Topic 4 VaR

Question 0

Name the distribution type where all observations are centred around the mean

Answer 0

degenerative

Question 1

VaR =

Answer 1

VaR = estimate with a predefined confidence level of how much one can lose from holding a position over a set horizon.
Downside risk proposition, how much could be lost

Question 2

compare normal distribution to a t distribution

Answer 2

t distribution: more chance of staying around the middle.

fatter tails, less chance of being within a certain sigma

Question 3

when calculating VaR:

Answer 3

1. Calculate slope of the line. Rise over run. This is DELTA, how much the portfolio drops by for a given confidence level.
2. sigma = vol; Z= confidence level stipulated by mgmt; h = time horizon, measured in units consistent with sigma.
3. Inputs: remember Z is the number from the area of the normal dist; eg Z = 1.645 for 95% confidence, 1 tail.
4. calc upper or lower bound (formula sheet) use Z. Difference from original value is the dP
5. Portfolio consequence = VaR = DELTA * dP

Question 4

Calculate rate of return on asset

• per annum
• if use normal distribution:

Answer 4

• rate of return p.a.: (Vt/Vo)^(1/T) - 1
• if use normal distribution; use log returns. (eliminates negative values). log return over time T = ln(Vt)-ln(Vo)=ln(Vt/Vo). Multiply by 100 for percent. **ln calculates log returns from stock prices
• Vt = Vo e ^ (CCRoR*T) ** exp calculates stock prices from log returns
• logreturn = ln (Vt/Vo) = CCRoR *T
• CCRORT is not always the same as ln(Vt/Vo) they are only the same if exactly one year.
• log returns are additive over contiguous time intervals.

Question 5

Bank capital

Answer 5

1. financial base/support for any commercial transactions the entity undertakes. Buffer against hits.
2. Capital is a liability to the bank. (eg debt capital borrowings, deposits, equity capital all = liabilities)

Question 6

debt ratio

Answer 6

debt ratio = (Vdep+Vd) / (Vdep+Vd+Ve)
= (Vdep+Vd)/Vassets

note: Vassets = Vdep+Vd+Ve

Question 7

insolvency

Answer 7

Insolvency: sell all assets at current market prices, and this wouldnt cover obligations.
Not far from bankruptcy.

Question 8

Capital structure question

How much capital

Answer 8

Capital structure: allocation of debt vs equity
How much capital: too little - not enough buffer against events that push it closer to insolvency. Too much capital: earn low return and the excess capital could be better deployed elsewhere.

Question 9

VaR

• definition
• uses

Answer 9

VaR Def: evaluate a distribution of possible outcomes with a focus on the worst that might happen
VaR uses: compute capital requirements, input to risk taking and risk management decisions, assess quality of bank’s models

Question 10

Two vital steps to VaR estimation

Answer 10

1. what are the potential bad moves in the underlying asset;s prices
2. how will portfolio change in value should this move occur

Note:
slope will tell you in which drection you will lose money. eg upward sloping, lose money on lower limit. Downward sloping: lose money on upper limit.

Question 11

Lognormal Distribution:

1. Describe shape
2. Describe relationship between mode, median and mean
3. Implications for stock prices

Answer 11

Lognormal distribution

1. Shape: assymetric with long, drawn out right hand tail (positive skew)
2. Mode < Median < Mean (being below the mean is a more probabilistic outcome)
3. Implications: Large downward movements are more likely than large upward movements

Question 12

Distributions:

1. describe N(u,var)
2. describe CCRoR
3. Describe N(uT, varT)

Answer 12

Distributions:

1. N(u,var) is the normal distribution. Note uses variance; therefore need SQRT (VAR) = sigma
2. CCRoR = continuously compounding rate of return; this is approximated by N(u, variance)
3. N(uT, Variance T) = distributional assumption approximation of log return on asset over time horizon T

V(t) = V(o) EXP ^(CCRoR * T)

Question 13

per annum CCRoR = ?

Distribution of CCRoR = ?

Answer 13

per annum CCRoR = ln (Pt/Po) divided by T

Distribution of CCRoR = approx. N(u, variance)

Question 14

VaR is used for (7)

Answer 14

VaR is used for
1. setting limits to constrain risks taken by traders
2. Determining economic capital to ensure that the firm has a buffer of equity in the event of a large adverse market move
3. Assessing possible margin calls both for oneself and also for clients when trading on a collateralised basis. This type of analysis is essential for monitoring and managing fund liquidity
4, VaR is an input to RAPM which is then used to determine appropriate pricing (eg spread on trade for risk adjusted return
5. RAPM (based on VaR) is used to allocate resrouces around/within the firm. Typically business units with highest RAPM argue for more resources
6. VaR used as risk measure to review portfolios, positions, changing spreads etc
7. risk disclosure to stakeholders including shareholders

Question 15

Can you add VaR numbers together

Answer 15

Generally not unless you know the correlation between the two is +1. Could be some diversification benefits. Could assume +1 to be conservative.

