ERM Chapter 14

Question 1

What are the axioms of coherence?

Answer 1

A list of properties that a good risk measure should have. A risk measure is coherent if it satisfies the following four axioms:

where Li = probability distribution of losses on a portfolio
F = amount of capital that should be added to a risk portfolio with distribution Li to make it acceptable to the risk controller

1. Monotonicity: if L1 <= L2 then F(L1) <= F(L2)
2. Subadditivity: F(L1+L2)<= F(L1) + F(L2)
3. Positive Homogeneity: F(k x L) = k x F(L)
4. Translation Invariance: F(L+k) = F(L) + k

Question 2

Describe the four axioms in words.

Answer 2

1. Monotonicity - If risk portfolio 2 exhibits greater or equal losses under all future scenarios than the losses on risk portfolio 1, then a monotonic risk measure will indicate that a greater or equal amount of capital should be held w.r.t. the former.
2. Subadditivity - A merger of risk situations does not increase the overall level of risk. It may decrease the overall level as a consequence of diversification.
3. Positive homogeneity - If we double the size of the loss situation then we double the risk - no reduction being given for non-existent diversification.
4. Translation invariance - If we add or deduct an amount to or from the loss, then the capital requirement needed to mitigate the impact of the loss increases by the same amount.

Question 3

What makes a risk measure convex?

Answer 3

F(aL1 + (1-a)L2) <= aF(L1) + (1-a)F(L2)

This is a desirable feature as it means diversification can reduce risk and the amount of capital needed. Convexity follows the axioms of subadditivity and positive homogeneity.

Question 4

Outline the nature of probabilistic and deterministic risk measures.

Answer 4

Deterministic risk measures are simplistic, giving broad indication of the level of risk.

Probabilistic risk measures are potentially more accurate, but are more complex, and can imply inappropriate levels of confidence.

Question 5

Outline the three deterministic approaches, and the advantages and disadvantages of each.

Answer 5

1. Notional approach - a broad-brush risk measure. For example, risk weightings might be applied to the market value of assets, then summed and this total compared to the value of liabilities in order to determine a notional financial position.
A: - simple to implement and interpret across a diverse range of organisations.
D: - potential undesirable use of a catch all weighting, for possibly undefined asset classes
- possible distortions to the market caused by increased demand for asset classes with high weightings
- treating short positions as if they were the exact opposite of equivalent long position
- no allowance for concentration of risk, as risk weightings are the same for an asset class regardless of whether it consists of a single security or variety of securities
- probability of changes considered is not quantified

Factor sensitivity approach - determines the degree to which an organisation’s financial position is affected by the impact that a change in a single underlying risk factor has on the value of assets and liabilities.
A: increased understanding of drivers of risk
D: - not assessing a wider range of risks by focusing upon a single risk factor
- being difficult to aggregate over different risk factors
- probability of changes considered is not considered quantified

Scenario sensitivity approach - similar to that of factor sensitivity, but instead of changing one factor the effect of changing a set of factors is considered.
D: probability of the changes considered is not quantified

Question 6

What are five probabilistic approaches to quantifying risk?

Answer 6

1. deviation
2. value at risk (VaR)
3. probability of ruin
4. tail value at risk (TVaR)
5. expected shortfall

Question 7

Define deviation.

Answer 7

Deviation is a measure of the spread from a given reference.

Standard deviation - where deviation is measured from the mean.

Tracking error - where deviation is measured relative to a benchmark other than the mean e.g. investment returns measured with reference to a benchmark portfolio

A: - simplicity of calculation

• applicable to a wide range of financial risks
• can be aggregated if correlations are known

D: - difficulty in interpreting comparisons, other than in terms of simple ranking

• potentially misleading if the underlying distributions are skewed
• do not focus on tail risk and underestimates tail risk if distributions are leptokurtic
• aggregations of deviations can be misleading

Question 8

Define value at risk (VaR).

Answer 8

The maximum potential loss which is not exceeded with a given high probability over a time period.

I.e. P(L<i></i>

Question 9

Outline the three general approaches to calculating VaR, along with their advantages and disadvantages.

Answer 9

Empirical approach - rank losses over T time periods from smallest to largest, and derive the VaR based on the confidence percentage required.

A: - it’s simplicity

• no requirement to specify the distribution of returns
• it’s realism, in that it focuses on the largest market movements observed

D: - it’s reliance on boot-strapping past data having captured all possible future scenarios

• the implication that past data is indicative of future experience
• it doesn’t facilitate stress or scenario testing
• practical difficulties and limitations of interpolation

Parametric approach - assume that losses follow some specified statistical distribution function. Estimates of distributions parameters might be obtained from past data or by using the future volatility implied by option prices.

