Mod 22: Assessment of market risks

Question 1

Comment on the observed degree of serial correlation in periodic returns on individual equities
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Answer 1

Serial correlation in periodic returns on individual equities
1. Returns are rarely independent and identically distributed.
2. There is little evidence of serial correlation, however there is some evidence of momentum effects (in the short term) and mean reversion (in the longer term).
3. Any such correlation is insufficient to support significant profit-taking because it is very difficult to predict the return in the next period given only the history of the process – our best estimate of tomorrow’s return, given the daily returns up to today, is therefore zero.
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Question 2

Comment on the observed variation in volatility of periodic returns on individual equities

Answer 2

Volatility of periodic returns on individual equities

1. Volatility (if defined to be the conditional standard deviation of returns given historical data) appears to vary over time.
2. Volatility clustering occurs when extreme values tend to be followed by other extreme values, although not necessarily of the same sign.
3. There is evidence of serial correlation between absolute or squared returns, which is consistent with the phenomenon of volatility clustering.
4. Volatility clustering supports the idea that conditional standard deviations are changing in a way that is to some extent predictable (hence exploring ARCH and GARCH models).
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Question 3

Comment on the observed degree of kurtosis in the distribution of periodic returns on individual equities

Answer 3

Kurtosis in the distribution of periodic returns on individual equities

1. Return series are leptokurtic, ie more peaked and fatter-tailed than a normal distribution.
2. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly sized deviations.
3. Kurtosis of the distribution of financial returns falls as the period over which we are measuring returns increases – consistent with volatility clustering.
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Question 4

Comment on the observed correlations within and volatility of returns on portfolios of equities
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Answer 4

Features of the returns on portfolios of equities For such multivariate returns:
1. correlations exist between returns of different series (eg between different equities) at the same point in time. (This is also true for return series of other asset classes and economic variables.)
2. correlations between different series vary over time
3. multivariate returns data show little evidence of cross-correlation (ie between time periods t and t + 1)
4. multivariate series of absolute or squared returns do show strong evidence of cross-correlation
5. extreme returns in one series often coincide with extreme returns in several other series – ie during periods of high volatility, the level of dependence between various returns appears to be higher

Question 5

Describe the three main approaches to modelling market risks ©

Answer 5

Approaches to modelling market risks
1.
historical simulations
−each simulation is generated by referencing historical data eg bootstrapping
2. forward-looking factor-based approaches
−where the complex links between a response variable and explanatory variables are described explicitly within a model
eg risk-free yield, coupon rates, credit spread etc
3. forward-looking data-based approaches −
− which focus on modelling the key variables that provide the best fit to the data rather than the factors which drive them
eg multivariate distributions or models ©

Question 6

Outline the steps required to apply a forward-looking data-based approach to modelling correlated, multivariate normal log-returns, ie
ln(s6/s5)

Answer 6

Modelling using the multivariate normal distribution
1. Decide on the frequency of calculation (daily, weekly, etc).
2. Decide on the timeframe of historic data to be used (bearing in mind the trade-off between volume of data and relevance).
3. For each asset class, choose the total return index to be used, say St .
4. For each asset class, calculate the log-returns, ie x6= ln(S6/S5)
5. Calculate the average returns, variance of returns for each asset class and the covariances of returns between each class (and sub-set of classes).
6.
Simulate a series of returns with the same characteristics based on a multivariate normal distribution using, eg Cholesky decomposition or PCA.
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Question 7

Outline the steps required (in addition to those on Flashcard 6) to complete the simulation of a series of correlated, multivariate normal log-returns using Cholesky decomposition

Answer 7

see 700

Question 8

Outline the steps required (in addition to those on Flashcard 6) to complete the simulation of a series of correlated, multivariate normal log-returns using PCA

Answer 8

see 702

Question 9

List the main disadvantages of modelling returns using the multivariate normal distribution and state how these might be overcome
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Answer 9

Disadvantages of modelling using the multivariate normal distribution
1. features of the distribution
- the tails of the univariate marginal distributions are too thin
- the joint tails do not assign enough weight to joint extreme outcomes (ie joint tails too thin)
- the distribution has a strong form of symmetry (known as elliptical symmetry)
2. the above might be avoided by either using a non-normal multivariate distribution, or combining non-normal marginals with an appropriate copula (but tricky to select if data is insufficient)

3.computation time
- can be significant, particularly as the number of asset classes increases
4. the above might be reduced by using PCA ©

Question 10

Define credit spread and describe the three factors that it reflects ©

Answer 10

Definition of credit spread and the factors it reflects

Consider two fixed interest bonds with identical characteristics (eg coupon and redemption rates) other than one was issued by the government (deemed risk-free) and the other by a corporation.

The credit spread is defined as the difference between the gross redemption yields (or, equivalently, yield-to-maturity) on each of these two similar bonds.

