FINANCIAL FINC3018
BANK MANAGEMENT
Week 4
BUSINESS
SCHOOL
Objectives of this Weeks Session
To examine the approaches to the assessment of
Risk
-
Daily Earnings at Risk (DEAR)
-
Various Value at Risk (VaR) Models
-
Their application to portfolios
-
The broader application of VaR
-
To examine some of the strengths and weaknesses of the
VaR approach
2
Recap - The Factors in Measuring Bank Risk
The likely Quantum or amount of the Loss arising from the
risk event;
Possibility or Probability of a risk event occurring; and
Measured in financial terms
Value at Risk can help with this analysis
3
Value at Risk (VAR)
Definition
- VAR measures the worst expected loss over a given
horizon under normal market conditions at a given
confidence level
- Value at Risk Phillipe Jorion
- Note this is not necessarily the worst actual loss
4
Phillipe Jorion on VaR
The greatest benefit of VAR lies in the imposition of
a structured methodology for critically thinking about
risk. Institutions that go through the process of
computing their VAR are forced to confront their
exposure to financial risks and to set up a proper
risk management function. Thus the process of
getting to VAR may be as important as the number
itself.
5
The Basis of VAR
Initially developed as a measure of market risk
- Has been extended to encompass other risks
Reflects the Portfolio Managers
- Beliefs about the future distribution of changes to the value of the
portfolio
- Relative aversion to the risk of loss
Actual future loss is unpredictable but expected losses can
be estimated
6
VAR Fundamentals
The expectation of loss relates to the probability of
certain events occurring in the future
The problem is that the future is unknowable
- but can possibly be estimated by reference to the past
VaR therefore observes and models past events and
assumes that the results reflect likely or possible
future events
7
VaR - The Basics
Observed Past Events and Consequent Changes in Value
Change In
Value ($)
-30 -20
-10
0
10
20
30
Probability
(%)
5
10
20
30
20
10
5
Cum.
Probability
(%)
5
15
35
65
85
95
100
If this represented all of the potential changes in the value of the portfolio over
a given timeframe what is the VaR with 95% confidence and 85% confidence?
8
The Resultant Loss Distribution
The Probability of Loss
100
90
80
Cumulative Probability
Distribution
Probability of Loss (%
)
70
60
50
40
30
20
10
0
-30
-20
-10
0
10
20
30
Amount of Loss
9
VAR Fundamentals
The expectation of loss relates to the probability of certain events
occurring in the future
Requires determining:
- 1. The Relevant Probability distribution to develop the probability
distribution
- VaR uses either or a combination of:
- Historical Simulation (historical observations)
- Estimated Variance - Co-Variance
- Monte Carlo Simulation
- We will focus on the historical method.
- 2. The appropriate time frame over which to consider potential
risk events
10
Historical Simulation
Estimate the value of the current portfolio on a number of dates in the
past or market rates (Vi)
Based on this estimate the distribution of value or market rate changes
from this sample by:
Vi = Vt + n Vt
Derive VaR either:
-
From cumulative distribution
- Percentile method
Via an assumption of normality (do some tests)
- Normal distribution (statistical) method (Riskmetrics)
Issues
-
Assumes that the past reflects the future
-
No estimate of parameters required
-
Reasonably heavy computational costs
11
Estimated Variance/Co-Variance
Standard deviations and correlations between instruments
obtained from historical data
Derive standardised risk instruments, eg the price change of a
range of corporate bonds could be related to the price change of
government bonds
Model the behaviour of the standardised instruments and relate to
all other instruments
Issues
- Assumes the future is like the past, therefore parameters can be
extrapolated into the future
- Less computationally intensive than historical method
- This is the approach pioneered by Riskmetrics
- Now rarely used
12
Monte Carlo Simulation
Generate a series of possible price changes based on:
- The type of process normal, lognormal etc
- Estimate parameters of price change
- Estimate correlations between securities
Maps a broader range of possible value changes
Issues
- Model must reflect reality
- Imposes quite heavy computational requirements
13
Monte Carlo Simulation
Simulates a range of possible outcomes based on
-
Current market rates
-
The volatility and the path by which values can migrate
-
Confidence intervals
$625
$620
$615
$610
VaR
$605
$600
$595
$590
588.06
$585
$580
Simulations below the red line represent values
outside or beyond the confidence interval
14
2. Determining the Time Frame
Depends on:
- The nature of the risks
- The likely impact of risk events
- How many such events could be borne in any one period (week,
month or year)
- Likely time to immunise the portfolio
Commonly VaR for market risk is:
- Daily (DEAR) or 1 week to one month
Commonly VaR for credit risk is:
- Assessed over one year (Week 10)
15
Daily Earnings at Risk and VaR
DEAR is VaR with a one day time horizon, ie where N = 1 below
If we are interested in assessing the potential change in value over
longer time horizons we can use the relationship:
VaRN = DEAR N
This however assumes that daily volatility is constant over the time
horizon
-
ie that a financial shock does not have compounding effects (ie
autocorrelation).
