7 Value at Risk Limit Systems

7 Value at Risk Limit Systems 

7.1 Necessity of Risk Limits 

7.2 Identification of a Bank’s Total Value at Risk 

7.3 Dimensions of Value at Risk Limits 

7.4 Functional and Interdivisional Limits 

7.5 Time-Related, Intertemporal Limits 

7.6 Aggregation Problems of Limit Allocation 

Prof. Dr. Hans-Peter Burghof, University of Hohenheim, Financial Intermediation 144

7.1 Necessity y of Risk Limits 

Derives from the internal and regulatory request that risks have to be covered 

with equity. q y 

Therefore, financial resources for risk coverage have to be allocated to the 

particular business units. This allocation is done by defining risk limits (resp. 

VaR limits). 

Moreover, risk cluster should be avoided. Within a risk limit system, e.g. for 

market risks, limits for interest rate risks, foreign exchange risks and stock price 

risks are conceivable. 

Information and specific knowledge about the risks of the particular business 

units exist primarily in this units, thus the delegation of decision-making 

authority to lower management levels is important. 

Central risk allocation according to VaR limits guarantees that a bank’s total risk 

position does not exceed her risk bearing capacity. 


7.2 Identification of a Bank’s Total Value at Risk 

Risk Bearing Capacity of a Bank 

The risk bearing capacity is defined as the maximum loss the bank can handle 

without threatening the bank’s further existence; this loss should not be exceeded 

under no circumstances. 

Its level depends mainly on the equity capitalization of a bank. 

The risk bearing capacity as defined above goes beyond the officially reported 

equity and the regulatory equity requirements and includes, for instance, parts of 

the operational results and other valuation reserves in the bank’s assets. 


Classification of Risk Capital and Risk Bearing Capacity 

The total VaR limit is set up once a year and presents the origin for the bank’s 

risk limit system. 

Risk Capital Source 

Primary Risk Capital Excess Profits 

Secondaryy Risk Capital p 

Inner Reserves 

Tertiary Risk Capital Minimum Profit 

Common Stock 

Quarternary Risk Capital Open Reserves 

Quintary Risk Capital 

Special Items for Common Bank Risks 

Tier 2 Capital (Without Inner Reserves) 

Tier 3 Capital 


Risk Preferences of a Bank 

Which sources of the risk bearing capacity are included in the indentification of 

the risk capital/the total value at risk limits, depends on the risk preferences of 

the bank management. 

But the risk capital should only be a subset of the risk bearing capacity of a bank 

and should generally not exceed the latter. 

The identification of the level of the total value at 

risk limit of a bank is mainly depending on the risk 

bearing capacity and the risk preference. 


7.3 Dimensions of Value at Risk Limits 

For the following explanation, a theoretical distinction between two dimensions 

is helpful. 

But: from a practical point of view and according to an efficient risk limitation 

and risk adjustment, an integrated consideration is required. 

Functional, interdivisional limits: 

Efficient distribution of a given total 

risk limit to business units/profit 

center/portfolios... in terms of value at 

risk ikli limits. i 

li limit i addressee. dd 

Time-related, intertemporal limits: 

Temporal dynamic adjustment of the 

value at risk limits according to the 

current P&L situation of the relevant 

“Passive” risk limitation “Active” risk management 


7.4 Functional, Interdivisional Limits 

Breakdown/Distribution of the Total Value at Risk Limits 

At first, the total value at risks limits have to be distributed to the bank’s 

most important risk categories. 

It is common to distinguish at least between market price risks and 

counterparty risks. Furthermore, operational risks, liquidity risks and 

strategic risks are possible. 

Th Then, th there is i a further f th breakdown b kd of f limits li it ffor e.g. market ktrisks ik in i interest it t 

rate risks, foreign exchange risks, stock price risks and commodity risks. 