The credit spread reflects the following three factors:
1. the expected probability of default and the expected loss given default (perhaps measured by considering the default history of bonds with similar credit ratings)
2. any risk premium (ie the uncertainty in the expected default probability and loss given default)
3. a liquidity premium (ie it may be more difficult to sell the corporate bond, when required, at an acceptable price).

Question 11

Outline three common measures of credit spread ©

Answer 11

Measure of credit spread
1. The nominal spread is simply the difference between the gross redemption yields of risky and risk-free bonds (or other reference bonds).
−an easy-to-calculate, rule-of-thumb measure
2. The static spread is the addition to the risk-free rate at which discounted cash flows from a risky bond will equate to its price.
−allows for the term structure of the underlying risk-free rate and the term premium
3. The option-adjusted spread further adjusts this discount rate through the use of stochastic modelling to allow for any options embedded in the bond.
− a more market-consistent measure ©

Question 12

Explain why observed market credit spreads are generally higher than can justified by the actual historic defaults on bonds
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Answer 12

Observed market credit spreads
The difference can be put down to risk premia in respect of:
1. the higher volatility of returns relative to the risk-free asset
2. the higher uncertainty of returns, particularly the possibility of unprecedented extreme events
3. the greater skewness of the potential future returns on corporate debt (more significant downside), due to the possibility of default
4. the lower liquidity of corporate debt
5. the lower marketability and the associated higher costs of trade 6. differences in taxation
6. the ‘risk-free’ asset is not actually risk-free! ©

Question 13

State the key difficulties in interpreting a quoted credit spread

Answer 13

Difficulties in interpreting a quoted credit spread
1.
uncertainty as to the reference asset −
 eg is it a government bond or a swap agreement?
2. uncertainty as to the credit worthiness of a government bond −
sovereign debt is not always ‘risk-free’! ©

Question 14

State the fundamental method by which market risk might be measured

Answer 14

Measuring market risk Market risk should be measured relative to a suitable benchmark.
Benchmarks are typically based on:

market indices or

the investor’s liabilities
−the reference point often being a set of cashflows (or notional assets) that represent the actual (uncertain) liabilities
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Question 15

Define:
1. strategic risk
2. active risk
3. active return ©

Answer 15

Strategic risk and active risk / return
1. strategic risk – the risk of poor performance of the benchmark against which the manager’s performance will be judged (the strategic benchmark) relative to the liability-based benchmark
2. active risk – the risk of poor performance of the manager’s actual portfolio relative to the manager’s (strategic) benchmark
3. active return – the difference between the return on the actual (active) portfolio and the return on the manager’s (strategic) benchmark

Question 16

List the general features of a good benchmark

Answer 16

General features of a good benchmark
1. unambiguous
2. investable and trackable
3. measurable on a reasonably frequent basis
4. appropriate, eg to the investor’s objectives
5. reflective of current investment opinion, eg positive, negative, neutral
6. specified in advance
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Question 17

List the desired features of a specific benchmark

Answer 17

Desired features of a specific benchmark

1. contains a high proportion of the assets held in the portfolio
2. has a similar investment style to the portfolio, eg growth or value 3. has a low turnover of constituents
3. has investable position sizes
4. behaves in a similar way to the portfolio, ie shows a strong positive correlation between the portfolio return, iR , and the benchmark return, RB , in excess of the market return, MR , ie  (
   RiM,) 0 R RR  B  
5. low correlation, ie    RiB B R RRM ( ,) 0
6. the variability of the portfolio returns relative to the benchmark returns should be lower than variability relative to the market return
   M

Question 18

Outline three main approaches to modelling interest rates

Answer 18

Approaches for modelling interest rates
1. single-factor models, eg Vasicek model
−inadequate for modelling multiple points on a yield curve or changes to the shape of a yield curve
2. two-factor models, eg Brennan and Schwartz model
−an improvement over single-factor models but still constrained and it is hard to select the appropriate parameters
3. PCA
- can be applied to GRYs (or their natural logs, to avoid negatives), forward yields or bond prices (by first calculating log returns)
eg model deviations from average GRYs across all durations
- reduces the dimensionality of the problem, eg focussing on the main factors that contribute to changes in the level and shape of the yield curve
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Question 19

Outline how foreign exchange risk might be modelled

Answer 19

see 724

Question 20

Outline how contagion or systemic risks (market, credit etc) might be modelled

Answer 20

Modelling contagion risks
Contagion risks can be modelled as the interaction between different financial series. Using a copula may be a sensible approach (assuming it can be suitably parameterised)

For example see 726

Alternatively, contagion can be considered as a feedback risk, ie that there is some serial correlation that can be modelled. However, this would present an arbitrage opportunity, which in theory should be eliminated by arbitrageurs. Hence such serial correlation effects are usually ignored when modelling.
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