Question 16

Risk adjusted returns (eg traders bonus question)

Answer 16

(revenue - cost) / VaR

Question 17

Distribution of real world returns

Answer 17

• not normally distributed
• can use ln function to transform, tends to induce symmetry
• assume cc per annum rates of return with normal distribution
• cc rates of return - some assymetry.
• market returns generally aren’t normal, exhibit both fat tails and higher peaks ie leptokurtosis

Question 18

volatility clustering

Answer 18

• bunching of observations together

- observe some dependence of market observations which violates an independent and identically distributed distribution

Question 19

issues with assumption of normality in probability distribution

Answer 19

1. real distributions may be assymetric and have either longer LHS or RHS tails
2. Variance is not a good way to measure risk if not normal: draws on mean and squares the differences from mean; therefore assumes observation is just as likely to be positive as it is negative.
3. Risk Adjusted Return on Capital: expected losses. If distribution is asymmetric, the expected value (mean) is not the highest point, it may be skewed to left or right of the middle.

Question 20

Methods for estimating distributionz

Answer 20

1. historic data (simple to implement, easy to understand, accommodates any kind of data - whether skewed or otherwise. Monte Carlo can be used to simulate) cons: assumes past is a guide to future, sample size,
   2, analytical method (likely range of price movements assuming lognormal distribution)

Question 22

stress test

Answer 22

• evaluate potential impact on portfolio value of unlikely but plausible events
• explore the tails of the distribution losses beyond threshold used in VAR
• dangerous to base stress tests on short data samples

Question 23

Types of stress test:

5

Answer 23

1. Historical (assumes historical event is repeated)
2. Hypothetical or forward looking stress tests (assess impact of hypothetical event)
3. Single factor (univariate) stress test (micro focus on only one risk factor_). Offers insights but ignores contagion effects
4. Multivariate stress test (vital for portfolios with multiple assets, correlations that may appear tranquil can change in stressed market conditions. Risk is contagious
5. Portfolio driven stress test (capture the specifics of the portfolio - eg straddle position

Question 24

GFC stress test lessons

Answer 24

1. problems in the manner the tests were conducted
2. problems with how senior management & supervisors responded to the process of stress testing
3. Board & senior mgmt. need to be involved in setting stress tests, objectives, defining scenarios, discussing results and decision making
4. Consider moral hazard, overconfidence, principal/agent conflict

Question 25

Risk Adjusted Return on Capital

- ideal uses

Answer 25

• calc precise amt of extra VaR the next deal will add
• Apply to that VaR the appropriate RARoC for the risk category of the trade
• convert that amt into extra basis points or margin on some interest rate or FX rate or other price assoc w/ the deal
• note some deals can decrease risk as they serve as a hedge.

Question 26

VaR: what does it mean if VaR is 50m at 99.9% confidence interval

Answer 26

• In extreme circumstances the institution is expected to lose more than $50m in a hyear. If it had $50m of capital it will have 99.9% probability of not running out of capital in the year.

Question 27

Name 3 conditions that VaR satisfies

Answer 27

1. monotonicity (if portfolio produces worse risk result than another in every state, its risk measure should be greater)
2. translation invariance: if cash K is added to portfolio, risk should go down by K
3. Homogeneity: change size of portfolio by factor x, should result in risk measure multiplied by x

Question 28

Name 1 condition that VaR does not satisfy

Answer 28

1. Subadditivity: risk measure of 2 portfolios after merging should be no greater than the sum of the risk measures before they were merged.
   - consider diversification

Question 29

Risk

Answer 29

= unexpected loss
= function of volatility of outcomes
= measured by SD of the outcomes

Question 30

Risk adjusted performance measurement

Answer 30

compare returns against capital invested by adopting some form of risk adjustment

Question 31

Return on Risk Adjusted Assets (RORAA)

Answer 31

take Return on Assets ratio, but instead of ranking all assets equally, adjust them for relative riskiness
- Basel 1998

Question 32

Risk Adjusted Return on Assets (RAROA)

Answer 32

Use ROA, but deduct a risk factor from return (eg if 1% chance of loan default, deduct 1% from return generated)

Question 33

Return on Risk Adjusted Capital (RORAC)

RORAC and RAROC most important

Answer 33

RoC, replace regulatory capital in the4 denominator by internal measure of capital at risk. Capital can be expanded to cover non balance sheet items

Question 34

Risk Adjusted Return on Capital (RAROC)

RORAC and RAROC most important

Answer 34

Similar to RORAC, but adjust the numerator rather than denominator

Question 35

Difference, RAROC (risk adjusted return on cap) vs RORAC (return on risk adjusted cap)

Answer 35

• RAROC adjusts numerator (return) for risk

- RORAC adjusts the denominator (capital)

Question 36

VaR

Answer 36

ascertain potential loss in value of assets and other exposures (or increase in value of liabilities) over a given time period, at a given confidence level.

Question 37

Adjust for expected losses

Answer 37

If you are in the business of credit, you need to adjust for expected losses. This is different to expected risk./ Consider selling insurance premiums & hoping there is never a claim
Expected loss is not a part of risk, it is a cost of doing business.

Question 38

3 components to assess expected loss

Answer 38

1. expected default frequency (EDF): prob that borrower will default over given period
2. Loss in the event of default: amt of credit exposure that will be lost given default
3. potential credit exposure: likely amount of credit granted outstanding at time of default

Question 39

Limitations of asset volatility approaches to risk management

Answer 39

• high reliances on statistics (correlations etc) can lead to big differences in the way risk is measured
• Aggregating risks can be difficult - eg holding period may differ depending on market risk vs credit risk, etc
• assumption that all risks can be identified & modelled. Mkt vol & liq can impact
• greater precision can be costly - computing power & maths experts