A: - ease of calculation

• reduces dependence on past data
• easy adjustment of parameters initially derived from past data

D: - more difficult to explain than empirical approach

• reliance on past data to the extent that parameters are derived from this data
• difficulty ensuring parameters chosen are consistent
• assuming that the parameter values remain constant
• risk of adopting an inappropriate statistical distribution
• difficulty in reflecting complex inter-dependencies

Stochastic approach - derivation is the same as the empirical approach, but the dataset is not the full set of observed past losses. The dataset may be simulated using a statistical distribution, or bootstrapped via random sampling of past observed returns.

A: - accommodates more complex features of the underlying loss distribution

• wider range of future possibilities than the empirical approach
• accommodates sensitivity testing

D: - more difficult to explain than the other two approaches

• subjective and difficult choices of distribution and parameter values
• it gives a different answer each time
• potentially high computation time

Question 10

Define probability of ruin.

Answer 10

The probability that the net financial position of an organisation falls below zero over a specified time horizon. It is linked closely to the VaR, for example, if the net financial position is below the 95% VaR then we can infer that the probability of ruin is greater than 5% over the same time horizon.

Question 11

Define tail value at risk (TVaR).

Answer 11

The expected loss given a loss over the specified VaR has occurred.

A: - considers the losses over the VaR
- it is a coherent risk measure
D: - the choice of distribution and parameter values is subjective and difficult
- it is highly sensitive to assumptions

Empirical approach - average of losses that are greater than of equal to the VaR.

Parametric approach - takes the mean through a statistical distribution.

Stochastic approach - based on data obtained by simulation or bootstrapping.

Question 12

Define expected shortfall

Answer 12

Same as TVaR but the average of all losses, not just those that are above the VaR threshold.

Can be calculated using an empirical, parametric or stochastic approach.

A: - considers losses beyond the rail
- is a coherent risk measure
D: - choice of distribution and parameters is subjective and difficult
- highly sensitive to assumptions
- has little intuitive meaning
- cannot next readily linked to the current valuation

Question 13

What does the ratio of TVaR to VaR indicate?

Answer 13

The skewness of the distribution. A higher ratio indicates the loss distribution is asymmetric with a fatter tail.

Question 14

Outline the effects of the time horizon, and describe two key factors influencing the choice of a suitable time horizon.

Answer 14

The longer the duration of exposure, the higher the level of the risk - both in terms of the possible outcome and what might happen in the intervening period.

The choice of a suitable time horizon will be influenced by expectations as to:

1. The time to recover from a loss event
2. The time to reinstate risk mitigation e.g. re-establish a derivatives hedge

Question 15

Outline the role of the risk discount rate and contributing factors.

Answer 15

The size of the discount rate will impact the appraised viability of projects. The higher the discount rate the lower the present value of future earnings.

Discount rates should take into account the sponsors cost of capital, the rate of inflation, interest rates, and returns on investments throughout the economy.

Ultimately the discount rate will depend on issues such as the company’s cost of capital, and any hurdle rates the company sets for investments. Some companies may wish to use higher or lower rates for projects them seem as having higher or lower inherent risk. Risks may change over time if the risk of a project varies significantly over different time periods. Artificially high discount rates chosen as a substitute for more detailed risk analysis should be avoided, as profitable projects may be rejected.

Question 16

What are the two rules of thumb?

Answer 16

1. The number of days that a mark-to-market loss might exceed VaR(a) might be estimated as [100%-a]x250, where a is the confidence level and 250 is the number of trading days in a year.
2. Let X be the loss over 1 day. If X~N(u,o) then the n-day loss is distributed as N(nu, no^2), and the n-day volatility is sqrt(n)o. Assuming u=0, an approximation of an n-day VaR can be obtained by multiplying the 1-day VaR by sqrt(n).

Question 17

Define CAPM.

Answer 17

rx = rf + Bx(ru - rf)
rx = expected return on a risky asset
rf = risk-free rate of return
ru = expected return on the universe of investment opportunities (existing project portfolio)
Bx = (ox/ou) x px,u
the beta of the asset where o are standard deviations and p is the correlation

On a graph with B on the x-axis and rx on the y-axis, SML has a y-axis intercept of rf and gradient ru - rf

PML is obtained by re-expressing rx as the required return on project X, and re-expressing U as the existing project portfolio. A higher return is required for a project which exposes the organisation to greater uncertainty (ox/ou) and/or has a lower diversification benefit (higher px,u)

Difficult to use PML to determine discount rate as returns on projects and correlations between them may not be stable over their respective lifetimes.