Now Now, these limits are broken down and distributed further to business units 

and within these even on individual persons (most likely on traders in the 

area of market risks). 

risks) 


The following figure is supposed to illustrate the prinicipal organisation 

of a limit structure, structure as described above: 

Stock Price Risks 

30 Mio. € 

Market Price Risks 

80 Mio. € 

Unit 

8 Mio. € 

TTotal lVl Value at Ri Risk k Li Limit i 

200 Mio. € 

… 

Foreign Exchange 

Risks 20 Mio. € 

… Team 

3 Mio. € 

Counterparty Risks 

100 Mio. € 

… … … 

Interest Rate Risks 

40 Mio. € 

… Trader 

1 Mio. € 

… 

Commodity Risks 

10 Mio. € 

Prof. Dr. Hans-Peter Burghof, University of Hohenheim, Financial Intermediation 151 

…

CConsideration id ti of f CCorrelations l ti 

FFor th the li limit it distribution, di t ib ti all ll operational ti l business b i units it hhave tto be b interpreted i t t d as 

one portfolio. 

The total value at risk limit has to be divided between the risk capital 

addressees according to the correlation of the potential loss factors. 

Hence Hence, diversification effects in the total portfolio must be considered considered. 

With regard of the correlations, the sum of the individual value at risk limits 

VL i of all different business units n is equal to or greater than the total value at 

risk limit VL total : 

n 

 

i1 

VL 

i 

VL 

total 


Regarding the limit structure illustrated above, above one recognizes that the sum 

of the limits per market risk category (100 million €) exceed the particular 

total value at risk limit (80 million €) for this unit . 

This can appears between each level of limit structures when considering 

correlations. 

The conservative manner of disregarding the risk compensating effects of 

correlation leads to lower value at risk limits on subordinate levels. 

Such an allocation leads to an under-utilization of limits on higher levels of 

aggregation; that is not conform to an efficient allocation (in terms of risk 

adjusted performance measures). 


Functionality of a Limit System 

The functionality of a limit system is based on the following restriction: 

every single transaction can only be executed within the scope of a value 

at risk ikli limit, i if there h iis a li limit i for f this hi kind ki d of f transactions i that h captures the h 

potential loss of that transaction with high probability. 

Withi Within the th given i value l at t risk i k li limits, it all ll business b i units it are allowed ll d tto 

execute deals freely and self dependent. 

But it must be secured that all transactions are taken into account of the 

corresponding limits immediately. 

The peripherally acting persons need to know their personal value at risk 

limits AND its current utilization. 


Violation of Limits 

Actions in the case of limit violations are different according to the affected 

management level. 

A violation on a subordinate level, e.g. a trader’s limit violation, can be tolerated if 

th the value l at t risk i k limit li it on a higher hi h level l l (team (t or bbusiness i unit it level) l l) is i not t in i 

danger. 

But if a limit on a critical level is violated (according to the bank’s risk profile), the 

senior management must define further proceedings. 

proceedings 


In the following, some proceedings for limit violations are illustrated 

for market risks: 

The affected limit is 

violated shortly. 

Some other 

unaffected units 

cannot exhaust their 

limits completely. 


violated shortly. 

Trading of the 

affected unit is 

cancelled temporally. 


permanently 

exceeded exceeded. 

More risk capital is 

admitted. More 

sources of the risk 

bearing capacity, e.g. 

inner reserves, are 

considered. 

The position that 

caused the violation 

is closed instantly instantly. 


Th The Allocation All ti of f Risk Ri k CCapital/Total it l/T t l Vl Value at tRikLi Risk Limits it 

Th The following f ll i table t bl gives i an overview i of f the th possible ibl proceedings: di 

Top-Down Bottom-Up Negotiation as an 

Iterative Process 

The business center 

simultaneously defines 

profit targets and the 

essential ti l risk i k capital it l tto 

achieve this targets. 

The feasibility y in banks 

is difficult, as the center 

does not have all 

detailed information 

about the profit p and 

risk situation of each 

risk addressee. 

