Module 22: Assessment of market risks

Question 1

4 Features of observed returns on individual equities

Answer 1

• rarely iid (inconsistent with random walk and Brownian motion models)
• volatility appears to vary over time
• volatility clustering (supporting use of ARCH & GARCH models)
• leptokurtic / excess kurtosis

Question 2

5 Features of observed returns on portfolios of equities

Answer 2

• correlations exist between returns of different series at the same point in time
• correlations between different series vary over time
• little evidence of cross-correlation (ie between time periods t and t + 1)
• multivariate series of absolute or squared returns do show strong evidence of cross-correlation
• during periods of high volatility, the level of dependence between various returns appears to be higher.

Question 3

5 Approaches to modelling market returns

Answer 3

• historical simulations, eg bootstrapping
• forward-looking data-based approaches, eg using a multivariate normal distribution (6-step process)
• forward-looking factor-based approaches, eg PCA (10-step process)

If there is sufficient data, alternatives to using the multivariate normal distribution or PCA include:

• use of a multivariate distribution other than normal
• combining non-normal marginal distributions using an appropriate copula distribution.

Question 4

Assessing market risk under the Basel accords

Answer 4

Under Basel II, market risk is typically quantified by using an internal model to model the assets (as described above) and then calculating a 10-day 99% (or 1% tail) Value at Risk (VaR). The regulatory capital requirement under Pillar 1 is a multiple of this VaR loss.

Question 5

A reasonable estimate of the expected return for risk-free government bonds (domestic and overseas)

Answer 5

Gross Redemption Yield (GRY)

Yield-to-Maturity (YTM) on a domestic government bond of a similar term as the projection period.

Question 6

A reasonable estimate for the expected return for risky bonds

Answer 6

Can be derived from an adjustment to the expected risk-free return. Adjustments should reflect:

• credit spread
• historical default rates
• taxation

Question 7

Credit spread reflects 3 factors

Answer 7

• expected profitability of, and loss given default - measurable, in principle, using the default history of similarly rated bonds
• the uncertainty surrounding the above (ie a risk premium)
• a liquidity premium

Question 8

3 Most common ways of measuring credit spread

Answer 8

• nominal spread - the difference between the GRYs of risky and risk-free bonds
• static spread - the addition to the risk-free rate at which discounted cashflows from a risky bond will equate to its price
• option-adjusted spread - further adjusts this discount rate (through the use of stochastic modelling) to allow for any options embedded in the bond.

Question 9

3 Main approaches to modelling interest rates

Answer 9

• single-factor models - eg for modelling short-term single interest rates
• two-factor models - eg Brennan-Schwartz
• PCA - eg modelling deviations from average GRYs for all durations

Question 10

Assessing exchange rate risk

Answer 10

Exchange rate risks can be modelled in terms of the returns on short-term interest-bearing deposits denominated in different currencies. There is no additional currency return to be gained (or modelled) if working in a single denomination (currency).

Question 11

Assessing contagion risks

Answer 11

Contagion (or systemic) risks are usually seen as an extension of market risks, but can, however, also apply to other risks, eg credit risk.

Contagion risks can be modelled as the interaction between different financial series. In particular, certain series may be linked for extreme negative values. This increased level of dependence suggests that using a copula may be a sensible approach (assuming it can be suitable parameterised).

Contagion can be considered as a feedback risk. However such (serial correlation) effects are usually ignored when modelling as it is assumed that the resulting arbitrage opportunities would be eliminated by arbitrageurs. Alternatively, some studies suggest fitting a t-copula using a situation-dependent correlation parameter.

Question 12

Volatility clustering

Answer 12

Occurs when extreme values tend to be followed by other extreme values, although not necessarily of the same sign.

Question 13

Kurtosis

Answer 13

Kurtosis measures the ‘peakedness’ of a distribution.

Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations.

Question 14

Leptokurtic

Answer 14

A leptokurtic distribution has more acute peaks around the mean (ie a higher probability than a normally distributed variable of values near the mean)

and fatter tails (ie a higher probability than a normally distributed variable of the extreme values).

Question 15

A forward-looking factor-based approach to modelling corporate bond yields might describe the complex links between variables such as: (3)

Answer 15

• the risk-free yield
• coupon rates
• credit spread

Question 16

Outline the steps involved to model log returns on a portfolio using a forward-looking data-based approach.

Answer 16

1. Decide on the frequency of calculation
2. Decide on the time-frame 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 Xt = ln(St / S_{t-1})
5. Calculate the average returns and variance of each asset class and the co-variances between each class (and subset of classes).
6. Simulate a series of returns with the same characteristics based on a multivariate normal distribution.

