3 is not the same as the unsecured funding we inserted in the 𝑇𝑆𝐸𝐶𝐹 and 𝑇𝑆𝐸𝐶𝐶𝐹, but is mainly related to the rollover of existing liabilities or the issuance of new debt to finance business expansion.

• The term structure of 𝐿𝐺𝐶 can be defined as the collection, at reference time 𝑡₀, of liquidity that can be generated at a given time 𝑡_j, by the sources of  liquidity, up to a terminal time 𝑡_b, i.e.

\( TSLGC(t_0, t_b) = \{ AS(t_0, t_1), RP(t_0, t_1), USF(t_0, t_1), \dots, AS(t_0, t_b), RP(t_0, t_b), USF(t_0, t_b) \} \)

where

\( AS(t_0, t_1) \) is the liquidity that can be generated by the sale of assets at time \(t_i\) , computed at the reference time \(t_0\), \(RP(t_0,t_1)\) and \(USF(t_0,t_b)\) are defined similarly.

• The term structure of cumulated 𝐿𝐺𝐶 is the collection, at the reference time 𝑡₀, of the cumulated liquidity that can be generated at a given time 𝑡_i, by the sources of liquidity, up to a terminal time 𝑡_b:

\( TSCLGC(t_0, t_b) = \left\{ \sum_{i=0}^{1} TSLGC(t_0, t_b), \sum_{i=0}^{2} TSLGC(t_0, t_i), \dots, \sum_{i=0}^{b} TSLGC(t_0, t_i) \right\} \)

The quantities entering in the 𝑇𝑆𝐿𝐺𝐶 and hence the 𝑇𝑆𝐶𝐿𝐺𝐶 are expected values, since they all depend on stochastic variables such as the price of assets and the haircut applied to repo transactions. Moreover, the stochasticity of the amount of unsecured funding that it is possible to raise in the market could and should be considered.