The individual 

addressees state their 

profit targets and the 

required i d amount t of f risk i k 

capital on their own. 

Here, it is essential that 

the addressees can 

estimate their profit and 

risk factors realistically. 

There is a common 

negotiation about the 

admitted risk capital 

and dth the risk i kadjusted dj t d 

performance that has 

to be reached. 

Hence Hence, the efficiency of 

capital utilization can 

be increased. 

This approach is very 

costly though. 

Internal Market Place 

The admitted risk 

capital and 

performance targets 

are dderived i d ffrom 

supply (of the center) 

and demand (of the 

risk capital 

addressees). 

dd ) 

With this peripheral 

process, there is no 

way to define the total 

risk limit ex-ante. 


IIn reality, li there h is i often f a mixture i between b the h first fi three h methods. h d 

PPartly, tl there th are allocation ll ti processes which hi h are not t yet tstringent t i t and d structured. t t d 

The reason for that is that the risk allocation cannot be based on a method like 

capital budgeting or net present value calculation. 

Cash flow forecasting for one year is much more difficult for a trading unit than 

for a classical investment project. 

To meet the standards of an efficient risk capital p allocation, , RAPM and share 

holder value aspects shall always be taken into account. 


7.5 Time-Related, Intertemporal Limits 

Time i Reference f of fValue at Risk i Limits, i i Using i the Example of fTrading i Units i 

UUsually, ll a value l at t risk i k limit li it is i admitted d itt d for f one year, although lth h trading t di strategies t t i 

are often pursued for shorter periods, e.g. for one day. 

Hence Hence, a time time-related related disaggregation of annual value at risk limits is needed as the 

calculation period of this limit does not match the one-day holding period of the 

trading position. position 

Thus, the value at risk of the trading position cannot be subtracted from the 

annual value at risk limit as both values are incomparable. 

The value at risk of a trading position is calculated relatively too low caused by 

the short holding gpperiod. 


HHence, the h question i comes up, which hi hvalue l at risk i kli limit i is i available il bl ffor a subperiod, b i d 

especially for one day. 

To translate an annual into a one day value at risk limit, the literature suggests the 

square square-root-T-rule: root T rule: 

DL 

YL 

T 

* 

The one-day limit, DL, has to be computed from the annual limit YL and the square- 

root of T, where T is the unit of the annual limit (e.g. 250 trading days). 

* Simplified with the assumption of an expected daily return = 0. 


Methods of Limit Adjustment Over Time Time, Using the Example of Trading Units 

A fixed limit represents the most simple case: an unchanged limit is set for the 

complete business year. 

By doing so, current dynamic profit and loss conditions of the value at risk addressee 

remain unconsidered. 

As there is no intertemporal adjustment of the value at risk limit throughout the year, 

there is actually no active risk management but only a risk limitation by the annual 

limit. 


Hence, there is only a transition of the annual limit into a daily limit according to 

square-root-T-rule. 

YL 

DL FL with FL = fixed f limit 

T 

For instance, there are T = 250 trading days and an annual limit of 2 million €, 

then the daily limit equals 126.491,11 € 


AApplying l i the h second d method, h d the h lloss li limitation it ti li limit, it the h trader d calculates l l with i h 

the whole annual limit of (e.g.) 2 million € from the very first trading day on. But 

thi this limit li it is i not t available il bl to t him hi daily. d il 

Realized losses over time reduce the annual limit for the following days of the 

year; realized profits can increase the VaR limit to the initial value of 2 million €. 

Regular accounting for losses cuts down the trader’s scope of activities. 

Accounting for profits can only happen if there had been limit cuts due to losses 

so far. 


Additionally Additionally, there is the opportunity to compensate losses with realized past 

profits. Hence, one can “save” for future losses. 

The profits or losses (market value changes) in t are derived from: 

LL 

t Vt 

1 

VL 

Vt 1 

V R 

t 

with LL = loss limit. 

stands for the invested position from t-1 to t and R t for the daily returns. 

