Class 23 - Chap 10
1
Purpose: to understand what market risk is and how it is
measured

Brief introduction to market risk

Measurement methods:
◦ RiskMetrics
◦ Historical Back Simulation
◦ Monte Carlo Simulation
2
Financial
Institution
Dealer
Trading Book
Investor
Banking Book
“Tradable” assets/liabilities
•Short horizon investments
Investment assets/liabilities
• Long horizon investments
• Liquid securities
• Illiquid securities
• Long and short positions in:
• Usually consist of:
•
•
•
•
•
•
Bonds
Commodities
FX Futures/Options
Equity Securities
Options
Securitizations
• CMO
• RMBS
•
•
•
•
Consumer loans
Commercial Loans
Retail Loans
Branches
Market Risk is the risk associated with daily
fluctuations in the price of actively traded assets,
liabilities and derivatives - i.e. the risk of losses in
value on an FIs trading book
3
4

FIs need an answer to the following question to understand their exposure to
market risk
How much money will the firm lose on its trading portfolio if the
market has a really bad day, month, year …?
How do we
define a bad day

Value at risk (VaR) is an essential tool used in answering this question

What horizon?
◦ Regulators usually consider “tradable” assets/liabilities as those held for horizons
less than 1 year – these assets/liabilities are included in the trading book and the VaR
◦ FIs usually consider “tradable” assets/liabilities as those held for a much shorter
horizon
5
Three Main Measurement Methods
1.
RiskMetrics (variance/covariance)
2.
Historical (Back Simulation)
3.
Monte Carlo Simulation
6

Developed by JPMorgan in 1994
◦ The object was to produce a single number that summarized the firms
market exposure across all markets in which it traded
◦ In 1994 JPMorgan had 120 independent units trading:






Fixed income
Foreign Exchange
Commodities
Derivatives
Emerging Markets Securities
Proprietary assets
◦ 2008 JPMorgan held a trading portfolio of $460 billion – typical value
for a major money center bank
7

RiskMetrics begins by measuring the FI’s Daily Earnings at Risk (DEAR)
DEAR =
Total position
Value
X
Extreme Loss
Per Unit
Example: if a financial institution has a DEAR of $2 mill at
For the
most
paraverage,
this
Howhas
do awe
calculate
the 95%
level
– on
that bank
5%
chance of
is
easy
to
calculate
this
piece?
losing $2 mill or more tomorrow

We are going to do this for three markets
1. Fixed Income
2. Foreign Exchange
3. Equity