• The sources of liquidity contributing to the 𝑇𝑆𝐿𝐺𝐶 belong either to the banking or the trading book.
• In the banking book the sources of liquidity are all the bonds available for sale (𝐴𝐹𝑆) and  other assets that can be sold and/or repoed relatively easily : they are referred to collectively as eligible assets. The repo can be seen as a collateralized loan that the bank may receive: as such it can be safely assumed that liquidity can be obtained more easily than unsecured funding. The funding cost is also dramatically reduced for the bank.
• Haircut is another factor determining the actual liquidity that can be obtained by a repo or a collateralized loan transaction. It is the cut in the market value of the bond indicating how much the counterparty is willing to lend to the bank, given an amount of bonds transferred as collateral. The haircut will depend on the volatility of the price of the collateral bond and on the probability of default of the issuer of the bond. Haircuts have to be modelled not only for unencumbered assets to assert their liquidity potential, but also for encumbered assets, involved in collateralized loans, to forecast possible margin calls and reintegration of the collateral when their prices decline by an appreciable amount.
• Finally, received committed credit lines, which are not exactly in the balance sheet until they are used (in which case they become liabilities) have to be included in the 𝑇𝑆𝐿𝐺𝐶 and their amount and future actual existence taken into account.
• In the trading book, bonds and other assets which are similar to those included in the  banking book can be identified as sources of liquidity. Stocks and some structured products, like ABSs or even more complex structures can also be included.
• Building the 𝑇𝑆𝐿𝐺𝐶 can be very difficult, since the assumptions made for a given period affect other periods. The process to build the actual 𝑇𝑆𝐿𝐺C should be carried out in a greater number of steps.
• It should also be stressed that the 𝑇𝑆𝐿𝐺𝐶 is intertwined with the 𝑇𝑆𝐸𝐶𝐹 (and thus the 𝑇𝑆𝐸𝐶𝐶𝐹): there are feedback effects when defining the 𝑇𝑆𝐿𝐺𝐶 that affect expected cash flows, which makes the building process a recursive procedure to be repeated until an equilibrium point is reached. For example, if the 𝑇𝑆𝐿𝐺𝐶 for a given period can be fed by a bond that can be pledged as collateral or repoed, then the cash flows of this bond should be excluded from the 𝑇𝑆𝐸𝐶𝐹 for the period of the loan or repo contract. Missing cash-flows for the corresponding period worsen the cash flow term structure (although marginally), but this means that they must also be included in the analysis.
• In the end, most of the problems in building the 𝑇𝑆𝐿𝐺𝐶 are caused by unencumbered  assets (most of which are what we earlier referred to as AFS bonds), both because the bank must keep track of how many are either sold or repoed out, and because the liquidity amount that can be extracted from these assets depends on several risk factors. The term structure of available assets (𝑇𝑆𝐴𝐴) is a useful tool that helps monitoring this part of the LGC more thoroughly, and gives an idea about the amount of asset that can be used to generate liquidity.
• When an asset, such as a bond, is purchased by a bank a corresponding outflow equal to the (dirty) price is recorded in the cash position of the bank. During the life of the bond coupon flows are received by the bank and finally, at expiry, the face value of the bond is reimbursed by the issuer. All these cash flows should be considered contract related, so that they are included in the 𝑇𝑆𝐸𝐶𝐹 and the 𝑇𝑆𝐸𝐶𝐶𝐹. The likelihood of the issuer defaulting should also be taken into account.
• The 𝑇𝑆𝐴𝐴 is affected by purchases, since it records increases in the security for the amount bought. When the asset expires, the 𝑇𝑆𝐴𝐴 records a reduction to zero of its availability, since it no longer exists. During the life of the asset, the availability is affected by total or partial selling of the position, and by repo transactions. The 𝑇𝑆𝐴𝐴 is also affected by other kinds  of transactions that can be loosely likened to repo agreements, but have different impacts. Buy/sellback (and sell/ buyback) transactions and security lending (and borrowing) should also be analyzed. What matters in terms of availability for liquidity purposes is the possession of the asset rather than its ownership.
• Following is the description of the effects of transactions on 𝑇𝑆𝐴𝐴 –
  • Repo transactions – Initially the bank receives cash for an amount equal to the price of the asset reduced by the haircut, and it delivers the asset to the counterparty. Although the asset is still owned by the bank, its possession passes to the counterparty. This asset becomes encumbered and cannot either be sold or used as collateral until the end of the repo agreement. Payments by the asset during the repo agreement belong to the bank since it is the owner, so that the 𝑇𝑆𝐸𝐶𝐹 and in the 𝑇𝑆𝐸𝐶𝐶𝐹 are not affected in any way. The 𝑇𝑆𝐴𝐴 of the asset is reduced by an amount equal to the notional of the repo agreement, whereas the cash flow received by the bank initially and the negative cash flow at the end are both entered in the 𝑇𝑆𝐿𝐺𝐶. Repo transactions produce a liability in the balance sheet, since they can be seen as collateralized debts of the bank.
  • Reverse repo transactions – Initially the bank pays cash for an amount equal to the  price of the asset reduced by the haircut and receives the asset. The asset is owned by the counterparty, but it is now in the possession of the bank, so that it can be used as collateral by the bank for other transactions until the end of the repo agreement. The payments by the asset during the repo agreement belong to the counterparty, so that they do not enter the 𝑇𝑆𝐸𝐶𝐹 or the 𝑇𝑆𝐸𝐶𝐶𝐹, but the cash flow paid by the bank at the start and the cash flow received at the end are included, just once. The 𝑇𝑆𝐴𝐴 of the asset is increased by an amount equal to the notional of the repo agreement. The 𝑇𝑆𝐿𝐺𝐶 is not affected but the asset can be repoed until the end. Reverse repo transactions are treated as assets in the balance sheet, as they can be seen as collateralized loans to the counterparty.
  • Sell/buyback transactions – These are similar to repo transactions in terms of the exchange of cash and of the asset, with the difference that ownership passes to the buyer (the counterparty) at the start of the contract together with the possession. All payments received for the asset before the buyback belong to the counterparty. The cash flows between the start and end of the contract will be removed from the 𝑇𝑆𝐸𝐶𝐹 and the 𝑇𝑆𝐸𝐶𝐶𝐹. The TSAA of the asset decreases by an amount equal to the notional of the sell/buyback contract. The 𝑇𝑆𝐿𝐺𝐶 is affected in the same way as in the repo agreement, since sell/buyback transactions are a way to generate BSL. Sell/buyback transactions represent a commitment for the bank at the end of the contract.
  • Buy/sellback transactions – These are similar to reverse repo transactions in terms of the exchange of cash and of the asset, but ownership passes to the buyer (the bank) initially along with possession. Hence payments received for the asset before the sellback belong to the bank, which enter the 𝑇𝑆𝐸𝐶𝐹 and the 𝑇𝑆𝐸𝐶𝐶𝐹, along with the cash flows at the start and end that relate to the purchase and sale. The 𝑇𝑆𝐴𝐴 of the asset is increased by an amount equal to the notional of the buy/sellback agreement. The 𝑇𝑆𝐿𝐺𝐶 is not affected but the asset can be repoed until the end, so that it can be altered until this date. Buy/sellback transactions represent an asset for the period of the contract.
  • Security lending – These are similar to sell/buyback transactions in terms of exchange of the asset, but no cash is paid by the counterparty to the bank (except a periodic fee as service remuneration). Only possession passes to the counterparty, so that the payments received for the asset before the end of the contract belong to the bank and they enter the 𝑇𝑆𝐸𝐶𝐹 and the 𝑇𝑆𝐸𝐶𝐶𝐹 along with interest paid by the counterparty at expiry when returning the asset to the bank. The 𝑇𝑆𝐴𝐴 of the asset decreases by an amount equal to the notional of the lending, since the bank cannot use it as collateral or sell it. The 𝑇𝑆𝐿𝐺𝐶 is not affected and the asset cannot produce any liquidity until the end of the contract. Security lending represents an asset of the bank for the period of the contract.
  • Security borrowing – These are similar to buy/sellback transactions in terms of exchange of the asset, but no cash is paid by the bank to the counterparty (except a periodic fee as service remuneration). Possession passes to the bank, so that the payments received for the asset before the end of the contract belong to the counterparty. The 𝑇𝑆𝐸𝐶𝐹 and the
  • 𝑇𝑆𝐸𝐶𝐶𝐹 are not affected apart from the interest paid by the bank at expiry of the borrowing. The 𝑇𝑆𝐴𝐴 of the asset increases by an amount equal to the notional of the borrowing, since the bank can use it as collateral provided it returns it to the counterparty at expiry. The TSLGC is not affected but the asset can produce liquidity until the end of the contract. Security borrowing represents a liability of the bank.
• Other assets, such as stocks, do not have a specific expiry date. In this case contract cash flows entering the 𝑇𝑆𝐸𝐶𝐹 and the 𝑇𝑆𝐸𝐶𝐶𝐹 are just the initial outflow representing the price paid to purchase the asset and the periodic dividend received. Moreover, non-maturing assets can be the underlying of repo transactions, buy/sellback (sell/buyback) contracts and security lending (and borrowing): the analysis is the same as above.

