CHAPTER 9
Calculating Capital for Market
Risk
INTRODUCTION
• VaR gives a solid foundation for assessing the
amount of capital that should be held by a bank
to protect it in the case of losses arising from
market risks
• There are three reasons for analyzing the capital
consumed by market risks
– Complying with industry regulations
– Calculating economic capital to control the bank's
default probability
– Measuring Risk-Adjusted Profitability
COMPLYING WITH INDUSTRY
REGULATIONS
• In 1996, the Basel Committee on Banking
Supervision recommended to national
regulators that the minimum capital to be
set aside for market risks should be based
on VaR.
• The regulatory capital should equal the
99% 1-day VaR, multiplied by 3, times the
square root of 10:
• (Regulatory) Capital =3x 100.5 x VaR99%
COMPLYING WITH INDUSTRY
REGULATIONS
• The square root of 10 represents the losses that
could occur over a 10-day holding period
– if there was a 10-day crisis in which the markets
became illiquid.
• The factor of 3 allows 3 such events to happen
each year
• If the back-testing shows an unusual number of
exceptions, the capital should be increased
according to Table 8-2 in the previous chapter
• In calculating the VaR, each bank is allowed to
use whatever method it thinks will best pass the
back-test.
CALCULATING ECONOMIC CAPITAL TO
CONTROL THEBANK'S DEFAULT
PROBABILITY
• To be allowed to stay in business by the
regulators, banks must hold at least the
capital discussed above
• However, if banks wish to maintain a high
credit rating and a low probability of
default, they may wish to hold more capital
• This amount is calculated using the
Economic Capital framework
CALCULATING ECONOMIC CAPITAL TO
CONTROL THEBANK'S DEFAULT
PROBABILITY
(1)the maximum probable loss such
that there is only probability p that
the profitability over a year will be
worse than wp
(2) p = probability [profit < Wp]
Discount value from
one year later
CALCULATING ECONOMIC CAPITAL TO
CONTROL THEBANK'S DEFAULT
PROBABILITY
• VaR gives us the maximum probable loss
that could happen with a 1% probability
over one day
• To get the economic capital, we need to
translate
– from 1% to the required confidence level
– from one day to one year
CALCULATING ECONOMIC CAPITAL TO
CONTROL THEBANK'S DEFAULT
PROBABILITY
• If we assume that the losses are Normally distributed,
then if a bank held capital equal to 2.32 times the
standard deviation of the possible annual losses, there
would be a 1% chance that the actual loss over 1 year
would exceed the capital
• If the bank held capital equal to 3.7 times the standard
deviation of losses, there would only be a 0.01% chance
that the actual loss would exceed the capital
– The bank could therefore expect to be rated AA or AAA
• Table 9-1 gives the relationship between the amount of
capital that a bank holds against market risks and the
consequent probability that the actual losses would be
greater than the capital
CALCULATING ECONOMIC CAPITAL TO
CONTROL THEBANK'S DEFAULT
PROBABILITY
CALCULATING ECONOMIC CAPITAL TO
CONTROL THEBANK'S DEFAULT
PROBABILITY
• Typically, A-rated institutions have a
default probability of around 0.1%
• From Table 9-1, we can say that an Arated institution should aim to hold capital
equal to 3.1 times the standard deviation
of the potential annual change in value
(assuming a Normal distribution)
• Economic Capital A = 3.1 xσ1year
CALCULATING ECONOMIC CAPITAL TO
CONTROL THE BANK'S DEFAULT
PROBABILITY
CALCULATING ECONOMIC CAPITAL TO
CONTROL THEBANK'S DEFAULT
PROBABILITY
• This is more than twice as high as the regulatory capital
• One reason for the discrepancy is that banks may wish
to be more creditworthy than is implied by the minimum
regulatory capital
• Another reason is that the regulatory capital implicitly
includes a reduction due to diversification between
market risks and the other risks in the bank
• A further reason is that we assumed that the standard
deviation of losses scales with the square root of time.
This assumption breaks down for two reasons
– The first is that over long periods, such as a year, effects such as
mean reversions manifest themselves.
– The second reason is that it assumes that the trader holds a
fixed position all year and takes whatever losses the market
brings. With the trader's skill, and with bank's policies to stop
losses, this should not be the case.
CALCULATING ECONOMIC CAPITAL TO
CONTROL THEBANK'S DEFAULT
PROBABILITY
• in general, the required economic capital for an A rating
was found to be between one and two times the
minimum regulatory capital
• Once the capital has been established for the portfolio
as a whole, it can be allocated to the individual traders,
subportfolios, transactions, or desks, using the VaRC
methodology
• The capital allocated to an individual transaction is the
VaRC for the transaction, divided by the VaR for the
whole portfolio, and multiplied by the capital for the
whole portfolio:
MEASURING RISK-ADJUSTED
PROFITABILITY
• Once we know the capital being consumed by a
particular transaction, we can calculate the RiskAdjusted Return on Capital (RAROC) and the
Shareholder Value Added (SVA)
• Knowing the risk-adjusted performance, we can
properly decide on whether the profit from a
transaction is worth the risk and whether a trader
is performing well or just risking a lot of the
bank's capital
• The RAROC is simply the net income from the
transaction divided by the capital consumed
MEASURING RISK-ADJUSTED
PROFITABILITY
If the transaction only
lasts a few days (T)
and only requires
capital for that time,
the
calculated RAROC
should be annualized
so that it can be
compared with the
RAROC
from other transactions
MEASURING RISK-ADJUSTED
PROFITABILITY
• The hurdle income is the amount of capital
consumed multiplied by the hurdle rate that the
shareholders require as a minimum return for
risking their capital
• The hurdle rate HT, for a short-term transaction
lasting T days can be calculated from annual
hurdle rate, HAnnual
MEASURING RISK-ADJUSTED
PROFITABILITY
• The required income is then the capital
consumed multiplied by HT:
– Required Income = Allocated Capital x HT
• The SVA is then the net income minus the
required income
– SVA = Net Income — Allocated Capital x HT