1. Types of risk Market risk
-Risk of losses for a portfolio that the firms owns.
-Risk factors: interest rate, foreign exchange, equity prices, commodity
prices...
-Estimation of market risk: VAR models
-IB manage these risks by diversifying exposures, controlling position
sizes, and establishing hedges.

2. Credit risk
- Possible loss that occurs when a counterparty or an issuer of
securities held by the firm fails to meet its contractual obligations.
- To reduce risk, IB establish limits of credit exposures, and require
collateral.
Operating risk
Risk resulting from failed internal process, people and systems.
Reputational risk
Risk of a financial loss resulting from the loss of business
attributable to a decrease in reputation.

3. Value at Risk (VAR)
The VAR of a portfolio is the loss in value in the portfolio that can be
expected over a given period of time (e.g., 1-Day) with a probability not
exceeding a given number (e.g., 5%).
Probability (Portfolio Loss < -VAR) = K
K = Given Probability

4. VAR is a measure of the losses due to “normal” market movements.
It gives an estimate of the likelihood of a loss greater than the VAR
VAR takes into account how prices changes of different assets are related to each other
VAR must be combined with other measures of risk to provide a comprehensive risk measure

5. VAR for portfolios
Consider assets A and B, and the combined portfolio A+B
VAR(A)+VAR(B)>VAR(A+B)
Two assets: Investor has proportion x of asset A and (1-x) of asset B.
Expected return:
Variance:

6. Diversification effect
The standard deviation of a portfolio composed of two securities is
less than a weighted average of the standard deviations of the two
securities
Correlation:
The correlation is between -1 and 1.
1: perfect positive correlation
0: no correlation
-1: perfect negative correlation

7. E(R) B
corr=-1
corr=1
corr=0 A

8. Consider a forward contract:
Example:
Consider a forward contract:
On delivery you will deliver $15m and receive £10m.
At date t the contract has 91 days remaining until delivery.
The 3-month $ and £ interest rates are 5.469% and 6.063% respectively.
The spot exchange rate is S=1.5335$/£. 8

9. The USD mark-to-market value is:
The risk factors are the exchange rate and the interest rates.
The distribution of changes in these variables allows us to determine the value at risk
See Figure 1

10. Variance-covariance approach
Based on the assumption that the underlying factors have a
multivariate Normal distribution.
Under this assumption, we can compute the distribution of portfolio
profits and losses.
Then, standard properties of the normal distribution determine the
VAR.
VAR= 1.65 * (standard deviation of change in portfolio value)

11. Identify the risk factors, and map the forward contract into
Step 1
Identify the risk factors, and map the forward contract into
standardized positions: 11

12. Assume that percentage changes in the risk factors have a
Step 2
Assume that percentage changes in the risk factors have a
multivariate Normal distribution with means of zero, and
estimate the parameters of the distribution (standard
deviations, correlation coefficients).
Step 3
Use the standard deviations and correlations of the risk
factors to determine the standard deviations and
correlations of changes in the value of the standardized
positions (using the sensitivities of standardized positions
to changes in the risk factors). 12

13. Percentage change in X1:
Hence, stdev of percentage change in X1:
where 13

14. Same for X2 and X3: 14

15. Calculate the portfolio standard deviation using the
Step 4
Calculate the portfolio standard deviation using the
properties of Normal distributions:
Then, the VAR is: 15

16. Historical simulation
Simple atheoretical approach, that requires relatively few assumptions.
It consists of using historical changes in market rates and prices to
construct a distribution of potential future profits and losses.
The distribution of profits and losses takes the current portfolio,
subjecting it to actual changes in risk factors experienced in the past N
periods (trade-off regarding N).
We calculate N historical percentage changes in risk factors, and see
how these changes would affect the portfolio value today. 16

17. Steps:
- Obtain historical values of the markets factors for the last N periods (100 days for instance)
- Subject the current portfolio to the changes in market factors in the last N periods, calculating the daily profits and losses that would occur if comparable daily changes in the market factors are experienced
- Order the results and select a loss that is equaled or exceeded 5% of the time. 17

18. Monte-Carlo simulation
Used when historical data is not sufficiently available.
Consists of choosing a (multivariate) statistical distribution that is
believed to adequately capture the possible changes in the risk factors.
The distribution is not necessarily normal.
Using this distribution, thousands of risk factor changes are
simulated.
Then, the possible losses are calculated using the simulated risk factors.

19. Comparing the methods
Ability to capture the risk of portfolios that include options
Variance-covariance approach and options
Problem with options: skewed distribution of options payoff.
The variance-covariance approach linearizes the option positions.
This leads to a normal payoff distribution, which biases the results.
The problem is more significant more for long holding periods.
Both historical simulation and MonteCarlo work with options because
they compute the value of the portfolio for each draw of the basic
market factors.

20. Price Frequency Exchange rate Exchange rate Frequency
Normal distribution
Payoff 20

21. OPTIONS Frequency Option price Exchange rate Exchange rate Frequency
Skewed distribution
Payoff 21

22. VAR and options: the Delta-Gamma estimation
Call S the dollar price of British pound and C(S) the option price as a
function of S.
For example, if Δ=0.51 and the price changes by $0.01, the predicted
change in the option price is $0.0051=0.51*$0.01
Problem: the Delta changes as the price of the underlying asset.
Gamma or Γ measures how Δ changes. Gamma is the partial derivative
of delta wrt the price of the underlying asset. Gamma is equivalently
the second partial derivative of the option price wrt the price of the
underlying asset.

23. With Gamma:
Change in price
Adding Gamma improves the Delta estimation for options.

24. Generalities on the reliability of the results
For historical simulation, there is the risk that recent history is not
typical (volatility).
Other methodologies also use historical data, but by assuming
particular distributions, limit the possible shape that the estimated distribution can have.
Variance-covariance and MonteCarlo approaches: the assumed
distributions do not necessarily correspond to the true distributions (fat tails...).

25. VAR performance Beder (1995) -Do VAR assumptions matter?
-Do parameters matter?
-Does the methodology matter?
Analysis of eight methods:
Historical, 100days, 1day holding period
Historical, 250days, 1day holding period
Historical, 100days, 2weeks holding period
Historical, 250days, 2weeks holding period
MonteCarlo (RiskMetrics), 1day holding period
MonteCarlo (RiskMetrics), 2weeks holding period
MonteCarlo (BIS), 1day holding period
MonteCarlo (BIS), 2weeks holding period 25

26. Portfolio 1: Consists of US Treasury strips only.
Portfolio 2: Consists of outright and options positions on the S&P 500.
Portfolio 3: Combination of portfolio 1 and 2.
Large difference between the methods (see Beder 1995) 26

27. Hendricks (1996) : Evaluation of VAR methods
Analyses the performances of 12 approaches on random portfolios:
Equally weighted moving average approaches (50d, 125d, 500d, 1250d)
Exponentially weighted weighted average (with different decay factors)
Historical simulation (125d, 250d, 500d, 1250d)
Performance measures:
Mean relative bias
Root mean squared relative bias
Annualized percentage volatility
Fraction of outcome covered
Multiple needed to attain desired coverage
Average multiple of tail event to risk measure
Correlation between risk measure and absolute value of outcome 27

28. What happens when price changes are extreme?
Stress testing
What happens when price changes are extreme?
Extreme movements in the basic market factors are more frequent than under Normal distribution
Uncertainty on the correlation between the basic market factors
Extreme events rarely repeat themselves in the same way
Changing distributions over time
- Set of hypothetical extreme markets scenarios, and they price effect 28