KV 

The accumulated profits or losses, , which occur between the first trading 

day of the year t=1 till t are calculated as follows: 

t 

LL 

LL 

KVt Vt 

s1 

Vt 

1 

Rt 

... 

V0 

 

s1 

LL 

where V0 is the amount of money invested between t=0 (the last trading 

day of the old year) and t=1 (the first trading day of the new year). 

t 

Prof. Dr. Hans-Peter Burghof, University of Hohenheim, Financial Intermediation 164 

R 

t

The h annual lvalue l at risk i kli limit, i YL LL 

t , computed d iin t equals l YL if 

KVt 0 

t V K YL 

0 K Vt 

and , if . 

YL defines the initial value at risk limit (2 million €) admitted at the 

beginning of the year. 

The daily value at risk limit that is available to the trader in t+1 is 

computed p by y the square-root-T-rule q again: g 

DL 

LL 

YLt 

T 

LL 


Th The third hi d method, h d the h ddynamic i limit, li it iis equal l to the h lloss limitation li i i limit, li i whereas h 

the available annual limit can exceed the 2 million € level by accounting for profits. 

Hence, the scope of the trader’s activities can be limited but it can also be enlarged 

infinitely (theoretically). 

(theoretically) 

YL 

YL K Vt 

with 

DL 

t 

KVt t 

DL 

DL 

Vt 

s1 

Vt 

1 

Rt 

... 

V0 

Rt 

s1 

The annual value at risk limit limit, , that is the basis for the trader’s trader s calculation calculation, is: 

YL 

DL 

t 

where V 0 DL is the amount invested on the last trading day of the passed year (t=0). 

The daily value at risk limit is calculated as follows: 

DL 

DL 

YLt 

T 

DL 


7.6 Aggregation Problems of Limit Allocation 

Two main aggregation problems have to be solved in the value at risk model as well 

as in a value at risk limit system. 

The aggregation of value at risk limits over the complete hierarchical limit structure 

respectively over the “limit portfolio”. 

The aggregation of value at risk limits over time (intertemporal scaling). 

Each problem can be allocated to one particular VaR limitation dimension mentioned 

above, which actually have to be viewed integrated in practice; that brings up even 

more problems. 


Problems of Integrating Both Dimensions of Value at Risk Limits 

The mentioned models for VaR limitation over time are only viable for one 

particular limit addressee. 

Within a hierarchical limit structure, the adjustment of individual value at risk limits 

cannot be done without interdependency p y effects on the whole value at risk limit 

system. 

If the value at risk limit of an addressee changes due to risk management actions, the 

aggregated value at risk limit changes immediately, too (ceteris paribus). 

This can lead to an over-utilization of the formerly utilized total value at risk limit. 


In a complex limit structure, the value at risk limits should never be adjusted on 

low hierarchy limit levels. 

Because of the correlations between the portfolios of the different limit addressees, 

the h risk i kmanagement actions i according di to the h operating i results l must bbe executed d on 

the highest level of aggregation, that is, on the total value at risk limit. 

Hence, this imperatively leads to a reconstruction of the value at risk limit system 

respectively to a reallocation of the modified risk capital. capital 

Finally Finally, due to the consideration of the operating results results, the profitability of the 

limit addressees changes. 


How Frequently Should/Could a Reallocation of Risk Capital Be Executed? 

A continuous value at risk limit management in combination with a continuous 

reallocation of all value at risk limits surely fails because of the 

technical/mathematical feasibility. 

A steady y confrontation of the trading g units with new limits does not seem 

reasonable. 

The same holds for a daily or weekly complete reconstruction of the limit system. 


Modifications should rather be executed if ongoing changes in the results and the 

profitability of trading units become apparent. apparent 

More realistic is a monthly or quarterly reconstruction of the value at risk limit 

system. 

In the short run, the methods of intertemporal limit management could be executed 

within predefined tolerance ranges by the corresponding limit addressees.

7 Value at Risk Limit Systems

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