The Term Structure of Cash Flows and Changes

Type	Ownership	Possession	Changes to
			TSECF/TSECCF	TSAA	TSLGC
Buy	Bank	Bank	Yes	Yes	No/Possible
Sell	Counterparty	Counterparty	Yes	Yes	Yes
Repo	Bank	Counterparty	No	Yes	Yes
Reverse repo	Counterparty	Bank	Yes	Yes	No/Possible
Sell/buyback	Counterparty	Counterparty	Yes	Yes	Yes
Buy/sellback	Bank	Bank	Yes	Yes	No/Possible
Security lending	Bank	Counterparty	Yes	Yes	No
Security borrowing	Counterparty	Bank	Yes	Yes	No/Possible

The TSAA can now be built keeping these results in mind, since for a given asset it is defined as the collection, for each date from an initial time 𝑡₀ to a terminal date 𝑡_b, of the quantity in possession of the bank, regardless of its ownership. In fact, the main purpose of the 𝑇𝑆𝐴𝐴 is to indicate how much of the asset can be used to extract liquidity and thus its contribution to global 𝐿𝐺𝐶.
In more formal terms, for an asset 𝐴₁ –

\( TSAA^{A1}(t_0, t_b) = \{ A_{1}^{P}(t_0), A_{1}^{P}(t_1), \dots, A_{1}^{P}(t_b) \} \)

where

\( A_{1}^{P}(t_i) \) is the quantity of the asset in possession of the bank at time \(t_i\). On an aggregated basis, regarding a set of 𝑀 securities in possession of the bank, the total 𝑇𝑆𝐴𝐴 including all the assets is:

\( TSAA^{A1}(t_0, t_b) = \left\{ \sum_{m=1}^{M} A_{m}^{P}(t_0), \sum_{m=1}^{M} A_{m}^{P}(t_1), \dots, \sum_{m=1}^{M} A_{m}^{P}(t_b) \right\} \)

The TSAA only shows how many single securities, or all of them, are available for inclusion in LGC. Obviously, this does not imply that the notional amount can be fully converted into liquidity.