The analysis is shown for the 1day horizon but it can be generalized to any
horizon
8
Dear for Fixed Income Portfolio
9
Suppose an FI has a position in BBB rated zero coupon bonds with total face value of $10,631,483 with
an average maturity of 26 years that it plans to hold for less then 2 months. The average yield to maturity
of the bonds is 13.5%. Find the 95% DEAR of the portfolio.
Step 1 find the extreme change in interest rates
Yield
Baathis so we need some historical data
We are going to use value at risk
toonfind
9.5
for the yield on BBB rated bonds – Federal Reserve
8.5
7.5
6.5
5.5
Change in Baa Yield
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
10
Suppose an FI has a position in BBB rated zero coupon bonds with total face value of $10,631,483 with
an average maturity of 26 years that it plans to hold for less then 2 months. The average yield to maturity
of the bonds is 13.5%. Find the 95% DEAR of the portfolio.
What is the change in bond YTM that
I would expect to be exceeded only
5% of the time Why is it positive?
Changes in Baa Yield
Frequency
80
6
60
σ = 0.069795%
4
40
20
2
0
0
Mean
= -0.00034%
Standard Deviation = 0.069795%
5%
1.6449 STDevs
-0.00034%
From the tables, 5% of the area under the curve is to the right of 1.6449
on the standard normal distribution. So, We know that 5% occurs 1.6449
standard deviations away from the mean (on any normal distribution)
Question: So how many standard deviations from the mean
will 5% occur on the distribution above?
Suppose an FI has a position in BBB rated zero coupon bonds with total face value of $10,631,483 with
an average maturity of 26 years that it plans to hold for less then 2 months. The average yield to maturity
of the bonds is 13.5%. Find the 95% DEAR of the portfolio.
Changes in Baa Yield
Frequency
80
6
60
4
40
20
2
0
0
Mean
= -0.00034
Standard Deviation = 0.069795
σ = 0.069795
5%
1.6449 STDevs
-0.00034
Find the change in interest rates under a really “bad case” scenario
X  z
 0.00034%  1.6449(0.06975%)  0.114466%
Based on historical data – the change in interest
rates will exceed 0.1145% only 5% of the time
12
Step #2 Calculate the daily earnings at risk DEAR
a)
Calculate the value of the bond position under the current YTM 13.5%
V
b)
10,631,483
 395,094.92
26
(1.135)
Calculate the value of the bond position under the new YTM
10,631,483
V
 384,874.78
(1.135 0.001145)26
c)
DEAR equals the difference or potential loss in value
DEAR  $384,874.78  $395,094.92  $10,220.15
Based on historical data – There is a 5% chance that the FI’s daily
losses on their fixed income portfolio will exceed $10,220.15
13
Dear for Foreign Exchange (FX)
Portfolio
14
Suppose an FI has a position in €1.4 million on their trading book currently the FX rate is 1.36 $/ €
find the 95% daily earnings at risk (DEAR) for the companies FX portfolio
Step 1 Find the extreme change in FX rates $/€
15
Suppose an FI has a position in €1.4 million on their trading book currently the FX rate is 1.36 $/ €
find the 95% daily earnings at risk (DEAR) for the companies FX portfolio
-0.06
300 45
250 40
35
200 30
150 25
20
100 15
50 10
5
0 0
-5
Frequency
Frequency
Step 1 Find the extreme change in FX rates $/€
Changes in $/€ FX rate
Changes
-0.04
-0.02 in 0$/€ FX
0.02 rate
0.04
0.06
0.08
45
40
35
30
25
20
15
10
5
0
5%
-1.6449
Will the FI lose money if this
goes up or down?
X  z
0.00  1.6449(.01)  0.016449
Based on historical data – the decrease in
exchange rates will exceed -0.016449
only 5% of the time
Mean = 0.00
Standard Dev. = .01
16
Step #2 Calculate the daily earnings at risk DEAR
a)
Calculate the dollar value of the euro position at the current FX rate 1.36 $/€
V = (€1,400,000)(1.36$/€) = $1,904,000
a)
Calculate the dollar value of the euro position at the extreme FX rate 1.344 $/€
V = (€1,400,000)(1.36-.016449$/€) = $1,880,971
b)
DEAR equals the difference or potential loss in value
V = $1,880,971 – 1,904,000 = – $23,028.60
Based on historical data – There is a 5% chance that the FI’s daily losses on its
FX portfolio will exceed $23,928.06.
17
Dear for Equity Portfolio
18
Suppose an FI holds an equity portfolio in their trading book with market value of $500,000. The
portfolio has a market beta of 1.3 the daily risk free rate is currently .001%. Calculate the 95% DEAR
on the FIs equity trading portfolio
Step 1 Find the extreme Market Return
Daily S&P 500 Returns
Daily-0.05
S&P 500
Returns
-0.10
0.00
0.05
-0.15
20
20
5%
15
15
-1.6449
0.12
0.10
0.09
0.07
0.06
0.04
0.03
0.01
0
0.00
0
-0.02
0
 2 2  
E[ RP ]  R f   E[ RM ]  R f 
10
-0.03
5
-0.09
50
E[ RP ]  z  2 2
2 2


Extreme
Return

R


E
[
R
]

R

Z


f
M
f
5
10
-0.05
100
25
-0.06
150
Frequency
200
-0.11
Frequency
250
X  z
0.15
25
-0.08
300
0.10
0.00001 1.3(.00023 0.00001)  1.6449 1.32  0.01582
 0.00031.64490.02183  0.0362
Mean = -0.00023
Standard dev = 0.0158
Based on historical data – there is a 5% chance
that losses on the portfolio will exceed -0.0362
tomorrow due to market exposure
19
Step #2 Calculate the daily earnings at risk DEAR
a)
Calculate the 95% DEAR – the extreme portfolio return times the total equity
position
DEAR = –0.0362($500,000) = –18,073.20
Based on historical data – There is a 5% chance that the FI’s
daily losses on its equity portfolio will exceed $18,073.20
20