Question 17

Factor-based modelling using principle component analysis (PCA)

Answer 17

Principle component analysis can reduce the computational overhead when compared to application of the multivariate normal distribution.

The aim of PCA is to determine only the main factors that contribute to deviations from the average return and ignore the factors with less influence, thereby reducing the complexity (dimensionality) of the analysis.

Question 18

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

Answer 18

PCA is particularly useful when projecting returns on bonds as changes in bond yields can be explained largely by shifts in just a couple of factors:
- the level and shape of the yield curve.

Question 19

Term premium

Answer 19

That part of the risk premium that is a function of term.

The term premium will vary from market to market, and investor to investor.

It may be that for some investors, the term premium will reduce with increasing term, eg for investors with long-term assets.

Question 20

Why would the domestic risk-free rate be suitable for risk-free overseas government bonds?

Answer 20

Because, in theory, purchasing power parity will compensate for any difference in yields.

Question 21

Credit spread

Answer 21

A measure of the difference between the yield on a risky and risk-free security, typically a corporate bond and a government bond respectively.

Question 22

Credit spread:

liquidity premium is to compensate for…

Answer 22

The fact that it may be more difficult to sell the corporate bond, when required, at an acceptable price.

Question 23

Define “loss given default”

Answer 23

The residual value of a bond after default has happened.

Occasionally, the loss might be total.

However, more often the loss will only be partial, especially if the corporation has not gone bankrupt, but has been forced, e.g., to reschedule payments.

Question 24

Outline 6 risk premia that help to explain why observed market credit spreads are generally higher than can be justified by the actual historic defaults on bonds.

Answer 24

• Higher volatility of returns relative to the risk-free asset (a credit beta)
• Higher uncertainty of returns, particularly the possibility of unprecedented extreme events.
• The greater skewness of the potential future returns on corporate debt (more significant downside), due to the possibility of default.
• The lower liquidity of corporate debt than government debt.
• The lower marketability of corporate debt, and the associated higher costs of trade.
• Differences in taxation.

Question 25

Outline how appropriate risk premiums might be determined

Answer 25

• Historical risk premiums can be calculated by deducting the observed return on a risk-free asset from the observed return on the risky asset, averaging over the periods that data is available.
• The average historical risk premium should be altered to reflect any changes that might be expected in the future. Such changes may be subjective or based on fundamental structural changes in the asset classes.
• When considering overseas investments, it is important to consider volatility in each asset’s domestic currency and allow separately for exchange rate risk in the correlation calculation.
• The capital asset pricing model (CAPM) gives a structure for analysing risk premiums and ensuring their consistency.

Question 26

CAPM

Answer 26

Links the expected return, rᵪ, on asset X, to:

• the risk free rate rᶠ
• the return on the universe of investment opportunities, rᵘ
• the associated variances and covariances of returns

as follows:

rᵪ = rᶠ + βᵪ ( rᵘ - rᶠ )

βᵪ = { σᵪ / σᵘ } ρᵪ,ᵘ

σᵪ and σᵘ are the respective standard deviations.

ρᵪ,ᵘ is the (linear) correlation between the returns on X and the universe of investment opportunities.

Question 27

Market risk should be measured relative to…

Answer 27

A suitable benchmark.

Benchmarks are typically based on market indices or the investor’s liabilities.

Question 28

Outline the features of a good benchmark

Answer 28

In general, a good benchmark is one that is:

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

It may also be appropriate to measure against a specific benchmark that:

• contains a high proportion of the assets held in the portfolio
• has a similar investment style to the portfolio (eg growth or value)
• has a low turnover of constituents
• has investable position sizes
• behaves in a similar way to the portfolio, ie shows a strong positive correlation between the portfolio return and the benchmark return in excess of the market return.
  ie ρ( rᵪ - rᵘ , rᵇ - rᵘ)&raquo_space; 0
• has low correlation between the difference of the portfolio return and the benchmark return, and benchmark return and the market return.
  ie ρ( rᵪ - rᵇ , rᵇ - rᵘ) ≈ 0
• the variability of the portfolio returns relative to the benchmark returns should be lower than the variability relative to the market return.

Question 29

Define strategic risk

Answer 29

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.

Question 30

Define active risk

Answer 30

Risk of poor performance of the manager’s actual portfolio relative to the manager’s (strategic) benchmark.

Question 31

Define active return

Answer 31

The difference between the return on actual (active) portfolio and the return on the manager’s (strategic) benchmark.