Impact Of Purchase Example

The bank buys a bond at time 0 for a notional amount of 1,000,000 at a price of 98.50; the payment is settled after 3 days, when the bond’s possession also passes to the bank. The bond pays a semiannual coupon of 10% p.a. and it expires in 2 years. This table shows what happens to the term structures The 𝑇𝑆𝐸𝐶𝐹 records an outflow equal to the notional amount of the bond times the price (we assume the bank buys the bond upon a coupon payment, so that the dirty price and the clean price are the same) occurring on the settlement date, 3 days after the reference time 0 (or 0.01 years). The 𝑇𝑆𝐴𝐴 records an increase of the quantity available to the bank of the bond until its expiry, when it is reset to zero. The 𝑇𝑆𝐿𝐺𝐶 is unaffected. The last two columns show the price and the haircut. They are expected values and for the moment we consider them as given, although they can be the output of some model or just assumptions of the bank.

Purchase of a Bond: Effects on Term Structures

Time	Operation	TSECF	TSECCF	TSAA	TSLGC	Price	Haircut (%)
0	Buy					99.85	15
0.01	Settlement	-985,000	-985,000	1,000,000		99.85	15
0.25	Coupon	50,000	-935,000	1,000,000		99.85	15
0.5	Coupon	50,000	-885,000	1,000,000		99.90	15
0.75				1,000,000		99.90	15
1	Coupon	50,000	-835,000	1,000,000		99.90	15
1.25				1,000,000		99.90	15
1.5	Coupon	50,000	-835,000	1,000,000		99.95	15
1.75				1,000,000		99.95	15
2	Coupon + Reimbursement	1,050,000	215,000			100.00	15

• Assume now the bank decides to sell a quantity of the bond equal to a notional of 500,000 after 9 months (or 0.75 years). This trade can be dealt to generate liquidity, so that the 𝑇𝑆𝐶𝐿𝐺𝐶 records an inflow equal to the amount times the price, including the accrued interests (500,000 × (99.90/100 + 10% × 0.25) = 512,000 as well. The 𝑇𝑆𝐸𝐶𝐹 and the 𝑇𝑆𝐸𝐶𝐶𝐹 are modified so as to show the reduced amounts of interest and capital received on the scheduled dates. The TSAA records a cut in the available amount for 500,000 until expiry of the bond, when it drops to zero. This table shows the results.

Selling of a Bond: Effects on Term Structures

Time	Operation	TSECF	TSECCF	TSAA	TSLGC	Price	Haircut (%)
0	Buy					99.85	15
0.01	Settlement	-985,000	-985,000	1,000,000		99.85	15
0.25						99.85	15
0.5	Coupon	50,000	-935,000			99.85	15
0.75	Sell	25,000	-910,000	500,000	512,000	99.90	15
1	Coupon	25,000	-885,000	512,000	512,000	99.90	15
1.25				512,000	512,000	99.90	15
1.5	Coupon	25,000	-835,000	512,250	512,250	99.95	15
1.75				512,250	512,250	99.95	15
2	Coupon + Reimbursement	525,000	-360,000		512,500	100.00	15

Impact Of Repo/Reverse Repo Example

• Assume the bank repoes the bond after 3 months (0.25 years) for a notional amount equal to 500,000 and a period of 6 months. Given the price and the haircut of the bond, and keeping accrued interest in mind, the amount of cash received by the bank is:

(500,000 × (99.85% + 10% × 0.25) × (1 – 15%) = 424,469.

The TSCLGC indicates an increase of liquidity, whereas the TSAA indicates that the available quantity of the bond dropped to 500,000. The bank pays 9% as interest on this repo transaction, so that the terminal price paid when getting the bond back is:

424,469 × (1 + 9% × 0.5) = 443,569.84

The difference 443,569.84 – 424,469 = 19,101.09 should be considered as a contract cash flow so that it enters the TSEC on the date at the end of the repo. The TSCLGC drops to zero and the TSAA returns the available amount back to 1,000,000.