The last step is to put it all together
◦ We cannot just add them up because that ignores diversification
◦ we need to account for how bonds, currency and stocks are related
(correlated)
Portfolio DEAR:
2
2
DEARFI
 DEARFX
 DEARE2
DEARP 



DEAR DEAR 
DEAR DEAR 
 2  FI , FX DEARFI DEARFX
 2  FI , E
 2  FX , E
FI
FX
E
E
21

Following our example suppose the following correlation matrix for S&P
returns, changes in FX rates and changes in Baa bond yields
Fixed income
FX
Market Returns
Fixed income
1
0.011877
0.269737
FX
Equity portfolio
1
0.202521
1
Portfolio DEAR:
10,220.152  23,028.602  18,073.202
DEARP 
 2(0.01188)10,220.1523,028.60
 2(0.2025)23,028.6018,073.20
 $35,145.76
 2(0.2697)10,220.1518,073.20
22

FI’s usually calculate their DEAR and work to reduce portfolio risk when
these DEARs are violated

We have done some pretty simple DEARs but in reality banks trade in
many different markets
In 2008 Citigroup’s DEAR calculation required updating 250,000
correlation and variance parameters

23
JP Morgan holds:
a) A BBB rated bond portfolio with $12M in face value that it plans to hold for less than 1 month. The portfolio
has an average time to maturity of 7.5 years, aggregate semiannual coupon of 8.3% and average YTM of 9.2%.
b) (ii) A $360.5M position in their equity trading portfolio. The portfolio has a market beta of .73 and the daily
risk free rate is currently 0.003%.
Find JP Morgan’s 99 % DEAR if the mean and standard deviation of daily changes in YTM for BBB rated bonds is
-0.0005 and 0.039 respectively over the last year. The daily mean and standard deviation of market returns is
0.00046 and 0.012 over the last year. The correlation between changes in YTM and market returns is 0.24
24

Market risk:
◦ A bank’s risk of experiencing losses (on their trading book)
due to market exposure.

Measurement
◦ RiskMetrics
25
Appendix
26
Historical Back Simulation
27

The biggest drawbacks of the RiskMetrics approach is that:
◦
It assumes a normal distribution
 This may not always be appropriate – for example options have a minimum negative return
but unlimited positive return
◦ Correlation must be calculated

The biggest change with historical back simulation is that it:
◦ Does not assume any distribution. It uses the empirical distribution to find the
daily earnings at risk (DEAR)
◦ Do not need to calculate correlations and variances when aggregating risks

Basic Idea
◦ We are going to use historical observations to simulate potential scenarios or
outcomes for tomorrow
28
Suppose an FI has a position in BBB rated zero coupon bonds with total face value of $1,631,483 with
an average maturity of 26 years that it plans to hold for less then 2 months. The average yield to maturity
of the bonds is 13.5%. Find the 95% DEAR of the portfolio.
Find the 5% DEAR for the fixed income portfolio
• We want to find the cutoff value where 5% of all
observations fall below
• Procedure
1. Collect historical changes in interests rates 4
year (1000 observations is a good number)
• We always calculate the change in value
in relation to a change in the market
(interest rates, market return, FX rate)
• The value of the portfolio could be
affected by other factors (liquidity) but
we just want to measure the exposure to
market risk
2. Calculate the change in value for each observation ie
if the interest rate is at 13.5% calculate:
ΔV = P(13.5+ΔI) – P(13.5%)
for each value of ΔI
3. Sort values from largest to smallest loss. Find the 5%
VaR i.e. 95% of all observations fall below this value
VaR(.95) = (1003)(0.05) = 50.15
We used 1003 historical observations
Observations Change in Yield
45
-56185.6
46
-56185.6
47
-56185.6
48
-56185.6
49
-56185.6
50
-56185.6
51
-56185.6
52
-56185.6
53
-56185.6
54
-56185.6
55
-55157.9
56
-55157.9
57
-55157.9
29


