1. Market-Risk Measurement
CHAPTER 2
Market-Risk Measurement

2. Introduction Market risk for a bank
Trading group in a bank can trade the financial instruments (for example, bonds and stocks) in the market
When the values of traced instruments changes, the bank might get a loss
The common approaches to measuring market risk
Standard deviation, variance: total risk
Capital Asset Pricing Model (CAPM): systematic risk
Value at Risk (VaR)
In this class, we focus on VaR

3. Risk vs. Uncertainty

4. Average (1/2) Equation: sum of probability * average dollars
Compute: 20%*-2+80%*+2=1.2 (hundred million)
20%
80%
-2 hundred million
+2 hundred million

5. Average(2/2) FX: Company A Company B
Determine : Low average dollars has higher risk
Advantage: easy to understand
Disadvantage:
FX: Company A Company B
The average dollars is the same (1.2 hundred million), but the risk is different
20%
80%
-2 hundred million
+2 hundred million
100% firm gets 1.2 hundred million

6. Variance (Standard Deviation)(1/2)
Equation: sum of probability * (dollars-average dollars)^2
Compute:
Variance: 20%*(-2-1.2)^2+80%*(+2-1.2)^2=2.56 (hundred million)
Standard deviation: 2.56^0.5=1.6 (hundred million)
20%
80%
-2 hundred million
+2 hundred million

7. Variance(Standard Deviation)(2/2)
Determine : high variance or standard deviation has higher risk
Disadvantage:
FX: Company C Company D
The variance or standard deviation is the same (1.6 hundred
million), but the risk is different
80%
20%
-2 hundred million
+2 hundred million
20%
80%
-2 hundred million
+2 hundred million

8. VALUE AT RISK
Value at Risk (VaR) is a measure of market risk that tries objectively to combine the sensitivity of the portfolio to market changes and the probability of a given market change.
In overall, VaR is the best single risk-measurement technique available
As such, VaR has been adopted by the Basel Committee to set the standard for the minimum amount of capital to be held against market risks

9. VALUE AT RISK
VaR summarizes the predicted maximum loss (or worst loss) over a target horizon within a given confidence interval. (Jorion（2000))
How to compute VaR
Know the distribution of the asset
Know the confidence interval or significant level
Know the time horizon (time period)
Typically, a severe loss is defined as a loss that has a 1% (significant level) chance of occurring on any given day

10. VALUE AT RISK
Feature
Unit: dollars. Not standard deviation or ratio (Sharpe ratio)
Estimate value. is computed by statistics, given the confidence interval to compute the estimate value, not certain value.
VaR is estimated in normal market. VaR fails to tell you the maximum loss in stress market, such as 2008 financial crisis.

11. VALUE AT RISK
A common assumption is that movements in the market have a Normal probability distribution, meaning there is a 1% (significant level) chance that losses will be greater than 2.32 standard deviations.
Assuming a Normal distribution, 99% (confidence interval) VaR can be defined as follows:
standard deviation of the portfolio's value
The subscript T in the VaR expression refers to the time period over which the standard deviation of returns is calculated. VaR can be calculated for any time horizon. For trading operations, a one-day horizon is typically used.

13. VALUE AT RISK
For an example of a VaR statement, consider an equity portfolio with a daily standard deviation of $10 million. Using the assumption of a Normal distribution, the 99% confidence interval VaR is $23 million. We would expect that the losses would be greater than $23 million on 1% of trading days.

14. VALUE AT RISK
Senior management should clearly understand that VaR is not the worst possible loss.
Losses equal to the size of VaR are expected to happen several times per year
VaR is therefore not equal to capital
We will discuss the relationship between VaR and capital in great depth in later chapters
but a very rough rule of thumb is that the capital should be 10 times VaR

15. VaR for Bonds
For a bond, VaR can be approximated by multiplying the dollar duration by the "worst-case" daily interest move. This gives the value change in the "worst case."

16. VaR for Bonds
the "worst-case" daily interest move for 1% chance in one day
If we assume that interest-rate movements have a Normal probability distribution, then the 1% worst case will correspond to2.32 standard deviations of the daily rate movements

17. VaR for Bonds
As an example. If the duration is 7 years (time duration in term of year), the current price is $100 (the dollar duration is 7*100), and the daily standard deviation in the absolute level of interest rates is 0.2%, then the VaR is approximately $3.24:
VaR = $100 *7 *2.32 * 0.2 = $3.24
The approximations that we made here were as follows:
the changes in the rate is Normally distributed
The change in the price can be well-approximated by the linear measure of duration (no consider the convexity).

18. VaR for Equities The VaR for an equity is easy to calculate
If we assume that equity prices have a Normal distribution. The VaR is then the number of shares held (N), multiplied by 2.32 and the standard deviation of the equity price (σE):
VaR = 2.32* σE * N
So, for example, if we held 100 shares of IBM, and the daily standard deviation of the price was 10 cents, the VaR would be
$23.2: VaR = 2.32 *$0.1 * 100 = $23.2

19. VaR for Options
A simple approximation of the VaR to an option can be obtained using the linear sensitivities
The standard deviation of the option price caused by changes in the stock price is simply the standard deviation of the stock price multiplied by delta

20. VaR for Options
Delta: the derivative of value of option with respect to the price of underlying stock
The standard deviation of the underlying stock’s price

21. How to calculate volatility of each asset?
JP Morgan's RiskMetrics system
Equally-Weighted Moving Average (that is Simple Moving Average (SMA); variance)
Exponentially-Weighted Moving Average (EWMA)
λ: decay rate, 0<λ<1. The more the λ value, the less last observation affects the current dispersion estimation.
The formula of the EWMA model can be rearranged to the following form:

22. How to calculate volatility of each asset?
The EWMA model has an advantage in comparison with SMA, because the EWMA has a memory.
Using the EWMA allows one to capture the dynamic features of volatility. This model uses the latest observations with the highest weights in the volatility estimate. However, SMA has the same weights for any observation.
JP Morgan suggests: the optimal value for current daily dispersion (volatility) is =0.94; the optimal value for current monthly dispersion (volatility) is =0.97
Using moving window method to calculate the daily volatility and then calculate the daily VaR

23. General Considerations in Using VaR
In the discussion above, we gave approximations for calculating the one-day 99% VaR.
However, there are several conventions in use for the VaR probability, which implies a different multiplication factor for the standard deviation.
The most common alternative is to set the tail probability at 2.5%. If a Normal distribution is assumed, this implies a multiplier of 1.96 rather than 2.32

24. General Considerations in Using VaR
In some cases, we may wish to know the VaR for the potential losses over multiple days.
A reasonable approximation to the multi day VaR is that it is equal to the one-day VaR multiplied by the square root of the number of days:

25. General Considerations in Using VaR
This relationship requires the following assumptions:
Changes in market factors are Normally distributed.
The one-day VaR is constant over the time period.
There is no serial correlation. Serial correlation is present if the results on one day are not independent of the results on a previous day.

26. General Considerations in Using VaR
In general, for trading operations it is safe to assume that if the term VaR is used without a specified time, it means one-day VaR.
the term VaR is also used to refer to the potential loss from asset liability management, in which case a monthly or yearly horizon is used.
Also, the term "credit VaR" is sometimes used to describe the loss distribution from a credit portfolio. This is quite different from the VaR used for trading portfolios.

27. General Considerations in Using VaR
The major limitation of VaR is that it describes what happens on bad days (e.g., twice a year) rather than terrible days (e.g., once every 10 years).
VaR is therefore good for avoiding bad days, but to avoid terrible days you still need stress and scenario tests.