• After 1 year and 3 months (1.25 years) the bank deals a 6-month reverse repo on this bond for a notional of 500,000. The price it pays to deliver the bond at inception is

500,000 × (99.90% + 10% × 0.25) × (1 – 15%) = -424,681

This amount enters the TSECF and alters the TSECCF as a consequence. The TSAA increases up to 1,500,000 since the bond is in possession of the bank. The TSCLGC is left unchanged. At the end of the reverse repo contract, assuming the interest rate paid by the counterparty is 11%, the inflow received by the bank is:
424,681 × (1 + 11% × 0.5) = 471,396

which should be considered fully as a contract cash flow, thus entering the TSECF; the bond is returned to the counterparty and consequently the TSAA is set back to 1,000,000.

• The effects discussed above are summarized in this table

Repo and Reverse Repo of a Bond: Effects on Term Structures

Time	Operation	TSECF	TSECCF	TSAA	TSLCGC	Price	Haircut (%)
0	Buy			1,000,000		99.85	15
0.01	Settlement	-985,000	-985,000	500,000	424,469	99.85	15
0.25	Repo	-985,000		500,000	424,469	99.85	15
0.5	Coupon	50,000	-935,000	500,000	424,469	99.85	15
0.75	End repo	-19,101	-954,101	1,000,000		99.90	15
1	Coupon	50,000	-904,101	1,000,000		99.90	15
1.25	Reverse repo	-424,681	-1,328,782	1,500,000		99.90	15
1.5	Coupon	50,000	-1,278,676	1,500,000		99.95	15
1.75	End reverse repo	471,396	-807,386	1,500,000		99.95	15
2	Coupon + Reimbursement	1,050,000	195,899			100.00	15

Impact Of Buy/Sellback And Sell/Buyback Example

• When the bank operates buy/sellback (or sell/buyback) operations, the effects are different. Assume that after 3 months (0.25 years) the bank buys 400,000 bonds and sells it back after 6 months at the forward price. At the start of the contract the bank pays:

400.000 × (99.85% + 10% × 0.25) = 409,400

• The effects of a sell/buyback of the bond starting after 1 year and 3 months (1.25 years) and terminating after 6 months (1.75 years) can also be demonstrated. The price received by the bank is:

300.000 × (99.90% + 10% × 0.25) = 307,200

which    is    included    in    the 𝑇𝑆𝐶𝐿𝐺𝐶 since the operation can be seen as a way to extract 𝐵𝑆𝐿 from the available assets; the 𝑇𝑆𝐴𝐴 indicates a reduction of the available quantity down to 700,000.

Impact Of Lending/Borrowing Example

• Assume that the bank lends 500,000 of the bond after 3 months for a period of 6 months. The 𝑇𝑆𝐸𝐶𝐹 does not record any cash flow at the inception of the contract, whereas the 𝑇𝑆𝐴𝐴 shows a reduction of the available quantity of 500,000. After 6 months the bond is returned to the bank (the 𝑇𝑆𝐴𝐴 increase) and the bank receives a fee for the lending, which is assumed to be 3% p.a.: 500,000 × (3% × 0.5) = 7,500
• The coupon paid during the lifetime of the contract are possessed by the legal owner (i.e., the bank).
• The bank borrows a quantity of 300,000 of the same bond at 1.25 years for a period of 6 Months. The TSAA is only affected at the start and end of the contract. The TSECF only records the borrowing fee paid by the bank: 300,000 × (3% × 0.5) = 4,500

The Term Structure Of Expected Liquidity

• The term structure of expected liquidity (𝑇𝑆𝐿 ) is basically a combination of the 𝑇𝑆𝐸𝐶𝐶𝐹 and the TSLGC. Formally, it can be written as:

\( TSL_e(t_0, t_b) = \{ TSECCF(t_0, t_0), TSECCF(t_0, t_1) + TSCLGC(t_0, t_1), TSECCF(t_0, t_2) + TSCLGC(t_0, t_2), \dots, TSECCF(t_0, t_b) + TSCLGC(t_0, t_b) \} \)

where

the term \(TSECCF(t_0,t_0)\) is simply the cash existing at the initial time in the balance sheet, so that:

\( TSECCF(t_0, t_0) = Cash(t_0) \)