16
DEAR and VaR (Cont)
An alternative technique for assessing VaR over longer time frames is to model
potential changes directly over those longer time periods
-
eg a one week VaR could be assessed by reference to value or market changes
over a series of one week observations, and these form the observations in the
distribution
Vi =Vt + n Vt
Where n = 7
17
Typical Risk/Return distribution
of market based securities
Normal Distribution of Returns
Expected Loss (mean)Potential Gain
Risk of Loss
Confidence Level
Catastrophic Loss
Unexpected Loss
The above segments of the probability distribution will be examined later in this session
18
VaR and Bank Financial Risks
VaR is not only applicable to market risks generally (as
defined in text) but also specifically to interest rate risk
and
-
As we will see later in the course, it is also applicable to other risks such
as credit risk.
For interest rate sensitive assets and liabilities we can:
-
Model the potential behaviour (volatility) of interest rates for various
terms to maturity
-
Apply a confidence level to this model to provide an estimate of the
potential change in interest rates
-
Apply this change in interest rates to the price sensitivity of the portfolio
(or individual assets) to assess the potential change in value
-
Price sensitivity could be measured in terms of either Duration or PVBP
19
Observed Interest Rate Changes
for 1 day for 2 year int. rates
Change In
Rates bp
-30 -20
-10
0
10
20
30
Probability
(%)
5
10
20
30
20
10
5
Cum.
Probability
(%)
5
15
35
65
85
95
100
now in Basis Points (bps)
20
Assessing the Potential Change in Interest Rates
There are two approaches to assessing the potential
change in interest based on the probability distribution
1. The Percentile method
2. The Statistical method
21
1. The Percentile Approach
Assumes the identified distribution is the only set of values
possible
VaR = VCI
Using the previous data:
-
The maximum likely change in interest rates at 95%
confidence interval is +20bps (VCI)
-
-
We +20bps use because an increase in interest rates is
always the basis of our risk assessment (it also usually
involves a loss)
The mean ( ) of the previous distribution is zero, thus the
VaR (the potential change in interest rates) is also 20bps
22
Standard Normal Distribution Confidence Levels
using the empirical rule, or the 3-sigma rule
Confidence level*
85%
90%
95%
97.5%
99%
99.9%
Approx Std Devs
1
1.25
1.65
1.95
2.33
3
* One Tailed
23
2. The Statistical Approach
Assuming a Normal Distribution (see previous)
VaR = l
Again, using the previous data
-
The Standard deviation is 14.564bp
-
Assuming normality a one-tailed confidence level of 95% equates to 1.65
standard deviations
-
The 95% 1 day potential rate change would therefore be:
1.6514.564= 24.03
-
24bps or 0.24%pa
Note: The value we are looking for in the
change in value ie a distance from the mean
Where
l = confidence level factor
= standard deviation
24
Calculating the Portfolio's VaR
using Duration
Applying to the security that we looked at last week (see
the following two slides):
-
Current Value = $100 (Par), say Face Value is $1,000,000
Dmod = 1.77(ie a 1.77% change in value for a 1% change in yield)
Using the VaR for one day as 24bp (previous slide)
-
One Day VaR (DEAR) using duration is:
$1,000,000 x 1.77% /100 x 24 = $4,248
-
Similarly a 5 day VaR could be assessed as:
VaR5 = $4,248 x 5 = $9,499
25
Duration and Price (Value) Change
from Week 3
Duration represents the value weighted term of the security
A small change in yield will result in a change in price (as a proportion) as
follows:
P = D (r / 1+ r / f )
Eg in the previous example where D=1.8616 and the price was 100, a
small change in yield (0.01%) would change price by 0.000177 or
0.0177%
-
Note a 1% change in yield (although large) would result in a 1.77% change in
price
Where: r = the market yield and f = the frequency of compounding (i.e. f = 2 is
equivalent to a semi-annual compounding frequency
26
Calculating Duration
from Week 3
Term
Years
0.5
1.0
1.5
2.0
Termx
PV
n
Cashflow
Discounted PV
1.0
2.0
3.0
4.0
5
5
5
105.0
4.7619
4.5351
4.3192
86.3838
2.3810
4.5351
6.4788
172.7675
100.0000
186.1624
Duration is :
M
odified
Duration
186.1624
100.0000
1.8616 years
1.77%
Eg 2 year Semi-annual Bond 10.00% coupon @ Par
27
An Alternative Approach to
Time Horizons Greater than 1 day
If using historical models set n>1 eg:
Vi = Vt + n Vt
This would obviate the need to assume no autocorrelation but
places greater reliance on the past reflecting the future
Requires the calculation of a sliding window of observations
-
eg a 5 day VaR would examine price changes from today to 5 days
ago (1 observation) and yesterday to 6 days ago (a 2nd observation),
and so on
28
Applying VaR to Portfolios using Duration
In using Dmod of the portfolio we are required to assume that
the interest rate change applies uniformly to all assets in the
portfolio
-
The portfolio that was examined in Week 3 has an assessed
change in value of -$981.31 per basis point.