We can do the same thing for the foreign currency position and the equity
position
Finally, to aggregate the risk we just sum up the change in value across all
portfolios and sort the total
Obs
45.00
46.00
47.00
48.00
49.00
50.00
51.00
52.00
53.00
54.00
55.00
56.00
Bonds
-56185.63
-56185.63
-56185.63
-56185.63
-56185.63
-56185.63
-56185.63
-56185.63
-56185.63
-56185.63
-55157.94
-55157.94
FX
-23800.00
-23660.00
-23520.00
-23520.00
-23380.00
-23240.00
-22960.00
-22680.00
-22540.00
-22540.00
-22400.00
-22400.00
Equity
-56385.64
-56358.65
-56250.42
-56118.47
-55837.17
-55021.25
-53718.18
-53211.42
-52851.18
-51687.01
-51646.37
-50888.61
Total
-68336.54
-67306.65
-67047.75
-66199.20
-66150.30
-65822.59
-65795.34
-65332.19
-65017.40
-64389.43
-63928.60
-63542.26
Aggregating each day and
then calculating the VaR
accounts for the correlation.
That is, the interactions
between assets is taken into
account when we create the
full portfolio of bonds stocks
and currency
30

Back simulation relies on prior data
◦ Because it uses historical data, there are relatively few observations.
This decreases the accuracy (statistical precision) of the estimate
◦ We can use more observations but the further back we go the less
relevant those observations become as potential outcomes for tomorrow
◦ We can try to weight prior observations less i.e. give them a lower
probability of occurring
◦ The other solution is just to make up numbers. However, we want to do
that in a reasonable way → Monte Carlo simulation
◦ We are going to generate observations such that the probability that they
occur tomorrow is the same as the probability that they have occurred in
the past
31
Monte Carlo Simulation
32


We will do this for the 2 asset case only – things get a little more
complicated for more than 2 assets
Procedure:
1. Generate 2 standard normal variables you can do this in excel using the
following command
=NORMINV(RAND(), 0, 1)
Transform the uniform
variable into a standard normal
Generates a uniform random
variable between 0-1
33

We will do this for the 2 asset case only – things get a little more complicated
for more than 2 assets

Procedure:
1. Generate 2 standard normal variables you can do this in excel using the
following command
=NORMINV(RAND(), 0, 1)
2.
Calculate the correlation between changes in asset prices or returns
1. MCS assumes a distribution (multivariate normal) so we want to makes sure the variables
we are modeling are normally distributed – prices and values are non-normal
3.
Transform variables using estimated parameters:
X 1  1   1 z1

X 2   2   2 z1   z 2 1   2
4.
Repeat for as many simulations as you want
5.
Calculate the simulated price and the change in value
6.
Calculate the DEAR using the simulated data

34
Example: Excel Spread Sheet
We can estimate the mean, standard deviation and correlation of the change
in FX and equity values calculated above.
◦ Note: with Monte Carlo simulation you could simulate anything prices, changes in
returns, FX rates, interest rates …
mean
st dev


FI
-0.0002
0.0698
FX
0.0000
0.0100
Correlation
0.0130
Pull 5,000 draws from the standard normal distribution
Convert the draws to draws from a bivariate normal


X 1  1   1 z1
X 2   2   2 z1   z 2 1   2
X1  0.0002 .0698(1.529)
X 2  0  .01(0.1419) 1.529 0.013 (0.1419) 1  0.0132

35



Using simulated values calculate the change in value of the portfolio
Now we just repeat the procedure for back simulation
◦ Calculate the change in value of the portfolios
◦ Sort the values from smallest to largest
◦ Calculate the 5% DEAR – (5000)(.05) = 250th observation
obs
247
248
249
250
251
252
253
254

FI
-55,454.6
-55,449.8
-55,437.8
-55,432.8
-55,430.1
-55,412.6
-55,399.2
-55,390.1
FX
-22,672.5
-22,628.4
-22,620.6
-22,470.3
-22,458.5
-22,453.8
-22,444.9
-22,435.6
total
-78,127.1
-78,078.2
-78,058.4
-77,903.1
-77,888.6
-77,866.4
-77,844.1
-77,825.7
If markets experience a really bad day, the FI will lose:
◦ $55,432.80 on their fixed income portfolio
◦ $22,470.30 on their currency portfolio
◦ 77,903.10 combine
36
Basel II Standardized Approach
37