• The 𝑇𝑆𝐿 is in practice a measure to check whether the financial institution is able to cover negative cumulated cash flows at any time in the future, calculated at the reference date 𝑡 . The 𝑇𝑆𝐿 must always be positive if the financial institution has to be solvent all the time. The 𝑇𝑆𝐿 includes all possible expected cash flows generated by ordinary business activity, new business, the liquidity policy operated and the measures taken to cope with negative cumulated cash flows. If in the end it is impossible to exclude negative expected cumulated cash flows, then it is also impossible to prevent the financial institution from becoming insolvent.
• The second example of this chapter where the interest payments were added and the TSECF was built in a table (containing 𝑐𝑓⁺ and 𝑐𝑓^–) be expanded it to take account of the 𝑇𝑆𝐴𝐴 and the 𝑇𝑆𝐶𝐿𝐺𝐶 with the objective of finally building a 𝑇𝑆𝐿_e. The main results are shown in this table. The 𝑇𝑆𝐸𝐶𝐶𝐹 is negative between the 7th and 8th year. This negative cumulated cash flow must be covered and in the balance sheet there is a bond that can be sold to create ( 𝐵𝑆𝐿 ) liquidity. In fact, the 𝑇𝑆𝐴𝐴 includes an amount for the bond equal to 30 until the 6th year, then in the 7th year an amount of 4 is sold at the (expected) price of 99.00, so as to generate a liquidity of 3.96, which is included in the 𝑇𝑆𝐶𝐿𝐺𝐶 thereafter. It should be noted that selling the bonds affects the TSECF, and hence the TSECCF, in the two ways shown in the previous section: there are fewer inflows for interest paid by the bond and the final reimbursement is lower as well. In this example the change in the TSECF does not produce other negative cumulated cash flows, so the LGC can be limited to selling the bond.
• The 𝑇𝑆𝐿_e is the sum of the 𝑇𝑆𝐸𝐶𝐶𝐹 and the 𝑇𝑆𝐶𝐿𝐺𝐶 at each period as shown in the previous table. It is always greater than or equal to zero, so the bank is in (expected) liquidity equilibrium. This figure shows how the 𝑇𝑆𝐸𝐶𝐶𝐹, the 𝑇𝑆𝐶𝐿𝐺𝐶 and the 𝑇𝑆𝐿_e have evolved: the first and the last term structure clash on the same line until the seventh year, when the TSECCF becomes negative and it has to be counterbalanced by the 𝑇𝑆𝐶𝐿𝐺𝐶.

The Term Structure of Expected Liquidity and Its Building Blocks

Years	TSECF	TSECCF	TSCLGC	TSLe	TSAA	Price
0	0	0	0	0	30	97.00
1	22.4	22.4	0	22.4	30	97.20
2	-8.6	13.8	0	13.8	30	97.45
3	1.8	15.6	0	15.6	30	97.60
4	1.8	17.4	0	17.4	30	98.00
5	51.8	69.2	0	69.2	30	98.20
6	-1.2	68	0	68	30	98.60
7	-71.2	-3.2	3.96	0.76	26	99.00
8	1.69	-1.51	3.96	2.45	26	99.50
9	1.69	0.18	3.96	4.14	26	99.75
10	27.69	27.87	3.96	31.83	—
>10	-20	7.87	0	7.8

• The fact that cash flows are stochastic suggests that not only one synthetic metric of the distribution (i.e., the expected value) should be taken into account, but also some other measure related to its volatility. In this way it is possible to build the same term structure from a different perspective showing the extreme values that both positive and negative cash flows may assume during the time of their occurrence. In order to achieve this result, the single cash flows originated by contracts on and off the balance sheet need to be linked to risk factors related to market, credit and behavioural variables.

• The first concept to introduce is the positive cash-flow-at-risk, at a given confidence level 𝛼, defined as

\( cfaR_{\alpha}^{+}(t_0, t_1) = cf_{\alpha}^{+}(t_0, t_i; x) – E[cf(t_0, t_i; x)] = cf_{\alpha}^{+}(t_0, t_i; x) – cf_{e}(t_0, t_i; x) \)

where, 𝑥 is an array of 𝑅 risk factors, which include market, credit and behavioural variables, i.e. 𝑥 = [𝑥₁, 𝑥₂ ,…, 𝑥_R ]. Here, on a given date 𝑡_i, determined by the reference date 𝑡₀, the maximum positive cash flow, which is computed at 𝛼, is reduced by an amount equal to the expected amount of the (sum of positive and negative) cash flows on the same date

• Analogously, a negative cash-flow-at-risk is defined as:

\( cfaR_{1-\alpha}^{-}(t_0, t_i) = cf_{1-\alpha}^{-}(t_0, t_i; x) – E[cf(t_0, t_i; x)] = cf_{1-\alpha}^{-}(t_0, t_i; x) – cf_{e}(t_0, t_i; x) \)

Here, on a given date 𝑡 , determined by the reference date 𝑡 , the minimum negative cash flow, which is computed at 1 – 𝛼, is netted with the expected cash flow occurring on the same date \( (cf_{e}(t_0, t_i; x)) \)

• The distribution of cash flows ranges from the smallest, possibly but not necessarily, negative ones to the largest, possibly but not necessarily, positive ones. Given a confidence level of 𝛼, on the right-hand side of the distribution all cash flows bigger than \( cf_{\alpha}^{+}(t_0, t_1; x) \), whose total probability of occurrence is 1 – 𝛼, are neglected. In the same way, on the left-hand side of the distribution all cash flows smaller than \( cf_{1-\alpha}^{-}(t_0, t_i; x) \), whose total probability of occurrence is still 1 – 𝛼, are not taken into account.
• The term structure of unexpected positive cash flows, given a confidence level of 𝛼, is the collection of positive 𝑐𝑓𝑎𝑅 for all the dates included between the start and the end of the observation period:

\( \overline{TSCF}_{\alpha}^{+}(t_0, t_b) = \{ cfaR_{\alpha}^{+}(t_0, t_0), cfaR_{\alpha}^{+}(t_0, t_1), \dots, cfaR_{\alpha}^{+}(t_0, t_b) \} \)

• Similarly, the term structure of unexpected negative cash flows, given a confidence level of 1 – 𝛼, is:

\( TSCF_{1-\alpha}^{-}(t_0, t_b) = \{ cfaR_{1-\alpha}^{-}(t_0, t_0), cfaR_{1-\alpha}^{-}(t_0, t_1), \dots, cfaR_{1-\alpha}^{-}(t_0, t_b) \} \)

• The \( TSCF_{\alpha}^{+} \) and \( TSCF_{1-\alpha}^{-} \) can be described as the upper and lower bound of the term structure of cash flows, centered around the expected level that is given by the 𝑇𝑆𝐸𝐶𝐹. It does not make much sense to build a cumulated \( TSCF_{\alpha}^{+} \) and \( TSCF_{1-\alpha}^{-} \) , since they will rapidly diverge upward or downward at unreasonable levels without providing accurate information for liquidity risk management. It is much more useful to build a term structure of unexpected liquidity that includes both the term structure of cash flows and liquidity generation capacity jointly computed at some confidence level 𝛼.
• The 𝑇𝑆𝐿 (term structure of liquidity) can be redefined at the maximum extreme at confidence level 𝛼

\( TSL_{\alpha}(t_0, t_b) = \{ cf_{\alpha}(t_0, t_1), \dots, cf_{\alpha}(t_0, t_b) \} \)

where the initial condition is that \( TSL_{\alpha}(t_0, t_0) = Cash(t_0) \).

Cash Flows At Risk And The Term Structure Of Liquidity At Risk

• Similarly, the TSL with minimum cash flows at confidence level \(1 – \alpha\) is:

\( TSL_{1-\alpha}(t_0, t_b) = \{ cf_{1-\alpha}(t_0, t_1), \dots, cf_{1-\alpha}(t_0, t_b) \} \)

• These two term structures of liquidity, together with the term structure of expected liquidity \(TSL_e\), help to define a term structure of liquidity-at-risk (TSLaR), which is a collection of unexpected cash flows at each date in a given period [to, tol, calculated as the difference between the minimum and the average level of cash flows. Although it is possible to compute the T’SLaR for both the unexpected maximum and minimum levels, for risk management purposes it is more sensible to refer to the minimum unexpected levels, since unexpected inflows should not bring about problems. Formally the TSLaR, at a confidence level of \(1 – \alpha\), is:

\( TSLaR_{1-\alpha}(t_0, t_b) = \{ cf_{1-\alpha}(t_0, t_1) – TSECCF(t_0, t_1) – TSCLGC(t_0, t_1), \dots, cf_{1-\alpha}(t_0, t_b) – TSECCF(t_0, t_b) – TSCLGC(t_0, t_b) \} \)

Each element of the \( TSLaR_{1-\alpha} \) is the difference between corresponding elements of the \( TSL_{1-\alpha} \) and the \( TSLaR_{1-\alpha} \) and the \(TSL_e\).