-
This was discussed in Week 3
See the following slide
If all interest rates change by 24bps this equates to a
change in the value of the portfolio (subject to convexity) of:
-
-$981.31 x 24 = $23,551.44
29
Using Duration Gap
from Week 3
Duration gap reflects the extent of leverage in the banks balance
sheet
- Via the ratio of the PVl to PVa (the difference is the PV (MV) of
Equity)
As with the Duration for an individual security;
- Dgap multiplied by the MVa of assets & the change in yields
provides an estimate of change in value of entire balance sheet,
using the data from the slide
- -0.05 x 210 million x 0.01/1.07 = -$98,131
- or a 0.01% (1bp) change (increase) is -$981.31
What conditions or assumptions are required?
- Uniform changes in yield
- A single common yield or the use of the principle that for a small
change r/1+r approximates r
Where: Dgap = .05 years, Assets = $210 million and current rates 7%pa, r =1%
30
Applying VaR to Portfolios using PVBP
Interest rates are, however, likely to change differently at different points
in the yield curve therefore we need to use PVBP
-
The yield curve may actually steepen or flatten
As noted on p295 of text this approach is used by Riskmetrics.
The previous results (slide 24) were 2 year rates, but ASSUME:
-
that 6 month rates are 10% less volatile (ie 22bps)
-
that 1 year rates are equally volatile as 2 yr rates calculated earlier in Slide 24 (ie 24bps)
that 3 months rates are 20% less volatile (ie 19bps);
Given that the changes in interest rates for the various maturities are not the same,
we would need to look at each cashflow separately
-
this can be done using PVBP, eg see next slide:
31
VaR Applied with PVBP (from Week3)
Total Net $m
Rates
PVBP
Var
VaR
$ mill
Cashlow
PV +0.01
$
bps
$
(ie 1 bp)
3 mth
-10.05
5.50%
-9.91344
244.48
19
4,645.12
6 mth
-13.95
6.50%
-13.5102
654.25
22
14,393.50
12 mth
21.15
7.52%
19.6430
-1,893.17
24
-45,436.08
Total
-2.85
-
-26,397.46
-5.13211
-994.44
Note: This was the portfolio examined in Week 3
32
Practical Issues to be Considered
-
Rates for different maturities do not move
independently of one another, they are correlated
-
-
The use of the maximum change for each maturity may
give rise to excessive results
Models will factor this in via examining the extent of
correlation between interest rates changes for varioous
maturities
Assets across diverse portfolios also do not move
independently of one another
-
-
Must therefore factor in correlations between portfolios
Positive correlation increases risk, while negative
correlation (a form of hedge) reduces risk
VaR cannot therefore be simply aggregated
33
Application of VaR to Other Instruments
Foreign Exchange
-
DEAR = Position X Price Volatility
-
Price volatility is assessed in the same way
See pp296-297
Equities
-
DEAR = Position X Equity Market Volatility
-
Assuming the portfolio has a beta of close to 1
See pp2970298
Can also be applied to other portfolios
-
Commodities and Credit
We will examine Credit Modelling later
VaR across portfolios can however only be aggregated
after allowing for the extent of correlation
34
What is Expected Loss?
Expected loss arises in the course of normal business
Statistically usually seen as the mean of the
distribution
The loss should be absorbed in the normal course of
business
- Usually as a direct charge against profit
35
What is Unexpected Loss?
Not unusual but their occurrence is outside the norm
Larger than the expected losses
Statistically is based on some level of confidence of
historical or simulated observations this is VaR
Should this loss also be absorbed in the normal
course of business?
- A charge against either reserves or capital
36
What are Catastrophic Losses?
Improbable extreme events which are rare
- LTCM
Statistically these are outlier observations. Stress testing
is intended to help identify these events
How should these losses be absorbed when they
occur?
- A charge against reserves or capital?
- Covered by re-insurance?
37
Typical Non-market risk distribution
Approach was borrowed from the insurance industry
and is how insurance companies manage their risks
38
Use of VaR
Traditionally VaR has been used to measure Market Risks
Increasingly applied to the measuring Credit Risks
- (Weeks 9 and 10)
The next stage, which is being developed, is to apply this
technique to the measurement of Operational Risks
Has potential for all risks for which value changes are
reasonably observable
39