ERM Chapter 22

Question 1

What are the key characteristics of a financial time series analysis of historic equity returns, for:

• individual equities
• portfolios of equities

Answer 1

Individual equities:

• returns are rarely independent and identically distributed
• volatility appears to vary over time
• extreme returns appear in clusters (volatility clustering)
• return series are leptokurtic (heavy-tailed) and non-symmetric

Portfolios of equities:

• correlations exist between returns of different series at the same point in time
• correlations between different series vary over time
• multivariate returns data shows little evidence of cross-correlation (i.e. between time periods t and t+1)
• multivariate series of absolute or squared returns do show evidence of cross-correlation
• extreme returns in one series often coincides with extreme returns in several other series

Question 2

What is volatility clustering?

Answer 2

• when extreme values tend to be followed by other extreme values, although not necessarily of the same sign
• as the time period over which we calculate the returns increases, volatility clustering is less marked and returns appear to be more iid and less heavy-tailed
• serial correlation between absolute or squared returns is consistent with the phenomenon of volatility clustering

Question 3

List three factors that may be linked within a factor-based approach to modelling corporate bond yields.

Answer 3

1. the risk-free yield
2. coupon rates
3. credit spread

Question 4

Outline the steps involved in modelling returns on a portfolio using a data-based forward-looking approach.

Answer 4

1. Decide on the frequency of calculation e.g. daily, weekly, monthly
2. Decide on the timeframe of historic data to be used (trade-off between volume of data and relevance)
3. For each asset class, choose the total return index to be used (St)
4. For each asset class, calculate the log-returns,
   Xt = ln[St/(St-1)]
5. Calculate the average returns and variance of each asset class and the covariances between each class
6. Simulate a series of returns with the same characteristics based on a multivariate normal distribution.

Question 5

Why is PCA particularly useful when projecting returns on bonds (with a variety of durations)?

Answer 5

• Changes in bond yields can be explained largely by shifts in just a couple of factors i.e. the level and shape of the yield curve

Question 6

What are the steps in the simulation process using PCA?

Answer 6

1. Decide on the frequency of calculation e.g. daily, weekly, monthly
2. Decide on the timeframe of historic data to be used (trade-off between volume of data and relevance)
3. For each asset class, choose the total return index to be used (St)
4. For each asset class, calculate the log-returns,
   Xt = ln[St/(St-1)]
5. Calculate the average returns and variance of each asset class and the covariances between each class
6. Derive the matrix deviations form average returns by deducting the average return in each period for each asset class.
7. Derive the principal components that explain a sufficiently high proportion of the deviations from average past returns.
8. Project this number of independent normally distributed random variables using the associated eigenvalues as the variances.
9. Weight these projected series of deviations by the appropriate elements of the relevant eigenvectors.
10. Add these weighted projected deviations to the expected returns from each asset class.

Question 7

Outline the relevance in the context of the analysis of risk-free government bond yields of the following:

• term premium
• purchasing power parity
• bootstrapping

Answer 7

Term Premium:

• Where comparison is being made between bonds of different terms, an allowance may be made for the part of the risk premium that is a function of the term.
• The term premium will vary by market and investor

Purchasing power parity:
- For risk-free overseas government bonds the domestic rate is suitable (as in theory, purchasing power parity will compensate for any differences in yield)

Bootstrapping:
- Implied forward spot yield curves can be constructed from the gross redemption yields of the bonds in a portfolio using a bootstrapping process.

Question 8

What is the credit spread of a corporate bond, and what factors contribute to the spread?

Answer 8

• The difference between the YTM of a corporate bond and a government bond
• Credit spread reflects:
  > expected probability of default, and expected loss given default
  > any risk premium attached to the risk of default (uncertainty regarding the expected probability and loss at default)
  > a liquidity premium

Question 9

What are the three common measures of credit spread?

Answer 9

1. Nominal spread - the difference between the gross redemption yields of risky and risk-free bonds.
2. Static spread - addition to the risk-free rate such that the discounted cash flows from a risky bond will equate to the price of that bond.
3. Option-adjusted spread - further adjusts this discount rate through the use of stochastic modelling to allow for options embedded in the bond.

Question 10

Outline risk premia that help to explain why observed market credit spread are generally higher than can be justified by the actual historic defaults on bonds.

Answer 10

• higher volatility of returns compared to the risk-free asset
• higher uncertainty of returns, particularly the possibility of unprecedented extreme events
• greater skewness of potential future returns on corporate debt, due to the possibility of default
• lower liquidity of corporate debt
• lower marketability of corporate debt, and associated higher costs of trade
• differences in taxation

Question 11

What are the features of a good benchmark?

Answer 11

• unambiguous
• investable and trackable
• measurable on a reasonably frequent basis
• appropriate e.g. to the investor’s objectives
• reflective of current investment opinion e.g. positive, negative, neutral
• specified in advance

Question 12

Define:

• strategic risk
• active risk
• active return

Answer 12

Strategic risk - the risk of poor performance of the benchmark against which the manager’s performance will be judged relative to the liability-based benchmark

Active risk - the risk of poor performance of the manager’s actual portfolio relative to the manager’s benchmark

Active return - the difference between the return on the actual portfolio and the return on the manager’s benchmark

Question 13

How are interest rates modelled?

Answer 13

Brennan-Schwartz model:
- considers changes in spot rates at two maturities e.g. short-term and long-term
- the model assumes:
> changes in short-term rates vary in line with the steepness of the yield curve
> the volatility of short-term rates varies in proportion to the most relevant short-term rates
> changes in long-term rates vary in proportion to the square of the level of long-term rates , but are also influenced by short-term rates through the product term
> the volatility of long-term rates varies in proportion to the level of long-term rates

The PCA approach can also be used to model interest rates, determining the main factors that contribute to changes in the level and shape of the yield curve. PCA can be applied to gross redemption yields, forward yields or bond prices.

Question 14

Describe two different approaches to modelling contagion risks.

Answer 14

Single financial series:
- contagion can be considered a feedback risk, that is, there is some serial correlation that can be modelled. However, any suggestion that serial correlation exists presents an arbitrage opportunity, which in theory should be eliminated by arbitageurs. Hence, the effects are usually ignored when modelling.

Related financial series:
Some studies suggest fitting a t-copula using a correlation parameter, p, which is situation dependent. For example, p = p0 + D1p1 + D2p2 where the normal level of dependency (p0) is adjusted by additional dependency (p1, p2) in different states (D1 = 1 in financial crisis and 0 otherwise, D2 = 1 in the aftermath of a financial crisis and 0 otherwise).